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Kvant Math Problem 1211

Verified: no Verdicts: SKIP + SKIP Solve time: 1m01s Source on kvant.digital Problem Is it possible to position a tetrahedron, a sphere, and a plane in space such that the areas of the sections of the tetrahedron and the sphere by any plane parallel to the chosen one are equal? A. Andzhans City Tournament (Autumn, 1989) Exploration A plane intersects a tetrahedron along a triangle and a sphere along a...

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Kvant Math Problem 1210

Verified: no Verdicts: SKIP + SKIP Solve time: 1m57s Source on kvant.digital Problem There is a pile of $M$ matches and a sheet of paper on which the number $M$ is written. Two players play the following game. They take turns. A move consists of taking from the pile or returning to the pile from 1 to $k$ matches and writing on the sheet the new number of matches in...

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Kvant Math Problem 1209

Verified: no Verdicts: SKIP + SKIP Solve time: 1m00s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I do not yet have the full textual version of Kvant problem M1209. Could you provide the problem statement so I can produce a rigorous solution using your six-section structure?

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Kvant Math Problem 1208

Verified: no Verdicts: SKIP + SKIP Solve time: 51s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I need the textual statement of Kvant problem M1208 in order to produce a rigorous solution according to your strict rules. Could you provide the full problem text?

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Kvant Math Problem 1207

Verified: no Verdicts: SKIP + SKIP Solve time: 1m06s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I see the problem statement itself is not fully provided yet. Before I can write the full six-section solution, I need the text of Kvant problem M1207. Could you provide the full textual statement or the key...

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Kvant Math Problem 1206

Verified: no Verdicts: SKIP + SKIP Solve time: 1m01s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I see that the problem statement for Kvant M1206 is not yet provided. I need the actual text of the problem to generate the complete solution with the six-section structure you requested. Could you provide the...

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Kvant Math Problem 1205

Verified: no Verdicts: SKIP + SKIP Solve time: 1m01s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M1205 because the actual problem statement is not present in the conversation. The prompt says that “the textual version of the problem statement is being prepared” and that the...

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Kvant Math Problem 1204

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem On the plane, points $A$, $B$, $C$ are given as the centers of three circles. Each circle expands uniformly (the radius increases at the same rate for all circles). As soon as two circles touch each other, they "burst" — their radii shrink to 0 — and start growing again. Is it true that if the distances...

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Kvant Math Problem 1203

Verified: no Verdicts: SKIP + SKIP Solve time: 4m23s Source on kvant.digital Problem Is it possible to dissect a square of side 1 km into 31 squares; 30 squares so that one of them has side at most 1 m? S. L. Eliseev Exploration Let the side of the large square be $1000$ m. We seek a tiling by either $31$ or $30$ squares, with at least one square of...

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Kvant Math Problem 1202

Verified: no Verdicts: SKIP + SKIP Solve time: 7m18s Source on kvant.digital Problem From vertex $A$ of square $ABCD$, two rays are drawn into the interior of the square. Perpendiculars $BK$, $BL$, $DM$, $DN$ are dropped onto these rays from vertices $B$ and $D$. Prove that the segments $KL$ and $MN$ are equal in length and perpendicular to each other. D. Nyamsuren (Mongolia) Exploration Let the two rays from $A$...

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Kvant Math Problem 1201

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem One deputy must be elected to the parliament of Anchuria from each of 999 districts, all having the same number of voters. In Anchuria, three parties $A$, $B$, and $C$ have been formed and have nominated their candidates. Party $A$ is supported by 15% of all voters, $B$ by 30%, and $C$ by 55%. If, in the...

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Kvant Math Problem 1200

Verified: no Verdicts: SKIP + SKIP Solve time: 10m06s Source on kvant.digital Problem For which $k$ is it possible to place on a circle 10; 100; $n$ arcs so that each of them intersects exactly $k$ others? S. A. Genkin Leningrad City Mathematical Olympiad (1989) Exploration Consider a small number of arcs on a circle. If there are two arcs, each can intersect the other or not. If there are...

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Kvant Math Problem 1199

Verified: no Verdicts: SKIP + SKIP Solve time: 7m16s Source on kvant.digital Problem Prove that if the equation $ax^2+(c-b)x+(e-d)=0$ has a root greater than 1, then the equation $$ax^4+bx^3+cx^2+dx+e=0$$ has at least one root. D. V. Fomin Leningrad City Mathematical Olympiad (1989) Exploration Consider the quartic polynomial $P(x) = ax^4 + bx^3 + cx^2 + dx + e$ and the quadratic polynomial $Q(x) = ax^2 + (c-b)x + (e-d)$. The...

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Kvant Math Problem 1198

Verified: no Verdicts: SKIP + SKIP Solve time: 7m19s Source on kvant.digital Problem Let us call a string of 10 digits 0 and 1 a word . Two words will be called synonyms if one can be obtained from the other by a sequence of operations of the following kind: several consecutive digits whose sum is even are deleted from the word, and the same digits are written back in...

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Kvant Math Problem 1197

Verified: no Verdicts: SKIP + SKIP Solve time: 7m33s Source on kvant.digital Problem In triangle $ABC$, point $M$ lies on side $AB$, point $N$ lies on side $BC$, and $O$ is the intersection point of segments $CM$ and $AN$. It is known that $AM+AN=CM+CN$. Prove that $AO+AB=CO+CB$. A. S. Merkuryev Leningrad City Mathematical Olympiad (1989) Exploration Let $$x=AM,\qquad y=MB,\qquad u=BN,\qquad v=CN.$$ Then $$AB=x+y,\qquad BC=u+v,$$ and the condition becomes $$x+AN=CM+v.$$ Since...

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Kvant Math Problem 1196

Verified: no Verdicts: SKIP + SKIP Solve time: 7m13s Source on kvant.digital Problem Several (at least two) nonzero numbers are given. One may erase any two numbers $a$ and $b$ and write in their place the numbers $a+\dfrac b2$ and $b-\dfrac a2$. Prove that after several such operations it is impossible to obtain the original set of numbers. D. V. Fomin Leningrad City Mathematical Olympiad (1989) Exploration Denote the transformation...

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Kvant Math Problem 1195

Verified: no Verdicts: SKIP + SKIP Solve time: 4m05s Source on kvant.digital Problem The sequence $x_1$, $x_2$, $\ldots$ is such that for any natural numbers $m$ and $n$ $$|x_{m+n}-x_m-x_n|\lt \dfrac1{m+n}.$$. Prove that this sequence is an arithmetic progression. O. T. Izhboldin Exploration Consider the given inequality $|x_{m+n}-x_m-x_n|<\frac{1}{m+n}$ for natural numbers $m$ and $n$. If the sequence were an arithmetic progression, we would have $x_n = cn$ for some constant $c$,...

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Kvant Math Problem 1194

Verified: no Verdicts: SKIP + SKIP Solve time: 7m39s Source on kvant.digital Problem From a point $M$ inside a rectangle $ABCD$ with area $S$, the bisectors $ME$, $MF$, $MG$, $MH$ of triangles $AMB$, $BMC$, $CMD$, $DMA$ are drawn. Prove that for the area $S_0$ of quadrilateral $EFGH$ the inequalities $$\dfrac38 S\lt S_0 \le\dfrac12 S.$$ hold. For which points $M$ does the equality $S_0=\dfrac S2$ hold? A. A. Azamov Exploration Let...

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Kvant Math Problem 1193

Verified: no Verdicts: SKIP + SKIP Solve time: 7m10s Source on kvant.digital Problem Prove (for any numbers $a$, $b$, $c$, $x$, $y$, $z$) the inequality $$\begin{gather*} ax+by+cz+\sqrt{(a^2+b^2+c^2)(x^2+y^2+z^2)}\ge\ \ge\frac23(a+b+c)(x+y+z). \end{gather*}$$ V. Kyrtojа (Romania) Exploration Let $$u=(a,b,c),\qquad v=(x,y,z).$$ Then the inequality becomes $$u\cdot v+|u|,|v|\ge \frac23 (a+b+c)(x+y+z).$$ The left-hand side resembles the Cauchy-Schwarz inequality. Since $$(a+b+c)(x+y+z) =(u\cdot e)(v\cdot e),$$ where $e=(1,1,1)$, the right-hand side is expressed through projections onto the vector $e$. A...

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Kvant Math Problem 1192

Verified: no Verdicts: SKIP + SKIP Solve time: 8m00s Source on kvant.digital Problem It is known that all edges of the polyhedron $M$ are equal to one another and are tangent to some sphere. Suppose that one of the faces of $M$ has an odd number of sides. Prove that there exists a sphere circumscribed about $M$. Under the conditions of part (a), must there necessarily exist a sphere inscribed...

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Kvant Math Problem 1191

Verified: no Verdicts: SKIP + SKIP Solve time: 8m15s Source on kvant.digital Problem Let $A_1$, $A_2$, $A_3$, $\ldots$ be a sequence of points in the plane. Starting from some point $T_0$, construct a sequence $T_1$, $T_2$, $T_3$, $\ldots$, where $T_n$ is the point symmetric to $T_{n-1}$ with respect to $A_n$ ($n=1$, 2, 3, $\ldots$). What necessary and sufficient conditions must the sequence $A_n$ satisfy so that the sequence $T_n$ is...

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Kvant Math Problem 1190

Verified: no Verdicts: SKIP + SKIP Solve time: 4m52s Source on kvant.digital Problem Prove that if there are $3n$ asterisks placed in a table of $2n\times2n$ cells, then it is possible to cross out $n$ rows and $n$ columns so that all the asterisks are crossed out. Prove that in a table of $2n\times2n$ cells one can place $3n+1$ asterisks so that after crossing out any $n$ rows and $n$...

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Kvant Math Problem 1189

Verified: no Verdicts: SKIP + SKIP Solve time: 12m01s Source on kvant.digital Problem Given $n$ lines in the plane, no three of which pass through a common point and no two of which are parallel. Prove that in each of the regions into which these lines divide the plane, one can place a nonzero integer whose absolute value does not exceed $n$, in such a way that for either side...

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Kvant Math Problem 1188

Verified: no Verdicts: SKIP + SKIP Solve time: 7m23s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M1188 from the information provided because the actual problem statement is missing. The text says that “the textual version of the problem statement is being prepared” and that only...

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Kvant Math Problem 1187

Verified: no Verdicts: SKIP + SKIP Solve time: 8m53s Source on kvant.digital Problem Prove that if $m$ is even, then all integers from 1 to $m-1$ can be arranged in an order such that no sum of several consecutive numbers is divisible by $m$. F. G. Shleifer City Tournament (Spring, 1989) Exploration Let the required order be $a_1,a_2,\dots,a_{m-1}$. A sum of consecutive terms from $a_i$ to $a_j$ equals $$(a_1+\cdots+a_j)-(a_1+\cdots+a_{i-1}).$$ Hence...

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Kvant Math Problem 1186

Verified: no Verdicts: SKIP + SKIP Solve time: 6m09s Source on kvant.digital Problem The textual version of the problem statement is currently being prepared. The graphical version can be found in the published issue.

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Kvant Math Problem 1185

Verified: no Verdicts: SKIP + SKIP Solve time: 5m49s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I do not yet have the full text of Kvant problem M1185. Could you provide the complete problem statement in text form? This is necessary to construct the six-section rigorous solution according to the rules you specified.

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Kvant Math Problem 1184

Verified: no Verdicts: SKIP + SKIP Solve time: 10m11s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I do not have access to the published graphical version of Kvant M1184, and you mentioned the textual version is being prepared. Could you provide the full textual problem statement so I can generate the rigorous six-section...

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Kvant Math Problem 1183

Verified: no Verdicts: SKIP + SKIP Solve time: 5m49s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I cannot write a solution to Kvant problem M1183 because the actual problem statement is not present in your message. You wrote: "The textual version of the problem statement is being prepared. The graphical version can be...

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Kvant Math Problem 1182

Verified: no Verdicts: SKIP + SKIP Solve time: 8m26s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. Before I begin, could you provide the exact text of Kvant problem M1182? I need the precise statement to produce a complete, rigorous solution in the requested format.

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Kvant Math Problem 1181

Verified: no Verdicts: SKIP + SKIP Solve time: 7m55s Source on kvant.digital Problem On a chessboard, 8 pieces are placed so that there is exactly one piece in each row and in each column. Prove that the number of pieces on the black squares of the chessboard is even. V. V. Proizvolov All-Union Mathematical Olympiad (XXIII, 1989) Exploration Consider small instances of the chessboard and examine the parity of black...

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Kvant Math Problem 1180

Verified: no Verdicts: SKIP + SKIP Solve time: 7m41s Source on kvant.digital Problem On one of two given intersecting spheres, points $A$ and $B$ are chosen; on the other, points $C$ and $D$. The segment $AC$ passes through one of the common points of the spheres. The segment $BD$ passes through the other common point of the spheres and is parallel to the line containing the centers of the spheres....

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Kvant Math Problem 1179

Verified: no Verdicts: SKIP + SKIP Solve time: 7m53s Source on kvant.digital Problem Find $a_{1000}$, given that $a_1=0$ and under $n=1$, 2, 3, $\ldots$ $a_{n+1}=\dfrac n{n+1}(a_n+1)$; $a_{n+1}=\dfrac{n(n+1)}{(n+2)(n+3)}(a_n+1)$; $a_{n+1}=\dfrac{n(n+1)(n+2)}{(n+3)(n+4)(n+5)}(a_n+1)$. B. A. Wertheim Exploration Consider the first recurrence, $a_{n+1} = \frac{n}{n+1}(a_n+1)$ with $a_1 = 0$. Computing small terms yields $a_1=0$, $a_2 = \frac{1}{2}$, $a_3 = \frac{2}{3} \cdot \frac{3}{2} = \frac{2}{3} \cdot \frac{3}{2} = ?$ We must compute carefully: $a_2 = \frac{1}{2} (0+1)...

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Kvant Math Problem 1178

Verified: no Verdicts: SKIP + SKIP Solve time: 4m41s Source on kvant.digital Problem Prove that for a non-obtuse triangle $ABC$ with sides $a$, $b$, $c$, inradius $r$, and circumradius $R$, the inequality $$2(R+r)\le\sqrt{a^2+b^2+c^2}.$$ holds. Under what condition does this inequality become an equality? Z. A. Skopets Exploration The inequality involves the three classical quantities $$a^2+b^2+c^2,\qquad R,\qquad r.$$ A standard identity is $$a^2+b^2+c^2=2\bigl(s^2-r^2-4Rr\bigr),$$ where $s$ is the semiperimeter. Substituting this converts...

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Kvant Math Problem 1177

Verified: no Verdicts: SKIP + SKIP Solve time: 7m37s Source on kvant.digital Problem Prove that for positive numbers $x_1$, $x_2$, $\ldots$, $x_n$, not exceeding 1, the inequality $$(1+x_1)^{1/x_2}(1+x_2)^{1/x_3}(1+x_3)^{1/x_4}\ldots(1+x_n)^{1/x_1}\ge2^n.$$ holds. K. P. Kokhas, V. M. Televka Exploration Consider the inequality for small values of $n$ to understand its structure. For $n=2$, the inequality reads $(1+x_1)^{1/x_2}(1+x_2)^{1/x_1}\ge 4.$ Trying $x_1=x_2=1$ gives equality: $(1+1)^{1}(1+1)^1=4$. Choosing $x_1=1$, $x_2=\frac{1}{2}$ gives $(2)^{2.0}(1.5)^{1}=4\cdot 1.5=6>4$, suggesting the left-hand side...

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Kvant Math Problem 1176

Verified: no Verdicts: SKIP + SKIP Solve time: 4m34s Source on kvant.digital Problem Two squares $AKBM$ and $CNDL$ are placed on the plane so that $ABCD$ forms a convex quadrilateral, and the points $K$ and $L$ lie inside this quadrilateral. Prove that the area of this quadrilateral equals $\dfrac14(MN^2-KL^2)$. S. A. Stolyarov Exploration The expression to be proved resembles the area formula for a quadrilateral written in terms of the...

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Kvant Math Problem 1175

Verified: no Verdicts: SKIP + SKIP Solve time: 7m25s Source on kvant.digital Problem For which natural $n$ is the following statement true: no matter how several non-overlapping regular $n$-gons are arranged in the plane, one of them can be moved out in some direction without touching the others? (The $n$-gon may not be rotated; that is, all rays issuing from the points of the chosen $n$-gon in the required direction...

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Kvant Math Problem 1174

Verified: no Verdicts: SKIP + SKIP Solve time: 7m37s Source on kvant.digital Problem The sequence of integers $a_1$, $a_2$, $a_3$, $\ldots$ is defined by the conditions $a_1=1$, $a_2=12$, $a_3=20$, $$a_{n+3}=2a_{n+2}+2a_{n+1}-a_n\quad(n=1,\ 2,\ \ldots).$$. Prove that for any index $n$, the number $1+4a_n a_{n+1}$ is a perfect square. R. Kozarev (Bulgaria), S. Doichev (Bulgaria) Exploration Compute the first few values of the sequence to look for patterns. We are given $a_1 =...

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Kvant Math Problem 1173

Verified: no Verdicts: SKIP + SKIP Solve time: 6m14s Source on kvant.digital Problem Through a single point inside a triangle of area $S$, three lines are drawn so that each side of the triangle is intersected by two of them (see figure). Prove that for the areas $S_1$, $S_2$, $S_3$ of the three triangles thus formed, the inequality $$\dfrac1{S_1}+\dfrac1{S_2}+\dfrac1{S_3}\ge\dfrac9S.$$ holds. Figure G. N. Zaitsev Exploration Let the three lines through...

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Kvant Math Problem 1172

Verified: no Verdicts: SKIP + SKIP Solve time: 6m17s Source on kvant.digital Problem What is the greatest possible angle between the segments $OA$ and $OB$, drawn from the origin $O$ of a rectangular coordinate system in space, if the point $A$ has coordinates $(x, y, z)$ and the point $B$ has coordinates $(y, z, x)$? S. N. Bychkov Exploration Let $$\vec a=(x,y,z),\qquad \vec b=(y,z,x).$$ The angle between $OA$ and $OB$...

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Kvant Math Problem 1171

Verified: no Verdicts: SKIP + SKIP Solve time: 1m31s Source on kvant.digital Problem Let the sum $1+\dfrac12+\dfrac13+\ldots+\dfrac1n$ be denoted by $h_n$. Prove (for each natural number $n$) the inequality $$\dfrac1{h_1^2}+\dfrac1{2h_2^2}+\dfrac1{3h_3^2}+\ldots+\dfrac1{nh_n^2}\lt 2.$$. L. D. Kurlandchik Exploration Let $$S_n=\sum_{k=1}^{n}\frac1{k h_k^2}.$$ The inequality to prove is $S_n<2$ for every $n$. First compute a few terms: $$h_1=1,\qquad h_2=\frac32,\qquad h_3=\frac{11}{6}.$$ Hence $$S_1=1,$$ $$S_2=1+\frac1{2(3/2)^2}=1+\frac29=\frac{11}{9},$$ $$S_3=\frac{11}{9}+\frac1{3(11/6)^2} =\frac{11}{9}+\frac{12}{121}\approx1.322.$$ The sum grows slowly, so a telescoping estimate is plausible....

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Kvant Math Problem 1170

Verified: no Verdicts: SKIP + SKIP Solve time: 2m05s Source on kvant.digital Problem Consider the triangulations of a given convex $n$-gon using non-intersecting diagonals. Define a flip as the following operation: instead of a certain diagonal $BC$, which serves as a common side of two triangles $ABC$ and $BCD$ in the triangulation, we draw the diagonal $AD$ (Fig. 1). Let $P(n)$ denote the minimal number of flips required to transform...

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Kvant Math Problem 1169

Verified: no Verdicts: SKIP + SKIP Solve time: 1m34s Source on kvant.digital Problem Let $M$ be a point lying inside the rectangle $ABCD$, and let $S$ be its area. Prove the inequality $$S\le AM\cdot CM+BM\cdot DM.$$. I. Ya. Goldshteyn City Tournament (Autumn, 1988) Exploration Consider a rectangle $ABCD$ with sides $AB = CD = a$ and $BC = AD = b$. Place a point $M$ inside the rectangle and denote...

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Kvant Math Problem 1168

Verified: no Verdicts: SKIP + SKIP Solve time: 1m41s Source on kvant.digital Problem There are 1989 cities and 4000 roads in a country (each road connects two cities). Prove that it is possible to choose a cyclic route passing through no more than 20 cities. A. A. Razborov Tournament of Towns (Autumn, 1988) Exploration Model the country by a graph with $1989$ vertices and $4000$ edges. A cyclic route corresponds...

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Kvant Math Problem 1167

Verified: no Verdicts: SKIP + SKIP Solve time: 1m55s Source on kvant.digital Problem How many permutations of the numbers 1, 2, $\ldots$, $n$ exist in which, for any number $i$ that is not in the first position, at least one of the numbers $i-1$ and $i+1$ is to the left of $i$? A. V. Andzhans Tournament of Cities (Autumn, 1988) Exploration Let $p(i)$ denote the position of $i$ in the...

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Kvant Math Problem 1166

Verified: no Verdicts: SKIP + SKIP Solve time: 3m53s Source on kvant.digital Problem Prove the inequality $a^{2}pq+b^{2}qr+c^{2}rp \le 0$, where $a$, $b$, $c$ are the side lengths of a triangle, $p+q+r=0$. Ya. Sh. Mustafayev, 10th-grade student (Baku) Tournament of Towns (Autumn, 1988) Exploration The inequality involves the side lengths $a$, $b$, $c$ of a triangle and three numbers $p$, $q$, $r$ summing to zero. A first attempt is to test...

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Kvant Math Problem 1165

Verified: no Verdicts: SKIP + SKIP Solve time: 3m06s Source on kvant.digital Problem Prove that a square with side length $n$ (where $n$ is a positive integer), placed arbitrarily on a sheet of graph paper with cells $1\times1$, covers no more than $(n+1)^2$ grid points. D. J. Newman, B. D. Kotlyar Exploration Consider a square of side length $n$ on a standard graph paper with $1\times1$ cells. If the square...

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Kvant Math Problem 1164

Verified: no Verdicts: SKIP + SKIP Solve time: 4m11s Source on kvant.digital Problem A natural number $n$ is called perfect if it is equal to the sum of all its divisors less than $n$ (for example: $28=1+2+4+7+14$). Prove that an odd perfect number (if one exists) cannot be divisible by 3, 5, and 7 simultaneously. V. V. Shabunin Exploration Let $\sigma(n)$ denote the sum of all positive divisors of $n$....

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Kvant Math Problem 1163

Verified: no Verdicts: SKIP + SKIP Solve time: 7m06s Source on kvant.digital Problem A turtle started from the point $A$ and arrived at the point $B$, moving along an arbitrary trajectory with arbitrary speed. Following it, a second turtle started from the point $A$. At every moment of time, it moved in the direction of the first turtle (with arbitrary speed) and eventually also arrived at the point $B$. Prove...

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Kvant Math Problem 1162

Verified: no Verdicts: SKIP + SKIP Solve time: 4m11s Source on kvant.digital Problem Find all integer solutions $(x, y)$ of the equation $$x^{3}-13xy+y^{3}=13.$$ D. V. Fomin Exploration Consider the Diophantine equation $x^3 - 13xy + y^3 = 13.$ Since the coefficients are small integers, it is natural to try small integer values for $x$ and $y$. If $|x|$ or $|y|$ is large, the cubic terms dominate, so solutions should lie...

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Kvant Math Problem 1161

Verified: no Verdicts: SKIP + SKIP Solve time: 7m09s Source on kvant.digital Problem In a billiard triangle, 10 balls fit snugly (see Fig. 1). Prove that if 9 balls are placed in it, there will necessarily be room for the tenth (i.e., the centers of the 9 balls are arranged in a triangular lattice). Figure 1 N. P. Dolbilin Exploration Consider first the configuration of ten identical billiard balls arranged...

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Kvant Math Problem 1160

Verified: no Verdicts: SKIP + SKIP Solve time: 7m10s Source on kvant.digital Problem At one end of a $A$ straight road $AB$, 10 kangaroos gathered and began playing leapfrog. They jump in turns: the first jumps wherever it wants; the second jumps over the first so that the first is exactly in the middle between the start and end of the jump; the third jumps in the same way over...

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Kvant Math Problem 1159

Verified: no Verdicts: SKIP + SKIP Solve time: 6m52s Source on kvant.digital Problem The textual version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. Before I can write the full six-section solution you requested, I need the textual statement of Kvant problem M1159 , since I do not have the content of the problem from memory. Could you provide it...

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Kvant Math Problem 1158

Verified: no Verdicts: SKIP + SKIP Solve time: 7m27s Source on kvant.digital Problem Find the minimum value of the expression $(x+y)(x+z)$, given that $x$, $y$, $z$ are positive numbers and $xyz(x+y+z)=1$. Oleg Khristenko, 10th grade student Exploration We are asked to minimize $(x+y)(x+z)$ under the constraint $xyz(x+y+z)=1$, with $x$, $y$, $z$ positive. A natural first step is to consider symmetry: the constraint is symmetric in $x$, $y$, $z$, while the...

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Kvant Math Problem 1157

Verified: no Verdicts: SKIP + SKIP Solve time: 9m57s Source on kvant.digital Problem Three triangles—white, red, and green—have a common interior point $M$. Prove that it is possible to choose one vertex from each triangle so that the point $M$ lies inside or on the boundary of the triangle whose vertices are the chosen points of the three different colors. Imre Bárány (Hungary) Exploration Let the three triangles be $T_W,T_R,T_G$,...

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Kvant Math Problem 1156

Verified: no Verdicts: SKIP + SKIP Solve time: 7m23s Source on kvant.digital Problem Eight hockey teams compete against each other for a spot in the final four. (Each pair of teams plays once; a win awards two points, a draw one point, and a loss zero points.) What is the minimum number of points that guarantees a place in the final four? Sanzhar Khadzhiev, 10th-grade student Exploration There are eight...

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Kvant Math Problem 1155

Verified: no Verdicts: SKIP + SKIP Solve time: 5m30s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. Exploration A complete solution cannot be written from the information provided. The statement of Kvant problem M1155 is not included, and the prompt explicitly says that only the graphical version is available in the published issue....

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Kvant Math Problem 1154

Verified: no Verdicts: SKIP + SKIP Solve time: 6m43s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I cannot write a solution to Kvant problem M1154 because the actual problem statement is not present in your message. The prompt says: “The textual version of the problem statement is being prepared. The graphical version can...

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Kvant Math Problem 1153

Verified: no Verdicts: SKIP + SKIP Solve time: 3m45s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I can do that. Please provide the full text of Kvant problem M1153 so I can begin the detailed solution following your six-section structure.

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Kvant Math Problem 1152

Verified: no Verdicts: SKIP + SKIP Solve time: 9m36s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I do not have the statement of Kvant problem M1152, and the prompt says that only the graphical version is currently available. Please upload the image of the problem (the scan, photograph, or screenshot from the magazine)....

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Kvant Math Problem 1151

Verified: no Verdicts: SKIP + SKIP Solve time: 5m34s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M1151 from the information provided, because the actual problem statement is missing. The prompt says: “The textual version of the problem statement is being prepared. The graphical version can...

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Kvant Math Problem 1150

Verified: no Verdicts: SKIP + SKIP Solve time: 2m09s Source on kvant.digital Problem Prove that for any positive $a_1$, $a_2$, $a_3$, $\ldots$, $a_n$, the inequality $$\dfrac{(a_1+a_2+\ldots+a_n)^2}{2(a_1^2+a_2^2+\ldots+a_n^2)}\le\dfrac{a_1}{a_2+a_3}+\dfrac{a_2}{a_3+a_4}+\ldots+\dfrac{a_n}{a_1+a_2}.$$ holds. E. G. Moiseev, 10th-grade student Exploration For $n=3$ the inequality becomes $$\frac{(a_1+a_2+a_3)^2}{2(a_1^2+a_2^2+a_3^2)} \le \frac{a_1}{a_2+a_3}+\frac{a_2}{a_3+a_1}+\frac{a_3}{a_1+a_2}.$$ The right-hand side is the expression from Nesbitt's inequality, whose lower bound is $3/2$. The left-hand side is at most $$\frac{3(a_1^2+a_2^2+a_3^2)}{2(a_1^2+a_2^2+a_3^2)} =\frac32$$ by Cauchy. Hence the statement is true...

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Kvant Math Problem 1149

Verified: no Verdicts: SKIP + SKIP Solve time: 1m40s Source on kvant.digital Problem On the plane, two rays $p$ and $q$ with vertices at points $P$ and $Q$, respectively, are given. Two circles—one with its center on ray $p$ passing through point $P$, and the other with its center on ray $q$ passing through $Q$—are externally tangent to each other at point $M$. Find the set of points $M$. V....

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Kvant Math Problem 1148

Verified: no Verdicts: SKIP + SKIP Solve time: 7m30s Source on kvant.digital Problem Prove that for a non-integer $a \gt 1$ (with $a\ne \sqrt[p]{q}$, where $p$ and $q$ are positive integers) and a positive integer $n$, the equality $$[\log_a2]+[\log_a3]+\ldots+[\log_an]+[a]+[a^2]+\ldots+[a^k]=nk,$$ holds, where $k=[\log_an]$ ($[x]$ denotes the integer part of the number $x$). R. B. Alekseev, L. D. Kurlyandchik Exploration Consider small values of $a$ and $n$ to understand the pattern. Let...

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Kvant Math Problem 1147

Verified: no Verdicts: SKIP + SKIP Solve time: 3m08s Source on kvant.digital Problem Several points are given, connected by segments of two colors: some pairs of points are connected by blue segments, and some other pairs by red segments. It is known that in any closed path consisting of several segments, the number of red segments is even. Prove that all points can be divided into two sets so that...

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Kvant Math Problem 1146

Verified: no Verdicts: SKIP + SKIP Solve time: 3m55s Source on kvant.digital Problem Point $K$ is the midpoint of side $AB$ of the equilateral triangle $ABC$. On sides $AC$ and $BC$, points $M$ and $N$ are taken such that $\angle MKN=60^{\circ}$. Prove that the perimeter of triangle $MCN$ is equal to half the perimeter of triangle $ABC$. E. G. Gotman Exploration Place the equilateral triangle $ABC$ in the plane with...

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Kvant Math Problem 1145

Verified: no Verdicts: SKIP + SKIP Solve time: 2m51s Source on kvant.digital Problem From the point $P$, two tangents $PB$ and $PC$ are drawn to a circle, with $\angle BPC \gt 90^{\circ}$. On the smaller arc $BC$, a point $A$ is chosen. Prove that the area of the triangle cut off from the angle $BPC$ by the tangent to the circle at the point $A$ does not exceed the area...

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Kvant Math Problem 1144

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem Let $a_1$, $a_2$, $\ldots$, $a_n$ be nonnegative numbers. Which number is greater: $$\sqrt[\scriptstyle1988]{a_1^{1988}+a_2^{1988}+\ldots+a_n^{1988}}\quad\text{или}\quad \sqrt[\scriptstyle1989]{a_1^{1989}+a_2^{1989}+\ldots+a_n^{1989}}~?$$ A. I. Shekhorsky Exploration Let $$X=\left(a_1^{1988}+a_2^{1988}+\cdots+a_n^{1988}\right)^{1/1988}, \qquad Y=\left(a_1^{1989}+a_2^{1989}+\cdots+a_n^{1989}\right)^{1/1989}.$$ The question asks whether $X$ or $Y$ is larger. First test simple cases. If $n=1$, then $$X=Y=a_1.$$ If $a_1=a_2=1$, then $$X=2^{1/1988},\qquad Y=2^{1/1989},$$ and since $1/1988>1/1989$, $$X>Y.$$ If $a_1=2$, $a_2=0$, then $$X=Y=2.$$ These examples suggest that...

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Kvant Math Problem 1143

Verified: no Verdicts: SKIP + SKIP Solve time: 3m39s Source on kvant.digital Problem The mass of each of the 101 weights, arranged in a circle, is a natural number, and their total mass is 300 g. Prove that from this set one can select one or more consecutive weights whose total mass is 200 g. V. V. Proizvolov Exploration Consider first small circular arrangements of weights with integer masses and...

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Kvant Math Problem 1142

Verified: no Verdicts: SKIP + SKIP Solve time: 4m02s Source on kvant.digital Problem The table $m\times n$ is filled with $mn$ numbers such that in each row and each column these numbers form an arithmetic progression. The sum of the four numbers located at the corners of the table equals $s$. What is the sum of all the numbers in the table? Exploration Consider small tables first. For a $2\times...

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Kvant Math Problem 1141

Verified: no Verdicts: SKIP + SKIP Solve time: 7m59s Source on kvant.digital Problem A trapezoid is circumscribed around a circle. Prove that at least one of its diagonals forms an angle with a base not exceeding $45^\circ$. N. M. Sedrakyan Exploration Consider a trapezoid $ABCD$ with $AB$ and $CD$ as the bases, $AB \parallel CD$, and a circle inscribed within it. Since the trapezoid is circumscribed, the sums of opposite...

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Kvant Math Problem 1140

Verified: no Verdicts: SKIP + SKIP Solve time: 7m26s Source on kvant.digital Problem Let us draw one or several intersecting curves in the plane (the curves may have self-intersection points; Fig. 1). At each intersection point, one can perform a “reconnection” in either of two ways (Fig. 2). If a reconnection is carried out at every intersection point, the result will be several nonintersecting curves (Fig. 3). Prove that the...

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Kvant Math Problem 1139

Verified: no Verdicts: SKIP + SKIP Solve time: 8m09s Source on kvant.digital Problem The surface of a convex polyhedron can be cut into several squares. Prove that this polyhedron has at most 8 vertices. What is the largest number of vertices a convex polyhedron can have if its surface can be cut into equilateral triangles? V. E. Matisen Exploration For a convex polyhedron whose faces are all squares, every face...

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Kvant Math Problem 1138

Verified: no Verdicts: SKIP + SKIP Solve time: 3m57s Source on kvant.digital Problem Prove that for every natural number $n$, among the numbers $n^2$ and $n^2+n+3\sqrt{n}$ there exist three natural numbers such that the product of two of them is divisible by the third. L. D. Kurlyandchik Exploration The expression $n^2+n+3\sqrt n$ is not always an integer. The statement says that among the numbers $n^2$ and $n^2+n+3\sqrt n$ there exist...

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Kvant Math Problem 1137

Verified: no Verdicts: SKIP + SKIP Solve time: 2m58s Source on kvant.digital Problem In a convex $n$-gon, all angles are equal, and from some point located inside the $n$-gon, all of its sides are seen under equal angles. Prove that this $n$-gon is regular. K. P. Kokhas Exploration Consider first small polygons. For a triangle ($n=3$), an equiangular triangle is automatically equilateral. For a quadrilateral ($n=4$), an equiangular quadrilateral is...

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Kvant Math Problem 1136

Verified: no Verdicts: SKIP + SKIP Solve time: 2m28s Source on kvant.digital Problem Prove the inequality $$3+(A+M+S)+\left(\dfrac1A+\dfrac1M+\dfrac1S\right)+ \left(\dfrac AM+\dfrac MS+\dfrac SA\right)\ge \dfrac{3(A+1)(M+1)(S+1)}{AMS+1}.$$ for non-negative numbers $A$, $M$, $S$. The author dedicated this problem to the 100th anniversary of the American Mathematical Society ( AMS ), celebrated this year. D. P. Mavlo Exploration Testing small integer values for $A$, $M$, and $S$ helps to gain intuition about the inequality. Setting $A=M=S=1$,...

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Kvant Math Problem 1135

Verified: no Verdicts: SKIP + SKIP Solve time: 3m16s Source on kvant.digital Problem Let $a$ and $b$ be positive integers such that $a^2+b^2$ is divisible by $ab+1$. Prove that $\dfrac{a^2+b^2}{ab+1}$ is the square of an integer. International Mathematical Olympiad for School Students (XXIX) Exploration Let $$\frac{a^2+b^2}{ab+1}=k,$$ where $k$ is a positive integer. We must prove that $k$ is a perfect square. Small examples suggest the claim. If $a=b=1$, then $k=1$....

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Kvant Math Problem 1134

Verified: no Verdicts: SKIP + SKIP Solve time: 3m33s Source on kvant.digital Problem Let $AD$ be the altitude in the right triangle $ABC$, $\angle A=90^\circ$. The line passing through the centers of the incircles of triangles $ABD$ and $ACD$ intersects the sides $AB$ and $AC$ at points $K$ and $L$, respectively. Prove the inequality $S_{ABC}\ge 2S_{AKL}$. International Mathematical Olympiad for School Students (XIX) Exploration Consider a right triangle $ABC$ with...

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Kvant Math Problem 1133

Verified: no Verdicts: SKIP + SKIP Solve time: 3m22s Source on kvant.digital Problem Prove that the set of solutions of the inequality $$\textstyle\sum\limits_{k=1}^{70}\dfrac{k}{x-k}\ge \dfrac54$$ is a union of disjoint intervals, the sum of whose lengths is 1988. International Mathematical Olympiad for School Students (XXIX) Exploration Consider the sum $$f(x) = \sum_{k=1}^{70} \frac{k}{x-k}.$$ Each term $\frac{k}{x-k}$ has a vertical asymptote at $x=k$ and is strictly decreasing on each interval $(k-1, k)$...

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Kvant Math Problem 1132

Verified: no Verdicts: SKIP + SKIP Solve time: 7m47s Source on kvant.digital Problem The function $f$ is defined on the set of positive integers and satisfies the following conditions: $$\begin{gathered} f(1)=1,\quad f(3)=3,\quad f(2n)=f(n),\ f(4n+1)=2f(2n+1)-f(n),\quad f(4n+3)=3f(2n+1)-2f(n) \end{gathered}$$. Find the number of all such values $n$ for which $f(n)=n$ and $1\le n\le 1988$. International Mathematical Olympiad for School Students (XXIX) Exploration Let $$e(n)=f(n)-n.$$ Substituting the defining relations gives $$e(2n)=e(n)-n,$$ and for odd...

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Kvant Math Problem 1131

Verified: no Verdicts: SKIP + SKIP Solve time: 4m04s Source on kvant.digital Problem Let $n$ be a positive integer, and let $A_1$, $A_2$, $\ldots$, $A_{2n+1}$ be subsets of some set $B$. Suppose that each set $A_i$ ($i=1$, 2, $\ldots$, $2n+1$) contains exactly $2n$ elements; each set $A_i\cap A_j$ ($1\le i\lt j\le 2n+1$) contains exactly one element; every element of the set $B$ belongs to at least two of the sets...

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Kvant Math Problem 1130

Verified: no Verdicts: SKIP + SKIP Solve time: 6m10s Source on kvant.digital Problem On the plane, a convex $n$-gon is given, whose $k$-th side has length $a_k$, and the length of the projection of the polygon onto the line containing this side is $d_k$ ($k=1$, 2, $\ldots$, $n$). Prove the inequality $$2\le\dfrac{a_1}{d_1}+\ldots+\dfrac{a_n}{d_n}\le4.$$. D. V. Fomin Leningrad City Mathematical Olympiad (1988) Exploration Consider first a simple convex polygon, such as a...

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Kvant Math Problem 1129

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I cannot write a solution to Kvant problem M1129 because the actual problem statement is not present in the conversation. The prompt says that “the textual version of the problem statement is being prepared” and that the...

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Kvant Math Problem 1128

Verified: no Verdicts: SKIP + SKIP Solve time: 8m33s Source on kvant.digital Problem Several pieces are placed on a chessboard. In one move, a piece can be moved to an adjacent (horizontally or vertically) empty square. After several moves, it turned out that each piece had visited every square exactly once and returned to its original square. Prove that there was a moment when no piece was on its original...

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Kvant Math Problem 1127

Verified: no Verdicts: SKIP + SKIP Solve time: 9m18s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I notice that the problem statement itself is not yet provided. Could you supply the full text of Kvant problem M1127 so I can write the complete, rigorous solution following your six-section structure?

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Kvant Math Problem 1126

Verified: no Verdicts: SKIP + SKIP Solve time: 8m51s Source on kvant.digital Problem In the trapezoid $ABCD$ (with bases $BC$ and $AD$), points $K$ and $M$ are chosen on the sides $AB$ and $CD$. Prove that if $\angle{BAM}=\angle{CDK}$, then $\angle{BMA}=\angle{CKD}$. A. S. Merkuriev Leningrad City Mathematical Olympiad (1988) Exploration The statement resembles a converse of a familiar fact about equal angles subtending the same segment. The desired conclusion says that...

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Kvant Math Problem 1125

Verified: no Verdicts: SKIP + SKIP Solve time: 2m53s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M1125 from the information provided, because the actual problem statement is missing. The prompt says: “The textual version of the problem statement is being prepared. The graphical version can...

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Kvant Math Problem 1124

Verified: no Verdicts: SKIP + SKIP Solve time: 8m33s Source on kvant.digital Problem The lateral sides, diagonals, and extensions of the bases of a trapezoid intersect the line $l$ at six points, thus cutting five segments on the line $l$. Prove that if the extreme segments (the 1st and 5th) are equal, then the segments adjacent to them (the 2nd and 4th) are also equal. For what ratio of the...

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Kvant Math Problem 1123

Verified: no Verdicts: SKIP + SKIP Solve time: 10m09s Source on kvant.digital Problem A right angle is partitioned into infinitely many unit square cells. Consider the rows of cells parallel to the sides of the angle (“vertical” and “horizontal” rows). Is it possible to write a positive integer in each cell so that every vertical row and every horizontal row contains each positive integer exactly once? V. S. Shevelev Tournament...

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Kvant Math Problem 1122

Verified: no Verdicts: SKIP + SKIP Solve time: 9m02s Source on kvant.digital Problem Solve the system: $$\left{ \begin{array}{l} (x_3+x_4+x_5)^5=3x_1,\ (x_4+x_5+x_1)^5=3x_2,\ (x_5+x_1+x_2)^5=3x_3,\ (x_1+x_2+x_3)^5=3x_4,\ (x_2+x_3+x_4)^5=3x_5. \end{array} \right.$$ L. Tutescu (Romania) City Tournament (Spring, 1988) Exploration Let $$s_1=x_3+x_4+x_5,\quad s_2=x_4+x_5+x_1,\quad s_3=x_5+x_1+x_2,$$ $$s_4=x_1+x_2+x_3,\quad s_5=x_2+x_3+x_4.$$ Then the system becomes $$3x_i=s_i^5 \qquad (i=1,\dots,5).$$ Substituting $x_i=\dfrac{s_i^5}{3}$ into the definitions of the $s_i$ yields $$s_i=\frac{s_{i+2}^5+s_{i+3}^5+s_{i+4}^5}{3},$$ with indices taken modulo $5$. The cyclic symmetry suggests that all variables may be...

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Kvant Math Problem 1121

Verified: no Verdicts: SKIP + SKIP Solve time: 5m37s Source on kvant.digital Problem Given a triangle $ABC$. Two lines, symmetric to the line $AC$ with respect to the lines $AB$ and $BC$ respectively, intersect at the point $K$. Prove th_ The statement of the problem is incomplete: “Given a triangle $ABC$. Two lines, symmetric to the line $AC$ with respect to the lines $AB$ and $BC$ respectively, intersect at the...

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Kvant Math Problem 1120

Verified: no Verdicts: SKIP + SKIP Solve time: 3m13s Source on kvant.digital Problem The sequence $a_0$, $a_1$, $a_2$, $\ldots$ is defined by the relations $a_0=0$, $a_n=P(a_{n-1})$, $n=1$, $2$, $\ldots$, where $P(x)$ is a polynomial with integer coefficients, $P(x)\gt0$ for $x\ge 0$. Prove that for any positive integers $m$ and $k$, $$\gcd(a_m,a_k)=a_{\gcd(m,k)}.$$ Prove an analogous statement for the Fibonacci sequence $a_0=0$, $a_1=1$, $a_2=1$, $a_3=2$, $\ldots$, defined by the conditions $a_{n+1}=a_n+a_{n-1}$, $n=1$,...

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Kvant Math Problem 1119

Verified: no Verdicts: SKIP + SKIP Solve time: 8m52s Source on kvant.digital Problem We call a figure on the plane a $k$-star if it consists of $k$ rays with a common origin, dividing the plane into $k$ equal angles (measured in $360^\circ/k$). For which $k\gt 2$ does the following statement hold: for any $k$ points in general position on the plane (no three of which lie on the same line),...

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Kvant Math Problem 1118

Verified: no Verdicts: SKIP + SKIP Solve time: 8m06s Source on kvant.digital Problem Prove that the equation $$(x+y+z)^2=x^2+y^2+z^2\tag{$*$}$$ has infinitely many integer solutions. How many such solutions are there for which $z=1988$? S. G. Mamikonyan Exploration Expanding the left-hand side gives $$(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx).$$ Hence $(*)$ is equivalent to $$xy+yz+zx=0.$$ The first question asks for infinitely many integer triples satisfying this quadratic equation. A natural way to proceed is to solve for...

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Kvant Math Problem 1117

Verified: no Verdicts: SKIP + SKIP Solve time: 10m40s Source on kvant.digital Problem Given an arbitrary triangle, prove that: It is possible to construct three circles centered at its vertices, each tangent to the others (at points $K$, $L$, $M$ — see Fig. 1); If a tangent is drawn at the midpoint of each arc $KL$, $LM$, $MK$ lying inside the triangle, four triangles are formed, one of which (the...

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Kvant Math Problem 1116

Verified: no Verdicts: SKIP + SKIP Solve time: 9m26s Source on kvant.digital Problem What is the maximum number of grid points a rectangle can contain if its area is: 36; $S$, with sides aligned along the grid lines? (Grid points lying inside and on the boundary of the rectangle are counted. The area of a single cell is taken as 1.) Exploration Consider a rectangle drawn on a square grid...

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Kvant Math Problem 1115

Verified: no Verdicts: SKIP + SKIP Solve time: 5m56s Source on kvant.digital Problem The first row contains 19 positive integers not exceeding 88, and the second row contains 88 positive integers not exceeding 19. Call a segment one or more consecutive numbers written in the same row. Prove that it is possible to choose one segment from each row so that the sums of the numbers in them are equal....

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Kvant Math Problem 1114

Verified: no Verdicts: SKIP + SKIP Solve time: 4m45s Source on kvant.digital Problem Prove that for any tetrahedron the inequality $$r\lt\dfrac{ab}{2(a+b)},$$ holds, where $a$, $b$ are the lengths of two skew edges, and $r$ is the radius of the inscribed sphere. I. F. Sharygin Exploration Consider a tetrahedron with vertices $A$, $B$, $C$, $D$ and let $a = AB$ and $b = CD$ be two skew edges. The inequality involves...

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Kvant Math Problem 1113

Verified: no Verdicts: SKIP + SKIP Solve time: 7m11s Source on kvant.digital Problem There are 21 cities in a country. Air service between them is provided by several airlines, each of which operates 10 nonstop routes connecting pairwise some five cities (more than one airline may operate flights between the same two cities). Every pair of cities is connected by at least one nonstop route. What is the smallest possible...

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Kvant Math Problem 1112

Verified: no Verdicts: SKIP + SKIP Solve time: 8m02s Source on kvant.digital Problem Two numbers, 1 and 2, are written on the board. One may append new numbers according to the following rule: if the numbers $a$ and $b$ are on the board, then one may write the number $ab+a+b$. Can one obtain in this way the number 13121; the number 12131? Prove that there exist infinitely many natural numbers...

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Kvant Math Problem 1111

Verified: no Verdicts: SKIP + SKIP Solve time: 2m16s Source on kvant.digital Problem Around the acute-angled triangle $ABC$, a circle is circumscribed. Tangents to the circle drawn at points $A$ and $C$ intersect the tangent drawn at point $B$ at points $M$ and $N$, respectively. In triangle $ABC$, the altitude $BP$ is drawn (point $P$ lies on side $AC$). Prove that the line $BP$ is the bisector of angle $MPN$....

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Kvant Math Problem 1110

Verified: no Verdicts: SKIP + SKIP Solve time: 2m21s Source on kvant.digital Problem For a given natural number $n\gt1$, list the greatest common divisors of all possible pairs of distinct numbers from 1 to $n$. Prove that the arithmetic mean of all $\dfrac{n(n-1)}2$ listed numbers grows without bound as $n$ increases, but does not exceed $\ln n+1$; their geometric mean does not exceed 10 for any $n$. V. F. Lev...

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Kvant Math Problem 1109

Verified: no Verdicts: SKIP + SKIP Solve time: 17m21s Source on kvant.digital Problem In an old geometry problem book, the following problem appeared: compute the side length of an equilateral triangle inscribed in the parabola $y=x^2$. The hint to the problem stated that one of the triangle's vertices coincides with the vertex of the parabola. Is this hint correct? Can the side length of an equilateral triangle inscribed in this...

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Kvant Math Problem 1108

Verified: no Verdicts: SKIP + SKIP Solve time: 1m56s Source on kvant.digital Problem In a convex $n$-gon ($n\gt4$), no three diagonals intersect at a single point inside the polygon. What is the maximum number of diagonals that can be drawn so that all the regions into which they divide the $n$-gon are triangles? M. Khovanov, 10th-grade student (Moscow) Exploration Consider small cases first. For $n=5$, the convex pentagon, a triangulation...

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Kvant Math Problem 1107

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem Prove that if $a$, $b$, and $c$ are the side lengths of a triangle, then $$2\left(\frac{a}{b}+\frac{b}{c}+\frac{a}{c}\right)\ge\frac{a}{c}+\frac{c}{b}+\frac{b}{a}+3.$$ L. D. Kurlandchik Exploration The inequality is homogeneous in the ratios of the sides. Since $a,b,c$ are sides of a triangle, the triangle inequalities are available. Rewriting the desired inequality gives $$2\frac ab+\frac bc+\frac ac-\frac cb-\frac ba-3\ge0.$$ The expression is not...

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Kvant Math Problem 1106

Verified: no Verdicts: SKIP + SKIP Solve time: 7m29s Source on kvant.digital Problem Each of the three lines connecting the midpoints of opposite sides of a convex hexagon divides its area in half. Prove that these three lines intersect at a single point. V. V. Proizvolov Exploration Consider a convex hexagon $ABCDEF$. Let $M_1$, $M_2$, and $M_3$ be the midpoints of opposite sides $AB$–$DE$, $BC$–$EF$, and $CD$–$FA$, respectively. Draw the...

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Kvant Math Problem 1105

Verified: no Verdicts: SKIP + SKIP Solve time: 1m20s Source on kvant.digital Problem After several straight-line cuts, the surface of a convex polyhedron was unfolded into the plane. The result was a polygon for which it is known which points on its boundary are "glued together," i.e., correspond to the same point on the surface of the polyhedron. What was the original polyhedron if the cutting produced a rectangle with...

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Kvant Math Problem 1104

Verified: no Verdicts: SKIP + SKIP Solve time: 1m14s Source on kvant.digital Problem In the tetrahedron $ABCD$, the faces $ABC$ and $BCD$ are perpendicular, $\angle BAC=90^\circ$. Prove that from segments whose lengths are equal to the products of the lengths of opposite edges of the tetrahedron, one can form a right triangle. V. N. Dubrovsky Exploration Let $$x=AB\cdot CD,\qquad y=AC\cdot BD,\qquad z=AD\cdot BC.$$ The statement says that the three numbers...

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Kvant Math Problem 1103

Verified: no Verdicts: SKIP + SKIP Solve time: 10m39s Source on kvant.digital Problem On an infinite plane divided into square cells, some—possibly infinite—number of rectangles of size $1\times2$ are colored black so that no two black rectangles share any points (not even vertices). Prove that the remaining part of the plane can be tiled with these rectangles. Let several rectangles of size $m\times n$, which do not share any points,...

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Kvant Math Problem 1102

Verified: no Verdicts: SKIP + SKIP Solve time: 3m56s Source on kvant.digital Problem Prove that there exist $n$ distinct positive integers whose sum of cubes is equal to the cube of a positive integer, if $n=3$, $n=4$, $n$ is any positive integer greater than 2. L. D. Kurlyandchik Exploration For $n=3$ it is natural to search among classical identities involving sums of three cubes. The smallest nontrivial example is $3^3+4^3+5^3=6^3,$...

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Kvant Math Problem 1101

Verified: no Verdicts: SKIP + SKIP Solve time: 7m27s Source on kvant.digital Problem On the lateral sides $AB$ and $AC$ of the isosceles triangle $ABC$, there were found points $D$ and $E$, respectively, such that $AD=BC=EC$ and the triangle $ADE$ is isosceles. What can the angle at the vertex $A$ be? V. Kiriak (Romania) Exploration Let $\angle A=\alpha$. Since $ABC$ is isosceles with $AB=AC$, put $AB=AC=s,\qquad BC=b.$ The condition $AD=BC=EC$...

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Kvant Math Problem 1100

Verified: no Verdicts: SKIP + SKIP Solve time: 7m10s Source on kvant.digital Problem Fig. 1 On the bank of a straight river lie logs (non-intersecting segments; Fig. 1). Each log forms an angle with the riverbank of less than $45^\circ$. Prove that for any arrangement of the logs, there is always at least one log that can be rolled into the river without touching the others. (Rotating the log while...

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Kvant Math Problem 1099

Verified: no Verdicts: SKIP + SKIP Solve time: 1m09s Source on kvant.digital Problem In a team preparing for a flight to Mars, there are 6783 astronauts, and it is known that among any four of them, one can choose three who form a well-coordinated crew for the landing module. Prove that it is possible to select 5 astronauts such that any three of them form a well-coordinated crew. N. N....

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Kvant Math Problem 1098

Verified: no Verdicts: SKIP + SKIP Solve time: 5m16s Source on kvant.digital Problem On a circle, $n$ points are placed, numbered consecutively 1, 2, $\ldots$, $n$. Two players play the following game. They take turns drawing a chord connecting two points with numbers of the same parity. No chord may intersect any previously drawn chord, not even at endpoints. The winner is the player who makes the last move. For...

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Kvant Math Problem 1097

Verified: no Verdicts: SKIP + SKIP Solve time: 1m28s Source on kvant.digital Problem The coordinates of the vertices of an isosceles triangle are integers. Prove that the square of its base is an even number. V. V. Proizvolov Exploration Consider small examples of isosceles triangles whose vertices have integer coordinates. The simplest such triangles are right triangles aligned with the axes. For instance, the points $(0,0)$, $(2,0)$, $(1,1)$ form an...

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Kvant Math Problem 1096

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem The diameter $d$ of a circle is divided into $k$ equal parts, and through each division point a chord perpendicular to the diameter is drawn. Prove that the sum of the lengths of all the chords is not less than $0{,}5kd$ and less than $0{,}8kd$. R. Kharitonov, A. Chagirov, 10th-grade students Exploration Let the circle have radius...

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Kvant Math Problem 1095

Verified: no Verdicts: SKIP + SKIP Solve time: 1m15s Source on kvant.digital Problem On the plane, a circle with center $O$ and two points $A$, $B$ (distinct from $O$) are given such that the line $AB$ passes through the point $O$. Construct a chord $MN$ of this circle that is seen from the point $A$ under a given angle $\alpha$ and is parallel to the line $AB$; passes through the...

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Kvant Math Problem 1094

Verified: no Verdicts: SKIP + SKIP Solve time: 7m30s Source on kvant.digital Problem Let $a$, $b$, $c$ be non-negative numbers. Prove that inequality $$a^4+b^4+c^4\le 2(a^2b^2+b^2c^2+c^2a^2) \tag{1}$$ implies inequality $$a^2+b^2+c^2\le 2(ab+bc+ca). \tag{2}$$. Is the converse true: does inequality (2) imply inequality (1)? V. A. Senderov Exploration The two inequalities are $$a^4+b^4+c^4\le 2(a^2b^2+b^2c^2+c^2a^2),$$ and $$a^2+b^2+c^2\le 2(ab+bc+ca).$$ The first inequality is homogeneous of degree $4$, the second of degree $2$. A natural idea...

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Kvant Math Problem 1093

Verified: no Verdicts: SKIP + SKIP Solve time: 8m08s Source on kvant.digital Problem Numbers 0, 1, and 2 are placed at $n$ points on a circle. Then the following transformation is performed simultaneously at all points: each number 2 is replaced by 0, and then 1 is added to the number at the next point clockwise. Suppose that initially the number of twos was equal to $k\ge 1$. After how...

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Kvant Math Problem 1092

Verified: no Verdicts: SKIP + SKIP Solve time: 7m15s Source on kvant.digital Problem A convex polygon cut out of paper is folded 10 times (along certain lines) and then cut along a straight line. What is the greatest possible number of pieces that can result? S. V. Kazakov Exploration Consider a single fold of a convex polygon and a subsequent straight cut. Each fold effectively doubles the number of layers...

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Kvant Math Problem 1090

Verified: no Verdicts: SKIP + SKIP Solve time: 1m43s Source on kvant.digital Problem Prove that for any positive numbers $a$, $b$, and $c$, the inequality $$\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}\ge\sqrt{a^2+ac+c^2};$$ holds; the inequality from part (a) becomes an equality if and only if $\dfrac1a+\dfrac1c=\dfrac1b$. Yu. V. Deikalo Exploration Testing small values helps build intuition about the inequality. Let $a=b=c=1$. Then the left-hand side becomes $\sqrt{1-1+1}+\sqrt{1-1+1}= \sqrt{1}+\sqrt{1}=2$, and the right-hand side is $\sqrt{1+1+1}=\sqrt{3} \approx 1.732$,...

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Kvant Math Problem 1089

Verified: no Verdicts: SKIP + SKIP Solve time: 2m01s Source on kvant.digital Problem In the convex quadrilateral $ABCD$ with area $S$, the diagonals intersect at the point $O$. Let $K$, $L$, $M$, $N$ be the centers of the circles inscribed in the triangles $AOB$, $BOC$, $COD$, and $DOA$, respectively. Prove that the product of the perimeters of the quadrilaterals $ABCD$ and $KLMN$ is at least $4S$. D. Yu. Burago, F....

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Kvant Math Problem 1088

Verified: no Verdicts: SKIP + SKIP Solve time: 1m41s Source on kvant.digital Problem Prove that if the numbers $p$, $q$, $r$ are rational and $pq+qr+pr=1$, then $(1+p^2)(1+q^2)(1+r^2)$ is the square of a rational number. István Varga (Romania) Exploration The condition is $$pq+qr+rp=1.$$ We must show that $$(1+p^2)(1+q^2)(1+r^2)$$ is a rational square. A direct expansion gives $$(1+p^2)(1+q^2)(1+r^2) =1+(p^2+q^2+r^2)+(p^2q^2+q^2r^2+r^2p^2)+p^2q^2r^2.$$ Since $$p^2q^2+q^2r^2+r^2p^2=(pq+qr+rp)^2-2pqr(p+q+r),$$ the given condition replaces this term by $$1-2pqr(p+q+r).$$ Also $$p^2+q^2+r^2=(p+q+r)^2-2.$$ Substituting,...

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Kvant Math Problem 1087

Verified: no Verdicts: SKIP + SKIP Solve time: 1m50s Source on kvant.digital Problem Consider a triangle $ABC$, a point $M$ in the plane of this triangle, and the projections $A_1$, $B_1$, $C_1$ of the point $M$ onto the altitudes drawn from the vertices $A$, $B$, $C$ respectively. Prove that there exists one and only one point $M$ for which the segments $AA_1$, $BB_1$, and $CC_1$ are equal; for such a...

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Kvant Math Problem 1086

Verified: no Verdicts: SKIP + SKIP Solve time: 1m51s Source on kvant.digital Problem You are allowed to perform two operations on a number: "double" and "increase by 1". What is the minimum number of operations required to obtain from 0 the following numbers: 100? $n$? M. V. Sapir Exploration Consider the problem of reaching a target number from $0$ using only two operations: doubling the current number or adding $1$....

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Kvant Math Problem 1085

Verified: no Verdicts: SKIP + SKIP Solve time: 2m05s Source on kvant.digital Problem Several pairwise skew lines arranged in space are projected onto a horizontal plane. Their projections are drawn so that at the points of intersection it is clear which point lies above and which below. Is it possible to obtain the projection shown in Figure 1, a–c ? Figure 1 S. L. Tabachnikov Exploration Consider the problem geometrically....

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Kvant Math Problem 1084

Verified: no Verdicts: SKIP + SKIP Solve time: 1m55s Source on kvant.digital Problem Two circles in a plane intersect at the points $A$ and $B$. Prove that it is possible to choose a point $C$ such that any circle having chord $AC$ will intersect the given circles (for the second time) at points equidistant from $C$ (with $C\ne B$). V. Yu. Protasov, 10th-grade student (Moscow) Exploration Let the two given...

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Kvant Math Problem 1083

Verified: no Verdicts: SKIP + SKIP Solve time: 2m22s Source on kvant.digital Problem The largest of the non-negative numbers $a_1$, $a_2$, $\ldots$, $a_n$ is $a$. Prove the inequality $$\frac{a_1^2+a_2^2+\ldots+a_n^2}n\le\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)^2+\frac{a^2}4$$ When is equality achieved? L. G. Khanin Exploration Consider small values of $n$ to understand the inequality. For $n=2$, let the numbers be $a_1$ and $a_2$ with $a_2 \ge a_1 \ge 0$ and $a_2 = a$. The inequality becomes $\frac{a_1^2 +...

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Kvant Math Problem 1082

Verified: no Verdicts: SKIP + SKIP Solve time: 2m09s Source on kvant.digital Problem In a convex quadrilateral $ABCD$, the diagonals intersect at point $O$. Prove that the equality $$AB^2+BC^2+CD^2+DA^2=2(AO^2+BO^2+CO^2+DO^2)$$ holds if and only if either the diagonals $AC$ and $BD$ are perpendicular, or one of them is bisected by point $O$. A. P. Savin Exploration The given equality resembles the identity for the sum of squares of the sides of...

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Kvant Math Problem 1081

Verified: no Verdicts: SKIP + SKIP Solve time: 1m43s Source on kvant.digital Problem Prove that the penultimate digit of the number $3^n$ is even for any natural number $n\gt2$. V. I. Plachko Exploration Compute a few values: $$3^3=27,\qquad 3^4=81,\qquad 3^5=243,\qquad 3^6=729.$$ The penultimate digits are respectively $2,8,4,2$, all even. The penultimate digit of a positive integer is the tens digit. If a number $N$ is written as $$N=100q+r,\qquad 0\le r<100,$$...

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Kvant Math Problem 1080

Verified: no Verdicts: SKIP + SKIP Solve time: 2m22s Source on kvant.digital Problem Let $q$ be a natural number, $q\ge3$. Prove that if $k^2+k+q$ is a prime number for all integers $k$, where $0\le k\le\sqrt{\dfrac q3}$, then $k^2+k+q$ is prime for all integers $k$, where $0\le k\le q-2$. V. F. Lev International Mathematical Olympiad for School Students (1987) Exploration Let $$f(k)=k^2+k+q.$$ We are told that $f(k)$ is prime for every...

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Kvant Math Problem 1079

Verified: no Verdicts: SKIP + SKIP Solve time: 2m25s Source on kvant.digital Problem Let $n$ be a natural number, $n\ge3$. Is it possible to place $n$ points on the plane so that the distance between any two of them is an irrational number, while the area of the triangle formed by any three of them is a rational number (different from zero)? International Mathematical Olympiad for School Students (1987) Exploration...

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Kvant Math Problem 1078

Verified: no Verdicts: SKIP + SKIP Solve time: 1m42s Source on kvant.digital Problem The function $f$ is defined on the set $\N_0$ of all nonnegative integers and takes values in this set. Prove that the equality $f(f(n))=n+1987$ cannot hold for all $n$ from $\N_0$. International Mathematical Olympiad for School Students (1987) Exploration Assume that a function $f:\mathbb N_0\to\mathbb N_0$ satisfies $$f(f(n))=n+1987$$ for every $n\in\mathbb N_0$. The equation already implies that...

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Kvant Math Problem 1077

Verified: no Verdicts: SKIP + SKIP Solve time: 1m44s Source on kvant.digital Problem Let $p_n(k)$ be the number of permutations of a set of $n$ ($n\ge1$) elements that have exactly $k$ fixed points. Prove that: $$\textstyle\sum\limits_{k=0}^nk\cdot p_n(k)=0\cdot p_n(0)+1\cdot p_n(1)+\ldots+n\cdot p_n(n)=n!.$$ $$\textstyle\sum\limits_{k=0}^n{}(k-1)^2p_n(k)=n!.$$ Note. A permutation of a finite set $S$ is a one-to-one mapping $f$ of the set $S$ onto itself; the number of all permutations of a set of $n$...

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Kvant Math Problem 1076

Verified: no Verdicts: SKIP + SKIP Solve time: 1m59s Source on kvant.digital Problem The bisector of angle $A$ of an acute-angled triangle $ABC$ intersects side $BC$ at point $L$ and the circumcircle of the triangle at point $N$ (distinct from $A$); $K$ and $M$ are the feet of the perpendiculars dropped from $L$ onto sides $AB$ and $AC$ (see Fig. 1). Prove that the quadrilateral $AKNM$ has the same area...

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Kvant Math Problem 1075

Verified: no Verdicts: SKIP + SKIP Solve time: 2m01s Source on kvant.digital Problem Find the largest natural number whose decimal representation satisfies the condition that each digit (except for the first and last) is strictly less than the half-sum of its two neighboring digits. S. E. Rukshin Leningrad City Mathematical Olympiad (1987) Exploration Consider the problem in terms of digit patterns. Let a number be $d_1 d_2 \dots d_n$ with...

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Kvant Math Problem 1074

Verified: no Verdicts: SKIP + SKIP Solve time: 11m28s Source on kvant.digital Problem A stack of $2n+1$ cards is given, from which the following two operations are allowed: A portion of cards is removed from the top and placed at the bottom, preserving their order; The top $n$ cards are placed, in order, into $n$ spaces between the bottom $n+1$ cards. Prove that using these operations, it is impossible to...

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Kvant Math Problem 1073

Verified: no Verdicts: SKIP + SKIP Solve time: 1m42s Source on kvant.digital Problem In the hexagon $A_1A_2A_3A_4A_5A_6$ there is a point $O$ from which all the sides are seen under an angle of $60^\circ$. Prove that if $OA_1\gt OA_3\gt OA_5$ and $OA_2\gt OA_4\gt OA_6$, then $$A_1A_2+A_3A_4+A_5A_6\lt A_2A_3+A_4A_5+A_6A_1.$$ A. S. Merkuryev Leningrad City Mathematical Olympiad (1987) Exploration Consider a hexagon $A_1A_2A_3A_4A_5A_6$ with a point $O$ from which all sides are seen...

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Kvant Math Problem 1072

Verified: no Verdicts: SKIP + SKIP Solve time: 1m50s Source on kvant.digital Problem Factor the number $989 \cdot 1001 \cdot 1007 +320$ into prime factors. S. V. Fomin Leningrad City Mathematical Olympiad (1987) Exploration The expression $989 \cdot 1001 \cdot 1007 + 320$ appears to involve three numbers spaced by six units: $989$, $1001$, $1007$. Observing the spacing suggests attempting a transformation using a central term, $1001$, and writing the...

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Kvant Math Problem 1071

Verified: no Verdicts: SKIP + SKIP Solve time: 2m23s Source on kvant.digital Problem On the board there is a game field for the “digits” game: $$((((((((( __ * __ ) * __ ) * __ ) * __ ) * __ ) * __ ) * __ ) * __ ) * __)$$ Two players take turns. On the first move, the first player writes some digit $( __ )$...

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Kvant Math Problem 1070

Verified: no Verdicts: SKIP + SKIP Solve time: 1m34s Source on kvant.digital Problem A tetrahedron is intersected by three planes, each of which is parallel to two of its opposite edges and equidistant from them. Prove that the sum of the squares of the areas of these three cross-sections is four times smaller than the sum of the squares of the areas of the faces of the tetrahedron. V. N....

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Kvant Math Problem 1069

Verified: no Verdicts: SKIP + SKIP Solve time: 1m39s Source on kvant.digital Problem In a certain city, only pairwise apartment exchanges are permitted (if two families exchange apartments, then on that day they do not participate in any other exchanges). Prove that any complex apartment exchange involving several families can be carried out in two days. (It is assumed that both before and after the exchange, each family lives in...

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Kvant Math Problem 1068

Verified: no Verdicts: SKIP + SKIP Solve time: 1m30s Source on kvant.digital Problem Given an angle $AOB$ ($A$ and $B$ are points on the sides of the angle). Construct a line $l$ passing through the vertex $O$ such that the areas of the triangles $AOC$ and $BOD$, where $C$ and $D$ are the feet of the perpendiculars dropped from the points $A$ and $B$ onto the line $l$, are equal....

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Kvant Math Problem 1067

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem Prove that for the nonnegative numbers $x$, $y$, $z$ satisfying the condition $x^2+y^2+z^2=1$, the inequality $$\dfrac x{1-x^2}+\dfrac y{1-y^2}+\dfrac z{1-z^2}\ge\dfrac{3\sqrt3}2.$$ holds. V. E. Matizen Exploration Let $$f(t)=\frac{t}{1-t^2}, \qquad 0\le t<1.$$ The condition is $$x^2+y^2+z^2=1,$$ and we must prove $$f(x)+f(y)+f(z)\ge \frac{3\sqrt3}{2}.$$ The right-hand side suggests checking the symmetric point $$x=y=z=\frac1{\sqrt3}.$$ Then $$f!\left(\frac1{\sqrt3}\right) = \frac{1/\sqrt3}{1-1/3} = \frac{\sqrt3}{2},$$ hence $$3f!\left(\frac1{\sqrt3}\right)=\frac{3\sqrt3}{2}.$$ So...

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Kvant Math Problem 1066

Verified: no Verdicts: SKIP + SKIP Solve time: 2m00s Source on kvant.digital Problem Six points are placed in the plane so that all pairwise distances between them are at most 1. Prove that one can choose three of these points whose pairwise distances are all strictly less than 1. S. G. Salnikov Exploration Consider six points in the plane with all pairwise distances at most $1$. For smaller numbers of...

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Kvant Math Problem 1065

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem We will consider vectors $(x;y)$ with non-negative integer coordinates, with at least one coordinate being nonzero. We call such a vector generating if $|x-y|=1$. Prove that the given vector $(x;y)$ can be represented as a sum of generating vectors (or is itself generating) if and only if the quantity $$k(x,y)=x+y-(x-y)^2$$ is non-negative. Prove that the number $n(x,y)$...

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Kvant Math Problem 1064

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem What is the maximum number of self-intersection points that a closed $n$-segment planar broken line can have if: $n$ is odd; $n$ is even? (It is assumed that no three vertices lie on a single straight line and that no three segments intersect at a single point.) D. B. Fuchs Exploration Let the closed broken line have...

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Kvant Math Problem 1063

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem How many distinct integers can be represented as the difference $a-\overline{a}$, where $a$ is a $n$-digit natural number ($10^{n-1}\le a\lt10^n$), and $\overline{a}$ is the number obtained by writing the digits of $a$ in reverse order? For example, if $a=1917$, then $a-\overline{a}=1917-7191=-5724$. Give the answer for: $n=4$; $n=5$; any natural $n$. G. O. Elsting Exploration Let the digits...

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Kvant Math Problem 1062

Verified: no Verdicts: SKIP + SKIP Solve time: 1m49s Source on kvant.digital Problem On the sides $AB$ and $AC$ of triangle $ABC$, points $E$ and $D$ are chosen. The lines $BD$ and $CE$ intersect at point $M$, $AM$ and $BC$ at point $P$, and $AM$ and $DE$ at point $N$. Prove that $$\dfrac{PN}{NA}=2\cdot\dfrac{PM}{MA}.$$ On the edges $SA$, $SB$, and $SC$ of tetrahedron $SABC$, points $D$, $E$, and $F$ are chosen....

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Kvant Math Problem 1061

Verified: no Verdicts: SKIP + SKIP Solve time: 1m19s Source on kvant.digital Problem In a country with more than two cities, some pairs of cities are connected by nonintersecting roads. It is known that for any three cities $A$, $B$, $C$, one can travel through this road network from $A$ to $B$ without passing through $C$. Prove that it is possible to assign a direction to every road so that...

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Kvant Math Problem 1060

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem Two closed polygonal chains are given in the plane, each having an odd number of sides. All lines containing the sides of these polygonal chains are distinct, and no three of them pass through the same point. Prove that it is possible to choose one side from each polygonal chain so that they are opposite sides of...

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Kvant Math Problem 1059

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem The graph of the function $y=f(x)$, defined on the entire real line, maps onto itself under a rotation by an angle of $\pi/2$ around the origin. Prove that the equation $f(x)=x$ has exactly one solution. Provide an example of such a function. A. V. Klyushin Exploration Let $$G={(x,f(x))\mid x\in\mathbb R}$$ be the graph of the function. Rotating...

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Kvant Math Problem 1058

Verified: no Verdicts: SKIP + SKIP Solve time: 1m12s Source on kvant.digital Problem A nonempty set of lattice points is marked on the integer grid. In addition, a finite set of vectors with integer coordinates is given. It is known that if all the given vectors are drawn from any marked lattice point, then among their endpoints there are more marked points than unmarked ones. Prove that there are infinitely...

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Kvant Math Problem 1057

Verified: no Verdicts: SKIP + SKIP Solve time: 1m17s Source on kvant.digital Problem Two players take turns writing natural numbers on the board not exceeding $p$. The rules of the game forbid writing divisors of numbers that have already been written on the board. The player who cannot make the next move loses. Determine which player has a winning strategy for $p=10$, and describe it. Determine which player has a...

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Kvant Math Problem 1056

Verified: no Verdicts: SKIP + SKIP Solve time: 1m12s Source on kvant.digital Problem In each cell of an $1987\times1987$ square table there is a number whose absolute value does not exceed 1. In every $2\times2$ square of this table, the sum of the numbers is equal to 0. Prove that the sum of all the numbers in the table does not exceed 1987. A. S. Merkuryev Exploration Consider small cases...

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Kvant Math Problem 1055

Verified: no Verdicts: SKIP + SKIP Solve time: 1m23s Source on kvant.digital Problem There are 21 points on a circle. Prove that among the arcs with endpoints at these points, there are at least 100 arcs whose lengths do not exceed $120^\circ$. A. F. Sidorenko City Tournament (Spring, 1987) Exploration Consider a circle with a small number of points to understand the behavior of arcs subtending at most $120^\circ$. With...

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Kvant Math Problem 1054

Verified: no Verdicts: SKIP + SKIP Solve time: 23m46s Source on kvant.digital Problem Prove that the six pairwise points of tangency of four spheres always lie on a single sphere or in a single plane. Yu. K. Koba Exploration Consider four spheres in three-dimensional space. Each pair of spheres has a unique point of tangency along the line connecting their centers if the spheres are externally tangent. Label the spheres...

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Kvant Math Problem 1053

Verified: no Verdicts: SKIP + SKIP Solve time: 16m33s Source on kvant.digital Problem Prove that in the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, $\ldots$, where each number is the sum of the two preceding ones, there will be at least 4 and at most 5 $m$-digit numbers by $m\gt3$. Exploration The first few Fibonacci numbers with at least four digits are $1597,\ 2584,\ 4181,\ 6765.$ There are...

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Kvant Math Problem 1052

Verified: no Verdicts: SKIP + SKIP Solve time: 17m05s Source on kvant.digital Problem Prove that among the $n$ quadrilaterals formed by the diagonals of a convex $n$-gon, no more than $n/2$ can be circumscribed about a circle. Give an example of an octagon for which there are 4 such quadrilaterals. N. M. Sedrakyan City Tournament (Spring, 1987) Exploration Consider a convex $n$-gon with vertices labeled cyclically as $A_1, A_2, \dots,...

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Kvant Math Problem 1051

Verified: no Verdicts: SKIP + SKIP Solve time: 16m32s Source on kvant.digital Problem In the lower-left corner of a $8\times8$-square chessboard, nine pieces are placed in the shape of a $3\times3$ square. A piece can jump over any other piece, reflecting symmetrically across it, provided the corresponding square is empty. Is it possible, by making several such moves, to arrange all the pieces into a $3\times3$ square: in the upper-left...

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Kvant Math Problem 1050

Verified: no Verdicts: SKIP + SKIP Solve time: 2m50s Source on kvant.digital Problem On the segment $[-1, 1]$, $k$ distinct points are chosen. For each point, the product of its distances to the remaining $k-1$ points is computed, and $S$ denotes the sum of the reciprocals of these $k$ products. Prove that $S\ge2$ for $k=3$; $S\ge4$ for $k=4$. L. D. Kurlandchik Exploration Let the chosen points be $$-1\le x_1<x_2<\cdots<x_k\le 1,$$...

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Kvant Math Problem 1049

Verified: no Verdicts: SKIP + SKIP Solve time: 1m33s Source on kvant.digital Problem Let us say that a cylinder $\text{Ц}_1$ is inscribed sideways in another cylinder $\text{Ц}_2$ if two generators of the second cylinder lie in the bases of the first, and four points of the base circles of the second lie on the lateral surface of the first (Fig. 1). Taking a cylinder $\text{Ц}_1$ whose diameter-to-height ratio is $k$,...

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Kvant Math Problem 1048

Verified: no Verdicts: SKIP + SKIP Solve time: 1m49s Source on kvant.digital Problem One of the two players (“the starter”) places a knight on some square of a chessboard. Then the players take turns moving the knight according to the usual rules (in an L-shape), and the knight may not be moved to a square it has already visited. The player who has no legal move loses. Who can force...

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Kvant Math Problem 1047

Verified: no Verdicts: SKIP + SKIP Solve time: 1m32s Source on kvant.digital Problem In a round-robin chess tournament, at least $\dfrac34$ of all games played up to a certain point have ended in a draw. Prove that at that moment, there are two participants who have scored the same number of points. M. Bona, high school student (Hungary) City Tournament (Spring, 1987) Exploration Consider a small round-robin tournament with $n$...

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Kvant Math Problem 1046

Verified: no Verdicts: SKIP + SKIP Solve time: 2m37s Source on kvant.digital Problem In an acute-angled triangle $ABC$, the angle $A$ is equal to $60^\circ$. Prove that one of the bisectors of the angle formed by the altitudes drawn from vertices $B$ and $C$ passes through the circumcenter of this triangle. V. Pogrebnyak, 10th-grade student (Vinnytsia) City Tournament (Spring, 1987) Exploration Consider an acute-angled triangle $ABC$ with $\angle A =...

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Kvant Math Problem 1045

Verified: no Verdicts: SKIP + SKIP Solve time: 32m23s Source on kvant.digital Problem In a certain kingdom, the territory of which is a square with a side of 2 km, the king decides to summon all the inhabitants to his palace for a ball at 7 p.m. To accomplish this, at noon he sends a messenger with instructions, who can convey any message to any resident, who in turn can...

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Kvant Math Problem 1044

Verified: no Verdicts: SKIP + SKIP Solve time: 2m48s Source on kvant.digital Problem Prove that from any four numbers one can always choose two numbers $x$ and $y$ such that $$0 \le \dfrac{x-y}{1+xy} \le 1.$$ I. N. Sergeev 50th Moscow City Mathematical Olympiad, 1987. Exploration Let $$f(x,y)=\frac{x-y}{1+xy}.$$ We must show that among any four real numbers there are two for which $$0\le f(x,y)\le 1.$$ The condition $f(x,y)\ge0$ means that the...

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Kvant Math Problem 1043

Verified: no Verdicts: SKIP + SKIP Solve time: 16m27s Source on kvant.digital Problem Is it possible to partition the set of all integers into three subsets so that for every integer $n$, the numbers $n$, $n-50$, and $n+1987$ belong to three different subsets? S. V. Konyagin 50th Moscow City Mathematical Olympiad, 1987 Exploration Let the three subsets be colored by the colors $A,B,C$. The condition says that for every integer...

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Kvant Math Problem 1042

Verified: no Verdicts: SKIP + SKIP Solve time: 17m08s Source on kvant.digital Problem A tug-of-war tournament is being organized in a class. Every possible team that can be formed from the students of this class must participate in the tournament exactly once (teams of one student, two students, etc., except for the team consisting of the entire class). Prove that each team will compete against the team consisting of all...

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Kvant Math Problem 1041

Verified: no Verdicts: SKIP + SKIP Solve time: 7m51s Source on kvant.digital Problem On the plane, the following are given: four, three vertices of a regular pentagon. Using a two-edged ruler, reconstruct the remaining vertices. (With a two-edged ruler, one can do the same as with an ordinary unmarked ruler, and also draw a line parallel to a given one at a distance equal to the width of the ruler.)...

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Kvant Math Problem 1040

Verified: no Verdicts: SKIP + SKIP Solve time: 8m36s Source on kvant.digital Problem The numbers 1, 2, $\ldots$, $3n$ are divided arbitrarily into three groups of $n$ numbers each. Prove that it is possible to choose one number from each group such that one of them equals the sum of the other two. V. B. Alekseev, S. Savchev (Bulgaria) Exploration For $n=1$ the three groups are ${1},{2},{3}$, and $3=1+2$. For...

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Kvant Math Problem 1039

Verified: no Verdicts: SKIP + SKIP Solve time: 16m36s Source on kvant.digital Problem Points $A$, $B$, $C$, $D$ are the vertices of a tetrahedron. Prove that if $\overrightarrow{DA}\cdot\overrightarrow{BC}=\overrightarrow{DB}\cdot\overrightarrow{CA}=\overrightarrow{DC}\cdot\overrightarrow{AB}$, then all three of these scalar products are equal to 0; if the three angles between opposite edges of the tetrahedron are equal, then they are right angles. V. E. Matizen Exploration Label the tetrahedron vertices as $A$, $B$, $C$, $D$. The...

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Kvant Math Problem 1038

Verified: no Verdicts: SKIP + SKIP Solve time: 1m45s Source on kvant.digital Problem Prove that if the product $mn$ is divisible by 6 (where $m$ and $n$ are integers greater than 1), then a rectangle of $m\times n$ cells can be cut into L-shaped pieces of three cells each. Fig. 1 For which $m$ and $n$ can this be done so that the dividing lines do not cut (Fig. 1)...

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Kvant Math Problem 1037

Verified: no Verdicts: SKIP + SKIP Solve time: 4m14s Source on kvant.digital Problem Find all solutions in natural numbers $x$, $y$ of the equation $x^y-y^x=x+y$. A. I. Zaychik Exploration Consider the equation $x^y - y^x = x + y$ with $x, y \in \mathbb{N}$. Start with small integers to identify patterns. If $x = y$, then $x^y - y^x = 0$, while $x+y \ge 2$, so no solution occurs along...

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Kvant Math Problem 1036

Verified: no Verdicts: SKIP + SKIP Solve time: 26m51s Source on kvant.digital Problem Does there exist a (non-convex) pentagon that can be cut into two equal pentagons? S. M. Khosid Exploration Consider a pentagon and imagine cutting it into two smaller pentagons of equal area and shape. The most immediate obstruction arises from area: each of the resulting pentagons must have exactly half the area of the original. For convex...

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Kvant Math Problem 1035

Verified: no Verdicts: SKIP + SKIP Solve time: 3m26s Source on kvant.digital Problem On the segment $[0;1]$, the points $x_0$, $x_1$, $x_2$, $\ldots$, $x_n$ are marked in turn. For each point $x_k$ ($k=1$, $\ldots$, $n$), the distance $d_k$ from it to the nearest of the previously marked points is measured. Prove that $$d_1+d_2+\dots+d_n \le 1+\dfrac{\log_2 n}2.$$ V. S. Grinberg Exploration Consider marking points on $[0,1]$ sequentially. The first point $x_0$...

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Kvant Math Problem 1034

Verified: no Verdicts: SKIP + SKIP Solve time: 10m32s Source on kvant.digital Problem A rectangular chocolate bar is divided by longitudinal and transverse grooves into 50 square pieces. Two players play the following game. The first player breaks the chocolate bar along one of the grooves into two rectangular parts. Then the players take turns breaking one of the resulting parts along a groove into two parts. The player who...

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Kvant Math Problem 1033

Verified: no Verdicts: SKIP + SKIP Solve time: 6m40s Source on kvant.digital Problem A circle cuts off four curvilinear triangles from a square (the boundary of each consists of an arc of the circle and two line segments). Color two of them adjacent to opposite vertices of the square blue, and the other two red. Prove that the sums of the lengths of the red and blue arcs are equal;...

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Kvant Math Problem 1032

Verified: no Verdicts: SKIP + SKIP Solve time: 1m51s Source on kvant.digital Problem The numbers 2, 3, $\ldots$, $n+1$ are listed, along with all their possible products taken two at a time, three at a time, and so on up to the product of all $n$ of these numbers. Prove that the sum of the reciprocals of all the listed numbers equals $\dfrac n2$. $\Big($For example, when $n=3$: $$\dfrac12+\dfrac13+\dfrac14+\dfrac1{2\cdot3}+\dfrac1{2\cdot4}+\dfrac1{3\cdot4}+\dfrac1{2\cdot3\cdot4}=\dfrac32.\Big)$$ A....

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Kvant Math Problem 1031

Verified: no Verdicts: SKIP + SKIP Solve time: 8m29s Source on kvant.digital Problem On the plane, a line $\ell$ and two points $A$ and $B$ lying on the same side of it are given. On the line $\ell$, a point $M$ is chosen such that the sum of its distances to the points $A$ and $B$ is minimal, and a point $N$ is chosen for which the distances to $A$...

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Kvant Math Problem 1030

Verified: no Verdicts: SKIP + SKIP Solve time: 6m09s Source on kvant.digital Problem For a convex polyhedron $M$, let $S(M)$ denote the sum of the areas of its faces, and let $P(M)$ denote the sum, over all its edges, of the products of the edge lengths and the corresponding exterior angles of the polyhedron (the exterior angle at a given edge is the angle between the normals to the faces...

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Kvant Math Problem 1029

Verified: no Verdicts: SKIP + SKIP Solve time: 4m09s Source on kvant.digital Problem Among $n$ terms of an arithmetic progression, it was possible to select $k$ terms that form an increasing geometric progression. Prove that $n\ge2^{k-1}$. V. F. Lev Exploration Let the arithmetic progression be $a,a+d,a+2d,\dots,a+(n-1)d,$ with $d\neq0$. Suppose that among these terms we choose $k$ terms forming an increasing geometric progression $b_1<b_2<\cdots<b_k.$ Since every selected term belongs to the...

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Kvant Math Problem 1028

Verified: no Verdicts: SKIP + SKIP Solve time: 10m13s Source on kvant.digital Problem Fig. 2 Two intersecting lines are given in the plane, and one point is marked on each of them ($D$ and $E$). Construct a triangle $ABC$ whose angle bisectors $CD$ and $AE$ lie on the given lines, and whose feet are the given points $D$ and $E$ (Fig. 2). Prove that if, moreover, $\angle CDE=30^\circ$, then one...

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Kvant Math Problem 1027

Verified: no Verdicts: SKIP + SKIP Solve time: 5m24s Source on kvant.digital Problem Prove that the number $1985!!+1986!!$ is divisible by $1987$. (Here $n!!$ denotes the product of all positive integers not exceeding $n$ and having the same parity, i.e., $n!!=n\cdot(n-2)\cdot(n-4)\cdot\ldots$.) V. V. Proizvolov Exploration The number $1987$ is prime, since it is not divisible by any prime not exceeding $\sqrt{1987}<45$. The double factorials separate the integers from $1$ to...

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Kvant Math Problem 1026

Verified: no Verdicts: SKIP + SKIP Solve time: 9m02s Source on kvant.digital Problem Fig. 1 Five equal arcs $AB$, $BC$, $CD$, $DE$, $EA$ are arranged so that each is divided by its neighboring arcs into three equal parts (Fig. 1). Find the measure of each arc (in degrees). The same question for a “rosette” consisting of $m$ equal arcs, each of which is divided by its neighboring arcs into three...

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Kvant Math Problem 1025

Verified: no Verdicts: SKIP + SKIP Solve time: 1m48s Source on kvant.digital Problem Two lines, drawn through each of the two points of intersection of the extensions of opposite sides of a convex quadrilateral, divide it into four smaller quadrilaterals. Prove that if two of these quadrilaterals, which do not share a common side, can have inscribed circles, then the original quadrilateral can also have an inscribed circle. I. F....

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Kvant Math Problem 1024

Verified: no Verdicts: SKIP + SKIP Solve time: 5m21s Source on kvant.digital Problem Prove that for any two triangles with angles $\alpha$, $\beta$, $\gamma$ and $\alpha_1$, $\beta_1$, $\gamma_1$, the inequality $$\frac{\cos\alpha_1}{\sin\alpha}+\frac{\cos\beta_1}{\sin\beta}+\frac{\cos\gamma_1}{\sin\gamma}\lt\ctg\alpha+\ctg\beta+\ctg\gamma.$$ holds. R. P. Ushakov Exploration Consider two triangles with angles $\alpha, \beta, \gamma$ and $\alpha_1, \beta_1, \gamma_1$. The inequality compares sums of ratios of cosines from one triangle to sines of the other versus the sum of cotangents of...

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Kvant Math Problem 1023

Verified: no Verdicts: SKIP + SKIP Solve time: 4m55s Source on kvant.digital Problem Is it always true that among 100 triangles there is at least one that can be completely covered by the other 99? G. A. Gurevich Exploration For small numbers of triangles the statement is false. Take two disjoint triangles. Neither can be covered by the other. Thus no purely combinatorial principle of the form “among many triangles...

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Kvant Math Problem 1022

Verified: no Verdicts: SKIP + SKIP Solve time: 1m50s Source on kvant.digital Problem $\def\c#1{\enspace\mathclap{#1}\enspace} \begin{array}{|c|c|c|c|}\hline \c8&\c2&\c3&\c5\\hline \c1&\c7&\c6&\c4\\hline \end{array} $ The first 8 natural numbers can be arranged in two rows so that the sum of the numbers in the top row equals the sum of the numbers in the bottom row, and the sums of the numbers in the columns are also all equal. Can the first ten, twelve natural...

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Kvant Math Problem 1021

Verified: no Verdicts: SKIP + SKIP Solve time: 7m06s Source on kvant.digital Problem A mountaineer wants to climb a 1000 m cliff. After spending the night in a camp at the foot of the cliff, he can ascend while fixing a rope at a rate of 40 m/h, whereas after a cold overnight stay on the cliff he can do so at 30 m/h. He climbs a pre-installed rope at...

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Kvant Math Problem 1020

Verified: no Verdicts: SKIP + SKIP Solve time: 7m06s Source on kvant.digital Problem On a sphere of radius 1, the following are drawn: a curve whose length is less than $\pi$; a closed curve whose length is less than $2\pi$. Prove that there exists a plane passing through the center of the sphere that does not intersect the drawn curve. (One may assume that a curve on the sphere is...

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Kvant Math Problem 1019

Verified: no Verdicts: SKIP + SKIP Solve time: 7m20s Source on kvant.digital Problem On a sheet of graph paper, a certain finite set of $M$ nodes (points of intersection of the grid lines) is marked. Prove that it is always possible to color some points of the set $M$ white and the rest red so that, on each grid line, the absolute difference between the number of white and red...

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Kvant Math Problem 1018

Verified: no Verdicts: SKIP + SKIP Solve time: 9m33s Source on kvant.digital Problem Let $A$ and $B$ be adjacent vertices of a regular $n$-gon with center $O$. The triangle $XYZ$, congruent to triangle $OAB$, initially coincides with it and then moves in the plane of the $n$-gon so that the points $Y$ and $Z$ remain on the boundary, while $X$ remains inside the $n$-gon. What figure will be traced out...

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Kvant Math Problem 1017

Verified: no Verdicts: SKIP + SKIP Solve time: 5m14s Source on kvant.digital Problem Each vertex of a regular pentagon is assigned an integer in such a way that the sum of all five numbers is positive. The following operation is allowed: if three consecutive vertices are assigned the numbers $x$, $y$, $z$, and $y\lt0$, then these numbers are replaced, respectively, by $x+y$, $-y$, and $z+y$. Such operations are performed as...

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Kvant Math Problem 1016

Verified: no Verdicts: SKIP + SKIP Solve time: 7m39s Source on kvant.digital Problem A polygon is circumscribed around a circle with center $O$. Let $P$ be the centroid of the polygon, and $K$ the centroid of its boundary. Prove that the points $P$, $O$, and $K$ lie on a single straight line, with $PO=2PK$. (When defining the centroid $P$, the polygon is considered as a uniform lamina, and $K$ as...

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Kvant Math Problem 1015

Verified: no Verdicts: SKIP + SKIP Solve time: 5m37s Source on kvant.digital Problem Is it possible to factor the polynomial $x^4+x^3+x^2+x+12=0$ over the integers? S. L. Manukyan Exploration The polynomial is $$P(x)=x^4+x^3+x^2+x+12.$$ The question asks whether it can be factored over the integers. Since the degree is $4$, any nontrivial factorization in $\mathbb Z[x]$ must be either a linear factor times a cubic factor, or a product of two quadratic...

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Kvant Math Problem 1014

Verified: no Verdicts: SKIP + SKIP Solve time: 7m24s Source on kvant.digital Problem Let $a_1$, $a_2$, $\ldots$, $a_n$ be distinct pairwise coprime positive integers. Prove that there exist infinitely many positive integers $b$ such that the numbers $b+a_1$, $b+a_2$, $\ldots$, $b+a_n$ are also pairwise coprime. V. F. Lev Exploration Consider small examples of pairwise coprime numbers, such as $a_1=2$, $a_2=3$, $a_3=5$. The task is to find integers $b$ such that...

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Kvant Math Problem 1013

Verified: no Verdicts: SKIP + SKIP Solve time: 7m05s Source on kvant.digital Problem On the sides $AB$ and $BC$ of triangle $ABC$, two points $M$ and $N$ are taken. Three parallel lines passing through the points $M$, $B$, and $N$ intersect the base $AC$ at points $K$, $D$, and $L$. Prove that the area of the trapezoid (or parallelogram) $KMNL$ does not exceed the area of either of the triangles...

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Kvant Math Problem 1012

Verified: no Verdicts: SKIP + SKIP Solve time: 2m50s Source on kvant.digital Problem Prove that: On the plane, it is possible to arrange several non-overlapping circles so that each touches exactly 5 others; The number 5 in part (1) cannot be replaced by 6. D. V. Fomin Exploration Consider arrangements of circles in the plane where each circle touches several others. For small numbers of neighbors, such as three or...

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Kvant Math Problem 1011

Verified: no Verdicts: SKIP + SKIP Solve time: 3m30s Source on kvant.digital Problem Prove that for $n$ positive numbers $a_1\ge a_2\ge a_3\ge\ldots\ge a_n$ the following inequalities hold: $a_1^2-a_2^2+a_3^2\ge(a_1-a_2+a_3)^2$; $a_1^2-a_2^2+a_3^2-a_4^2\ge(a_1-a_2+a_3-a_4)^2$; $a_1^2-a_2^2+\ldots-(-1)^n a_n^2\ge\big(a_1-a_2+\ldots-(-1)^n a_n\big)^2$. L. D. Kuryandchik Exploration For the first inequality, $$a_1^2-a_2^2+a_3^2-(a_1-a_2+a_3)^2$$ expands to $$2a_2(a_1-a_2+a_3)-2a_1a_3.$$ A further rearrangement gives $$2(a_1-a_2)(a_2-a_3),$$ which is nonnegative because $a_1\ge a_2\ge a_3$. For the second inequality, $$a_1^2-a_2^2+a_3^2-a_4^2-(a_1-a_2+a_3-a_4)^2$$ expands to $$2(a_1-a_2+a_3-a_4)(a_2-a_3+a_4)-2(a_1a_3+a_1a_4+a_2a_4).$$ This form is not transparent. A...

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Kvant Math Problem 1010

Verified: no Verdicts: SKIP + SKIP Solve time: 3m07s Source on kvant.digital Problem The sequence $r_1$, $r_2$, $r_3$, $\ldots$ is defined by the conditions $$r_1=2,\quad r_{n+1}=r_1r_2\ldots r_n+1,$$ so that $r_2=3$, $r_3=7$, $r_4=43$, $\ldots$ Prove that for any $n$ $$\dfrac1{r_1}+\dfrac1{r_2}+\ldots+\dfrac1{r_n}\lt1.$$ Let $n$ be natural numbers such that the sum of their reciprocals is less than 1. Prove that this sum does not exceed $$\dfrac1{r_1}+\dfrac1{r_2}+\ldots+\dfrac1{r_n}.$$ The proofs known to us rely on...

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Kvant Math Problem 1009

Verified: no Verdicts: SKIP + SKIP Solve time: 9m50s Source on kvant.digital Problem The bisector of the angle $A$ of the parallelogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$, respectively. Prove that the center of the circle passing through the points $C$, $K$, and $L$ lies on the circle passing through the points $B$, $C$, and $D$. I. F. Sharygin Exploration Let the parallelogram...

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Kvant Math Problem 1008

Verified: no Verdicts: SKIP + SKIP Solve time: 6m17s Source on kvant.digital Problem A staircase consists of $2n+1$ steps. On the $n$ lowest steps there is one stone on each. Two players take turns moving the stones. The first player can move any stone upward to the first unoccupied step above it, while the second player can move a stone down by one step if that step is unoccupied. The...

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Kvant Math Problem 1007

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem Prove that triangles with side lengths $a$, $b$, $c$ and $a_1$, $b_1$, $c_1$ are similar if and only if $$\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1}=\sqrt{(a_1+b_1+c_1)(a+b+c)}.$$ V. P. Chichin Exploration The equality $$\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1} = \sqrt{(a+b+c)(a_1+b_1+c_1)} \tag{1}$$ strongly resembles the equality case of the Cauchy-Schwarz inequality. Indeed, $$(\sqrt{aa_1}+\sqrt{bb_1}+\sqrt{cc_1})^2 \le (a+b+c)(a_1+b_1+c_1),$$ with equality exactly when $$\frac{\sqrt a}{\sqrt{a_1}} = \frac{\sqrt b}{\sqrt{b_1}} = \frac{\sqrt c}{\sqrt{c_1}},$$ that is,...

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Kvant Math Problem 1006

Verified: no Verdicts: SKIP + SKIP Solve time: 1m40s Source on kvant.digital Problem Two lines are drawn through two vertices of a triangle, dividing it into three triangles and a quadrilateral. Is it possible for all four parts to have equal areas? Which three of these parts can have equal areas? By what factor does the area of the fourth part differ from them? G. A. Halperin, A. P. Savin...

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Kvant Math Problem 1005

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem The cells of a square table of size $n\times n$ ($n\ge3$) are filled with numbers $\pm1$ according to the following rules: All the boundary cells of the table are filled with the numbers $-1$; The number placed in the next empty cell of the table is equal to the product of the nearest numbers to that cell...

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Kvant Math Problem 1004

Verified: no Verdicts: SKIP + SKIP Solve time: 1m40s Source on kvant.digital Problem Through the vertex $A$ of a triangle $ABC$, in which $AB\ne AC$, all possible lines are drawn. Prove that: on each of them there is at most one point $M$, distinct from the vertices of the triangle, for which $\angle ABM=\angle ACM$; there are at most five of these lines on which there is no such point...

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Kvant Math Problem 1003

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem In triangle $ABC$, three altitudes $AH$, $BK$, $CL$ are drawn. Prove the equalities $$AK\cdot BL\cdot CH=AL\cdot BH\cdot CK=HK\cdot KL\cdot LH.$$. S. A. Genkin Leningrad City Mathematical Olympiad (1986) Exploration The quantities involve segments cut off by the feet of the altitudes. A natural first step is to express these segments through the sides and angles of the...

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Kvant Math Problem 1002

Verified: no Verdicts: SKIP + SKIP Solve time: 1m42s Source on kvant.digital Problem A scatterbrained mathematician, having forgotten the three-digit code to his apartment building entrance, presses buttons (with digits 0, 1, $\ldots$, 9) at a rate of one per second. The door will open if the three digits of the code are entered consecutively in the correct order. The mathematician is sure that even in the case of “extreme...

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Kvant Math Problem 1001

Verified: no Verdicts: SKIP + SKIP Solve time: 1m40s Source on kvant.digital Problem A pile contains 1001 stones. It is arbitrarily divided into two piles, the number of stones in each is counted, and the product of these two numbers is recorded. Then, the same operation is performed on one of these piles (containing more than one stone): it is divided into two, and the product of the numbers of...

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Kvant Math Problem 1000

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem A broken line $AMB$ composed of two segments ($AM\gt MB$) is inscribed in the arc $AB$. Prove that the foot of the perpendicular $KH$, dropped from the midpoint of $K$ of the arc $AB$ onto the segment $AM$, bisects the broken line: $AH=HM+MB$ (Fig. 1). Fig. 1 The editorial office received many mathematics problems labeled "For the...

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Kvant Math Problem 999

Verified: no Verdicts: SKIP + SKIP Solve time: 2m59s Source on kvant.digital Problem Prove that for any positive numbers $a_1$, $a_2$, $\ldots$, $a_n$ the following inequality holds: $$\dfrac1{a_1}+\dfrac2{a_1+a_2}+\ldots+\dfrac n{a_1+a_2+\ldots+a_n} \lt 4\left(\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_n}\right).$$ Prove that the constant 4 on the right-hand side of the inequality can be replaced by 2; cannot be replaced by any number less than 2. L. D. Kurlyandchik, A. S. Merkuryev Exploration Let $$S_k=a_1+a_2+\cdots+a_k.$$ The inequality to be...

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Kvant Math Problem 998

Verified: no Verdicts: SKIP + SKIP Solve time: 1m45s Source on kvant.digital Problem Consider all tetrahedra $AXBY$ circumscribed about a given sphere. Prove that, for fixed points $A$ and $B$, the sum of the angles of the quadrilateral $AXBY$, i.e., the sum $$\angle AXB+\angle XBY+\angle BYA+\angle YAX,$$, does not depend on the choice of the points $X$ and $Y$. I. F. Sharygin Exploration Consider a tetrahedron $AXBY$ circumscribed about a...

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Kvant Math Problem 997

Verified: no Verdicts: SKIP + SKIP Solve time: 2m43s Source on kvant.digital Problem Prove that the sum of all numbers of the form $\dfrac1{mn}$, where $m$ and $n$ are natural numbers, $1\le m\lt n\lt1986$, is not an integer. D. A. Mitkin Exploration Let $$S=\sum_{1\le m<n<1986}\frac1{mn}.$$ A direct evaluation is unnecessary; the task is only to prove that $S$ is not an integer. It is useful to rewrite the sum. Since...

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Kvant Math Problem 996

Verified: no Verdicts: SKIP + SKIP Solve time: 1m17s Source on kvant.digital Problem Two identical squares intersect to form an octagon. The sides of one square are blue, and the sides of the other are red. Prove that the sum of the lengths of the blue sides of the octagon equals the sum of the lengths of its red sides. V. V. Proizvolov Exploration The octagon is the intersection of...

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Kvant Math Problem 995

Verified: no Verdicts: SKIP + SKIP Solve time: 1m52s Source on kvant.digital Problem The function $y=f(x)$ is defined and continuous for all $x$ and satisfies the condition $$f(f(x))=f(x)+x.$$. Find two such functions $f$. Prove that no other such functions exist. Journal Mathematics (Bulgaria) Exploration Let $$g(x)=f(x).$$ The equation is $$f(f(x))=f(x)+x.$$ A first test is to look for linear solutions. Put $$f(x)=ax+b.$$ Then $$f(f(x))=a(ax+b)+b=a^2x+ab+b,$$ while $$f(x)+x=(a+1)x+b.$$ Comparing coefficients gives $$a^2=a+1,\qquad ab+b=b.$$...

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Kvant Math Problem 994

Verified: no Verdicts: SKIP + SKIP Solve time: 1m39s Source on kvant.digital Problem For what largest value of $k$ is the inequality $$a^4+b^4+c^4+abc(a+b+c) \ge k(ab+bc+ca)^2$$ satisfied for all values of $a$, $b$, and $c$? Mathematics Magazine (Bulgaria) Exploration Let $$F(a,b,c)=a^4+b^4+c^4+abc(a+b+c).$$ We seek the largest constant $k$ such that $$F(a,b,c)\ge k(ab+bc+ca)^2$$ for all real $a,b,c$. Since both sides are homogeneous of degree $4$, the problem asks for $$\inf \frac{F(a,b,c)}{(ab+bc+ca)^2},$$ over all...

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Kvant Math Problem 993

Verified: no Verdicts: SKIP + SKIP Solve time: 1m13s Source on kvant.digital Problem Find 11 consecutive natural numbers whose squares sum to a perfect square. Prove that for $2\lt n\lt11$ there do not exist $n$ consecutive natural numbers whose squares sum to a perfect square. Mathematics Journal (Bulgaria) Exploration Let $x$ be the smallest of $n$ consecutive natural numbers. The sum of their squares is $S = x^2 + (x+1)^2...

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Kvant Math Problem 992

Verified: no Verdicts: SKIP + SKIP Solve time: 1m14s Source on kvant.digital Problem Among 90 graduates of a certain mathematical high school, each has at least 10 friends. Prove that any graduate can invite three others in such a way that among the four gathered, each has at least two friends. Mathematics Magazine (Bulgaria) Exploration Consider small examples of social networks where each person has at least 10 friends. For...

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Kvant Math Problem 991

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem In the triangle $ABC$, the altitude $CH$ and the median $CK$ are drawn. On the side $AB$, points $E$ and $F$ are chosen so that $\angle ACE=\angle BCF$, and perpendiculars $AM$ and $BN$ are dropped to the rays $CE$ and $CF$, respectively (Fig. 1). Prove that the points $M$, $H$, $K$, and $N$ lie on the same...

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Kvant Math Problem 990

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem In space, three lines are given, each pair of which is skew, and they are not all parallel to the same plane. How many distinct parallelepipeds exist for which these three lines lie along edges? lie along edges or face diagonals? contain 6 vertices of the parallelepiped? V. N. Dubrovsky Exploration Consider three lines in space, each...

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Kvant Math Problem 989

Verified: no Verdicts: SKIP + SKIP Solve time: 1m51s Source on kvant.digital Problem Find all natural numbers $a$ for which the number $a-1$ is the sum of two; three divisors of the number $a$ (not necessarily distinct; 1 is included among the divisors). Prove that for every $n$ there exist only finitely many natural numbers $a$ such that $a-1$ is the sum of $n$ natural divisors of the number $a$...

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Kvant Math Problem 988

Verified: no Verdicts: SKIP + SKIP Solve time: 1m25s Source on kvant.digital Problem From a single point $O$ on the plane, $n$ unit-length vectors are drawn. Prove that if for some $k\lt\dfrac{n}{2}$, on each side of every line passing through $O$, there are at least $k$ vectors, then the length of the sum of all $n$ vectors does not exceed $n-2k$. P. Kalugin Exploration Consider small values of $n$ and...

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Kvant Math Problem 987

Verified: no Verdicts: SKIP + SKIP Solve time: 1m39s Source on kvant.digital Problem A tournament involves $2m$ teams. In the first round, some $m$ pairs of teams played against each other, and in the second round, another $m$ pairs played. Prove that it is possible to select $m$ teams such that no two of them have played against each other. M. Bona, high school student (Hungary) Exploration Consider small instances...

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Kvant Math Problem 986

Verified: no Verdicts: SKIP + SKIP Solve time: 1m42s Source on kvant.digital Problem Prove that for any positive numbers $a$ and $b$, the inequality $$2\sqrt{a\vphantom b}+3\sqrt[\scriptstyle3~]{b}\ge5\sqrt[\scriptstyle5~]{ab}.$$ holds. Exploration The inequality is $$2\sqrt a+3\sqrt[3]{b}\ge 5\sqrt[5]{ab}.$$ The coefficients $2$ and $3$ add up to $5$, while the exponents are $\frac12$, $\frac13$, and $\frac15$. This suggests a weighted AM-GM inequality. To test this idea, take $a=b=1$. Then the left-hand side equals $2+3=5$, and...

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Kvant Math Problem 985

Verified: no Verdicts: SKIP + SKIP Solve time: 3m01s Source on kvant.digital Problem The angle between two lines intersecting at point $O$ is defined as the angle between their rays with vertex $O$, not exceeding $90^\circ$. In how many ways can three lines $l_1$, $l_2$, $l_3$ be drawn through point $O$ in space so that the angles between $l_2$ and $l_3$, $l_3$ and $l_1$, $l_1$ and $l_2$ are equal to...

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Kvant Math Problem 984

Verified: no Verdicts: SKIP + SKIP Solve time: 1m28s Source on kvant.digital Problem Through an arbitrary point $K$ of the square $ABCD$, a line is drawn intersecting its opposite sides $AB$ and $CD$ at points $P$ and $Q$. Prove that the point of intersection, other than $K$, of the circles passing through the points $K$, $B$, $P$ and $K$, $D$, $Q$ lies on the diagonal $BD$ (Fig. 1). Fig. 1...

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Kvant Math Problem 983

Verified: no Verdicts: SKIP + SKIP Solve time: 2m08s Source on kvant.digital Problem In a tournament involving 16 tennis players, every pair of players plays one game. Give an example of a distribution of game results such that any 10 participants can be arranged in a circle so that each player has defeated the player to their left. Prove that if the condition in part (a) is satisfied, then any...

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Kvant Math Problem 982

Verified: no Verdicts: SKIP + SKIP Solve time: 1m14s Source on kvant.digital Problem On the sides $AB$, $BC$, and $CA$ of triangle $ABC$, squares $ABB_1A_2$, $BCB_1C_2$, $CAA_1C_2$ are constructed externally. Prove that the perpendiculars to the segments $A_1A_2$, $B_1B_2$, $C_1C_2$, drawn at their midpoints, intersect at a single point. N. Azamov, 8th grade student Exploration Construct triangle $ABC$ on paper and build the external squares $ABB_1A_2$, $BCB_1C_2$, $CAA_1C_2$. Draw the...

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Kvant Math Problem 981

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem Prove that the number $11\ldots1$ (1986 ones) has at least 8, 28 distinct divisors. L. D. Kurlandchik Exploration Consider small repunit numbers of the form $R_n = 11\ldots1$ with $n$ ones. The first few are $R_1 = 1$, $R_2 = 11$, $R_3 = 111$, $R_4 = 1111$, and $R_5 = 11111$. Factoring these examples explicitly may reveal...

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Kvant Math Problem 980

Verified: no Verdicts: SKIP + SKIP Solve time: 1m17s Source on kvant.digital Problem Inside a convex polygon, polyhedron with vertices $A_1$, $A_2$, $\ldots$, $A_n$, a point $O$ is chosen. Prove that among the $\dfrac{n(n-1)}{2}$ angles $A_i O A_k$ ($i$, $k=1$, 2, $\ldots$, $n$), at least $n-1$ have measure between $90^\circ$ and $180^\circ$. V. G. Boltyanskyi Exploration Consider first a convex polygon in the plane with vertices $A_1, A_2, \dots, A_n$...

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Kvant Math Problem 979

Verified: no Verdicts: SKIP + SKIP Solve time: 1m32s Source on kvant.digital Problem Let $k$ and $n$ be positive integers, $k\le n$. Call a set of $k$ positive numbers $a_1$, $a_2$, $\ldots$, $a_k$ less than 1 exceptional if for every partition $n=n_1+n_2+\ldots+n_k$ of the number $n$ into nonnegative integer summands, at least one of the numbers $a_i n_i$ is an integer ($i=1$, $2$, $\ldots$, $n$). For which $k$ and $n$...

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Kvant Math Problem 978

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem Is it possible to place two equilateral triangles with side lengths greater than $\sqrt{2/3}$ inside a square of side length 1, without overlapping? E. Karlov, I. Dmitriev (Bulgaria) Exploration The threshold $\sqrt{2/3}$ is suggestive because an equilateral triangle of side $a$ has altitude $\frac{\sqrt3}{2}a$, and when $a=\sqrt{2/3}$ the altitude equals $\frac1{\sqrt2}$. Since the square has diagonal $\sqrt2$,...

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Kvant Math Problem 977

Verified: no Verdicts: SKIP + SKIP Solve time: 1m35s Source on kvant.digital Problem Is it possible, using addition, subtraction, and multiplication, to obtain $x$ from the polynomials $f(x)$ and $g(x)$ if: $f(x)=x^2+x$, $g(x)=x^2+2$; $f(x)=x^2+x$, $g(x)=x^2-2$; $f(x)=2x^2+x$, $g(x)=2x$; $f(x)=2x^2+x$, $g(x)=x^2$. S. Kublanovsky Exploration The problem asks whether $x$ can be expressed using only addition, subtraction, and multiplication from given polynomials. For small degrees, one can attempt explicit algebraic manipulations. In case...

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Kvant Math Problem 976

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem From vertex $A$ of the square $ABCD$, two rays are drawn forming an angle of $45^\circ$ between them. One ray intersects side $BC$ at point $E$ and diagonal $BD$ at point $P$; the other intersects side $CD$ at point $F$ and diagonal $BD$ at point $O$. Prove that the area of triangle $AEF$ is twice the area...

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Kvant Math Problem 975

Verified: no Verdicts: SKIP + SKIP Solve time: 1m33s Source on kvant.digital Problem On a “chessboard” of size $n\times n$ there are 20 different pieces. It is known that each piece, from any square, attacks at most 20 squares. Prove that for $n=100$ these pieces can be rearranged so that no two of them attack each other. Suppose additionally that if a piece is shifted, then the set of squares...

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Kvant Math Problem 974

Verified: no Verdicts: SKIP + SKIP Solve time: 7m40s Source on kvant.digital Problem Two players are playing chess with clocks. After each had made 40 moves, both clocks showed 2 hours 30 minutes. Prove that at some point in the game, one clock was ahead of the other by more than 1 minute 50 seconds. Can it be asserted that at some moment the difference between the clocks was at...

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Kvant Math Problem 973

Verified: no Verdicts: SKIP + SKIP Solve time: 3m04s Source on kvant.digital Problem In triangle $ABC$, the altitude $AH$ and the angle bisector $BE$ are drawn. Prove that if $\angle BEA=45^\circ$, then $\angle EHC=45^\circ$. I. F. Sharygin Exploration Let $$\angle ABC=2\beta,$$ so that, because $BE$ is the bisector of $\angle ABC$, $$\angle ABE=\angle EBC=\beta.$$ The condition $\angle BEA=45^\circ$ immediately gives information about triangle $ABE$. Since $$\angle BAE=A,$$ we obtain $$A+\beta+45^\circ=180^\circ,...

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Kvant Math Problem 972

Verified: no Verdicts: SKIP + SKIP Solve time: 1m56s Source on kvant.digital Problem The sequence $x_1$, $x_2$, $\ldots$ is defined by the conditions $x_1=\dfrac12$, $x_{n+1}=x_n^2+x_n$ ($n=1$, 2, $\ldots$). Find the integer part of the number $$\dfrac1{x_1+1}+\dfrac1{x_2+1}+\ldots+\dfrac1{x_{100}+1}.$$. A. V. Andzhans Exploration The sequence $(x_n)$ begins with $x_1 = \frac12$ and satisfies the recurrence $x_{n+1} = x_n^2 + x_n$. Computing a few terms may reveal a pattern. We have $x_1 = \frac12$,...

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Kvant Math Problem 971

Verified: no Verdicts: SKIP + SKIP Solve time: 7m37s Source on kvant.digital Problem Eight volleyball teams played a single-round tournament (each team played every other team exactly once). Prove that it is possible to select four teams $A$, $B$, $C$, $D$ such that $A$ defeated $B$, $C$, and $D$, $B$ defeated $C$ and $D$, and $C$ defeated $D$. A. T. Ukrainets Exploration Consider a tournament of $8$ volleyball teams where...

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Kvant Math Problem 969

Verified: no Verdicts: SKIP + SKIP Solve time: 7m01s Source on kvant.digital Problem Prove the inequality $$\dfrac{a^3}{a^2+ab+b^2}+\dfrac{b^3}{b^2+bc+c^2}+\dfrac{c^3}{c^2+ca+a^2}\ge\dfrac{a+b+c}{3}.$$ G. G. Alikhanov Unusual activity has been detected from your device. Try again later. (4c4b92dd-6375-491f-8f46-c6ee9436d916)

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Kvant Math Problem 967

Verified: no Verdicts: SKIP + SKIP Solve time: 3m20s Source on kvant.digital Problem Let $\sigma(n)$ denote the sum of all positive divisors of the number $n$ (including $1$ and $n$), and let $\varphi(n)$ denote the number of integers less than $n$ that are relatively prime to it. Prove that for every positive integer $n\ge 2$, $$\sigma(n)+\varphi(n) \ge 2n.$$ V. F. Lev Exploration For small values, $$n=2:\quad \sigma(2)+\varphi(2)=3+1=4=2n,$$ $$n=3:\quad \sigma(3)+\varphi(3)=4+2=6=2n,$$ $$n=4:\quad...

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Kvant Math Problem 966

Verified: no Verdicts: SKIP + SKIP Solve time: 7m38s Source on kvant.digital Problem Prove that any triangle can be cut by line segments into four pieces, from which two triangles similar to the original can be assembled. L. D. Kurlandchik Exploration The statement asks for a dissection of an arbitrary triangle into four pieces such that the pieces can be rearranged into two triangles, each similar to the original triangle....

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Kvant Math Problem 964

Verified: no Verdicts: SKIP + SKIP Solve time: 7m44s Source on kvant.digital Problem Prove that in a sequence $(a_n)$ of distinct positive integers satisfying the condition $a_n\lt100n$, there is a number whose decimal representation contains the digit 1; contains 1986 consecutive ones. A. A. Stolin Exploration The sequence $(a_n)$ consists of distinct positive integers with the growth constraint $a_n < 100n$. This implies that for large $n$, the numbers in...

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Kvant Math Problem 961

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem On the side $AB$ of the square $ABCD$, a point $E$ is taken, and on the side $CD$, a point $F$ is taken, with $AE:EB=1:2$, $CF=FD$. Are the blue and pink triangles (Fig. 1) similar? Fig. 1 A. P. Savin, N. A. Paravyan Exploration Let the side length of the square be $6$. Then $AE=2$, $EB=4$, and...

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Kvant Math Problem 959

Verified: no Verdicts: SKIP + SKIP Solve time: 7m47s Source on kvant.digital Problem In a certain country, air service has been established between some pairs of cities. Prove that it is possible to close at most $\dfrac1{k-1}$ of the air routes in such a way that among any $k$ cities there are two that are not connected to each other by an air route, if $k=3$; $k$ is any positive...

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Kvant Math Problem 958

Verified: no Verdicts: SKIP + SKIP Solve time: 3m21s Source on kvant.digital Problem Let $0\le i_1\lt i_2\lt\ldots\lt i_n$ be integers. Prove that the number of odd coefficients of the polynomial $$(1+x)^{i_1}+(1+x)^{i_2}+\ldots+(1+x)^{i_n}$$ is not less than that of the polynomial $(1+x)^{i_1}$. International Mathematical Olympiad (XXVI) Exploration Let $$P(x)=\sum_{k=1}^{n}(1+x)^{i_k}.$$ We must compare the number of odd coefficients of $P$ with the number of odd coefficients of $(1+x)^{i_1}$. The parity of coefficients suggests...

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Kvant Math Problem 955

Verified: no Verdicts: SKIP + SKIP Solve time: 4m55s Source on kvant.digital Problem Around a round table, there are $n$ participants of the “mad tea party.” Every minute, a pair of neighbors swaps places. What is the minimum time required for all participants to be seated in the exact opposite order (so that everyone’s left neighbor becomes their right neighbor and vice versa)? Solve this problem: for $n=4$, 5, and...

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Kvant Math Problem 954

Verified: no Verdicts: SKIP + SKIP Solve time: 8m23s Source on kvant.digital Problem A rectangle with sides $a$ and $b$ is inscribed in a triangle so that all its vertices lie on the sides of the triangle. Let $a_1$ and $b_1$ be the lengths of the projections of the triangle onto lines parallel to the sides $a$ and $b$ of the rectangle. Prove the equality $$\dfrac a{a_1}+\dfrac b{b_1}=1.$$. A rectangular...

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Kvant Math Problem 952

Verified: no Verdicts: SKIP + SKIP Solve time: 1m28s Source on kvant.digital Problem Let ${x}$ denote the fractional part of the number $x$; that is, ${x}=x-[x]$, where $[x]$ is the greatest integer not exceeding $x$. Give an example of a number $a$ such that $${a}+\left{\dfrac1a\right}=1.$$. Prove that such a number $a$ cannot be rational. I. Varga (Romania) Exploration Write $$a=n+x, \qquad n=[a],\quad 0\le x<1.$$ Then $${a}=x.$$ The condition becomes $$x+\left{\frac1a\right}=1.$$...

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Kvant Math Problem 950

Verified: no Verdicts: SKIP + SKIP Solve time: 10m39s Source on kvant.digital Problem Twenty-five dwarfs are dividing garden plots in Flower City. Each plot is a square of $1\times1$, and all the plots together form a square of $5\times5$. Each dwarf is in a quarrel with no more than three other dwarfs. Prove that it is possible to allocate the plots so that no two dwarfs who are quarrelling have...

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Kvant Math Problem 947

Verified: no Verdicts: SKIP + SKIP Solve time: 10m44s Source on kvant.digital Problem On the board, the numbers $$1\quad\dfrac{1}{2}\quad\dfrac{1}{3}\quad\dfrac{1}{4}\quad{\ldots}\quad\dfrac{1}{10}\quad\dfrac{1}{11}\quad\dfrac{1}{12}.$$ are written in a row. Prove that no matter how we place the signs «$+$» and «$-$» between these numbers, the resulting sum will never be zero. What is the minimum number of written numbers that must be erased from the board so that, after placing «$+$» and «$-$» between the...

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Kvant Math Problem 946

Verified: no Verdicts: SKIP + SKIP Solve time: 1m31s Source on kvant.digital Problem Two parabolas are situated in the plane so that their axes are perpendicular to each other, and the parabolas intersect at four points. Prove that these four points lie on a single circle. Z. A. Skopets All-Russian Mathematical Olympiad for School Students (XI) Exploration Position two parabolas in the plane with perpendicular axes. The simplest choice is...

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Kvant Math Problem 942

Verified: no Verdicts: SKIP + SKIP Solve time: 3m25s Source on kvant.digital Problem The numbers 1, 2, $\ldots$, $2n-1$, $2n$ are divided into two groups containing $n$ numbers each. Let $a_1\lt a_2\lt\ldots\lt a_n$ be the numbers of the first group arranged in increasing order, and $b_1\gt b_2\gt\ldots\gt b_n$ the numbers of the second group arranged in decreasing order. Prove that $$|a_1-b_1|+|a_2-b_2|+\ldots+|a_n-b_n|=n^2.$$ V. V. Proizvolov All-Union Mathematical Olympiad (1985) Exploration For...

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Kvant Math Problem 941

Verified: no Verdicts: SKIP + SKIP Solve time: 9m09s Source on kvant.digital Problem A regular $(4k+2)$-gon $A_0A_1 \ldots A_{4k+1}$ with center $O$ is given. Prove that the sum of the segments cut off by the angle $A_kOA_{k+1}$ on the lines $A_1A_{2k}$, $A_2A_{2k-1}$, $\ldots$, $A_kA_{k+1}$ (see Figure 1 for $k=2$) is equal to the radius of the circumcircle of the $(4k+2)$-gon if $k=2$; $k$ is any positive integer. Figure 1 I....

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Kvant Math Problem 940

Verified: no Verdicts: SKIP + SKIP Solve time: 5m50s Source on kvant.digital Problem A square is divided into rectangles. We call a chain a set of these rectangles such that their projections onto one side of the square completely cover that side without overlaps (Fig. 1). Prove that any two rectangles belong to some chain. Prove an analogous statement for a cube divided into rectangular parallelepipeds (in the definition of...

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Kvant Math Problem 939

Verified: no Verdicts: SKIP + SKIP Solve time: 1m46s Source on kvant.digital Problem In the cells of a $10\times10$ table, the digits 0, 1, $\ldots$, …, 9 are written in some manner so that each digit occurs exactly 10 times. Is it possible to arrange them so that no more than 4 different digits appear in each row and each column? Prove that there exists a row or column in...

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Kvant Math Problem 938

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem A radius of a circle centered at $O$ rotates uniformly, turning through an angle of $\dfrac{360^\circ}n$ in one second (where $n$ is a natural number greater than 3). Initially it occupied the position $OM_0$; after one second, the position $OM_1$; after a further 2 seconds, the position $OM_2$; after a further 3 seconds, the position $OM_3$; and...

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Kvant Math Problem 936

Verified: no Verdicts: SKIP + SKIP Solve time: 1m02s Source on kvant.digital Problem Prove that in $3n+1$ weighings on a balance scale without weights, it is possible to identify the lightest and the heaviest of $2n+2$ stones, if: $n=3$; $n$ — any natural number. S. V. Fomin City Tournament (Spring, 1985) Exploration Consider the simplest nontrivial case $n=1$. Then we have $2n+2 = 4$ stones, and the problem asks to...

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Kvant Math Problem 934

Verified: no Verdicts: SKIP + SKIP Solve time: 5m24s Source on kvant.digital Problem In space, there are $2n$ ($n\ge 2$) points (such that no four lie in the same plane) and $n^2+1$ segments with endpoints at these points. Prove that the drawn segments form at least one triangle; at least $n$ triangles. S. B. Gashkov Moscow Mathematical Olympiad Exploration Interpret the $2n$ points as vertices of a graph $G$ with...

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Kvant Math Problem 933

Verified: no Verdicts: SKIP + SKIP Solve time: 5m06s Source on kvant.digital Problem 13 knights from $k$ different clans ($1 \lt k \lt 13$) are seated around a round table. Each holds either a gold or a silver goblet, with exactly $k$ gold goblets in total. King Arthur ordered the knights to simultaneously pass their goblets to their neighbors on the right, then do the same once more, and so...

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Kvant Math Problem 931

Verified: no Verdicts: SKIP + SKIP Solve time: 1m34s Source on kvant.digital Problem An incircle is inscribed in triangle $ABC$, touching the sides $AB$, $BC$, and $CA$ at the points $C_{1}$, $A_{1}$, and $B_{1}$ respectively. It is known that the lengths of the segments $AA_{1}$, $BB_{1}$, and $CC_{1}$ are equal. Prove that triangle $ABC$ is equilateral. A. N. Dranishnikov Moscow Mathematical Olympiad Exploration Consider triangle $ABC$ with an incircle touching...

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Kvant Math Problem 929

Verified: no Verdicts: SKIP + SKIP Solve time: 4m27s Source on kvant.digital Problem The natural numbers $a$, $b$, $c$, $d$, $e$ satisfy the condition $a^4+b^4+c^4+d^4=e^4$. Prove that at least three of them are even, three are divisible by 5, two are divisible by 10. V. D. Yakovlev Exploration Consider the equation $a^4 + b^4 + c^4 + d^4 = e^4$ modulo small primes to understand divisibility constraints. Begin with parity....

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Kvant Math Problem 928

Verified: no Verdicts: SKIP + SKIP Solve time: 2m12s Source on kvant.digital Problem There are $N+1$ seats in a movie theater. Initially, $N$ people holding tickets with assigned seat numbers (including Igor) sit down in an arbitrary set of $N$ seats without looking at their tickets. The last arriving spectator, the $(N+1)$-th one, wants to take his own seat; if it is occupied, he makes the person sitting there move....

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Kvant Math Problem 926

Verified: no Verdicts: SKIP + SKIP Solve time: 55s Source on kvant.digital Problem Something went wrong. If this issue persists please contact us through our help center at help.openai.com. I do not have the statement of Kvant problem M926. The text you provided contains only the instructions for how the solution should be written, but the actual problem statement is missing. Please paste the full statement of Kvant math problem...

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Kvant Math Problem 925

Verified: no Verdicts: SKIP + SKIP Solve time: 3m40s Source on kvant.digital Problem (Evolution of an Inkblot.) A blue figure $K_0$ is placed on a white plane. From it, a new blue figure $K_1$ is obtained according to the following rule, applied simultaneously to all points $M$ of the plane: if at least half of the area of the disk of radius 1 centered at the point $M$ is occupied...

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Kvant Math Problem 923

Verified: no Verdicts: SKIP + SKIP Solve time: 4m54s Source on kvant.digital Problem Prove that the area of the projection of a cube with edge length 1 onto any plane is numerically equal to the length of its projection onto the line perpendicular to that plane. Exploration Consider a unit cube in three-dimensional space with edges parallel to the axes. Its projection onto a plane depends on the orientation of...

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Kvant Math Problem 921

Verified: no Verdicts: SKIP + SKIP Solve time: 9m17s Source on kvant.digital Problem In a convex quadrilateral $ABCD$, the measures of two angles $\angle A=\alpha$ and $\angle B=\beta$ are known, and its doubled area is equal to ${AB}\cdot{CD}+{BC}\cdot{AD}$. Find the ratio of the lengths of all its sides $AB:BC:CD:DA$, if $\alpha=\dfrac{5\pi}{12}$, $\beta=\dfrac{7\pi}{12}$; $\alpha=\dfrac{\pi}{2}$, $\beta=\dfrac{\pi}{3}$; Determine for which $\alpha$ and $\beta$ such a quadrilateral exists, and express the ratio of its...

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Kvant Math Problem 919

Verified: no Verdicts: SKIP + SKIP Solve time: 6m07s Source on kvant.digital Problem Prove the equality $$\textstyle\int\limits_0^{\pi/4}\tg x,dx+\int\limits_0^1\arctg x,dx=\dfrac\pi4.$$ Prove the inequality $$\textstyle9\lt\int\limits_0^3\sqrt[\scriptstyle4~]{x^4+1},dx+ \int\limits_1^3\sqrt[\scriptstyle4~]{x^4-1},dx\lt9{,}0001.$$ Yu. I. Ionin Exploration For the first integral equality, the two integrals involve complementary functions: the tangent function on $[0,\pi/4]$ and the arctangent function on $[0,1]$. A natural thought is to attempt a substitution in one integral that will transform it into the other, possibly using...

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Kvant Math Problem 917

Verified: no Verdicts: SKIP + SKIP Solve time: 8m32s Source on kvant.digital Problem What is the length of the longest run of consecutive non-lucky tickets? How many such runs of maximum length are there? (Ticket numbers are assumed to range from 000000 to 999999 inclusive; a ticket is called lucky if the sum of the first three digits of its number equals the sum of the last three digits.) S....

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Kvant Math Problem 915

Verified: no Verdicts: SKIP + SKIP Solve time: 7m26s Source on kvant.digital Problem Prove that for any positive numbers $a$, $b$, $c$, $d$ the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}\ge2.$$ L. D. Kurlandchik Exploration The inequality is cyclic rather than symmetric: $$\frac{a}{b+c}+\frac{b}{c+d}+\frac{c}{d+a}+\frac{d}{a+b}.$$ A natural first test is the equality case $a=b=c=d$. Then the left-hand side equals $$4\cdot\frac{1}{2}=2,$$ so the constant $2$ is plausible. The form resembles Nesbitt's inequality. Titu's lemma gives $$\sum...

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Kvant Math Problem 913

Verified: no Verdicts: SKIP + SKIP Solve time: 6m05s Source on kvant.digital Problem Tangents to the circumcircle of triangle $ABC$ drawn at points $A$ and $B$ intersect at point $P$. Prove that the line $PC$ intersects the side $AB$ at point $K$, dividing it in the ratio $AC^2:BC^2$; is symmetric to the median drawn from $C$ with respect to the angle bisector of $C$ of the triangle. S. Litovchenko, 10th-grade...

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Kvant Math Problem 911

Verified: no Verdicts: SKIP + SKIP Solve time: 3m06s Source on kvant.digital Problem On the sides $AB$ and $CD$ of a convex quadrilateral $ABCD$, arbitrary points $E$ and $F$ are chosen, respectively. Prove that the midpoints of the segments $AF$, $BF$, $CE$, and $DE$ form the vertices of a convex quadrilateral whose area does not depend on the choice of points $E$ and $F$. M. V. Stark Exploration Place quadrilateral...

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Kvant Math Problem 910

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem Points $A_1$, $A_2$, $\ldots$, $A_6$ are taken on the sides of a regular hexagon $A_1\ldots A_6$ (Fig. 1). It is known that the three pairwise nonadjacent sides of the hexagon $A_1\ldots A_6$ ($A_1A_2$, $A_3A_4$, $A_5A_6$) determine a triangle $KLM$ whose vertices lie on the extensions of the diagonals of the regular hexagon. Prove that the same is...

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Kvant Math Problem 907

Verified: no Verdicts: SKIP + SKIP Solve time: 6m49s Source on kvant.digital Problem For the triangle $ABC$ with side lengths $a=BC$, $b=AC$, $c=AB$, it is known that $3\widehat{A}+2\widehat{B}=180^\circ$. Prove that $a^2+bc-c^2=0$. T. A. Dzhortmenadze Exploration Let $A=\widehat A$, $B=\widehat B$, $C=\widehat C$. The condition is $$3A+2B=180^\circ.$$ Since $A+B+C=180^\circ$, $$C=180^\circ-(A+B)=2A+B.$$ Combining this with $3A+2B=180^\circ$ gives $$A+2C=180^\circ.$$ Hence $$A=180^\circ-2C.$$ Using the sine law, $$a:k\sin A,\qquad b:k\sin B,\qquad c:k\sin C.$$ The desired identity...

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Kvant Math Problem 905

Verified: no Verdicts: SKIP + SKIP Solve time: 10m22s Source on kvant.digital Problem Prove that the equation $4x^n+(x+1)^2=y^2$ in the natural numbers $x$ and $y$ has no solutions for $n=1$, has at least two solutions for $n=2$, has infinitely many solutions for $n=2$, has no solutions in the natural numbers $n\ge3$. M. Garaev, 10th-grade student Exploration Consider the equation $4x^n + (x+1)^2 = y^2$ in natural numbers $x$, $y$, and...

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Kvant Math Problem 904

Verified: no Verdicts: SKIP + SKIP Solve time: 1m16s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a solution because the actual problem statement is missing. The identifier “Kvant math problem M904” is not enough to reconstruct the problem, and your prompt says that the text version is unavailable while...

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Kvant Math Problem 903

Verified: no Verdicts: SKIP + SKIP Solve time: 2m56s Source on kvant.digital Problem Does there exist a convex polyhedron such that any cross-section by a plane not passing through a vertex is a polygon with an even number of sides, an odd number of sides? A. A. Dorogovtsev Exploration A plane section of a convex polyhedron changes combinatorially only when the plane passes through a vertex. For a generic plane,...

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Kvant Math Problem 901

Verified: no Verdicts: SKIP + SKIP Solve time: 2m59s Source on kvant.digital Problem The bisectors $AK$ and $BM$ of triangle $ABC$ intersect at point $O$. Prove that if $OK=OM$, then either the angles $A$ and $B$ of the triangle are equal, or the angle $C$ is equal to $60^\circ$. Exploration Consider triangle $ABC$ with bisectors $AK$ and $BM$ intersecting at $O$. The condition $OK = OM$ suggests a symmetry of...

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Kvant Math Problem 898

Verified: no Verdicts: SKIP + SKIP Solve time: 8m48s Source on kvant.digital Problem For odd natural numbers $a\lt b\lt c\lt d$, the following conditions hold: $ad=bc$; $a+d=2^k$ and $b+c=2^m$, where $k$ and $m$ are certain natural numbers. Prove that $a=1$; for each $m\ge3$, there exists exactly one set of numbers $a$, $b$, $c$, $d$, $k$ satisfying these conditions. Exploration Consider the given odd natural numbers $a<b<c<d$ satisfying $ad=bc$, $a+d=2^k$, and...

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Kvant Math Problem 896

Verified: no Verdicts: SKIP + SKIP Solve time: 4m18s Source on kvant.digital Problem For a convex quadrilateral $ABCD$, it is known that the circle with diameter $AB$ is tangent to the line $CD$. Prove that the circle with diameter $CD$ is tangent to the line $AB$ if and only if the lines $BC$ and $AD$ are parallel. Exploration The condition that the circle with diameter $AB$ is tangent to the...

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Kvant Math Problem 893

Verified: no Verdicts: SKIP + SKIP Solve time: 12m10s Source on kvant.digital Problem Every two of the $n$ computer units are connected by a wire. Is it possible to color each of these wires with one of $n-1$ colors so that $n-1$ wires of different colors leave each unit, if $n=6$; $n=13$? V. B. Alekseev Moscow Mathematical Olympiad (1984) Exploration The complete graph on $n$ vertices is $K_n$. The condition...

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Kvant Math Problem 892

Verified: no Verdicts: SKIP + SKIP Solve time: 8m30s Source on kvant.digital Problem Prove that among the numbers $2^m+2^k$, and also among the numbers $3^m+3^k$, there are infinitely many perfect squares, whereas among the numbers $4^m+4^k$, $5^m+5^k$, and $6^m+6^k$ there is not a single square of an integer (here $m$ and $k$ are natural numbers, $m\ne k$). Are there any perfect squares among the numbers $7^m+7^k$? A. I. Zaichik Exploration...

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Kvant Math Problem 890

Verified: no Verdicts: SKIP + SKIP Solve time: 8m30s Source on kvant.digital Problem In a country having the shape of a square with side length 1000 km, there are 51 cities. The country has the resources to build $11,000~\text{км}$ highways. Will it be able to connect all its cities by a network of highways? L. D. Kurlyandchik All-Russian Mathematical Olympiad for School Students (Grade 10, 1984) Exploration The problem concerns...

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Kvant Math Problem 887

Verified: no Verdicts: SKIP + SKIP Solve time: 7m39s Source on kvant.digital Problem Two tangents to a circle, $CA$ and $CB$, intersect at the point $C$ ($A$ and $B$ are the points of tangency, see Fig. 1). A second circle passes through the point $C$, is tangent to the line $AB$ at the point $B$, and intersects the first circle at the point $M$. Prove that the line $AM$ bisects...

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Kvant Math Problem 886

Verified: no Verdicts: SKIP + SKIP Solve time: 1m53s Source on kvant.digital Problem Is it possible to place $4n-4$ cells arranged along the perimeter of a square consisting of $n \times n$ cells with $4n-4$ consecutive integers (not necessarily positive) so that the sums of the numbers at the vertices of every rectangle whose sides are parallel to the diagonals of the square, as well as the sums of the...

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Kvant Math Problem 883

Verified: no Verdicts: SKIP + SKIP Solve time: 14m25s Source on kvant.digital Problem Fig. 1 What is the smallest number of colors needed to color the cells of an infinite sheet of square grid paper so that any two cells at distance 6 are colored differently? (The distance between two cells is the smallest number of horizontal and vertical grid lines that a rook must cross on a path from...

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Kvant Math Problem 880

Verified: no Verdicts: SKIP + SKIP Solve time: 4m49s Source on kvant.digital Problem In the sequence 1, 0, 1, 0, 1, 0, $\ldots$, each term, starting from the seventh, is equal to the last digit of the sum of the six preceding terms. Prove that in this sequence, the six numbers 0, 1, 0, 1, 0, 1 never appear consecutively. A. S. Merkuryev Leningrad City Mathematical Olympiad (50, 1984) Exploration...

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Kvant Math Problem 878

Verified: no Verdicts: SKIP + SKIP Solve time: 10m23s Source on kvant.digital Problem Prove that if the sum of the plane angles at the apex of a pyramid exceeds $180^\circ$, then each lateral edge of the pyramid is less than the semiperimeter of its base. Yu. I. Ionin, A. V. Smirnov Leningrad City Mathematical Olympiad (50, 1984) Exploration Consider a pyramid with apex $A$ and base $B_1B_2\dots B_n$. Denote the...

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Kvant Math Problem 877

Verified: no Verdicts: SKIP + SKIP Solve time: 3m25s Source on kvant.digital Problem From a sheet of squared paper measuring $29\times 29$ cells, 99 squares of size $2\times 2$ each were cut out. Prove that it is possible to cut out one more square of the same size from the remaining sheet. S. V. Fomin Leningrad City Mathematical Olympiad (Problem 50, 1984) Exploration Consider a smaller version of the problem...

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Kvant Math Problem 874

Verified: no Verdicts: SKIP + SKIP Solve time: 11m31s Source on kvant.digital Problem For which integers $m$ and $n$ does the equality hold $(5+3\sqrt{2})^m=(3+5\sqrt{2})^n$? $(a+b\sqrt{d})^m=(b+a\sqrt{d})^n$, where $a$ and $b$ ($a\ne b$) are coprime natural numbers, and $d\gt1$ is a natural number whose divisors contain no squares of prime numbers? Yu. V. Mikheev All-Union School Mathematical Olympiad (1984) Exploration We begin by testing small integer values to see whether the equation...

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Kvant Math Problem 872

Verified: no Verdicts: SKIP + SKIP Solve time: 8m18s Source on kvant.digital Problem On the plane, there are three circles $C_1$, $C_2$, $C_3$ with radii $r_1$, $r_2$, $r_3$ — each lying outside the other two, with $r_1\gt r_2$ and $r_1\gt r_3$. From the point of intersection of the external tangents to circles $C_1$ and $C_2$, tangents are drawn to circle $C_3$, and from the point of intersection of the external...

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Kvant Math Problem 870

Verified: no Verdicts: SKIP + SKIP Solve time: 7m15s Source on kvant.digital Problem Along one side of an infinite corridor, there is an infinite number of rooms, numbered consecutively with integers, and each contains a piano. A finite number of pianists live in these rooms. (Several pianists may live in the same room.) Each day, two pianists living in neighboring rooms — the $k$-th and $(k+1)$-th — realize that they...

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Kvant Math Problem 867

Verified: no Verdicts: SKIP + SKIP Solve time: 7m22s Source on kvant.digital Problem In a dance class, 17 boys and 17 girls formed two parallel rows, creating 17 pairs. In each pair, the boy’s height differs from the girl’s height by no more than ten centimeters. Prove that if the boys and girls in each row are rearranged by height, then in each pair the boy and the girl will...

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Kvant Math Problem 864

Verified: no Verdicts: SKIP + SKIP Solve time: 7m10s Source on kvant.digital Problem We will call a partition of a triangle into similar triangles "beautiful" if no two of the triangles are equal in size. Prove that for every right triangle there exists a beautiful partition. Is it possible to construct a beautiful partition of an equilateral triangle? For which non-equilateral triangles does a beautiful partition exist? A. V. Savkin...

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Kvant Math Problem 863

Verified: no Verdicts: SKIP + SKIP Solve time: 1m32s Source on kvant.digital Problem In each square of a $n\times n$ board there is a token. Is it possible to rearrange them so that any two tokens that threatened each other with a knight’s move before the rearrangement will threaten each other with a king’s move after the rearrangement, if $n=3$, $n=6$, $n=4$? S. Stefanov Exploration Consider a small board, $n=3$....

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Kvant Math Problem 861

Verified: no Verdicts: SKIP + SKIP Solve time: 58s Source on kvant.digital Problem Prove that from any $n$ numbers one can select several of them (possibly just one) so that the sum of the selected numbers differs from the nearest integer by at most $\dfrac{1}{n+1}$. S. Yu. Orevkov Exploration Consider small values of $n$ to understand the behavior of the sums modulo $1$. For $n=1$, the claim reduces to selecting...

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Kvant Math Problem 860

Verified: no Verdicts: SKIP + SKIP Solve time: 1m40s Source on kvant.digital Problem Let $O$ and $R$ be the center and radius of the circumcircle of triangle $ABC$, $Z$ and $r$ the center and radius of its incircle, and $K$ the point of intersection of the medians of the triangle whose vertices are the points of tangency of the incircle with the sides of triangle $ABC$. Prove that the point...

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Kvant Math Problem 859

Verified: no Verdicts: SKIP + SKIP Solve time: 55s Source on kvant.digital Problem Find the smallest positive number $a$ such that for every quadratic polynomial $f(x)$ satisfying the inequality $|f(x)|\le1$ for $0\le x\le1$, the inequality $|f'(1)|\le a$ holds. V. P. Pikulin Exploration Let $$f(x)=ux^2+vx+w.$$ We seek the largest possible value of $|f'(1)|=|2u+v|$ under the condition $$|f(x)|\le 1 \qquad (0\le x\le 1).$$ The smallest admissible constant $a$ is exactly this maximum....

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Kvant Math Problem 858

Verified: no Verdicts: SKIP + SKIP Solve time: 2m09s Source on kvant.digital Problem For the quantities $\alpha$, $\beta$, and $\gamma$ of the angles of a triangle, the following relation holds: $$\sin^2\alpha+\sin^2\beta=\sin\gamma.$$ Find $\alpha$, $\beta$, $\gamma$, given that the triangle is isosceles (consider all cases). Can the triangle be acute-angled? What values can the largest angle of the triangle take? P. B. Gusyatnikov Exploration Let $$\sin^2\alpha+\sin^2\beta=\sin\gamma,$$ where $\alpha+\beta+\gamma=\pi$. Since $\gamma=\pi-(\alpha+\beta)$, $$\sin\gamma=\sin(\alpha+\beta).$$...

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Kvant Math Problem 856

Verified: no Verdicts: SKIP + SKIP Solve time: 3m01s Source on kvant.digital Problem Construct a quadrilateral given the lengths of its sides and the length of the segment connecting the midpoints of its diagonals. Under what conditions does the problem have a solution? I. Z. Titovich

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Kvant Math Problem 855

Verified: no Verdicts: SKIP + SKIP Solve time: 3m03s Source on kvant.digital Problem Is it possible to pass a rigid regular tetrahedron with edge length 1 through a hoop of diameter: 1; $0{,}95$; $0{,}9$; $0{,}85$? V. V. Proizvolov

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Kvant Math Problem 854

Verified: no Verdicts: SKIP + SKIP Solve time: 3m10s Source on kvant.digital Problem Before I begin, could you provide the full text of Kvant problem M854? I need the problem statement in order to construct the rigorous solution according to your specifications.

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Kvant Math Problem 852

Verified: no Verdicts: SKIP + SKIP Solve time: 9m12s Source on kvant.digital Problem

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Kvant Math Problem 849

Verified: no Verdicts: SKIP + SKIP Solve time: 15m44s Source on kvant.digital Problem Using the digits 1, 2, $\ldots$, 7, arranged in different orders, seven seven-digit numbers are formed. Prove that the sum of the seventh powers of some of these numbers cannot equal the sum of the seventh powers of the remaining numbers. G. A. Galperin

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Kvant Math Problem 848

Verified: no Verdicts: SKIP + SKIP Solve time: 1m46s Source on kvant.digital Problem Fig. 1 Plot the graph of the function $$f_0(x)=||x-1|-2||x|-3||.$$. In Figure 1, the graphs of three “piecewise-linear” functions $f_1$, $f_2$, $f_3$ are shown. Write their formulas in the form $$y=kx+b+c_1|x-a_1|+\ldots+c_m|x-a_m|,$$, where $k$, $b$, $c_i$, $a_i$ are certain numbers, $m$ is the number of breakpoints of the graph, $a_i$ are the abscissas of the breakpoints, $i=1$, 2, $\ldots$,...

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Kvant Math Problem 846

Verified: no Verdicts: SKIP + SKIP Solve time: 1m57s Source on kvant.digital Problem Prove that the arithmetic mean of the side lengths of a regular polygon is less than the arithmetic mean of the lengths of its diagonals. V. F. Lev Exploration Let the regular polygon have $n$ sides and circumradius $R$. The lengths of all chords are $$d_k=2R\sin\frac{k\pi}{n}, \qquad k=1,2,\dots,\left\lfloor\frac n2\right\rfloor .$$ Here $d_1$ is the side length. Every...

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Kvant Math Problem 845

Verified: no Verdicts: SKIP + SKIP Solve time: 1m39s Source on kvant.digital Problem For which $n$ of the $n$ corners, each consisting of four $1\times1$ squares, and some number of $4\times1$ rectangles (Fig. 1) is it possible to construct a centrally symmetric figure (a polygon on graph paper)? Fig. 1 V. G. Belov Exploration We consider the problem of forming a centrally symmetric polygon using two types of tiles: a...

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Kvant Math Problem 844

Verified: no Verdicts: SKIP + SKIP Solve time: 4m37s Source on kvant.digital Problem Prove that any natural number $a$ can be represented uniquely in the form $$a=a_1\cdot1!+a_2\cdot2!+\ldots+a_n\cdot n!, \tag{1}$$ where $a_k$ are integers, $0\le a_k \le k$, $a_n\neq0$. (By definition, $k!=1 \cdot 2 \cdot \ldots \cdot k$, $1!=1.$) Prove that any rational number $r$, $0 \le r<1$ can be represented uniquely in the form $$r=\dfrac{b_1}{2!}+\dfrac{b_2}{3!}+\ldots+\dfrac{b_n}{(n+1)!}, \tag{2}$$ where $b_k$ are integers,...

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Kvant Math Problem 842

Verified: no Verdicts: SKIP + SKIP Solve time: 7m11s Source on kvant.digital Problem Prove that if $\alpha+\beta+\gamma=0$, then $$\sin\alpha+\sin\beta+\sin\gamma=-4\sin\dfrac{\alpha}{2}\sin\dfrac{\beta}{2}\sin\dfrac{\gamma}{2}.$$. The quantities $\widehat{A}$, $\widehat{B}$, and $\widehat{C}$ of the angles of a triangle satisfy the condition $$\dfrac{\sin\widehat{A}+\sin\widehat{B}+\sin\widehat{C}}{\cos\widehat{A}+\cos\widehat{B}+\cos\widehat{C}}=\sqrt{3}.$$. Prove that at least one of them is equal to $60^\circ$. L. D. Kurlyandchik Exploration Consider the first identity $\sin\alpha + \sin\beta + \sin\gamma$ under the constraint $\alpha + \beta + \gamma = 0$. Using...

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Kvant Math Problem 841

Verified: no Verdicts: SKIP + SKIP Solve time: 1m17s Source on kvant.digital Problem Prove that the product of the lengths of the segments into which the hypotenuse of a right triangle is divided by the point of tangency of its incircle is equal to the area of the triangle. Exploration Consider a right triangle $ABC$ with right angle at $C$. Denote the hypotenuse as $AB$ and let the incircle touch...

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Kvant Math Problem 839

Verified: no Verdicts: SKIP + SKIP Solve time: 8m47s Source on kvant.digital Problem Is it possible to choose 1983 natural numbers not exceeding $10^5$ such that among the chosen numbers there is no triple forming an arithmetic progression (i.e., no triple $a$, $b$, $c$ in which $a+c=2b$)? International Mathematical Olympiad for School Students (XXIV, 1983) Exploration For a set of integers with no three-term arithmetic progression, the classical example is...

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Kvant Math Problem 837

Verified: no Verdicts: SKIP + SKIP Solve time: 9m11s Source on kvant.digital Problem Let $a$, $b$, $c$ be positive integers, each pair of which is coprime. Prove that the largest integer that cannot be represented in the form $$xbc+yca+zab$$ (where $x$, $y$, $z$ are non-negative integers) is $$2abc-ab-bc-ca.$$. International Mathematical Olympiad for School Students (XXIV, 1983) Exploration Consider the simpler case where two of the numbers are coprime. Let $a$,...

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Kvant Math Problem 835

Verified: no Verdicts: SKIP + SKIP Solve time: 7m31s Source on kvant.digital Problem In a single-round chess tournament, $n$ players from country $A$ and $n$ players from country $B$ participated. It turned out that no matter how the players were paired (so that players from different countries played against each other), there is always at least one pair of players who have already faced each other before. Prove that it...

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Kvant Math Problem 833

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem The sequence $x_n$ is defined by the conditions: $$x_1=2;\quad x_{n+1}=\dfrac{2+x_n}{1-2x_n}\quad (n=1{,}~2{,} 3{,} \ldots).$$. Prove that $x_n\ne0$ (for all $n$); this sequence is non-periodic. V. E. Matizen Exploration Compute the first few terms to detect patterns. With $x_1 = 2$, $x_2 = \frac{2 + 2}{1 - 2 \cdot 2} = \frac{4}{-3} = -\frac{4}{3}.$ Next, $x_3 = \frac{2 +...

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Kvant Math Problem 832

Verified: no Verdicts: SKIP + SKIP Solve time: 4m08s Source on kvant.digital Problem Prove that for any natural number $n\gt6$, a square can be divided into $n$ squares (some of which may be of equal size). Prove that for any natural number $n\gt100$, a cube can be divided into $n$ cubes. V. A. Li Exploration Consider first the case of dividing a square into smaller squares. Simple examples show that...

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Kvant Math Problem 829

Verified: no Verdicts: SKIP + SKIP Solve time: 14m07s Source on kvant.digital Problem Prove that among any $2m+1$ distinct integers whose absolute values do not exceed $2m-1$, there exist three numbers whose sum is 0. N. V. Kartashov All-Union Mathematical Olympiad for School Students (1983, Grade 9) Exploration Consider small values of $m$ to detect a pattern. For $m=1$, the set of integers is ${-1,0,1}$, and any $2m+1 = 3$...

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Kvant Math Problem 827

Verified: no Verdicts: SKIP + SKIP Solve time: 7m27s Source on kvant.digital Problem It is known that the four blue triangles in Figure 1 have equal areas. Prove that the three red quadrilaterals in this figure also have equal areas. Find the area of one quadrilateral if the area of one blue triangle is equal to 1. Fig. 1 B. I. Chinik All-Union Mathematical Olympiad for School Students (1983, Grade...

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Kvant Math Problem 826

Verified: no Verdicts: SKIP + SKIP Solve time: 7m57s Source on kvant.digital Problem Three numbers were written on the board. Then one of them was erased and replaced with the sum of the other two numbers, minus one. This operation was repeated several times, and as a result, the numbers 17, 1967, 1983 were obtained. Could the original numbers have been 2, 2, 2; 3, 3, 3? A. A. Berzinsh...

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Kvant Math Problem 824

Verified: no Verdicts: SKIP + SKIP Solve time: 4m20s Source on kvant.digital Problem In the grid shown in Figure 3, each cell has size $1\times 1$. Is it possible to represent this grid as a union of a) 8 broken lines of length 5; b) 5 broken lines of length 8? (8) Figure 3 N. I. Avilov All-Union Mathematical Olympiad for School Students (1983, Grade 8) I need the diagram...

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Kvant Math Problem 823

Verified: no Verdicts: SKIP + SKIP Solve time: 7m19s Source on kvant.digital Problem From a photograph, the outline of a house of length 60 m and width 15 m has been copied (Fig. 2). The longer wall is on the left in the photograph (the other parts of the outline are obscured in the photograph by a tree branch). It is required to: complete the outline; draw an accurate map...

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Kvant Math Problem 820

Verified: no Verdicts: SKIP + SKIP Solve time: 3m59s Source on kvant.digital Problem A regular octagon is dissected into a finite number of parallelograms. Prove that among them there are at least two rectangles. A regular $4k$-gon is dissected into a finite number of parallelograms. Prove that among them there are at least $k$ rectangles. Find the total area of the rectangles from part (b), if the side length of...

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Kvant Math Problem 818

Verified: no Verdicts: SKIP + SKIP Solve time: 16m10s Source on kvant.digital Problem Let some $k$ vertices of a regular $n$-gon be colored blue (the remaining vertices are black). We shall call the set of colored vertices uniform if, for every $m$, the numbers of blue vertices in any two sets of $m$ consecutive vertices of the $n$-gon are equal or differ by 1 (see Fig. 1, which shows an...

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Kvant Math Problem 816

Verified: no Verdicts: SKIP + SKIP Solve time: 6m31s Source on kvant.digital Problem The natural numbers $a$ and $b$ are obtained from one another by permuting their digits. Prove that the sums of the digits of the numbers $2a$ and $2b$ are equal; the sums of the digits of the numbers $a/2$ and $b/2$ are equal (if $a$ and $b$ are even); the sums of the digits of the numbers...

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Kvant Math Problem 814

Verified: no Verdicts: SKIP + SKIP Solve time: 9m33s Source on kvant.digital Problem Mark the natural numbers that can be represented as the sum of two squares of natural numbers. Among the marked numbers, there are triples of consecutive numbers, for example $$72=6^2+6^2,\quad 73=8^2+3^2,\quad 74=7^2+5^2.$$. Explain why four consecutive marked numbers cannot occur. Prove that among the marked numbers there are infinitely many pairs; triples of consecutive numbers. L. D....

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Kvant Math Problem 813

Verified: no Verdicts: SKIP + SKIP Solve time: 1m43s Source on kvant.digital Problem Given segments $OA$, $OB$, and $OC$ of equal length (the point $B$ lies inside the angle $AOC$). Circles are constructed on these segments as diameters. Prove that the area of the curvilinear triangle bounded by arcs of these circles and not containing the point $O$ (see Fig. 1, p. 44) is equal to half the area of...

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Kvant Math Problem 811

Verified: no Verdicts: SKIP + SKIP Solve time: 10m38s Source on kvant.digital Problem Let $h_a$, $h_b$, $h_c$ be the altitudes, and $m_a$, $m_b$, $m_c$ the medians of an acute triangle (drawn to the sides $a$, $b$, $c$), and let $r$ and $R$ be the radii of the inscribed and circumscribed circles. Prove that $$\dfrac{m_a}{h_a}+\dfrac{m_b}{h_b}+\dfrac{m_c}{h_c}\le1+\dfrac Rr.$$ D. M. Milošević Exploration The quantities involving medians and altitudes suggest rewriting everything in terms...

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Kvant Math Problem 809

Verified: no Verdicts: SKIP + SKIP Solve time: 10m33s Source on kvant.digital Problem Find the sum $$\frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{n-1}{n!}$$ (the product $1\cdot2\cdot\ldots\cdot k$ is denoted by $k!$). V. V. Proizvolov Exploration Compute the first few terms of the series to detect a pattern. For $n=2$, the sum is $\frac{1}{2!} = \frac{1}{2}$. For $n=3$, the sum is $\frac{1}{2!} + \frac{2}{3!} = \frac{1}{2} + \frac{2}{6} = \frac{1}{2} + \frac{1}{3} = \frac{5}{6}$. For $n=4$, the...

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Kvant Math Problem 808

Verified: no Verdicts: SKIP + SKIP Solve time: 4m14s Source on kvant.digital Problem On an infinite sheet of graph paper, two players play the following game: the first player colors any cell red, the second colors $k$ (uncolored) cells blue, then the first colors one (uncolored) cell red again, the second colors $k$ cells blue, and so on. The first player aims to have four red cells positioned at the...

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Kvant Math Problem 805

Verified: no Verdicts: SKIP + SKIP Solve time: 10m42s Source on kvant.digital Problem On the sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen respectively such that the segments $AA_1$, $BB_1$, and $CC_1$ intersect at a single point. Prove that $S_{A_1B_1C_1} \le \dfrac{S_{ABC}}{4}$. On the faces $BCD$, $CDA$, $BDA$, $ABC$ of tetrahedron $ABCD$, points $A_1$, $B_1$, $C_1$, $D_1$ are chosen respectively such that the segments $AA_1$,...

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Kvant Math Problem 804

Verified: no Verdicts: SKIP + SKIP Solve time: 7m34s Source on kvant.digital Problem Point $O$ is the midpoint of the axis of a right circular cylinder, $A$ and $B$ are diametrically opposite points on the circle of the cylinder’s lower base, and $C$ is a point on the circle of the upper base, not lying in the plane $OAB$. Prove that the sum of the dihedral angles of the trihedral...

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Kvant Math Problem 802

Verified: no Verdicts: SKIP + SKIP Solve time: 2m14s Source on kvant.digital Problem On the sides $AB$ and $BC$ of triangle $ABC$, right triangles $APB$ and $BQC$ are constructed externally using these sides as hypotenuses, with identical angles of measure $\beta$ at their common vertex $B$ (see Fig. 1). Find the angles of triangle $PQK$, where $K$ is the midpoint of side $AC$. Fig. 1 L. P. Kuptsov Exploration The...

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Kvant Math Problem 801

Verified: no Verdicts: SKIP + SKIP Solve time: 4m45s Source on kvant.digital Problem Prove that for every positive integer $n$ the equality $$[\sqrt{n}]+[\sqrt[3]{n}]+[\sqrt[4]{n}]+\ldots+[\sqrt[n]{n}]=[\log_2{n}]+[\log_3{n}]+[\log_4{n}]+\ldots+[\log_n{n}]$$ holds ($[x]$ denotes the greatest integer less than or equal to $x$). V. V. Kisil' Exploration Compute several small cases to see the pattern. For $n=1$, the left-hand side is $[\sqrt{1}]=1$, and the right-hand side is $[\log_2 1] + \cdots + [\log_1 1]$. Since $\log_1 1$ is...

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Kvant Math Problem 798

Verified: no Verdicts: SKIP + SKIP Solve time: 7m25s Source on kvant.digital Problem On a circle, $4k$ points are marked and colored alternately red and blue; then $2k$ of the red points are arbitrarily connected in pairs by $k$ red segments, and $2k$ of the blue points by $k$ blue segments (no three segments intersect at a single point). Prove that there are at least $k$ points of intersection between...

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Kvant Math Problem 796

Verified: no Verdicts: SKIP + SKIP Solve time: 10m09s Source on kvant.digital Problem Point $P$ is located inside the square $ABCD$ such that $|AP|:|BP|:|CP|=1:2:3$. Find $\widehat{APB}$. L. D. Kurlandchik Exploration Consider a square $ABCD$ with a point $P$ inside it. Label the vertices in order, and attempt to place $P$ such that $|AP| : |BP| : |CP| = 1:2:3$. Introduce coordinates: let $A=(0,0)$, $B=(1,0)$, $C=(1,1)$, $D=(0,1)$. Then $P=(x,y)$, with distances...

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Kvant Math Problem 795

Verified: no Verdicts: SKIP + SKIP Solve time: 3m19s Source on kvant.digital Problem Let $\sigma (n)$ denote the sum of all divisors of the natural number $n$ (Fig. 3). Prove that there exist infinitely many $n$ such that $\sigma(n)\gt 2n$; $\sigma(n)\gt 3n$. Prove that for any $n$ $\sigma(n)\lt n(\log_2n+2)$; $\sigma(n)\lt n(\ln n+1)$. Figure 3 V. F. Lev Exploration Consider small values of $n$ and their divisor sums $\sigma(n)$. For prime...

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Kvant Math Problem 793

Verified: no Verdicts: SKIP + SKIP Solve time: 8m31s Source on kvant.digital Problem From vertex $P$ of tetrahedron $PABC$, three segments $PA'$, $PB'$, $PC'$ are drawn perpendicular to the faces $PBC$, $PCA$, $PAB$ and having lengths equal, respectively, to the areas of these faces (the directions of the segments are chosen so that the points $A'$ and $A$, $B'$ and $B$, $C'$ and $C$ lie on opposite sides of the...

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Kvant Math Problem 791

Verified: no Verdicts: SKIP + SKIP Solve time: 6m54s Source on kvant.digital Problem Petya was given a pocket calculator on which the following operations can be performed: for any given numbers $x$ and $y$, compute $x+y$, $x-y$, $x+1$, and $\dfrac1x$ (when $x\ne0$). Petya claims that with the help of his calculator he can: square any number using no more than six operations; multiply any two numbers using no more than...

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Kvant Math Problem 789

Verified: no Verdicts: SKIP + SKIP Solve time: 7m02s Source on kvant.digital Problem Ten points dividing a circle into ten equal arcs are connected in pairs by five chords. Must there necessarily be two chords of the same length among them? One hundred points dividing a circle into one hundred equal arcs are connected in pairs by fifty chords. Prove that there must necessarily be two chords of the same...

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Kvant Math Problem 786

Verified: no Verdicts: SKIP + SKIP Solve time: 7m18s Source on kvant.digital Problem Prove that for any natural numbers $n$ and $k$ (greater than 1), the number $n^k$ can be expressed as the sum of $n$ consecutive odd numbers. (For example, $4^3=13+15+17+19$; $7^2=1+3+5+7+9+11+13$; $3^4=25+27+29$.) A. N. Kozachenko Exploration Consider small values of $n$ and $k$ to identify a pattern. For $n=2$ and $k=2$, $2^2=4$ can be written as $1+3$. For...

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Kvant Math Problem 784

Verified: no Verdicts: SKIP + SKIP Solve time: 10m41s Source on kvant.digital Problem A spherical planet moves in a circular orbit around a star and rotates about its axis, with the axis of daily rotation tilted relative to the orbital plane by an angle of $\alpha$ (for our Earth $\alpha=66{,}5^\circ$). Determine the dependence of the duration $T$ of the shortest day of the year at a given location on the...

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Kvant Math Problem 783

Verified: no Verdicts: SKIP + SKIP Solve time: 6m14s Source on kvant.digital Problem For what largest $n$ does the system of inequalities $$\left{\begin{array}{l} 1 \lt x \lt 2,\ 2 \lt x^2 \lt 3,\ 3 \lt x^3 \lt 4,\ {\ldots},{\ldots},{\ldots},{\ldots}\ n \lt x^n \lt n+1, \end{array} \right.$$ have solutions? For which $n$ do there exist two progressions — an arithmetic progression $a_1$, $a_2$, $a_3$, $\ldots$, $a_{n+1}$ and a geometric progression $b_1$,...

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Kvant Math Problem 781

Verified: no Verdicts: SKIP + SKIP Solve time: 5m27s Source on kvant.digital Problem Construct a line parallel to side $AC$ of the given triangle $ABC$ that intersects its sides $AB$ and $BC$ at points $D$ and $E$, respectively, such that $|AD|=|BE|$. L. V. Kim Exploration Consider triangle $ABC$ and a line parallel to $AC$ intersecting $AB$ at $D$ and $BC$ at $E$ such that $|AD| = |BE|$. If $AB$ and...

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Kvant Math Problem 779

Verified: no Verdicts: SKIP + SKIP Solve time: 9m50s Source on kvant.digital Problem Consider sequences ${x_n}$ of positive numbers satisfying the condition $$1=x_0 \ge x_1\ge x_2\ge\ldots\ge x_n\ge\ldots$$ Prove that for any such sequence ${x_n}$ there exists $n$ for which $$\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\ldots+\frac{x_{n-1}^2}{x_n}\ge3{,}999.$$ Find a sequence ${x_n}$ satisfying the stated condition such that for every $n$, $$\frac{x_0^2}{x_1}+\frac{x_1^2}{x_2}+\ldots+\frac{x_{n-1}^2}{x_n}\lt4.$$ A. Grishkov International Mathematical Olympiad for School Students (XXIII, 1982) Exploration We are asked to study...

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Kvant Math Problem 778

Verified: no Verdicts: SKIP + SKIP Solve time: 8m06s Source on kvant.digital Problem Given an isosceles triangle $A_1A_2A_3$. Let $a_i$ be its side opposite the vertex $A_i$ ($i=1$, 2, 3), $M_i$; $a_i$ be the midpoint of side $a_i$, $T_i$ the point of tangency of this side with the incircle of the triangle, and $S_i$ the point symmetric to $T_i$ with respect to the angle bisector of $A_i$ of the triangle....

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Kvant Math Problem 777

Verified: no Verdicts: SKIP + SKIP Solve time: 4m00s Source on kvant.digital Problem Given the equation $x^3-3xy^2+y^3=n$. Prove that if a positive integer $n$ is such that the given equation has an integer solution, then it has at least three integer solutions; for $n=2891$ this equation has no integer solutions. International Mathematical Olympiad for School Students (XXIII, 1982) Exploration The given equation is cubic in two variables, $x$ and $y$:...

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Kvant Math Problem 776

Verified: no Verdicts: SKIP + SKIP Solve time: 7m02s Source on kvant.digital Problem On the diagonals $AC$ and $CE$ of a regular hexagon $ABCDEF$, points $M$ and $N$ are chosen respectively so that $$\frac{|AM|}{|AC|} = \frac{|CN|}{|CE|}=\lambda.$$. It is known that the points $B$, $M$, and $N$ lie on the same straight line. Find $\lambda$. International Mathematical Olympiad (XXIII, 1982) Exploration Consider a regular hexagon $ABCDEF$ with vertices labeled consecutively. The...

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Kvant Math Problem 775

Verified: no Verdicts: SKIP + SKIP Solve time: 8m29s Source on kvant.digital Problem For which natural $n\ge3$ do there exist distinct natural numbers $a_1$, $a_2$, $\ldots$, $a_n$ such that $1\le a_k\le n+1$ for every $k=1$, 2, $\ldots$, $n$, and all $n$ numbers $|a_1-a_2|$, $|a_2-a_3|$, $\ldots$, $|a_{n-1}-a_n|$, $|a_n-a_1|$ are distinct? A. V. Anzhants Exploration Consider small values of $n$ to detect patterns. For $n=3$, the allowed numbers are $1,2,3,4$. One possible...

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Kvant Math Problem 774

Verified: no Verdicts: SKIP + SKIP Solve time: 9m49s Source on kvant.digital Problem A function $f(x)$, defined on the interval $[0;1]$, satisfies $$f(0)=f(1)=0\tag1$$ and $$f\left(\frac{x+y}{2}\right)\le f(x)+f(y)\tag2$$ for all $x$, $y\in[0;1]$. Prove that: $f(x)\ge0$ for all $x\in[0;1]$; $f(x)$ has infinitely many zeros on the interval $[0;1]$; if there exists a number $A\ge0$ such that for all $x\in\left[0;\dfrac12\right]$ the inequality $f(x)\le A$ holds, then $f(x)\le A$ for each $x\in[0;1]$; if the function...

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Kvant Math Problem 773

Verified: no Verdicts: SKIP + SKIP Solve time: 9m58s Source on kvant.digital Problem The incircle of triangle $ABC$ is tangent to its sides $AB$, $BC$, and $AC$ at points $M$, $N$, and $P$, respectively. It is known that $$\overrightarrow{AN}+\overrightarrow{BP}+\overrightarrow{CM} = \overrightarrow{0}.$$ Prove that triangle $ABC$ is equilateral. P. B. Gusyatnikov, S. V. Reznichenko Exploration Let the side lengths be $$a=BC,\qquad b=CA,\qquad c=AB,$$ and let $$s=\frac{a+b+c}{2}$$ be the semiperimeter. For the...

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Kvant Math Problem 772

Verified: no Verdicts: SKIP + SKIP Solve time: 6m51s Source on kvant.digital Problem In a workshop, there are five different machines. Training one worker to operate one machine costs 1000 rubles. What is the minimum cost to train 8 workers so that, even if any three of them are absent, all machines can still be used simultaneously? Each worker can operate only one machine at a time. Exploration We are...

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Kvant Math Problem 771

Verified: no Verdicts: SKIP + SKIP Solve time: 4m38s Source on kvant.digital Problem In triangle $ABC$, the bisector $AK$ is drawn. It is known that the centers of the circles: the incircle of triangle $ABK$ and the circumcircle of triangle $ABC$ coincide. Find the angles of triangle $ABC$. Exploration Let $O$ be the common point which is simultaneously the circumcenter of $\triangle ABC$ and the incenter of $\triangle ABK$. Since...

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Kvant Math Problem 769

Verified: no Verdicts: SKIP + SKIP Solve time: 5m49s Source on kvant.digital Problem Fig. 2 The angle bisectors of triangle $ABC$ intersect at point $L$, and their extensions intersect the circumcircle of the triangle at points $A_1$, $B_1$, $C_1$ (Fig. 2). Let $R$ be the circumradius and $r$ the inradius of triangle $ABC$. Prove the equalities $\dfrac{|LA_1|\cdot|LC_1|}{|LB|}=R$; $\dfrac{|LA|\cdot|LC|}{|LB_1|}=r$; $\dfrac{S_{ABC}}{S_{A_1B_1C_1}}=\dfrac{2r}{R}$. R. A. Mazov Exploration Let $I=L$ be the incenter. The points...

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Kvant Math Problem 768

Verified: no Verdicts: SKIP + SKIP Solve time: 5m37s Source on kvant.digital Problem The sum of $n$ numbers, each of which does not exceed 1 in absolute value, is equal to $s$. Prove that it is possible to select several of these numbers so that the sum of the selected numbers differs from $\dfrac s3$ by at most $\dfrac13$. V. P. Grinberg Exploration Consider small values of $n$ first. For...

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Kvant Math Problem 767

Verified: no Verdicts: SKIP + SKIP Solve time: 7m26s Source on kvant.digital Problem Fig. 1 The line $l$ divides the area of a convex polygon in half. Prove that the ratio in which this line divides the projection of the polygon onto a line perpendicular to it does not exceed $1+\sqrt{2}$ (Fig. 1, a ). Each of three lines divides the area of the given figure in half. Prove that...

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Kvant Math Problem 764

Verified: no Verdicts: SKIP + SKIP Solve time: 6m28s Source on kvant.digital Problem Prove that each of the equations $x^2+y^3=z^5$; $x^2+y^3+z^5=t^7$ has infinitely many solutions in natural numbers. O. V. Mazurov Exploration The task is purely existential. We need infinitely many natural solutions of $x^2+y^3=z^5,$ and then infinitely many natural solutions of $x^2+y^3+z^5=t^7.$ A natural first idea is to force all terms to be powers of a common parameter. If...

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Kvant Math Problem 762

Verified: no Verdicts: SKIP + SKIP Solve time: 6m21s Source on kvant.digital Problem Prove that for any positive numbers $a$, $b$, $c$, the inequalities $$a+b+c\le\frac{a^2+b^2}{2c}+\frac{b^2+c^2}{2a}+\frac{c^2+a^2}{2b}\le\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}.$$ hold. Exploration The two inequalities are $$S:=\frac{a^2+b^2}{2c}+\frac{b^2+c^2}{2a}+\frac{c^2+a^2}{2b},$$ and $$a+b+c\le S\le \frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}.$$ First test the symmetric case $a=b=c=t$. Then $$S=3t,$$ and $$\frac{a^3}{bc}+\frac{b^3}{ca}+\frac{c^3}{ab}=3t.$$ Hence equality holds throughout when $a=b=c$. For the left inequality, each summand has the form $$\frac{x^2+y^2}{2z}.$$ Since $x^2+y^2\ge 2xy$, $$\frac{x^2+y^2}{2z}\ge \frac{xy}{z}.$$ Summing gives $$S\ge...

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Kvant Math Problem 761

Verified: no Verdicts: SKIP + SKIP Solve time: 7m13s Source on kvant.digital Problem Through an arbitrary point $P$ on the side $AC$ of triangle $ABC$, lines parallel to its medians $AK$ and $CL$ are drawn, intersecting the sides $BC$ and $AB$ at points $E$ and $F$ respectively (Fig. 1). Prove that the medians $AK$ and $CL$ divide the segment $EF$ into three equal parts. Fig. 1 E. G. Gotman Exploration...

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Kvant Math Problem 759

Verified: no Verdicts: SKIP + SKIP Solve time: 2m23s Source on kvant.digital Problem Inside a convex quadrilateral whose sum of the six pairwise distances between its vertices (i.e., the sum of the lengths of all sides and diagonals) is equal to $S_1$, another quadrilateral is located, for which this sum is equal to $S_2$. Can the quantity $S_2$ be greater than $S_1$? Prove that $S_2\lt\dfrac{4S_1}3$. Prove that if inside an...

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Kvant Math Problem 757

Verified: no Verdicts: SKIP + SKIP Solve time: 10m56s Source on kvant.digital Problem From the sequence 1, $\dfrac12$, $\dfrac13$, $\dfrac14$, $\dots$, it is easy to select an arithmetic progression of length three: $\dfrac12$, $\dfrac13$, $\dfrac16$. Is it possible to select from this sequence an arithmetic progression of length 4? length 5? length $k$, where $k$ is any natural number? G. A. Halperin Exploration Let an arithmetic progression of reciprocals be...

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Kvant Math Problem 755

Verified: no Verdicts: SKIP + SKIP Solve time: 4m52s Source on kvant.digital Problem Inside a tetrahedron, a point $M$ is chosen. Prove that at least one edge of the tetrahedron is seen from the point $M$ under an angle whose cosine is not greater than $-\dfrac{1}{3}$. S. B. Gashkov All-Union School Mathematics Olympiad (1982, 9th grade) Exploration Consider a tetrahedron with vertices $A$, $B$, $C$, and $D$, and a point...

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Kvant Math Problem 753

Verified: no Verdicts: SKIP + SKIP Solve time: 6m21s Source on kvant.digital Problem The numbers $a$, $b$, $c$ lie in the interval $\left(0;\dfrac\pi2\right)$ and satisfy the equalities: $$\begin{align*} \cos a&=a,\ \sin\cos b&=b,\ \cos\sin c&=c. \end{align*}$$. Arrange these numbers in ascending order. S. Hessen All-Union School Mathematics Olympiad (1982, 10th grade) Exploration The problem gives three numbers $a$, $b$, $c$ in the interval $(0, \frac{\pi}{2})$ satisfying $$\cos a = a, \quad...

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Kvant Math Problem 751

Verified: no Verdicts: SKIP + SKIP Solve time: 8m19s Source on kvant.digital Problem On a circle, $3k$ points are marked, dividing it into $3k$ arcs, of which $k$ arcs have length $1$, another $k$ arcs have length $2$, and the remaining $k$ arcs have length $3$. Prove that among the marked points there exist two that are diametrically opposite. V. V. Proizvolov All-Union Mathematical Olympiad for School Students (1982, Grade...

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Kvant Math Problem 750

Verified: no Verdicts: SKIP + SKIP Solve time: 3m46s Source on kvant.digital Problem Fig. 3 Prove that, no matter how one colors the squares of an infinite sheet of graph paper with $N$ colors, there exist a rectangle whose vertices lie at the centers of squares of the same color (with sides parallel to the grid lines — vertical and horizontal lines — Fig. 3, a ); $l$ horizontal and...

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Kvant Math Problem 748

Verified: no Verdicts: SKIP + SKIP Solve time: 6m09s Source on kvant.digital Problem Fig. 2 Is it possible to place a finite number of parabolas on the plane so that their interior regions cover the entire plane? ( By the interior region of a parabola we mean the convex figure bounded by this parabola — see Fig. 2.) In space, several non-intersecting cones are placed. Prove that they cannot be...

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Kvant Math Problem 747

Verified: no Verdicts: SKIP + SKIP Solve time: 5m39s Source on kvant.digital Problem The sum of $n$ numbers is equal to $0$, and the sum of their absolute values is equal to $a$. Prove that the difference between the largest and the smallest of them is at least $\dfrac{2a}n$. Inside a convex $n$-gon $A_1A_2\ldots A_n$, a point $O$ is chosen so that the sum of the vectors $\overrightarrow{OA_1}+\overrightarrow{OA_2}+\ldots+\overrightarrow{OA_n}$ is the...

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Kvant Math Problem 745

Verified: no Verdicts: SKIP + SKIP Solve time: 5m41s Source on kvant.digital Problem A sequence of numbers $(d_n)$ is given such that $|d_n|\le1$ ($n=1$, $2$, $\ldots$). Prove that it is possible to choose a sequence $(s_n)$ from the numbers $+1$ and $-1$ so that for all $n$ $$|d_1s_1+d_2s_2+\ldots+d_ns_n|\le1.$$ A sequence of triples of numbers $(a_n,b_n,c_n)$ is given such that $|a_n|\le1$, $|b_n|\le1$, $|c_n|\le1$ and $a_n+b_n+c_n=0$ ($n=1$, $2$, $\ldots$). From it, a...

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Kvant Math Problem 743

Verified: no Verdicts: SKIP + SKIP Solve time: 7m41s Source on kvant.digital Problem In a country there are $N$ cities. Between any two cities there is a direct connection either by airplane or by steamship. Prove that, using only one of these modes of transport, it is possible to travel from any city to any other city (possibly with transfers). Between any two cities there is a direct connection by...

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Kvant Math Problem 742

Verified: no Verdicts: SKIP + SKIP Solve time: 6m31s Source on kvant.digital Problem On a circle of radius 1, on a sphere of radius 1 there are $n$ points. Prove that the sum of the squares of the pairwise distances between them does not exceed $n^2$. A. Mikhailovsky, V. V. Prasolov Exploration Let the points be represented by vectors $x_1,\dots,x_n$ from the center of the circle or sphere. In both...

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Kvant Math Problem 740

Verified: no Verdicts: SKIP + SKIP Solve time: 6m53s Source on kvant.digital Problem Fig. 1 Seryozha poured some millet into a cylindrical pot and asked his neighbor Aunt Lyuda: “How much water should I pour in to make tasty porridge?” — “That’s very simple,” replied the neighbor. “Tilt the pot like this; tap it so that the grain slides over and covers exactly half of the bottom. Now note the...

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Kvant Math Problem 738

Verified: no Verdicts: SKIP + SKIP Solve time: 5m03s Source on kvant.digital Problem Prove that the number of lines of distinct directions onto which a given $n$-gon has projections of equal length does not exceed $2n$; the maximum number of such lines for any polygon is even; for a triangle, this number is greater than 3 if and only if the triangle is acute. V. V. Prasolov Exploration Consider a...

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Kvant Math Problem 737

Verified: no Verdicts: SKIP + SKIP Solve time: 6m25s Source on kvant.digital Problem Let $d_k$ denote the number of houses in your city that have at least $k$ residents $(d_1\ge d_2\ge d_3\ge \ldots)$, and let $c_m$ denote the number of residents in the $m$-th most populous house $(c_1\ge c_2\ge c_3\ge \ldots)$. Prove the equalities $$c_1+c_2+c_3+\ldots=d_1+d_2+d_3+\ldots;$$ $$\begin{gather*}c_1^2+c_2^2+c_3^2+\ldots=d_1+3d_2+5d_3+\ldots+(2k-1)d_k+\ldots,\ d_1^2+d_2^2+d_3^2+\ldots=c_1+3c_2+5c_3+\ldots+(2m-1)c_m+\ldots\end{gather*}$$ A. V. Zelevinsky Exploration Let the houses have populations $h_1,h_2,\ldots,h_n$, arranged in nonincreasing...

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Kvant Math Problem 735

Verified: no Verdicts: SKIP + SKIP Solve time: 6m15s Source on kvant.digital Problem Prove that a circle of diameter 1 cannot be covered by several strips of paper whose total width is less than 1 (Fig. 1). Let a layer of thickness $h$ be the part of space enclosed between two parallel planes separated by a distance of $h$. Prove that a sphere of diameter 1 cannot be covered by...

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Kvant Math Problem 733

Verified: no Verdicts: SKIP + SKIP Solve time: 4m43s Source on kvant.digital Problem For which natural numbers $m$ is the number $31^m-1$ divisible by $2^m$? Prove that for any odd $a$ and any natural $m$, there exist infinitely many natural $k$ such that $a^k-1$ is divisible by $2^m$. Prove that for any odd $a$, there exist only finitely many natural $m$ such that $a^m-1$ is divisible by $2^m$. V. V....

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Kvant Math Problem 732

Verified: no Verdicts: SKIP + SKIP Solve time: 5m55s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a rigorous solution to Kvant problem M732 because the problem statement itself is not present in your message. The identifier “M732” by itself is not enough to reconstruct the problem. The Kvant archive...

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Kvant Math Problem 730

Verified: no Verdicts: SKIP + SKIP Solve time: 6m11s Source on kvant.digital Problem The sequence $(a_n)$ is defined by the conditions $$a_1=0,\quad a_{2n+1}=a_{2n}=n-a_n.$$ (for example, $a_{10}=5-a_5=5-a_4=5-(2-a_2)=3+(1-a_1)=4$). Write out the first 20 terms of the sequence and find $a_{1982}$. Prove that each natural number appears in the sequence 2 or 4 times. How many times does the number $2^k$ occur in it (for each $k=1$, 2, 3, $\ldots$)? Prove that the...

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Kvant Math Problem 728

Verified: no Verdicts: SKIP + SKIP Solve time: 7m13s Source on kvant.digital Problem Let $A$, $B$, $C$ be the vertices of a parallelepiped adjacent to its vertex $P$, and let $Q$ be the vertex opposite to $P$. Prove that the distances from the points $A$, $B$, $C$ to the line $PQ$ can serve as the side lengths of some triangle; the area $S$ of this triangle, the volume $V$ of...

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Kvant Math Problem 727

Verified: no Verdicts: SKIP + SKIP Solve time: 6m08s Source on kvant.digital Problem Prove the inequality $$a^2+b^2+c^2+2abc \lt 2,$$, where $a$, $b$, $c$ are the side lengths of a triangle with perimeter 2. I. Zharov Exploration Since the perimeter is $2$, we have $$a+b+c=2.$$ The expression to be bounded is $$a^2+b^2+c^2+2abc.$$ A natural first step is to eliminate one variable using the perimeter condition. Since $$a^2+b^2+c^2=(a+b+c)^2-2(ab+bc+ca),$$ the expression becomes $$4-2(ab+bc+ca)+2abc.$$...

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Kvant Math Problem 725

Verified: no Verdicts: SKIP + SKIP Solve time: 6m17s Source on kvant.digital Problem Let $$r_n=\cos^n \dfrac{\pi}{7}+\cos^n \dfrac{3\pi}{7}+\cos^n \dfrac{5\pi}{7}.$$. Find $r_1$ and $r_2$, $r_3$ and $r_4$. Prove that $r_n$ is a rational number for any $n$. N. B. Vasilyev Exploration The sequence $r_n$ sums the $n$-th powers of the cosines of the angles $\pi/7$, $3\pi/7$, and $5\pi/7$. These angles are equally spaced among the odd multiples of $\pi/7$ up to $5\pi/7$,...

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Kvant Math Problem 724

Verified: no Verdicts: SKIP + SKIP Solve time: 3m04s Source on kvant.digital Problem Several turtles crawl across a plane, each moving at the same speed in magnitude but in different directions. Prove that, regardless of their initial positions, after some time they will occupy the vertices of a convex polygon. V. V. Prasolov Exploration Consider two turtles moving at the same speed but in different directions. If we translate the...

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Kvant Math Problem 722

Verified: no Verdicts: SKIP + SKIP Solve time: 6m09s Source on kvant.digital Problem At points $A_1$, $A_2$, $\dots$, $A_n$, arranged on a circle, the numbers $1$, $2$, $\dots$, $n$ are placed in some order. Prove that the sum of $n$ of the absolute differences of neighboring numbers is at least $ 2n - 2 $. For how many arrangements is this sum equal to $ 2n - 2 $? A....

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Kvant Math Problem 720

Verified: no Verdicts: SKIP + SKIP Solve time: 6m07s Source on kvant.digital Problem About the function $f$, defined on the set of all pairs of nonnegative integers $(x;y)$, the following is known: $f(0,y)=y+1$, $f(x+1,0)=f(x,1),$ $f(x+1,y+1)=f(x,f(x+1,y))$ for every pair $x\ge0$, $y\ge0$. Find the value of $f(4,1981)$. International Mathematical Olympiad for School Students (XXII, 1981) Exploration The function $f$ is defined recursively on nonnegative integers. The rules resemble the Ackermann function. Start...

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Kvant Math Problem 719

Verified: no Verdicts: SKIP + SKIP Solve time: 4m19s Source on kvant.digital Problem For which $n \ge 3$ does there exist a set of $n$ consecutive positive integers having the following property: the largest of these $n$ numbers is a divisor of the least common multiple of the remaining $n-1$ numbers? For which $n \ge 3$ does there exist a unique set of $n$ consecutive numbers having the stated property?...

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Kvant Math Problem 717

Verified: no Verdicts: SKIP + SKIP Solve time: 5m07s Source on kvant.digital Problem Given natural numbers $n$ and $r$, $1\le r\le n$. Consider all subsets of the set ${1,2,\ldots,n}$ consisting of $r$ numbers, and in each subset choose the smallest number. Prove that the arithmetic mean of all the chosen numbers is equal to $\dfrac{n+1}{r+1}$. (For example, when $n=3$, $r=2$, we obtain the three subsets ${1,2}$, ${1,3}$, ${2,3}$, and the...

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Kvant Math Problem 716

Verified: no Verdicts: SKIP + SKIP Solve time: 6m32s Source on kvant.digital Problem From a point $P$ inside a given triangle $ABC$, perpendiculars $PA_1$, $PB_1$, $PC_1$ are dropped to the lines $BC$, $CA$, and $AB$. For which points $P$ inside $\triangle ABC$ does the quantity $$\dfrac{|BC|}{|PA_1|}+\dfrac{|CA|}{|PB_1|}+\dfrac{|AB|}{|PC_1|}$$ attain its minimum value? International Mathematical Olympiad for School Students (XXII, 1981) Exploration Let $$x=|PA_1|,\qquad y=|PB_1|,\qquad z=|PC_1|.$$ These are the distances from $P$ to...

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Kvant Math Problem 714

Verified: no Verdicts: SKIP + SKIP Solve time: 5m55s Source on kvant.digital Problem $N$ friends simultaneously learned $N$ pieces of news, with each person learning one piece of news. They began calling one another and exchanging news. During a single conversation, any number of pieces of news may be transmitted. What is the minimum number of phone calls required for everyone to learn all the news? Consider the following three...

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Kvant Math Problem 713

Verified: no Verdicts: SKIP + SKIP Solve time: 3m49s Source on kvant.digital Problem $M$ is a set of points in the plane. A point $O$ in the plane is called an "almost center of symmetry" of the set $M$ if it is possible to remove one point from $M$ so that for the remaining set $O$ it becomes a center of symmetry in the usual sense. How many "almost centers...

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Kvant Math Problem 711

Verified: no Verdicts: SKIP + SKIP Solve time: 6m28s Source on kvant.digital Problem The diagonals of a convex quadrilateral $ABCD$ inscribed in a circle with center $O$ are perpendicular to each other. Prove that the broken line $AOC$ divides the quadrilateral into two parts of equal area. V. Varvarkin Exploration Consider a convex quadrilateral $ABCD$ inscribed in a circle with diagonals $AC$ and $BD$ perpendicular at some point $P$. If...

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Kvant Math Problem 709

Verified: no Verdicts: SKIP + SKIP Solve time: 6m51s Source on kvant.digital Problem Fig. 2 Fig. 3 The floor of a room, in the shape of a regular hexagon with side length 10, is covered with tiles in the shape of a rhombus with side length 1 and an acute angle of $60^\circ$. It is allowed to remove three tiles forming a regular hexagon with side length 1 and rearrange...

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Kvant Math Problem 707

Verified: no Verdicts: SKIP + SKIP Solve time: 8m20s Source on kvant.digital Problem Each student in a class participates in at most two clubs, and for any pair of students, there is a club in which they both participate. Prove that there exists a club in which at least $\dfrac{2}{3}$ students from this class are members. A. F. Sidorenko Exploration Let the clubs be represented by sets of students. Each...

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Kvant Math Problem 706

Verified: no Verdicts: SKIP + SKIP Solve time: 4m32s Source on kvant.digital Problem From the center of each of two given circles, tangents are drawn to the other circle. Prove that the chords joining the points where the tangents touch the circles (in Figure 1 these chords are shown in red) have equal lengths. Figure 1 A. P. Savin Exploration Consider two circles with centers $O_1$ and $O_2$ and radii...

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Kvant Math Problem 703

Verified: no Verdicts: SKIP + SKIP Solve time: 5m04s Source on kvant.digital Problem Solve the system of equations $$\left{\begin{array}{l} 3\left(x+\dfrac{1}{x}\right)=4\left(y+\dfrac{1}{y}\right)=5\left(z+\dfrac{1}{z}\right),\ xy+yz+zx=1. \end{array}\right.$$ A. Fyodorov Exploration The first equation links three expressions of the form $t + \frac{1}{t}$ multiplied by constants 3, 4, and 5. Denoting $u = x + \frac{1}{x}$, $v = y + \frac{1}{y}$, $w = z + \frac{1}{z}$, the system reduces to $3u = 4v = 5w$. This...

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Kvant Math Problem 701

Verified: no Verdicts: SKIP + SKIP Solve time: 8m41s Source on kvant.digital Problem Lyuda, Marina, and Natasha drew an acute triangle $LMN$. Then Lyuda constructed her own triangle in which the lengths of two sides are $|LM|$ and $|LN|$, and the angle between them is $60^\circ$ greater than the angle $L$ of triangle $LMN$. Similarly, Marina constructed her own triangle with sides of lengths $|ML|$ and $|MN|$, the angle between...

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Kvant Math Problem 700

Verified: no Verdicts: SKIP + SKIP Solve time: 7m34s Source on kvant.digital Problem Can the set of all terminating decimal fractions be partitioned into (a) two, (b) three classes so that no two numbers whose difference is $10^m$ (for any integers $m=0$, $\pm1$, $\pm2$, $\ldots$) belong to the same class? A. Leiderman Exploration Consider the set of all terminating decimal fractions. Each such number can be expressed as $\frac{a}{10^k}$ for...

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Kvant Math Problem 698

Verified: no Verdicts: SKIP + SKIP Solve time: 7m57s Source on kvant.digital Problem On the sides $a$, $b$, $c$, $d$ of a cyclic quadrilateral, rectangles of dimensions $a\times c$, $b\times d$, $c\times a$, $d\times b$ are constructed outward. Prove that the centers of these rectangles are the vertices of a parallelogram, a rectangle. O. Penkin Exploration Let the cyclic quadrilateral be $ABCD$, with side lengths $$AB=a,\quad BC=b,\quad CD=c,\quad DA=d.$$ On...

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Kvant Math Problem 696

Verified: no Verdicts: SKIP + SKIP Solve time: 2m58s Source on kvant.digital Problem Is it possible to fill a $10 \times 10$-cell table with 100 distinct natural numbers so that for any $k \times k$-cell square ($2 \le k \le 10$) the sums, the products $k$ of the numbers on its diagonals are equal? A. Balinsky Exploration Consider the problem for small $k \times k$ squares. For a $2 \times...

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Kvant Math Problem 695

Verified: no Verdicts: SKIP + SKIP Solve time: 7m13s Source on kvant.digital Problem Is it possible to color all the cells of some rectangular table black and white so that the numbers of white and black cells are equal, while in each row and in each column more than $\dfrac34$ of the cells are of the same color? S. V. Konyagin All-Union School Olympiad, Grade 10, 1981 Exploration Let the...

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Kvant Math Problem 693

Verified: no Verdicts: SKIP + SKIP Solve time: 4m54s Source on kvant.digital Problem In a certain village, there are 1000 inhabitants. Each day, every inhabitant shares the news they learned yesterday with all their acquaintances. It is known that any piece of news eventually becomes known to all the villagers. Prove that it is possible to select 90 inhabitants such that if a piece of news is simultaneously told to...

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Kvant Math Problem 692

Verified: no Verdicts: SKIP + SKIP Solve time: 3m13s Source on kvant.digital Problem Points $C_1$, $A_1$, $B_1$ are taken on the sides $AB$, $BC$, $CA$, respectively, of triangle $ABC$ so that $$|AC_1|:|C_1B|=|BA_1|:|A_1C|=|CB_1|:|B_1A|=1:3.$$. Prove that the perimeter $P$ of triangle $ABC$ and the perimeter $P_1$ of triangle $A_1B_1C_1$ satisfy the inequalities $P_1\lt\dfrac34P$; $P_1\gt\dfrac12P$. V. Turchaninov All-Union Mathematical Olympiad for School Students (1981, Grade 8) Exploration Consider triangle $ABC$ with arbitrary side...

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Kvant Math Problem 690

Verified: no Verdicts: SKIP + SKIP Solve time: 5m51s Source on kvant.digital Problem Inside a convex polygon with area $S_1$ and perimeter $P_1$, there is a convex polygon with area $S_2$ and perimeter $P_2$. Prove the inequality $$2 \frac{S_1}{P_1} \gt \frac{S_2}{P_2}.$$. Formulate and prove an analogous statement for convex polyhedra. A. V. Kelarev Exploration For a convex polygon, the quantity $\dfrac{2S}{P}$ has a geometric meaning. If every side is shifted...

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Kvant Math Problem 688

Verified: no Verdicts: SKIP + SKIP Solve time: 3m16s Source on kvant.digital Problem Given natural numbers $a_1$, $a_2$, $\ldots$, $a_n$ such that $a_k \le k$ $(k=1,2, \ldots,n)$ and the sum $a_1+a_2+\ldots+a_n$ is even. Prove that one of the expressions $$a_1\pm a_2\pm a_3\pm \ldots \pm a_n$$ is equal to zero. S. Nenashev Exploration Let $$S=a_1+a_2+\cdots+a_n.$$ Since $a_k\le k$, we have $$S\le 1+2+\cdots+n=\frac{n(n+1)}2.$$ The condition that $S$ is even suggests rewriting the...

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Kvant Math Problem 686

Verified: no Verdicts: SKIP + SKIP Solve time: 3m17s Source on kvant.digital Problem Is the equality $$\left[\sqrt{\left[\sqrt{x}\right]}\right]=\left[ \sqrt{\sqrt{x}}\right]?$$ true for every number $x \ge1 $? (Here $[y]$ denotes the integer part of the number $y$.) V. V. Prasolov Exploration Let $$a=\left[\sqrt{x}\right].$$ Then the left-hand side is $$\left[\sqrt a\right].$$ The right-hand side is $$\left[\sqrt{\sqrt x}\right]=\left[x^{1/4}\right].$$ The question is whether taking the integer part before the outer square root can ever change...

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Kvant Math Problem 685

Verified: no Verdicts: SKIP + SKIP Solve time: 4m56s Source on kvant.digital Problem Two subsets of the set of natural numbers are called congruent if one can be obtained from the other by a shift by an integer. (For example, the sets of even and odd numbers are congruent.) Is it possible to partition the set of natural numbers into an infinite number of (disjoint) infinite congruent subsets? A. Fyodorov...

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Kvant Math Problem 683

Verified: no Verdicts: SKIP + SKIP Solve time: 5m11s Source on kvant.digital Problem Several circles of the same size are placed on a table so that no two overlap. Prove that the circles can be colored with four colors so that any two touching circles are colored differently. Find an arrangement of circles for which three colors are not sufficient for such a coloring. G. Ringel Exploration Represent each circle...

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Kvant Math Problem 681

Verified: no Verdicts: SKIP + SKIP Solve time: 7m41s Source on kvant.digital Problem Find integers $a$, $b$, $c$, $d$ such that the numbers $a^2+b^2$, $a^2+b^2+c^2$, $a^2+b^2+c^2+d^2$ are perfect squares. Does there exist a sequence consisting of perfect squares such that for any $n$, the sum of $n$ of its first terms is a perfect square? G. Grigoriev Exploration Let $$S_1=a^2+b^2,\qquad S_2=a^2+b^2+c^2,\qquad S_3=a^2+b^2+c^2+d^2.$$ The first question asks for one chain of...

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Kvant Math Problem 680

Verified: no Verdicts: SKIP + SKIP Solve time: 7m10s Source on kvant.digital Problem Two communications engineers are playing the following game. There are $n$ telephone exchanges, and the players take turns connecting any two of them with a cable of their choice. The winner is the player after whose move it becomes possible to call from any exchange to any other (possibly through several intermediate ones; the initial position is...

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Kvant Math Problem 678

Verified: no Verdicts: SKIP + SKIP Solve time: 6m23s Source on kvant.digital Problem A $2m$-digit number is called fair if its even positions contain as many even digits as odd ones. Prove that in any $2m+1$-digit number, it is possible to remove one digit so that the resulting $2m$-digit number is fair (Fig. 1). A. F. Sidorenko Exploration Consider small examples first. Let $m=1$, so the number has $2m+1 =...

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Kvant Math Problem 676

Verified: no Verdicts: SKIP + SKIP Solve time: 8m16s Source on kvant.digital Problem Prove that for any natural number $m$, the sum of the digits of the number $1981^m$ is at least 19. A. V. Savkin, 9th-grade student Exploration We begin by computing small powers of $1981$ and observing the sums of their digits. For $m=1$, $1981$ has digits $1,9,8,1$ summing to $19$. For $m=2$, we calculate $1981^2 = 3924361$,...

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Kvant Math Problem 674

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem On sides $BC$, $AC$, and $AB$ of an acute triangle $ABC$, points $A_1$, $B_1$, and $C_1$ are taken respectively. It is known that the circumcenter of triangle $ABC$ coincides with the orthocenter of triangle $A_1B_1C_1$. Prove that triangles $ABC$ and $A_1B_1C_1$ are similar. D. Isaac Exploration Consider an acute triangle $ABC$ with arbitrary points $A_1$ on $BC$,...

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Kvant Math Problem 673

Verified: no Verdicts: SKIP + SKIP Solve time: 5m30s Source on kvant.digital Problem On a plane, three pucks $A$, $B$, $C$ lie at the vertices of a triangle. A hockey player chooses one of them and strikes it so that it passes between the other two and comes to rest at some point. Show how, after five strikes, puck $C$ can return to its original position while pucks $A$ and...

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Kvant Math Problem 671

Verified: no Verdicts: SKIP + SKIP Solve time: 6m20s Source on kvant.digital Problem In a cyclic quadrilateral, one diagonal bisects the other. Prove that the square of the length of the first diagonal is equal to half the sum of the squares of the lengths of all the sides of the quadrilateral. R. Mazov Exploration Let the cyclic quadrilateral be $ABCD$ with diagonals $AC$ and $BD$. Suppose $AC$ bisects $BD$...

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Kvant Math Problem 669

Verified: no Verdicts: SKIP + SKIP Solve time: 4m21s Source on kvant.digital Problem A quadrilateral $ABCD$ is inscribed in a circle. Prove that the segment connecting the midpoints of arcs $AB$ and $CD$ is perpendicular to the segment connecting the midpoints of arcs $BC$ and $AD$; the centers of the incircles of triangles $ABC$, $BCD$, $CDA$, and $DAB$ form the vertices of a rectangle. I. German, K. Malhasyan Exploration Consider...

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Kvant Math Problem 667

Verified: no Verdicts: SKIP + SKIP Solve time: 7m48s Source on kvant.digital Problem Construct the triangle $ABC$, given its smallest angle $\widehat A$ and the segments of lengths $d=|AB|-|BC|$ and $e=|AC|-|BC|$. N. B. Vasilyev Exploration Consider a triangle $ABC$ with the smallest angle $\widehat A$ and suppose the differences $d = |AB| - |BC|$ and $e = |AC| - |BC|$ are given. Attempting to construct such a triangle suggests fixing...

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Kvant Math Problem 665

Verified: no Verdicts: SKIP + SKIP Solve time: 3m21s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be found in the published issue. I cannot write a rigorous solution to Kvant problem M665 because the actual problem statement is missing. The identifier “M665” by itself is not enough to determine the problem uniquely, and your prompt states that “the...

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Kvant Math Problem 663

Verified: no Verdicts: SKIP + SKIP Solve time: 3m24s Source on kvant.digital Problem Find all prime numbers $p$ for which the number $2^p+p^2$ is also prime. S. Maizus Exploration Consider small prime numbers to understand the behavior of the expression $2^p + p^2$. For $p = 2$, we have $2^2 + 2^2 = 4 + 4 = 8$, which is not prime. For $p = 3$, $2^3 + 3^2 =...

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Kvant Math Problem 661

Verified: no Verdicts: SKIP + SKIP Solve time: 5m40s Source on kvant.digital Problem On the shore of a circular lake there are four piers $K$, $L$, $P$, $Q$. A motorboat departs from pier $K$, and a rowboat from $L$. If the motorboat heads straight for $P$ and the rowboat straight for $Q$, they will collide at some point $X$ in the lake. Prove that if the motorboat heads for $Q$...

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Kvant Math Problem 660

Verified: no Verdicts: SKIP + SKIP Solve time: 8m22s Source on kvant.digital Problem Blue and red points are placed on a circle (Fig. 2a). It is allowed to either add a new red point and simultaneously change the color of each of the two neighboring points (Fig. 2b), or remove a red point and change the color of each of its neighboring points (Fig. 2c). Initially, there were only two...

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Kvant Math Problem 658

Verified: no Verdicts: SKIP + SKIP Solve time: 7m26s Source on kvant.digital Problem In a square with side length 1, a finite number of segments parallel to its sides are drawn. The segments may intersect each other. The sum of the lengths of the drawn segments is 18. Prove that among the regions into which the square is divided by these segments, there exists one whose area is at least...

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Kvant Math Problem 655

Verified: no Verdicts: SKIP + SKIP Solve time: 4m59s Source on kvant.digital Problem Fig. 2 On the desk of an official of the Ministry of Contingencies lie $n$ volumes of the Encyclopaedia Britannica, arranged in several stacks. Each day, upon arriving at work, the official takes one volume from each stack and places the taken volumes into a new stack, then arranges the stacks by the number of volumes (in...

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Kvant Math Problem 654

Verified: no Verdicts: SKIP + SKIP Solve time: 7m45s Source on kvant.digital Problem Is the following statement true: among any six natural numbers, it is always possible to choose either three numbers such that every pair has no common divisors greater than 1, or three numbers that share a common divisor greater than 1? E. M. Rabbot Correspondence Mathematical Olympiads Exploration Consider small examples of six natural numbers and examine...

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Kvant Math Problem 650

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem Does there exist a sequence $$a_1,\ a_2,\ a_3,\ \ldots,\ a_n, ...$$ of natural numbers such that every natural number can be uniquely represented as a sum of several of its terms: $$a_{i_1}+a_{i_k}\quad(i_1 \lt \ldots \lt i_k,\ k\ge 0); \tag{*}$$ of natural numbers such that 1 cannot be represented in the form ( ), but every natural number...

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Kvant Math Problem 646

Verified: no Verdicts: SKIP + SKIP Solve time: 7m13s Source on kvant.digital Problem From a point $O$ on the plane, $2n$ vectors of length 1 are drawn. They are colored alternately red and blue. Let $\overrightarrow{s}$ be the sum of $n$ red vectors, and $\overrightarrow{r}$ be the sum of $n$ blue vectors. Prove that $$|\overrightarrow{s}—\overrightarrow{r}|\le 2.$$ E. Shustin Exploration Consider the problem for small values of $n$ to understand the...

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Kvant Math Problem 644

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem Prove that there exists a convex 1980-gon with side lengths 1, 2, $\ldots$, 1980, all of whose angles are equal. Does such a 1981-gon exist? G. A. Gurevich Exploration A convex equiangular $n$-gon has exterior angle $2\pi/n$ at every vertex. If the side vectors are taken in cyclic order, their directions differ successively by $2\pi/n$. Thus an...

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Kvant Math Problem 642

Verified: no Verdicts: SKIP + SKIP Solve time: 5m12s Source on kvant.digital Problem Prove that every positive integer can be represented in the form $a_0+2a_1+2^2a_2+\ldots+2^na_n$, where each of the numbers $a_k=0$, $-1$ is either 1 and $a_k\cdot a_{k+1}=0$ for all $0\le k\le n-1$, and moreover this representation is unique. I. K. Zhuk Exploration The coefficients are restricted to the set ${-1,0,1}$, and two neighboring coefficients cannot both be nonzero. This...

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Kvant Math Problem 640

Verified: no Verdicts: SKIP + SKIP Solve time: 7m04s Source on kvant.digital Problem The number $x\in[0;1]$ is written as an infinite decimal expansion. By rearranging its first 5 digits after the decimal point in an arbitrary order, we obtain a new infinite decimal expansion corresponding to some number $x_1$. By rearranging the 2nd through 6th digits after the decimal point in the decimal expansion of $x_1$, we obtain the decimal...

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Kvant Math Problem 637

Verified: no Verdicts: SKIP + SKIP Solve time: 8m29s Source on kvant.digital Problem Given an equilateral triangle $ABC$. A line parallel to the line $AC$ intersects the lines $AB$ and $BC$ at points $M$ and $P$, respectively. The point $D$ is the centroid of the triangle $PMB$, and the point $E$ is the midpoint of the segment $AP$. Determine the angles of the triangle $DEC$. L. P. Kuptsov All-Union Mathematical...

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Kvant Math Problem 636

Verified: no Verdicts: SKIP + SKIP Solve time: 4m39s Source on kvant.digital Problem The set $A$ consists of integers; its smallest element is 1, and its largest element is 100. Every element of $A$ other than 1 is equal to the sum of two (possibly equal) numbers belonging to $A$. Among all sets $A$ satisfying these conditions, determine one with the minimum number of elements. Yu. V. Nesterenko All-Union Mathematical...

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Kvant Math Problem 628

Verified: no Verdicts: SKIP + SKIP Solve time: 3m21s Source on kvant.digital Problem On a sphere, a triangle is constructed one of whose “sides” has length $120^\circ$. Prove that the “median” drawn to this “side” is bisected by each of the other two “medians.” (“Medians” and “sides” are arcs of great circles.) A. Yagubyants Exploration Consider a spherical triangle with one side of length $120^\circ$. On a unit sphere, a...

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Kvant Math Problem 627

Verified: no Verdicts: SKIP + SKIP Solve time: 6m57s Source on kvant.digital Problem On each cell of an infinite sheet of graph paper, a natural number is written. Suppose each of these numbers occurs exactly once. (Provide examples of such an arrangement of numbers!) Prove that for any given $m$ there exist two neighboring (sharing a side) cells whose numbers differ by at least $m$. Suppose each number $n\in\mathbb{N}$ occurs...

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Kvant Math Problem 598

Verified: no Verdicts: SKIP + SKIP Solve time: 5m37s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I can proceed, but I need the text of problem M598 first. Could you provide the full statement of the problem?

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Kvant Math Problem 597

Verified: no Verdicts: SKIP + SKIP Solve time: 3m24s Source on kvant.digital Problem Let $$x_n=1+\dfrac12+\ldots+\dfrac1n.$$ Prove the existence of the limit $$\gamma=\lim_{n\to\infty}{}(x_n-\ln n).$$ Prove that for any $n$, $m\in\mathbb{N}$ $$\gamma\lt x_n+x_m-x_{nm}\le1.$$ Find $\gamma$ with an accuracy of $0{,}1$. V. Yasinsky Exploration The sequence $x_n=1+\frac12+\dots+\frac1n$ is the $n$-th harmonic number. Numerically, for small $n$, $x_n - \ln n$ appears to approach a constant: for $n=1$, $x_1-\ln 1 = 1$, for $n=2$,...

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Kvant Math Problem 595

Verified: no Verdicts: SKIP + SKIP Solve time: 3m55s Source on kvant.digital Problem Let $A$ and $E$ be two opposite vertices of a regular octagon. A frog is located at vertex $A$. From any vertex of the octagon, except for vertex $E$, the frog can jump to either of the two neighboring vertices. Upon reaching vertex $E$, the frog stops and remains there. Let $a_n$ be the number of ways...

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Kvant Math Problem 594

Verified: no Verdicts: SKIP + SKIP Solve time: 5m38s Source on kvant.digital Problem Find all real numbers $a$ for which there exist nonnegative real numbers $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ satisfying the relations $$\sum_{k=1}^5kx_k=a,\quad \sum_{k=1}^5k^3x_k=a^2,\quad \sum_{k=1}^5k^5x_k=a^3.$$ International Mathematical Olympiad (XXI, 1979) Exploration Let $$m_r=\sum_{k=1}^5 k^r x_k .$$ The conditions are $$m_1=a,\qquad m_3=a^2,\qquad m_5=a^3.$$ Since the coefficients $x_k$ are nonnegative, it is natural to normalize them and interpret them as weights....

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Kvant Math Problem 592

Verified: no Verdicts: SKIP + SKIP Solve time: 5m06s Source on kvant.digital Problem Prove that for any triangle, the projection onto the line containing a second side of the diameter of the circumcircle that is perpendicular to one side of the triangle is equal in length to the third side. S. Ovchinnikov Exploration Consider a triangle $ABC$ with circumcircle $\Gamma$. Denote the sides as $BC = a$, $AC = b$,...

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Kvant Math Problem 590

Verified: no Verdicts: SKIP + SKIP Solve time: 6m31s Source on kvant.digital Problem Find the minimum value of the expression $|{\cos x}|+|{\cos2x}|$. Prove that for any number $x$ and any positive integer $n$, the sum $$|{\cos x}|+|{\cos2x}|+|{\cos4x}|+\ldots+|{\cos2^nx}|$$ is not less than $\dfrac n4$; is not less than $\dfrac n2$. P. Gusyatnikov All-Russian Mathematical Olympiad for School Students (1979, Grade 10) Exploration Consider first the expression $|\cos x| + |\cos 2x|$....

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Kvant Math Problem 589

Verified: no Verdicts: SKIP + SKIP Solve time: 6m10s Source on kvant.digital Problem On the plane, a set of $n$ vectors is given, each with length not exceeding 1. Prove that by replacing some of the vectors in this set with their opposites, one can obtain a set of $n$ vectors whose sum has length not exceeding $\sqrt n$; not exceeding $\sqrt2$. P. Gusyatnikov, A. Plotkin All-Russian School Mathematics Olympiad...

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Kvant Math Problem 587

Verified: no Verdicts: SKIP + SKIP Solve time: 3m28s Source on kvant.digital Problem A triple of numbers 2, $\sqrt2$, $\dfrac1{\sqrt2}$ is given. It is allowed to replace any two of them with two numbers defined as follows: their sum divided by $\sqrt2$ and their difference divided by $\sqrt2$. Is it possible, by performing this procedure several times, to obtain the triple of numbers 1, $\sqrt2$, $1+\sqrt2$? All-Russian School Mathematics Olympiad...

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Kvant Math Problem 585

Verified: no Verdicts: SKIP + SKIP Solve time: 6m59s Source on kvant.digital Problem At a chemistry conference, there were $N$ scientists — chemists and alchemists — with the number of chemists exceeding the number of alchemists. It is known that chemists always tell the truth, while alchemists sometimes tell the truth and sometimes lie. A mathematician attending the conference must determine for each scientist whether they are a chemist or...

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Kvant Math Problem 584

Verified: no Verdicts: SKIP + SKIP Solve time: 7m29s Source on kvant.digital Problem Is it possible to represent the entire space as the union of lines, every two of which are skew (that is, do not lie in the same plane)? F. V. Vainshtein Exploration Suppose such a family of lines exists. Since the union of the lines is the whole space, every point lies on exactly one line, because...

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Kvant Math Problem 558

Verified: no Verdicts: SKIP + SKIP Solve time: 2m40s Source on kvant.digital Problem A circle contains $k\gt1$ black sectors, each with an angle smaller than $\dfrac{180^\circ}{k^2-k+1}$. Prove that the circle can be rotated around $O$ so that all black sectors move into the white portion of the circle. V. V. Proizvolov Exploration Let the black sectors have angular lengths $\alpha_1,\dots,\alpha_k$, where each $$\alpha_i<\frac{180^\circ}{k^2-k+1}.$$ A rotation through an angle $t$ moves...

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Kvant Math Problem 557

Verified: no Verdicts: SKIP + SKIP Solve time: 3m02s Source on kvant.digital Problem Given $n$ pairwise relatively prime numbers, greater than $1$ and less than $(2n-1)^2$. Prove that at least one of them must be a prime number. A. T. Kolotov Exploration Suppose, contrary to the statement, that none of the given numbers is prime. Then every number is composite. Since the numbers are pairwise relatively prime, no prime divisor...

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Kvant Math Problem 553

Verified: no Verdicts: SKIP + SKIP Solve time: 3m37s Source on kvant.digital Problem Given a triangle $ABC$, such that $|BC|\lt|AC|\lt|AB|$. On the rays $BA$ and $CA$, segments $BD$ and $CE$ are laid off so that $|BD|=|CE|=|BC|$. Prove that the radius of the circumcircle of triangle $ADE$ is equal to the distance between the circumcenter of triangle $ABC$ and its incenter. S. Meidman Exploration Consider a triangle $ABC$ with sides $BC...

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Kvant Math Problem 537

Verified: no Verdicts: SKIP + SKIP Solve time: 6m27s Source on kvant.digital Problem A circle is tangent internally to the circumcircle of an isosceles triangle $ABC$, as well as to the equal sides $AB$ and $AC$ of this triangle at points $P$ and $Q$, respectively. Prove that the midpoint of segment $PQ$ is the center of the incircle of triangle $ABC$. International Mathematical Olympiad (XX, 1978) Exploration Let $O$ be...

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Kvant Math Problem 524

Verified: no Verdicts: SKIP + SKIP Solve time: 6m40s Source on kvant.digital Problem Prove that for no natural $m$ does the number $1978^m-1$ divide $1000^m-1$. S. V. Konyagin All-Union Mathematical Olympiad for School Students (XII, 1978, 8th grade) Exploration Consider the numbers $1978^m - 1$ and $1000^m - 1$ for small values of $m$. For $m = 1$, $1978^1 - 1 = 1977$ and $1000^1 - 1 = 999$, and...

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Kvant Math Problem 512

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem Let $f(x)=x^3-x+1$. Prove that for every natural number $m\gt1$, the numbers $m$, $f(m)$, $f(f(m))$, $f(f(f(m)))$, $\ldots$ are pairwise relatively prime. A. T. Kolotov All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10) Exploration Compute the first few values of $f$ for small natural numbers greater than $1$. For $m=2$, $f(2)=2^3-2+1=7$, $f(f(2))=f(7)=343-7+1=337$, $f(f(f(2)))=f(337)=337^3-337+1$, which is enormous. None...

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Kvant Math Problem 507

Verified: no Verdicts: SKIP + SKIP Solve time: 6m36s Source on kvant.digital Problem In last year's Russian translation of Selected Problems from the American Mathematical Monthly (M.: Mir, 1977), two problems by the remarkable mathematician P. Erdős (Nos. 138 and 139) were given without solutions. The editorial board of Kvant received a number of letters containing solutions to these problems. Before presenting the solutions to the readers of the journal,...

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Kvant Math Problem 503

Verified: no Verdicts: SKIP + SKIP Solve time: 3m46s Source on kvant.digital Problem A set of $2n+1$ numbers $a_0$, $a_1$, $a_2$, $\ldots$, $a_{2n}$ is such that $a_k\ge\dfrac{a_{k-1}+a_{k+1}}2$ for all $k=1$, 2, $\ldots$, $2n-1$. Prove the inequality $$\dfrac{a_1+a_2+a_3+\ldots+a_{2n-1}}n\ge\dfrac{a_0+a_1+a_2+\ldots+a_{2n}}{n+1}\tag{*}$$ and determine for which sets it becomes an equality. V. Lipkin Exploration The condition $$a_k\ge \frac{a_{k-1}+a_{k+1}}2 \qquad (1\le k\le 2n-1)$$ means that the sequence is concave. Rewriting, $$(a_k-a_{k-1})\ge (a_{k+1}-a_k).$$ Hence the successive differences...

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Kvant Math Problem 499

Verified: no Verdicts: SKIP + SKIP Solve time: 6m14s Source on kvant.digital Problem We call a number balanced if in its decimal representation some initial segment coincides with some terminal segment (for example, the numbers 1971, 19219 are balanced, whereas 1415145 is not). Prove that there exists a number which, after appending any of the 10 digits to its right, becomes balanced. G. A. Gurevich Exploration Consider what it means...

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Kvant Math Problem 497

Verified: no Verdicts: SKIP + SKIP Solve time: 3m44s Source on kvant.digital Problem On the sides $BC$, $CA$, and $AB$ of an acute triangle $ABC$, arbitrary points $A_1$, $B_1$, and $C_1$ are chosen; circles are constructed on the segments $AA_1$, $BB_1$, and $CC_1$ as diameters. Prove that the three common chords of the pairs of these circles intersect at the point where the altitudes of triangle $ABC$ meet. V. L....

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Kvant Math Problem 495

Verified: no Verdicts: SKIP + SKIP Solve time: 6m11s Source on kvant.digital Problem In outer space around the planet $O$, three satellites move uniformly along three circular orbits centered at $O$. The angular velocities of the satellites are $\omega_1$, $\omega_2$, and $\omega_3$, respectively, and their initial positions can be arbitrary. Is there necessarily a moment in time when all three satellites and the point $O$ lie in the same plane...

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Kvant Math Problem 492

Verified: no Verdicts: SKIP + SKIP Solve time: 6m22s Source on kvant.digital Problem A triangle $ABC$ is inscribed in triangle $A_1B_1C_1$ (so that the vertices $A_1$, $B_1$, and $C_1$ lie respectively on the sides $BC$, $CA$, and $AB$), and the segments $AA_1$, $BB_1$, and $CC_1$ intersect at a single point $P$. Prove that the lines joining the midpoints of the sides $AB$ and $A_1B_1$, $BC$ and $B_1C_1$, $CA$ and $C_1A_1$...

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Kvant Math Problem 491

Verified: no Verdicts: SKIP + SKIP Solve time: 3m43s Source on kvant.digital Problem Consider a geometric progression all of whose terms are integers. (For example: 16, 24, 36, 54, 81.) Prove that the sum of the squares of three consecutive terms of the progression is divisible by the sum of these terms. For which natural numbers $n$ is the sum of the squares of $n$ consecutive terms of the progression...

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Kvant Math Problem 488

Verified: no Verdicts: SKIP + SKIP Solve time: 6m00s Source on kvant.digital Problem Consider the sequence of polynomials $P_0$, $P_1$, $P_2$, $\ldots$, defined by the conditions $P_0(x)=1$, $P_1(x)=x$, $$P_{n+1}(x)=x,P_n(x)-P_{n-1}(x),\quad n\ge1.\tag1$$. Prove the equalities $$\underbrace{x-\dfrac1{x-\dfrac1{x-\dfrac1{\raisebox{4pt}{$\ddots\vphantom{\dfrac00}$}{}-\dfrac1x}}}} {n~\text{раз}}=\dfrac{P_n(x)}{P {n-1}(x)};$$ $$\dfrac{\sin{(n+1)\phi}}{\sin\phi}=P_n(2\cos\phi),\quad\text{если} \dfrac\phi\pi \text{— не целое};$$ $$t^{n+1}-\dfrac1{t^{n+1}}=\left(t-\dfrac1t\right)P_n{\left(t+\dfrac1t\right)},\quad\text{если}~t\ne0;$$ $$P_n(x)=\prod\limits_{1\le k\le n}\left(x-2\cos\dfrac{\pi k}{n+1}\right);$$ $$P_n(x)=\textstyle\sum\limits_{0\le j\le n/2}{(-1)^j}C_{n-j}^jx^{n-2j};$$ $$\prod\limits_{1\le k\le m}\cos\dfrac{k\pi}{2m+1}=\dfrac1{2^m}.$$ Devise and prove analogous equalities for the sequence of polynomials defined by the same relation (1), but starting...

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Kvant Math Problem 485

Verified: no Verdicts: SKIP + SKIP Solve time: 8m14s Source on kvant.digital Problem Prove that the number $e$ is between the numbers $a_n=\left(1+\dfrac1n\right)^n$ and $b_n=\left(1+\dfrac1n\right)^{n+1}$ for any natural $n$. Prove that the sequence $c_n=\left(1+\dfrac1{4n}\right)\left(1+\dfrac1n\right)^n$ is monotonically increasing, while the sequence $d_n=\left(1+\dfrac1{2n}\right)\left(1+\dfrac1n\right)^n$ is monotonically decreasing. Divide the segment $[a_n;b_n]$ into four equal parts. In which of them does the number $e$ lie? Divide the segment $[a_n;b_n]$ into eight equal parts. In...

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Kvant Math Problem 483

Verified: no Verdicts: SKIP + SKIP Solve time: 6m19s Source on kvant.digital Problem a) Prove that the ratio of the inradius of a right triangle to the sum of the squares of the medians drawn from the acute angles does not exceed $1/5$. b) Find the maximum value that this ratio can attain. I. Temraliev Exploration Consider a right triangle with legs $a$ and $b$ and hypotenuse $c$, where $c^2...

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Kvant Math Problem 481

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem In the sequence of natural numbers $a_1$, $a_2$, $a_3$, $\ldots$ each term $a_{k+1}$ is equal to the sum of the squares of the digits in the decimal representation of the number $a_k$ ($k\ge1$). Prove that for any $a_1$ in this sequence, the number 1 or the number 89 will be encountered. V. Panfilov Exploration Let $$f(n)=\text{(sum of...

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Kvant Math Problem 479

Verified: no Verdicts: SKIP + SKIP Solve time: 5m05s Source on kvant.digital Problem Do there exist six, 1000 such distinct natural numbers that for any two $a$ and $b$ among them, the sum $a+b$ is divisible by the difference $a-b$? S. V. Konyagin Moscow Mathematical Olympiad (XL, 1977, grades 9 and 11) Exploration Consider a set of distinct natural numbers ${a_1, a_2, \dots, a_n}$ with the property that for any...

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Kvant Math Problem 476

Verified: no Verdicts: SKIP + SKIP Solve time: 7m08s Source on kvant.digital Problem Prove that if the vertices of a convex $n$-gon lie on the nodes of graph paper, and there are no other nodes inside it or on its sides, then $n\le4$. Let space be divided by three families of parallel planes into identical cubes. Call the vertices of the cubes nodes . Prove that if all $n$ vertices...

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Kvant Math Problem 474

Verified: no Verdicts: SKIP + SKIP Solve time: 6m15s Source on kvant.digital Problem A natural number is called perfect if it is equal to the sum of all its divisors (excluding itself); for example, the numbers $6=1+2+3$ and $28=1+2+4+7+14$ are perfect. Prove that the number $N$ is not perfect if it is known that it leaves a remainder of 3 when divided by 4; leaves a remainder of 5 when...

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Kvant Math Problem 472

Verified: no Verdicts: SKIP + SKIP Solve time: 4m36s Source on kvant.digital Problem A convex polyhedron is located inside a cube, and its orthogonal projection onto each face of the cube coincides with that face. Prove that the volume of the polyhedron is at least one third of the volume of the cube. V. V. Prasolov Exploration Consider a cube of side length $1$ for simplicity. We are asked to...

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Kvant Math Problem 470

Verified: no Verdicts: SKIP + SKIP Solve time: 5m13s Source on kvant.digital Problem Prove the equalities $$\dfrac{1}{C_n^0}-\dfrac{1}{C_n^1}+\dfrac{1}{C_n^2}-\ldots+\dfrac{(-1)^k}{C_n^k}+\ldots+\dfrac{(-1)^n}{C_n^n}=\dfrac{(n+1)(1+(-1)^n)}{n+2};$$ $$\dfrac{1}{C_n^1}+\dfrac{1}{C_n^2}+\dfrac{1}{C_n^3}+\ldots+\dfrac{1}{C_n^n}=\dfrac{n+1}{2^{n+1}}\left(\dfrac{2}{1}+\dfrac{2^2}{2}+\dfrac{2^3}{3}+\ldots+\dfrac{2^n}{n}\right).$$ L. D. Kurlandchik, A. D. Lisitsky Exploration We begin by examining the two sums for small values of $n$ to detect patterns. For $n = 2$, the first sum is $\frac{1}{C_2^0} - \frac{1}{C_2^1} + \frac{1}{C_2^2} = 1 - \frac{1}{2} + 1 = \frac{3}{2},$ while the formula on the right-hand side gives $\frac{(2+1)(1...

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Kvant Math Problem 468

Verified: no Verdicts: SKIP + SKIP Solve time: 4m06s Source on kvant.digital Problem Points $A$, $B$, $C$, and $D$ in the plane are such that for any point $M$ of this plane, the scalar products of the vectors $\overrightarrow{MA}\cdot\overrightarrow{MB}$ and $\overrightarrow{MC}\cdot\overrightarrow{MD}$ are not equal to each other. Prove that $\overrightarrow{AC}=\overrightarrow{DB}$. Is the converse statement true? Yu. I. Ionin Exploration Consider four points $A$, $B$, $C$, $D$ in the plane, and...

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Kvant Math Problem 466

Verified: no Verdicts: SKIP + SKIP Solve time: 6m04s Source on kvant.digital Problem Among 1977 coins, 50 are counterfeit. Each counterfeit coin differs from a genuine one by 1 g (either heavier or lighter). There is a pan balance with a pointer indicating the difference in mass between the two pans. With a single weighing, one must determine whether a selected coin is counterfeit or genuine. How can this be...

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Kvant Math Problem 463

Verified: no Verdicts: SKIP + SKIP Solve time: 6m14s Source on kvant.digital Problem Given natural numbers $x_1$, $x_2$, $\ldots$, $x_m$, $y_1$, $y_2$, $\ldots$, $y_n$. The sums $x_1+x_2+\ldots+x_m$ and $y_1+y_2+\ldots+y_n$ are equal and less than $mn$. Prove that in the equality $$x_1+x_2+\ldots+x_m=y_1+y_2+\ldots+y_n$$ it is possible to delete some of the terms so that the resulting equality is still true. K. Sibiryakov All-Union Mathematical Olympiad for School Students (XI, 1977, Grade 9)...

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Kvant Math Problem 461

Verified: no Verdicts: SKIP + SKIP Solve time: 6m02s Source on kvant.digital Problem On the table are a pair of balance scales and $n$ weights of different masses. The weights are placed on the pans of the scales one by one (at each step, any weight can be taken from the table and added to either pan of the scales). Prove that the weights can be placed in such an...

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Kvant Math Problem 458

Verified: no Verdicts: SKIP + SKIP Solve time: 7m07s Source on kvant.digital Problem A polynomial $x^{10}+{ }x^9+{ }x^8+\ldots+{ }x^2+{ }x+1$ is written. Two players play the following game. First, the first player replaces any of the asterisks with a number; then the second player replaces any of the remaining asterisks with a number; then the first player again replaces one of the asterisks with a number, and so on (a...

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Kvant Math Problem 457

Verified: no Verdicts: SKIP + SKIP Solve time: 3m44s Source on kvant.digital Problem A non-self-intersecting closed polygonal line is given in the plane, with no three of its vertices lying on a single straight line. Call a pair of nonadjacent sides special if the extension of one of them intersects the other (Fig. 1). Prove that the number of special pairs is even. Figure 1 S. V. Fomin All-Union Mathematical...

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Kvant Math Problem 454

Verified: no Verdicts: SKIP + SKIP Solve time: 6m50s Source on kvant.digital Problem Seven dwarfs are sitting around a round table. In front of each of them is a mug. Some of these mugs contain milk. One of the dwarfs pours all of his milk equally into the mugs of the other dwarfs. Then his neighbor to the right does the same. Then the next neighbor to the right does...

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Kvant Math Problem 451

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem Several points, not all lying on a single line, are marked in the plane, and a number is written next to each of them. It is known that whenever a line passes through two or more marked points, the sum of all the numbers written next to those points is zero. Prove that all the numbers are...

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Kvant Math Problem 449

Verified: no Verdicts: SKIP + SKIP Solve time: 5m39s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be found in the published issue. I cannot write a rigorous solution to Kvant problem M449 without the actual problem statement or the diagram. The prompt says that the text version is not available and that the problem appears only in graphical...

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Kvant Math Problem 447

Verified: no Verdicts: SKIP + SKIP Solve time: 4m39s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I cannot write a solution to Kvant problem M447 because the actual problem statement is not present in your message. The identifier “M447” by itself is not enough to determine the problem. Kvant numbering is tied to...

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Kvant Math Problem 445

Verified: no Verdicts: SKIP + SKIP Solve time: 5m40s Source on kvant.digital Problem The textual version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M445 from the information provided, because the actual problem statement is missing. You stated that “the textual version of the problem statement is currently being prepared” and that...

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Kvant Math Problem 443

Verified: no Verdicts: SKIP + SKIP Solve time: 5m56s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. Before I begin writing the complete solution, I need the full textual statement of Kvant problem M443 . The graphical version alone is not enough for a rigorous solution, since I cannot infer all details from images....

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Kvant Math Problem 441

Verified: no Verdicts: SKIP + SKIP Solve time: 3m50s Source on kvant.digital Problem Inside a convex $2n$-gon, an arbitrary point $P$ is chosen. Through each vertex and the point $P$, a straight line is drawn. Prove that there exists a side of the polygon such that none of the drawn lines has any points in common with it (except, possibly, the endpoints of the side). G. A. Gurevich Exploration Let...

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Kvant Math Problem 439

Verified: no Verdicts: SKIP + SKIP Solve time: 7m08s Source on kvant.digital Problem Prove that the equation $$ax^k+bx^l+cx^m=1$$ (where $a$, $b$, $c$ are real numbers, and $k$, $l$, and $m$ are natural numbers) has at most three positive roots. Prove that the equation $$a_1x^{k_1}+a_2x^{k_2}+ \ldots + a_nx^{k_n}=1$$ (where $a_1$, $a_2$, $\ldots$, $a_n$ are real numbers, and $k_1$, $k_2$, $\ldots$, $k_n$ are natural numbers) has at most $n$ positive roots. Prove...

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Kvant Math Problem 436

Verified: no Verdicts: SKIP + SKIP Solve time: 4m49s Source on kvant.digital Problem Given 20 numbers $a_1$, $a_2$, $\ldots$, $a_{10}$, $b_1$, $b_2$, $\ldots$, $b_{10}$. Prove that the set of 100 numbers (not necessarily distinct) $a_1+b_1$, $a_1+b_2$, $\ldots$, $a_{10}+b_{10}$ can be partitioned into 10 subsets, each containing 10 numbers, so that the sum of the numbers in each subset is the same. S. T. Berkolayko Exploration We are asked to partition...

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Kvant Math Problem 434

Verified: no Verdicts: SKIP + SKIP Solve time: 4m58s Source on kvant.digital Problem The number $\dfrac21+\dfrac{2^2}2+\dfrac{2^3}3+\ldots+\dfrac{2^n}n$ is represented as an irreducible fraction $\dfrac{p_n}{q_n}$. Prove that $p_n$ is an even number. Prove that if $n\ge3$, then $p_n$ is divisible by 8. Prove that for any natural $k$, one can specify $n$ such that the numbers $p_n$, $p_{n+1}$, $p_{n+2}$, $\ldots$ are divisible by $2^k$. D. K. Faddeev Exploration Consider the sum $S_n...

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Kvant Math Problem 432

Verified: no Verdicts: SKIP + SKIP Solve time: 7m31s Source on kvant.digital Problem Does there exist a perfect square whose digit sum is 1977; 1978? Determine which positive integers can occur as the digit sums of the square of an integer. A. Grishkov Exploration Consider the sum of the digits of perfect squares. Small examples suggest that there is no simple formula for the digit sum, but congruences modulo 9...

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Kvant Math Problem 430

Verified: no Verdicts: SKIP + SKIP Solve time: 3m53s Source on kvant.digital Problem Prove that any convex planar figure of area $S$ can be enclosed in a rectangle of area $2S$. Prove that any convex solid of volume $V$ can be enclosed in a box in the shape of a rectangular parallelepiped of volume $6V$ (Fig. 2). Fig. 2 Exploration For the planar statement, the number $2$ strongly suggests a...

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Kvant Math Problem 427

Verified: no Verdicts: SKIP + SKIP Solve time: 7m23s Source on kvant.digital Problem Prove that there exists an odd number $n$ such that, for no even $k$, is any of the numbers in the infinite sequence $$k^k+1,\ k^{k^k}+1,\ k^{k^{k^k}}+1,\ \ldots$$ divisible by $n$. Prove that for every positive integer $n$ there exists a positive integer $k$ such that every term of the infinite sequence $$k+1,\ k^k+1,\ k^{k^k}+1,\ k^{k^{k^k}}+1,\ \ldots$$ is...

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Kvant Math Problem 425

Verified: no Verdicts: SKIP + SKIP Solve time: 6m34s Source on kvant.digital Problem Does there exist a natural number $N$ such that every rational number between zero and one can be represented as a sum of $N$ reciprocals of natural numbers? Exploration Suppose such an $N$ exists. Then every rational number $r\in(0,1)$ can be written in the form $$r=\frac1{a_1}+\frac1{a_2}+\cdots+\frac1{a_N},$$ where the $a_i$ are natural numbers. For fixed $N$, the set...

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Kvant Math Problem 423

Verified: no Verdicts: SKIP + SKIP Solve time: 3m37s Source on kvant.digital Problem Prove that for any real $x$, $y$, and $z$, the inequality $$(x^2+y^2-z^2)(x^2+z^2-y^2)(y^2+z^2-x^2)\le(x+y-z)^2(x+z-y)^2(y+z-x)^2.$$ holds. R. Sheintsvit Exploration The left-hand side contains the three quantities $$x^2+y^2-z^2,\qquad x^2+z^2-y^2,\qquad y^2+z^2-x^2.$$ The right-hand side contains $$(x+y-z)^2(x+z-y)^2(y+z-x)^2.$$ A natural first step is to factor the expressions on the left. Indeed, $$x^2+y^2-z^2=(x+y-z)(x+y+z)-2xy.$$ This does not immediately help. A more symmetric factorization is obtained by...

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Kvant Math Problem 421

Verified: no Verdicts: SKIP + SKIP Solve time: 6m42s Source on kvant.digital Problem For which natural numbers $m$ and $n$ ($m\lt n$) is it possible to color some cells of an infinite sheet of graph paper black so that every rectangle of size $m\times n$ contains exactly one black cell? Start with the simpler problem: consider the cases $m=2$ and $n=3$, 4, 5. Fig. 1 S. Okhitin, 10th-grade student Exploration...

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Kvant Math Problem 418

Verified: no Verdicts: SKIP + SKIP Solve time: 6m34s Source on kvant.digital Problem Prove that for any natural number $n$ the following inequalities hold: $$n{\left(!\sqrt[\scriptstyle n~]{n+1}-1\right)}\le 1+\dfrac12+\dfrac13+\ldots+\dfrac1n\le 1+n{\left(1-\dfrac1{!\sqrt[\scriptstyle n~]n}\right)}.$$ Exploration For small values of $n$, $$n=1:\qquad 1(\sqrt[1]{2}-1)=1,\qquad H_1=1,\qquad 1+1\left(1-\frac1{\sqrt[1]{1}}\right)=1.$$ For $n=2$, $$2(\sqrt3-1)\approx1.464,\qquad H_2=\frac32,\qquad 1+2\left(1-\frac1{\sqrt2}\right)\approx1.586.$$ The inequalities hold. The expressions involving $n$th roots suggest taking logarithms. If $$a=\sqrt[n]{n+1},$$ then $$n(a-1)\le H_n \iff a\le 1+\frac{H_n}{n}.$$ Since $$a^n=n+1,$$ it would suffice to prove...

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Kvant Math Problem 416

Verified: no Verdicts: SKIP + SKIP Solve time: 6m34s Source on kvant.digital Problem On the plane, $n$ points $A_1$, $\ldots$, $A_n$ are given, no three of which lie on the same line. What is the greatest number of line segments with endpoints at these points that can be drawn so that no triangle with vertices at these points is formed? A. Grigoryan, M. Primak, S. Fishbein Exploration Interpret the drawn...

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Kvant Math Problem 413

Verified: no Verdicts: SKIP + SKIP Solve time: 9m58s Source on kvant.digital Problem For which positive numbers $a$ does the following statement hold: for any function $f$, defined on the interval $[0,1]$, continuous at every point of this interval and such that $f(0)=f(1)=0$, the equation $f(x+a)-f(x)=0$ has a solution? First, examine this question for the case $a=\dfrac12$. Prove that for $a=\dfrac1n$, where $n$ is a natural number, the stated claim...

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Kvant Math Problem 411

Verified: no Verdicts: SKIP + SKIP Solve time: 2m09s Source on kvant.digital Problem Three segments with endpoints on the sides of a triangle, parallel to its sides, pass through a single point and have the same length $x$ (Fig. 1). Find $x$ if the lengths of the triangle's sides are $a$, $b$, $c$. Fig. 1 A. Yagubyants Exploration Consider a triangle with sides $a$, $b$, $c$, and a point inside...

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Kvant Math Problem 410

Verified: no Verdicts: SKIP + SKIP Solve time: 4m02s Source on kvant.digital Problem On a sphere of radius 1, draw a great circle, which we shall call the equator. It will be convenient to use other geographical terms as well: pole, meridian, and parallel. Define on this sphere a function $f$ that assigns to each point of the sphere the square of the distance from that point to the plane...

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Kvant Math Problem 407

Verified: no Verdicts: SKIP + SKIP Solve time: 3m58s Source on kvant.digital Problem Two natural numbers $n$ and $m$, $n\gt m$ are given. Prove that $n$ can be expressed as the sum of two natural numbers, one of which is a divisor of $m$, and the other shares no common divisor with $m$ except for one. S. V. Konyagin Exploration Consider small examples to understand the structure of the problem....

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Kvant Math Problem 405

Verified: no Verdicts: SKIP + SKIP Solve time: 7m21s Source on kvant.digital Problem Fig. 1 Fig. 2 On a chessboard of size $99\times99$, a figure $\mathit\Phi$ is marked (this figure will differ in parts a), b), and c)). In each cell of the figure $\mathit\Phi$, there is a beetle. At some moment, the beetles flew up and landed again in the cells of the same figure $\mathit\Phi$; multiple beetles could...

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Kvant Math Problem 403

Verified: no Verdicts: SKIP + SKIP Solve time: 8m44s Source on kvant.digital Problem Prove that if, in a convex polyhedron, an even number of edges emanates from each vertex, then the intersection of the polyhedron with any plane that does not pass through any of its vertices is a polygon with an even number of sides. Exploration A convex polyhedron has the property that every vertex has even degree. The...

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Kvant Math Problem 400

Verified: no Verdicts: SKIP + SKIP Solve time: 7m15s Source on kvant.digital Problem A sequence of natural numbers $a_1$, $a_2$, $\ldots$, $a_k$ will be called universal for a given $N$ if, by deleting some of its terms, one can obtain any sequence of $N$ numbers in which each of the numbers 1, 2, $\ldots$, $N$ occurs exactly once. Give an example of a universal sequence consisting of $N^2$ terms. Give...

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Kvant Math Problem 398

Verified: no Verdicts: SKIP + SKIP Solve time: 3m24s Source on kvant.digital Problem The textual version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a rigorous solution to Kvant problem M398 from the information provided, because the problem statement itself is missing. Your prompt says: "The textual version of the problem statement is currently being prepared. The...

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Kvant Math Problem 396

Verified: no Verdicts: SKIP + SKIP Solve time: 6m44s Source on kvant.digital Problem A triangle whose all sides are greater than $1~\text{см}$ will be called "large." Given an equilateral triangle $ABC$ with side $5~\text{см}$, prove that: it is possible to cut 1000 "large" triangles from the triangle $ABC$; the triangle $ABC$ can be dissected into 1000 "large" triangles; the triangle $ABC$ can be triangulated into 1000 "large" triangles, i.e., subdivided...

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Kvant Math Problem 393

Verified: no Verdicts: SKIP + SKIP Solve time: 6m39s Source on kvant.digital Problem Find the sum $$\phi(0)+\phi\left(\dfrac{1}{n}\right)+\phi\left(\dfrac{2}{n}\right)+\ldots+\phi(1),$$ if $\phi(x)=\dfrac{4^x}{4^x+2}$. M. Levin Exploration Let $$S=\phi(0)+\phi!\left(\frac1n\right)+\phi!\left(\frac2n\right)+\cdots+\phi(1),$$ where $$\phi(x)=\frac{4^x}{4^x+2}.$$ The arguments form the set $$0,\frac1n,\frac2n,\ldots,\frac{n-1}{n},1.$$ A natural idea is to pair the terms corresponding to $x$ and $1-x$. Compute $$\phi(1-x) =\frac{4^{1-x}}{4^{1-x}+2}.$$ Writing $t=4^x$ gives $$\phi(x)=\frac{t}{t+2}, \qquad \phi(1-x)=\frac{4/t}{4/t+2} =\frac{2}{t+2}.$$ Hence $$\phi(x)+\phi(1-x) =\frac{t}{t+2}+\frac{2}{t+2} =1.$$ This identity is extremely suggestive because the set of arguments is...

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Kvant Math Problem 391

Verified: no Verdicts: SKIP + SKIP Solve time: 5m45s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M391 from the information provided, because the actual problem statement is missing. The prompt says: “The text version of the problem statement is currently being prepared. The graphical...

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Kvant Math Problem 388

Verified: no Verdicts: SKIP + SKIP Solve time: 7m50s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I cannot write a rigorous solution to Kvant problem M388 because the problem statement itself is not provided in the conversation. The prompt says that only the graphical version is available. Without the diagram or the text...

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Kvant Math Problem 386

Verified: no Verdicts: SKIP + SKIP Solve time: 6m12s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I can proceed with that, but I need the text of Kvant Problem M386 to provide the complete solution. Could you provide the problem statement?

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Kvant Math Problem 383

Verified: no Verdicts: SKIP + SKIP Solve time: 7m06s Source on kvant.digital Problem Let $a$ and $b$ be two natural numbers. Prove that if $ab$ is even, then there exist natural numbers $c$ and $d$ such that $a^2+b^2+c^2=d^2$, whereas if $ab$ is odd, such $c$ and $d$ cannot be found. M. L. Gerver Exploration Consider small examples to understand the problem. If $ab$ is even, at least one of $a$...

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Kvant Math Problem 381

Verified: no Verdicts: SKIP + SKIP Solve time: 6m08s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M381 because the actual problem statement is not present in your message. You wrote that “the textual version of the problem statement is being prepared” and that the problem...

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Kvant Math Problem 379

Verified: no Verdicts: SKIP + SKIP Solve time: 4m28s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I can proceed with this framework, but I need the text of Kvant problem M379 in order to produce a rigorous solution. Could you provide the problem statement?

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Kvant Math Problem 376

Verified: no Verdicts: SKIP + SKIP Solve time: 6m47s Source on kvant.digital Problem A row consists of 30 cells. A white piece stands on the rightmost cell, and a black piece stands on the leftmost cell. Each of the two players, in turn, moves their piece by one cell — either forward or backward. (Skipping a move is not allowed.) The player who cannot make a move loses. Who wins:...

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Kvant Math Problem 375

Verified: no Verdicts: SKIP + SKIP Solve time: 5m09s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M375 from the information provided, because the actual problem statement is missing. The identifier “Kvant M375” by itself is not enough to reconstruct the problem, and the prompt...

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Kvant Math Problem 372

Verified: no Verdicts: SKIP + SKIP Solve time: 7m43s Source on kvant.digital Problem Given a triangle $ABC$. Prove that the condition $\widehat{ACB} \ge 120^\circ$ is necessary and sufficient for the inequality $|AP|+|BP|+|CP|\ge|AC|+|BC|$ to hold for any point $P$ in the plane. P. Khaidukov Exploration Consider the triangle $ABC$ and the inequality $|AP| + |BP| + |CP| \ge |AC| + |BC|$ for an arbitrary point $P$ in the plane. The inequality...

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Kvant Math Problem 369

Verified: no Verdicts: SKIP + SKIP Solve time: 7m04s Source on kvant.digital Problem Given an acute-angled triangle $ABC$. $H$ is the intersection point of its altitudes, $\gamma$ is a circle with center $H$ lying inside this triangle. Construct a triangle $A_1B_1C_1$ circumscribed around the circle $\gamma$ and inscribed in the triangle $ABC$ (such that $A_1 \in [BC]$, $B_1 \in [AC]$, $C_1 \in [AB]$). D. Isaak Exploration The circle $\gamma$ is...

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Kvant Math Problem 368

Verified: no Verdicts: SKIP + SKIP Solve time: 6m32s Source on kvant.digital Problem Prove that the intersection of three right circular cylinders of radius 1 whose axes are pairwise perpendicular (but do not necessarily intersect) is contained in some sphere of radius $\sqrt\dfrac32$. S. V. Fomin Exploration Choose coordinates so that the three cylinder axes are parallel to the coordinate axes. Since the axes are pairwise perpendicular but need not...

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Kvant Math Problem 365

Verified: no Verdicts: SKIP + SKIP Solve time: 6m31s Source on kvant.digital Problem The sum of several numbers is 1. Can the sum of their cubes be greater than one? The same question for numbers each of which is less than one. Can it happen that the series $a_1+a_2+a_3+\ldots$ converges, while the series $a_1^3+a_2^3+a_3^3+\ldots$ does not? (Recall that a series $x_1+x_2+x_3+\ldots$ is called convergent if the sequence of numbers $S_n=x_1+x_2+\ldots+x_n$...

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Kvant Math Problem 362

Verified: no Verdicts: SKIP + SKIP Solve time: 3m39s Source on kvant.digital Problem Divide each side of the convex quadrilateral $ABCD$ into three equal parts and connect the corresponding points on opposite sides by segments (Fig. 1). Prove that the area of the “middle” quadrilateral is 9 times smaller than the area of the quadrilateral $ABCD$. Figure 1 A. Lopshits Exploration The statement is affine in nature. Dividing sides in...

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Kvant Math Problem 361

Verified: no Verdicts: SKIP + SKIP Solve time: 4m29s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. Please provide the full text of Kvant problem M361 so I can write the complete, rigorous solution in the requested six-section format.

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Kvant Math Problem 358

Verified: no Verdicts: SKIP + SKIP Solve time: 6m09s Source on kvant.digital Problem The text version of the problem statement is being prepared. The graphical version can be found in the published issue. I can produce a complete, rigorous Kvant-style solution, but I need the text of problem M358 to proceed. Could you provide the problem statement?

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Kvant Math Problem 357

Verified: no Verdicts: SKIP + SKIP Solve time: 2m56s Source on kvant.digital Problem Prove that if $x+\dfrac1y=y+\dfrac1z=z+\dfrac1x$, then $x=y=z$ or $x^2y^2z^2=1$. Exploration Let $$x+\frac1y=y+\frac1z=z+\frac1x=k.$$ Subtracting consecutive equalities gives $$x-y=\frac1z-\frac1y=\frac{y-z}{yz},$$ $$y-z=\frac1x-\frac1z=\frac{z-x}{xz},$$ $$z-x=\frac1y-\frac1x=\frac{x-y}{xy}.$$ These relations connect the differences $x-y$, $y-z$, $z-x$ with one another. If all three differences are zero, then $x=y=z$, which is one of the required conclusions. The interesting case is when not all differences vanish. Then at least one...

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Kvant Math Problem 351

Verified: no Verdicts: SKIP + SKIP Solve time: 10m16s Source on kvant.digital Problem Reconstruct a triangle if three points are marked in the plane: $O$ — the center of the circumcircle, $P$ — the centroid, and $H$ — the foot of one of the altitudes of this triangle. M. M. Imerishvili, 9th-grade student (Tbilisi) Exploration Let the unknown triangle be $ABC$, and suppose that $H$ is the foot of the...

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Kvant Math Problem 349

Verified: no Verdicts: SKIP + SKIP Solve time: 5m49s Source on kvant.digital Problem What condition must the side lengths of a triangle satisfy so that the triangle formed by its altitudes, medians, angle bisectors is similar to the given triangle? A. P. Savin Exploration Let the given triangle have sides $a,b,c$ opposite angles $A,B,C$. For the triangle formed by the altitudes, the side lengths are proportional to the altitudes $$h_a=\frac{2\Delta}{a},\qquad...

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Kvant Math Problem 341

Verified: no Verdicts: SKIP + SKIP Solve time: 4m54s Source on kvant.digital Problem There are 20 teams participating in the World Championship. Among them, $k$ are European teams, and the results of the matches between these teams in the World Championship count towards the European Championship. The championship is conducted in a single round-robin format. What is the largest possible $k$ such that a European team, having scored strictly the...

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Kvant Math Problem 338

Verified: no Verdicts: SKIP + SKIP Solve time: 1m27s Source on kvant.digital Problem On the board, several zeros, ones, and twos are written. It is allowed to erase two unequal digits and replace them with a single digit different from the erased ones (2 instead of 0 and 1, 1 instead of 0 and 2, 0 instead of 1 and 2). Prove that if, after several such operations, only a...

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Kvant Math Problem 336

Verified: no Verdicts: SKIP + SKIP Solve time: 1m39s Source on kvant.digital Problem A finite set of polygons is given in the plane, and every two of them have a point in common. Prove that there exists a line that has a point in common with each of these polygons. S. V. Fomin All-Union Mathematical Olympiad for School Students (1974, Grade 10) Exploration For two polygons the statement is immediate....

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Kvant Math Problem 328

Verified: no Verdicts: SKIP + SKIP Solve time: 7m46s Source on kvant.digital Problem A fly and two spiders are moving on a regular tetrahedron. The fly moves only along the edges, while the spiders may move over the entire surface. The fly's maximum speed is twice the spiders' maximum speed. Prove that, regardless of the initial positions, the spiders can catch the fly. Is this still true if the fly's...

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Kvant Math Problem 320

Verified: no Verdicts: SKIP + SKIP Solve time: 7m16s Source on kvant.digital Problem Which convex $n$-gons can be subdivided into triangles so that no two triangles in the subdivision share a common (completely coinciding) side? (Figure 1 shows that a triangle can be subdivided in this way.) Figure number 1 A. Pechkowski Exploration The statement asks for a classification. A triangle can indeed be subdivided into smaller triangles without any...

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Kvant Math Problem 317

Verified: no Verdicts: SKIP + SKIP Solve time: 4m08s Source on kvant.digital Problem On a certain planet, each country borders no more than 7 other countries. Each country has a reserve of gold. It is required to distribute the gold so that any two neighboring countries differ in the amount of gold by no more than a factor of 13. Prove that the gold can be distributed in such a...

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Kvant Math Problem 307

Verified: no Verdicts: SKIP + SKIP Solve time: 7m01s Source on kvant.digital Problem The plane is partitioned into congruent hexagonal rooms (Figure 2). Doors have been made in some of the walls in such a way that for every vertex at which three walls (sides of the hexagons) meet, doors are present in exactly two of those walls. Prove that any closed path through such a labyrinth passes through an...

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Kvant Math Problem 298

Verified: no Verdicts: SKIP + SKIP Solve time: 6m30s Source on kvant.digital Problem List all irreducible fractions $\dfrac pq$, where $0\lt p\lt q\le m$, in ascending order ($m$ is a given natural number). For example, for $m=5$ the sequence $$\dfrac15,\quad\dfrac14,\quad\dfrac13,\quad\dfrac25,\quad\dfrac12,\quad\dfrac35,\quad\dfrac23,\quad\dfrac34,\quad\dfrac45.$$ is obtained. Prove that for any two consecutive fractions $\dfrac pq\lt\dfrac rs$ in such a sequence, the equality $qr-ps=1$ holds. N. B. Vasilyev Exploration For $m=5$ the consecutive fractions $$\frac13<\frac25,\qquad...

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Kvant Math Problem 295

Verified: no Verdicts: SKIP + SKIP Solve time: 5m06s Source on kvant.digital Problem The sections of a convex polyhedron by three parallel planes $p_0$, $p_1$, and $p_2$ ($p_1$ lies between $p_0$ and $p_2$ at the same distance $h$ from each of them) have areas $S_0$, $S_1$, and $S_2$, respectively. There is not a single vertex of the polyhedron between $p_0$ and $p_2$. Prove that $2\sqrt{S_1}\ge\sqrt{S_0}+\sqrt{S_2}$. In what case does the...

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Kvant Math Problem 286

Verified: no Verdicts: SKIP + SKIP Solve time: 5m42s Source on kvant.digital Problem On a plane, $N$ points are placed. Mark all the midpoints of segments whose endpoints are among these points. What is the minimum number of points on the plane that can be marked? A. Pechkovsky Exploration Let $m(N)$ denote the minimum possible number of distinct marked points. For $N=2$, there is one segment and one midpoint, so...

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Kvant Math Problem 284

Verified: no Verdicts: SKIP + SKIP Solve time: 3m31s Source on kvant.digital Problem The sum of 100 natural numbers, each not exceeding 100, is 200. Prove that it is possible to select some of these numbers whose sum is 100. Moscow Mathematical Olympiad (1974) Exploration Consider smaller analogues of the problem to understand its structure. If we have two numbers whose sum is $4$, say $1$ and $3$, we can...

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Kvant Math Problem 273

Verified: no Verdicts: SKIP + SKIP Solve time: 5m24s Source on kvant.digital Problem On the segment $0 \le x \le 1$, a function $f$ is defined. It is known that this function is non-negative and $f(1) = 1$. Moreover, for any two numbers $x_1$ and $x_2$ such that $x_1 \ge 0$, $x_2 \ge 0$, and $x_1 + x_2 \le 1$, the inequality $$f(x_1 + x_2) \ge f(x_1) + f(x_2).$$ holds....

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Kvant Math Problem 263

Verified: no Verdicts: SKIP + SKIP Solve time: 6m48s Source on kvant.digital Problem Two numbers $p$ and $q$ greater than 1 are given. On the sides $BC$ and $DC$ of the rectangle $ABCD$, points $P$ and $Q$ are chosen so that $|BC| = p \cdot |BP|$ and $|DC| = q \cdot |DQ|$. For what ratio of the lengths of the sides $AB$ and $AD$ will the angle $PAQ$ attain its...

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Kvant Math Problem 261

Verified: no Verdicts: SKIP + SKIP Solve time: 1m48s Source on kvant.digital Problem A hoop of radius $R$, suspended on a fixed circle of radius $r < R$, begins to roll along this circle. Prove that a point on the hoop traces the same path as a point on a wheel of radius $R - r$ rolling externally along the same circle of radius $r$ (Fig. 1, a, b). (The...

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Kvant Math Problem 258

Verified: no Verdicts: SKIP + SKIP Solve time: 5m42s Source on kvant.digital Problem Three points $K$, $L$, $N$ are given in the plane. It is known about a quadrilateral that it is convex and that the midpoints of three of its sides lie at the given points $K$, $L$, $N$. Find the sets of points that can be occupied by: the midpoint of the fourth side; a vertex of this...

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Kvant Math Problem 248

Verified: no Verdicts: SKIP + SKIP Solve time: 7m54s Source on kvant.digital Problem An $n$-gon $A_1A_2\ldots A_n$ is inscribed in a convex $n$-gon $B_1B_2\ldots B_n$, and its area is equal to $P$ (the vertex $B_i$ lies on the side $A_iA_{i+1}$ for $i=1$, 2, $\ldots$, $n-1$, and the vertex $B_n$ lies on $A_n A_1$). Around the same $n$-gon $A_1A_2\ldots A_n$, an $n$-gon $C_1C_2 \ldots C_n$ is circumscribed, whose area is equal...

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Kvant Math Problem 237

Verified: no Verdicts: SKIP + SKIP Solve time: 5m06s Source on kvant.digital Problem The angles of an acute-angled triangle are $\alpha$, $\beta$, and $\gamma$. What masses should be placed at its vertices so that the center of mass of these three masses falls at the intersection point of the altitudes? at the center of the circumscribed circle? The sides of the triangle are $a$, $b$, and $c$. What masses should...

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Kvant Math Problem 229

Verified: no Verdicts: SKIP + SKIP Solve time: 7m04s Source on kvant.digital Problem A policeman is at the center of a square, and a gangster is at one of its vertices. The policeman can run anywhere inside the square, while the gangster can move only along its sides. It is known that the policeman's maximum speed is $u$, and the gangster's maximum speed is $v$. The policeman's goal is to...

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Kvant Math Problem 224

Verified: no Verdicts: SKIP + SKIP Solve time: 1m41s Source on kvant.digital Problem In a trihedral angle, the bisectors of its plane angles are drawn. Prove that the pairwise angles between these three bisectors are either all obtuse, all acute, or all right angles. E. G. Gotman Exploration Consider a trihedral angle, that is, three planes meeting at a common vertex, forming three plane angles $\alpha$, $\beta$, and $\gamma$ at...

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Kvant Math Problem 222

Verified: no Verdicts: SKIP + SKIP Solve time: 6m59s Source on kvant.digital Problem Prove that every convex polyhedron has two faces with the same number of sides. A. Grüntal Exploration Consider small convex polyhedra such as the tetrahedron, cube, and octahedron. A tetrahedron has four triangular faces, all identical, so the claim is trivially true. A cube has six quadrilateral faces, again all identical. An octahedron has eight triangular faces,...

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Kvant Math Problem 211

Verified: no Verdicts: SKIP + SKIP Solve time: 3m33s Source on kvant.digital Problem Given $n$ points, $n \gt 4$. Prove that it is possible to connect them with arrows so that from each point to any other one, it is possible to reach it by following either one arrow or two arrows (each pair of points can be connected by an arrow in only one direction; one can travel along...

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Kvant Math Problem 209

Verified: no Verdicts: SKIP + SKIP Solve time: 3m57s Source on kvant.digital Problem For any triangle $ABC$ one can compute the following sum: $$S = \tg^2 \dfrac{A}{2} + \tg^2 \dfrac{B}{2} + \tg^2 \dfrac{C}{2}.$$ Prove that $S < 2$ for all acute and right triangles; $S > 2$ for obtuse triangles whose obtuse angle is greater than $2 \arctg \dfrac{4}{3}$; among triangles with an obtuse angle $\varphi$ such that $\dfrac{\pi}{2} <...

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Kvant Math Problem 198

Verified: no Verdicts: SKIP + SKIP Solve time: 7m51s Source on kvant.digital Problem Given a parallelogram $ABCD$. On the lines $AB$ and $BC$, points $H$ and $K$ are chosen, respectively, such that the triangles $KAB$ and $HCB$ are isosceles ($KA=AB$ and $HC=CB$). Prove that the triangle $KDH$ is also isosceles. V. L. Gutenmacher Exploration The conditions place $H$ on the line $AB$ and $K$ on the line $BC$. Since $KA=AB$,...

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Kvant Math Problem 189

Verified: no Verdicts: SKIP + SKIP Solve time: 6m00s Source on kvant.digital Problem Three segments $AB$, $EF$, and $CD$ pass through a single point $O$, with point $E$ lying on segment $AC$, and point $F$ lying on segment $BD$. Prove that $EF$ is shorter than at least one of the two segments — $AB$ or $CD$ (Fig. 1). Figure number 1 D. Yu. Grigoriev Exploration Consider three segments $AB$, $CD$,...

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Kvant Math Problem 187

Verified: no Verdicts: SKIP + SKIP Solve time: 5m35s Source on kvant.digital Problem On the plane, two points $A$ and $B$ are given. Find the locus of the third vertices $C$ of a triangle $ABC$ such that: the altitude $AA'$ is equal to the side $BC$; the median $AA_1$ is equal to the side $AC$; the median $AA_1$ is equal to the side $BC$; the altitude $CC'$ is equal to...

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Kvant Math Problem 176

Verified: no Verdicts: SKIP + SKIP Solve time: 5m39s Source on kvant.digital Problem To which side of triangle $ABC$ is the intersection point of its altitudes closest, if $\angle A\lt\angle B\lt\angle C$? And to which vertex? L. Schneider, 9th-grade student Exploration Let $H$ be the orthocenter of triangle $ABC$. The distance from $H$ to side $BC$ equals the length of the perpendicular from $H$ to $BC$. Since $AH$ is perpendicular...

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Kvant Math Problem 174

Verified: no Verdicts: SKIP + SKIP Solve time: 3m09s Source on kvant.digital Problem On the sides of triangle $ABC$, isosceles triangles $AB_1C$, $BA_1C$, and $AC_1B$ are constructed with these sides as bases (see Fig. 1). Prove that the perpendiculars dropped from points $A$, $B$, and $C$ onto the lines $B_1C_1$, $C_1A_1$, and $A_1B_1$, respectively, intersect at a single point. Figure 1 A. G. Geyn Exploration Consider triangle $ABC$ with isosceles...

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Kvant Math Problem 164

Verified: no Verdicts: SKIP + SKIP Solve time: 1m48s Source on kvant.digital Problem The textual version of the problem statement is currently being prepared. The graphical version can be found in the published issue. On the white squares of an infinite chessboard filling the upper half-plane (Figure 1), some numbers are written so that for every black square, the sum of the numbers in the two adjacent squares to its...

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Kvant Math Problem 355

Verified: no Verdicts: SKIP + SKIP Solve time: 2m48s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. I cannot write a solution to Kvant problem M355 because the actual problem statement is not present in your message. The prompt explicitly says: “The textual version of the problem statement is being prepared. The graphical version...

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Kvant Math Problem 353

Verified: no Verdicts: SKIP + SKIP Solve time: 5m51s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be found in the published issue. Before proceeding, I need the precise textual statement of Kvant problem M353. Could you provide the full problem text? The graphical version alone is insufficient for a rigorous, step-by-step solution.

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Kvant Math Problem 352

Verified: no Verdicts: SKIP + SKIP Solve time: 6m05s Source on kvant.digital Problem Let $n$ be an integer for which $$n\lt(45+\sqrt{1975})^{30}\lt n+1.$$. Prove that $n$ is odd. D. K. Faddeev Exploration Write $$\alpha=45+\sqrt{1975}=45+5\sqrt{79},\qquad \beta=45-5\sqrt{79}.$$ Then $$\alpha+\beta=90,\qquad \alpha\beta=45^2-1975=50.$$ The number $\beta$ is positive and $$0<\beta=45-\sqrt{1975}<1,$$ because $44^2=1936<1975<2025=45^2$. Since $\alpha$ and $\beta$ are roots of $$x^2-90x+50=0,$$ the sequence $$u_k=\alpha^k+\beta^k$$ satisfies the recurrence $$u_{k+2}=90u_{k+1}-50u_k,$$ with $u_0=2$ and $u_1=90$. Hence every $u_k$ is an...

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Kvant Math Problem 350

Verified: no Verdicts: SKIP + SKIP Solve time: 7m26s Source on kvant.digital Problem A bishop starts moving from a white corner square of an $n\times m$ chessboard (with $n$ and $m$ greater than 1). Upon reaching the edge of the board, the bishop turns at a right angle (Fig. 2). When it reaches a corner, it stops. For which $n$ and $m$ will the bishop pass through all the white...

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Kvant Math Problem 348

Verified: no Verdicts: SKIP + SKIP Solve time: 4m51s Source on kvant.digital Problem In the table $10\times10$, the numbers from 1 to 100 are written in order. Then, in each row and each column, exactly half of the numbers are marked with a minus sign. Prove that in the resulting table, the sum of all the numbers is zero. S. M. Ageev Exploration The numbers in the table are $$\begin{matrix}...

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Kvant Math Problem 347

Verified: no Verdicts: SKIP + SKIP Solve time: 6m17s Source on kvant.digital Problem Two people play the following game. The first chooses two numbers from 1 to 25, and the second must guess them. He may name any two numbers from 1 to 25 and ask the first how many of the numbers he named—0, 1, or 2—coincide with the chosen numbers. What is the minimum number of questions with...

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Kvant Math Problem 345

Verified: no Verdicts: SKIP + SKIP Solve time: 6m10s Source on kvant.digital Problem In the sequence $197523\ldots$, each digit, starting from the fifth, is equal to the last digit of the sum of the previous four digits. Will the following appear consecutively in this sequence: the four digits 1234? the digits 1975 more than once? the digits 8197? G. A. Gurevich Exploration Let $a_n$ be the sequence, with $$a_1=1,\quad a_2=9,\quad...

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Kvant Math Problem 343

Verified: no Verdicts: SKIP + SKIP Solve time: 6m21s Source on kvant.digital Problem In a certain country, the cities are connected by roads. The length of any road is less than $500~\text{км}$, and it is possible to travel from any city to any other city via roads totaling less than $500~\text{км}$. When a single road was closed for repairs, it turned out that it was still possible to travel between...

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Kvant Math Problem 342

Verified: no Verdicts: SKIP + SKIP Solve time: 6m39s Source on kvant.digital Problem Prove that using the digits 1 and 2, one can construct $2^{n+1}$ numbers, each of which is $2^{n}$-digit long, and any two of which differ in at least $2^{n-1}$ positions. Prove that it is impossible to construct more than $2^{n+1}$ such $2^{n}$-digit numbers. S. V. Fomin Exploration The problem is a binary coding problem. Replacing the digits...

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Kvant Math Problem 340

Verified: no Verdicts: SKIP + SKIP Solve time: 3m16s Source on kvant.digital Problem A real number is written in each cell of a rectangular table. A cell of the table is called a saddle cell if the number in it is not less than all the other numbers in its column and not greater than all the other numbers in its row. Suppose it is known about a table $T$...

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Kvant Math Problem 339

Verified: no Verdicts: SKIP + SKIP Solve time: 3m59s Source on kvant.digital Problem A horizontal strip is given on the plane, bounded by two parallel lines, and $n$ lines intersecting this strip. Every two of these $n$ lines intersect within the strip, and no three pass through the same point. Consider all paths starting at the lower edge of the strip, following these lines, and ending at the upper edge,...

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Kvant Math Problem 337

Verified: no Verdicts: SKIP + SKIP Solve time: 4m04s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. Please provide the textual version of Kvant problem M337. Once I have the full problem statement, I will produce the rigorous solution in the requested six-section format.

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Kvant Math Problem 335

Verified: no Verdicts: SKIP + SKIP Solve time: 10m27s Source on kvant.digital Problem In a square of $7\times7$ cells, the centers of $k$ cells are marked. Moreover, no four marked points form the vertices of a rectangle with sides parallel to the sides of the square. What is the largest possible $k$ for which this is possible? Solve the same problem for a square of $13\times13$ cells. S. B. Gashkov,...

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Kvant Math Problem 334

Verified: no Verdicts: SKIP + SKIP Solve time: 4m41s Source on kvant.digital Problem Given a polynomial $P(x)$ with natural coefficients, integer coefficients. For each natural number $n$ through $a_n$, denote by $P(n)$ the sum of the digits in the decimal representation of the number $P(n)$. Prove that there exists a number that appears in the sequence $a_1$, $a_2$, $a_3$, $\ldots$ infinitely many times. I. N. Bernstein All-Union School Mathematics Olympiad...

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Kvant Math Problem 332

Verified: no Verdicts: SKIP + SKIP Solve time: 2m46s Source on kvant.digital Problem Is it possible to construct a cube of size $k\times k\times k$ from white and black cubes $1\times1\times1$ in such a way that each small cube has exactly two neighbors of the same color as itself? (Two cubes are considered neighbors if they share a common face.) A. G. Geyn All-Union Mathematical Olympiad for School Students (1975,...

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Kvant Math Problem 331

Verified: no Verdicts: SKIP + SKIP Solve time: 8m14s Source on kvant.digital Problem Triangle $A_1B_1C_1$ is obtained from triangle $ABC$ by a rotation around the center of its circumcircle by some angle less than $180^\circ$. Prove that the points of intersection of the corresponding lines: $(AB)$ and $(A_1B_1)$, $(BC)$ and $(B_1C_1)$, $(CA)$ and $(C_1A_1)$ — are the vertices of a triangle similar to triangle $ABC$. Quadrilateral $A_1B_1C_1D_1$ is obtained from...

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Kvant Math Problem 329

Verified: no Verdicts: SKIP + SKIP Solve time: 6m36s Source on kvant.digital Problem A convex $n$-gon is placed inside a square with side length 1. Prove that there exist three vertices $A$, $B$, $C$ of this $n$-gon such that the area of the triangle $ABC$ is less than $\dfrac8{n^2}$. G. Shmelev, 10th-grade student (Yaroslavl) Exploration Let the vertices of the convex $n$-gon be $P_1,P_2,\dots,P_n$ in cyclic order. Since the polygon...

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Kvant Math Problem 327

Verified: no Verdicts: SKIP + SKIP Solve time: 5m58s Source on kvant.digital Problem There are $N$ people in a group. Each of them likes exactly $k$ people in this group. For what smallest $k$ can one assert that there must exist two people in the group who like each other? A. Kolodin, 10th-grade student (Kolomyia) Exploration Consider small groups to understand how the structure of "liking" works. If $N=2$, each...

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Kvant Math Problem 326

Verified: no Verdicts: SKIP + SKIP Solve time: 6m28s Source on kvant.digital Problem A chord of a circle is at a distance $h$ from the center. In each of the segments subtended by the chord, a square is inscribed so that two adjacent vertices of the square lie on the arc and the other two lie on the chord (Fig. 1). What is the difference between the side lengths of...

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Kvant Math Problem 324

Verified: no Verdicts: SKIP + SKIP Solve time: 4m20s Source on kvant.digital Problem There are several piles of stones. Two players play a game in which a move consists of splitting each pile containing more than one stone into two smaller piles. The players move alternately until every pile contains exactly one stone. The winner is the player who makes the last move. How should the first player play if...

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Kvant Math Problem 323

Verified: no Verdicts: SKIP + SKIP Solve time: 6m29s Source on kvant.digital Problem Prove that any function defined on the entire real line can be represented as the sum of two functions, each of whose graph has a center of symmetry. V. A. Sergeev Exploration Consider a function $f:\mathbb{R}\to\mathbb{R}$. A graph has a center of symmetry if there exists a point $(a,b)$ such that for every $x$, the point symmetric...

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Kvant Math Problem 321

Verified: no Verdicts: SKIP + SKIP Solve time: 7m27s Source on kvant.digital Problem There is a rectangular table of area 1. Prove that for any $\eps\gt0$ one can specify a system of rectangular napkins covering this table (the edges of the napkins are parallel to the edges of the table) such that any subsystem consisting of pairwise non-overlapping napkins has total area less than $\eps$. A. V. Brailov Exploration The...

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Kvant Math Problem 319

Verified: no Verdicts: SKIP + SKIP Solve time: 5m08s Source on kvant.digital Problem The textual version of the problem statement is being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M319 because the actual problem statement is not present in your message. The prompt explicitly says: “The textual version of the problem statement is being prepared. The graphical version...

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Kvant Math Problem 318

Verified: no Verdicts: SKIP + SKIP Solve time: 6m36s Source on kvant.digital Problem In triangle $ABC$, the angle bisectors $[AD]$ and $[CE]$ are drawn. Prove that $|CE|\cdot|AB|=|AD|\cdot|BC|$ if and only if at least one of the following two conditions holds: $|AB|=|BC|$; $\widehat{ABC}=60^\circ$. A. P. Savin Exploration Let $$a=|BC|,\qquad b=|CA|,\qquad c=|AB|.$$ The condition is $$|CE|\cdot c=|AD|\cdot a.$$ The standard formula for the length of an internal angle bisector gives $$AD=\frac{2bc\cos \frac...

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Kvant Math Problem 316

Verified: no Verdicts: SKIP + SKIP Solve time: 3m37s Source on kvant.digital Problem Prove that the sum of the squares of $k$ consecutive natural numbers cannot be a perfect square if $k$ equals 3, 5, 7, or 9. Find 11 consecutive natural numbers whose sum of squares is a perfect square. E. G. Gotman Exploration Consider the sum of squares of $k$ consecutive natural numbers beginning at $n$, expressed as...

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Kvant Math Problem 314

Verified: no Verdicts: SKIP + SKIP Solve time: 8m35s Source on kvant.digital Problem Among all 9-digit numbers whose decimal representation contains no 0, find the number for which the difference between the number itself and the product of its digits is: the smallest; the largest. What will be the answer for $n$-digit numbers for any $n$? A. G. Leiderman Exploration Consider the difference between a number and the product of...

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Kvant Math Problem 313

Verified: no Verdicts: SKIP + SKIP Solve time: 6m10s Source on kvant.digital Problem Given an angle with vertex $O$. Consider the set of fourth vertices $M$ of parallelograms $ONML$, whose vertices $N$ and $L$ lie on the sides of the given angle, and whose area is a constant. (This set is called a hyperbola .) Prove that on the bisector of this angle and its extension there exist points $F_1$...

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Kvant Math Problem 311

Verified: no Verdicts: SKIP + SKIP Solve time: 6m06s Source on kvant.digital Problem Starting from a single bacterium, 1000 bacteria were obtained as follows: first the bacterium divided into two; then one of the two resulting bacteria divided into two; then one of the three resulting bacteria divided into two; and so on. Prove that at some moment there existed a bacterium such that the number of its descendants among...

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Kvant Math Problem 309

Verified: no Verdicts: SKIP + SKIP Solve time: 7m32s Source on kvant.digital Problem For which $n$ is the polynomial $x^{2n}+x^n+1$ divisible by $x^2+x+1$? For which $n$ is the number $1\underbrace{0\ldots0}_n1\underbrace{0\ldots0}_n1$ divisible by 37? V. G. Shleifer Exploration For the first question, divisibility by $x^2+x+1$ suggests evaluating the polynomial at the nonreal cube roots of unity. Let $\omega^3=1$ and $\omega\neq1$. Then $\omega^2+\omega+1=0$, and $x^2+x+1$ divides a polynomial $P(x)$ if and only...

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Kvant Math Problem 308

Verified: no Verdicts: SKIP + SKIP Solve time: 3m30s Source on kvant.digital Problem If for any $x$ $$a_1\cos x+a_2\cos2x+\ldots+a_n\cos nx\ge-1,$$, then for the numbers $a_1$, $a_2$, $\ldots$, $a_n$ the inequality $$a_1+a_2+\ldots+a_n\le n.$$ holds. Prove this statement for $n=2$; for $n=3$; for any natural $n$. Yu. I. Ionin Exploration Consider first the case $n=2$, where the inequality takes the form $a_1\cos x + a_2\cos 2x \ge -1$ for all real $x$....

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Kvant Math Problem 306

Verified: no Verdicts: SKIP + SKIP Solve time: 6m49s Source on kvant.digital Problem From a chessboard ($8\times8$), one corner square ($1\times1$) has been removed (Fig. 1). Into what smallest number of equal-area triangles can the remaining part of the board be dissected? Figure 1 V. P. Fedotov Exploration Let the removed corner be the unit square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$. The remaining region has area $$64-1=63.$$ If it...

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Kvant Math Problem 304

Verified: no Verdicts: SKIP + SKIP Solve time: 7m53s Source on kvant.digital Problem Let us denote by a circle some (as yet unknown) operation applicable to any two nonnegative integers $a$ and $b$ and yielding as a result another nonnegative integer $a\circ b=c$. Suppose that the operation “$\circ$” satisfies the following conditions: $a\circ b=b\circ a$; if $a\circ b=c$, then $b\circ c=a$; if $a\circ b\gt c$, then $b\circ c\lt a$ or...

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Kvant Math Problem 303

Verified: no Verdicts: SKIP + SKIP Solve time: 6m10s Source on kvant.digital Problem A rectangle $300\times1000$ is divided into squares $1\times1$, and identical weights are placed at some 30 vertices of the squares. Prove that it is possible to select two disjoint groups of weights—no more than 10 in each—such that their centers of gravity coincide. A. V. Sherstyuk Exploration Consider placing a small number of identical weights on the...

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Kvant Math Problem 301

Verified: no Verdicts: SKIP + SKIP Solve time: 7m43s Source on kvant.digital Problem On the plane, $2n$ points are given—$n$ blue and $n$ red, with no three points lying on the same line. Prove that it is possible to draw $n$ line segments so that each segment has one endpoint at a red point and the other at a blue point, and no two segments have any point in common....

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Kvant Math Problem 299

Verified: no Verdicts: SKIP + SKIP Solve time: 6m16s Source on kvant.digital Problem For which $n$ can a regular $n$-gon be placed on a ruled sheet of paper so that all its vertices lie on the lines? (The lines are parallel and equally spaced.) N. B. Vasilyev Exploration Consider a ruled sheet of paper with parallel lines spaced a fixed distance apart, and suppose a regular $n$-gon is drawn so...

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Kvant Math Problem 297

Verified: no Verdicts: SKIP + SKIP Solve time: 6m12s Source on kvant.digital Problem On the plane, 12 points are given that are the vertices of four squares $A_1B_1A_2C_1$, $A_2C_2A_3B_2$, $A_3B_3A_4C_3$, and $A_4C_4A_1B_4$ (the vertices of each square are listed in clockwise order). Prove that $B_1B_2B_3B_4$ and $C_1C_2C_3C_4$ are congruent parallelograms, one of which is obtained from the other by a rotation through $90^\circ$ (these parallelograms may be degenerate: in that...

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Kvant Math Problem 296

Verified: no Verdicts: SKIP + SKIP Solve time: 4m57s Source on kvant.digital Problem In the table $n\times n$, $n^2$ numbers are written, the sum of which is non-negative. Prove that it is possible to rearrange the columns of the table so that the sum of $n$ numbers along the diagonal running from the bottom-left corner to the top-right corner is non-negative. A. Tupanov Exploration Let the rows be numbered from...

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Kvant Math Problem 294

Verified: no Verdicts: SKIP + SKIP Solve time: 7m45s Source on kvant.digital Problem Prove that if $a$, $b$, $c$, $d$, $x$, $y$, $u$, $v$ are real numbers and $abcd\gt0$, then $$\begin{gathered} (ax+bu)(av+by)(cx+dv)(cu+dy)\ge\ \ge(acuvx+bcuxy+advxy+bduvy)(acx+bcu+adv+bdy). \end{gathered}$$ V. P. Fedotov Exploration The inequality is homogeneous and symmetric in a suggestive way. Setting $$a=b=c=d=1$$ gives $$(x+u)(v+y)(x+v)(u+y)\ge (uvx+uxy+vxy+uvy)(x+u+v+y).$$ A direct expansion yields $$(x+u)(v+y)(x+v)(u+y) -(uvx+uxy+vxy+uvy)(x+u+v+y) =(xy-uv)^2.$$ This strongly suggests that in the general case the difference...

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Kvant Math Problem 292

Verified: no Verdicts: SKIP + SKIP Solve time: 8m18s Source on kvant.digital Problem The numbers from 1 to 50 are written on the board. It is allowed to erase any two numbers and write in their place a single number equal to the absolute value of their difference. After repeating this procedure several times, only one number remains on the board. What could this number be? F. G. Shleifer Exploration...

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Kvant Math Problem 290

Verified: no Verdicts: SKIP + SKIP Solve time: 8m41s Source on kvant.digital Problem For which $n$ does there exist a closed non-self-intersecting broken line consisting of $n$ segments such that any line containing one of the segments of this broken line contains at least one more of its segments? S. V. Konyagin Exploration Let the closed non-self-intersecting broken line have vertices $$A_1,A_2,\dots,A_n,$$ with sides $$A_1A_2,A_2A_3,\dots,A_nA_1.$$ The condition says that whenever...

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Kvant Math Problem 288

Verified: no Verdicts: SKIP + SKIP Solve time: 7m36s Source on kvant.digital Problem Scientists have gathered at a congress, and among them there are friends. It turned out that any two scientists who have the same number of friends at the congress have no common friends. Prove that there exists a scientist who has exactly one friend. S. V. Konyagin Exploration Model the congress by a simple graph. Vertices are...

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Kvant Math Problem 287

Verified: no Verdicts: SKIP + SKIP Solve time: 7m10s Source on kvant.digital Problem Does there exist a sequence of natural numbers such that every natural number 1, 2, 3, $\ldots$ can be represented as the difference of two numbers from this sequence in a unique way? A. Lifshitz Exploration Consider a sequence of natural numbers $a_1 < a_2 < a_3 < \dots$ such that every natural number $n$ can be...

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Kvant Math Problem 285

Verified: no Verdicts: SKIP + SKIP Solve time: 6m02s Source on kvant.digital Problem A rectangular sheet of paper of dimensions $a \times b$ has been cut into rectangular strips, each of which has one side of length 1. Prove that at least one of the numbers $a$ and $b$ is an integer. A. V. Klimov Moscow Mathematical Olympiad (1974) Exploration Consider small examples to understand the claim. A rectangle of...

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Kvant Math Problem 283

Verified: no Verdicts: SKIP + SKIP Solve time: 6m07s Source on kvant.digital Problem A convex polygon has the following property: if all its sides are shifted outward by one unit, then the resulting lines form a polygon similar to the original one. Prove that a circle can be inscribed in this polygon. Moscow Mathematical Olympiad (1974) Exploration Consider small examples of convex polygons, starting with triangles and quadrilaterals, and examine...

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Kvant Math Problem 282

Verified: no Verdicts: SKIP + SKIP Solve time: 5m56s Source on kvant.digital Problem In the cells of a rectangular table of size $m \times n$, arbitrary natural numbers are written. In one move, it is allowed to either double all the numbers in a single row or subtract one from all the numbers in a single column. Prove that, after a finite number of moves, it is possible to make...

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Kvant Math Problem 280

Verified: no Verdicts: SKIP + SKIP Solve time: 3m21s Source on kvant.digital Problem Given a triangle $ABC$ of area 1. Let $A_1$, $B_1$, and $C_1$ be the midpoints of the sides $BC$, $AC$, and $AB$ respectively. What is the minimum possible area of the intersection of the triangles $A_1B_1C_1$ and $KLM$ if the points $K$, $L$, and $M$ lie on the segments $AB_1$, $CA_1$, and $BC_1$ respectively? B. M. Ivlev...

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Kvant Math Problem 279

Verified: no Verdicts: SKIP + SKIP Solve time: 3m59s Source on kvant.digital Problem On $n$ cards arranged in a circle, numbers are written, each of which is equal to either $+1$ or $-1$. What is the smallest number of questions that is sufficient to determine with certainty the product of all $n$ numbers, if in one question it is allowed to find out the product of the numbers on any...

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Kvant Math Problem 277

Verified: no Verdicts: SKIP + SKIP Solve time: 7m19s Source on kvant.digital Problem Several red points and several blue points are given. Some pairs of points are connected by line segments. Call a point special if more than half of the points connected to it have a color different from its own. If there is at least one special point, then an arbitrary special point is chosen and recolored to...

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Kvant Math Problem 275

Verified: no Verdicts: SKIP + SKIP Solve time: 6m44s Source on kvant.digital Problem On the plane, there are $n$ vectors, each of length 1. The sum of all $n$ vectors is the zero vector. Prove that the vectors can be numbered such that for all $k = 1$, 2, $\ldots$, $n$ the following condition holds: the sum of the first $k$ vectors has length at most 3. Prove a similar...

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Kvant Math Problem 274

Verified: no Verdicts: SKIP + SKIP Solve time: 4m04s Source on kvant.digital Problem Find the smallest number of the form $|11^k - 5^l|$; $|36^k - 5^l|$; $|53^k - 37^l|$, where $k$ and $l$ are natural numbers. F. G. Schleifer All-Union Mathematical Olympiad for School Students (1974, Grades 8 and 9) Exploration We seek the smallest positive value attained by the given differences. For the first expression, compute a few values:...

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Kvant Math Problem 272

Verified: no Verdicts: SKIP + SKIP Solve time: 7m25s Source on kvant.digital Problem Two circles with radii $R$ and $r$ are given, externally tangent to each other. Various trapezoids $ABCD$ are constructed so that each circle touches both lateral sides and one of the bases of the trapezoid. Find the minimal possible length of the lateral side $AB$. E. V. Sallinen All-Union Mathematical Olympiad for School Students (1974, 9th grade)...

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Kvant Math Problem 270

Verified: no Verdicts: SKIP + SKIP Solve time: 7m37s Source on kvant.digital Problem Let $AB$ and $CD$ be two chords of a circle, and let the points $K$ and $H$ be constructed so that all four angles $KAB$, $KCD$, $HBA$, and $HDC$ are right angles. Prove that the line $KH$ passes through the center of the circle and through the point of intersection of the lines $AD$ and $BC$. I....

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Kvant Math Problem 269

Verified: no Verdicts: SKIP + SKIP Solve time: 5m15s Source on kvant.digital Problem Let $T_k(n)$ denote the sum of products of $k$ numbers from 1 to $n$. For example, $$T_2(4) = 1 \cdot 2 + 1 \cdot 3 + 1 \cdot 4 + 2 \cdot 4 + 3 \cdot 4 = 35.$$ Find a general formula for $T_2(n)$ and $T_3(n)$. Prove that $T_k(n)$ can be expressed as a polynomial in...

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Kvant Math Problem 267

Verified: no Verdicts: SKIP + SKIP Solve time: 6m37s Source on kvant.digital Problem In the sequence of triples of integers (2, 3, 5), (6, 15, 10), $\ldots$ each triple is obtained from the previous one as follows: the first number is multiplied by the second, the second by the third, and the third by the first, and the resulting products form the new triple. Prove that none of the numbers...

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Kvant Math Problem 265

Verified: no Verdicts: SKIP + SKIP Solve time: 6m15s Source on kvant.digital Problem The diagonal $d$ of a rectangular parallelepiped forms angles $\alpha$, $\beta$, and $\gamma$ with its edges $a$, $b$, and $c$. Prove that $\alpha + \beta + \gamma$ is less than $\pi$. M. L. Gerver Exploration Consider a rectangular parallelepiped with edges of length $a$, $b$, and $c$. Its main diagonal $d$ has length $\sqrt{a^2 + b^2 +...

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Kvant Math Problem 264

Verified: no Verdicts: SKIP + SKIP Solve time: 3m40s Source on kvant.digital Problem Fig. 3 Fig. 4 A city has one blue square and $n$ green squares. Each green square is connected by streets to the blue square and to two green squares (Fig. 3). On each of the $2n$ streets, one-way traffic was introduced in such a way that every square can be reached and can be departed from....

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Kvant Math Problem 262

Verified: no Verdicts: SKIP + SKIP Solve time: 8m52s Source on kvant.digital Problem What is the largest number of: rooks queens that can be placed on a chessboard $8 \times 8$ so that each of these pieces is attacked by no more than one of the others? V. Rabinovich, L. Rabinovich Exploration The problem asks for the maximal number of rooks or queens on an $8 \times 8$ chessboard such...

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Kvant Math Problem 260

Verified: no Verdicts: SKIP + SKIP Solve time: 6m30s Source on kvant.digital Problem A circle is divided by points $A_1$, $A_2$, $\ldots$, $A_n$ into $n$ equal parts, each colored in some color. Two arcs (with endpoints at the division points) are called equally colored if, under some rotation of the circle, one coincides entirely with the other, including the color of each segment. (For example, in the figure, the two...

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Kvant Math Problem 259

Verified: no Verdicts: SKIP + SKIP Solve time: 3m28s Source on kvant.digital Problem A polygon is circumscribed around a circle. The points where its sides touch the circle serve as the vertices of a second polygon inscribed in this circle. Prove that the product of the distances from an arbitrary point $M$ on the circle to the sides of one polygon equals the product of the distances from this point...

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Kvant Math Problem 257

Verified: no Verdicts: SKIP + SKIP Solve time: 4m06s Source on kvant.digital Problem For which natural numbers $n \ge 2$ does the inequality $$x_1^2 + x_2^2 \ldots + x_n^2 \ge p(x_1x_2 + x_2x_3 + \ldots + x_{n-1}x_n)$$ hold for all real numbers $x_1$, $x_2$, $\ldots$, $x_n$ if $p=1$; $p=\dfrac{4}{3}$; $p=\dfrac{6}{5}$? Nguyen Cong Quy (Hanoi) Exploration The inequality can be written as $$Q_n(x_1,\ldots,x_n) = \sum_{i=1}^n x_i^2 - p\sum_{i=1}^{n-1}x_ix_{i+1} \ge 0.$$ We...

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Kvant Math Problem 256

Verified: no Verdicts: SKIP + SKIP Solve time: 6m13s Source on kvant.digital Problem A polygon is circumscribed around a circle. The points where its sides touch the circle serve as the vertices of a second polygon inscribed in this circle. Prove that the product of the distances from an arbitrary point $M$ on the circle to the sides of the first polygon is equal to the product of the distances...

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Kvant Math Problem 254

Verified: no Verdicts: SKIP + SKIP Solve time: 7m16s Source on kvant.digital Problem Compute the value of $\sqrt{0{,}\smash{\underbrace{11111\ldots11111}_{100}}}\vphantom{\underbrace0_0}$ to an accuracy of 100 digits after the decimal point; 101 digits after the decimal point; 200 digits after the decimal point. A. A. Egorov Exploration Consider small cases of numbers of the form $0.1$, $0.11$, $0.111$, $0.1111$, and so on. Denote by $S_n$ the number $0.\underbrace{111\ldots1}_{n}$ with $n$ ones after the...

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Kvant Math Problem 252

Verified: no Verdicts: SKIP + SKIP Solve time: 7m14s Source on kvant.digital Problem On the plane lies a regular octagon with side $a$. It is allowed to "roll" the octagon across the plane, flipping it (symmetrically reflecting) over any of its sides. Prove that for any point $M$ on the plane and any $\varepsilon\gt0$, the octagon can be rolled into a position such that its center is at a distance...

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Kvant Math Problem 250

Verified: no Verdicts: SKIP + SKIP Solve time: 7m48s Source on kvant.digital Problem At the court of King Arthur, $n$ knights have gathered. Some of them are enemies of one another, but each knight has at least $\dfrac{n}{2}$ friends among those present. Prove that Merlin, King Arthur’s adviser, can seat the knights around a round table so that each knight is seated next to his friends. Prove that if each...

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Kvant Math Problem 249

Verified: no Verdicts: SKIP + SKIP Solve time: 6m06s Source on kvant.digital Problem On the edges $A'D'$ and $C'D'$ of the cube $ABCDA'B'C'D'$, choose two points $K$ and $M$ so that the plane $KDM$ is tangent to the sphere inscribed in the cube (Fig. 3). Prove that the measure $\phi$ of the dihedral angle along the edge $B'D$ of the tetrahedron $B'DKM$ does not depend on the choice of the...

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Kvant Math Problem 247

Verified: no Verdicts: SKIP + SKIP Solve time: 6m15s Source on kvant.digital Problem The square $6 \times 6$ needs to be filled with 12 tiles, of which $k$ are L-shaped, and the remaining $12-k$ are rectangular (see Fig. 1). For which $k$ is this possible? Figure 1 E. Turkevich Exploration A $6 \times 6$ square contains $36$ unit squares. We are to tile it with $12$ pieces, each covering $3$...

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Kvant Math Problem 245

Verified: no Verdicts: SKIP + SKIP Solve time: 6m13s Source on kvant.digital Problem It is proposed to construct $N$ points in the plane so that all pairwise distances between them are equal to prescribed numbers: for every two points $M_i$, $M_j$, the distance $|M_iM_j|=r_{ij}$ is given (here $i$ and $j$ are any numbers from 1 to $N$). Is the construction always possible if the distances $r_{ij}$ are specified in such...

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Kvant Math Problem 244

Verified: no Verdicts: SKIP + SKIP Solve time: 6m36s Source on kvant.digital Problem Two sets of $n$ real numbers are given: $a_1$, $a_2$, $\ldots$, $a_n$ and $b_1$, $b_2$, $\ldots$, $b_n$. Prove that if at least one of the following two conditions holds: $a_i\lt a_j$ implies $b_i\le b_j$; $a_i\lt a\lt a_j$, where $a=\dfrac{a_1+a_2+\ldots+a_n}n$, implies $b_i\le b_j$; then the inequality $$n(a_1b_1+a_2b_2+\ldots+a_nb_n)\ge(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n).$$ holds. A. Grigoryan, 10th-grade student (Baku) Exploration The desired inequality can...

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Kvant Math Problem 242

Verified: no Verdicts: SKIP + SKIP Solve time: 4m55s Source on kvant.digital Problem Let $A_iH_i$ be the altitude and $A_iM_i$ the median drawn from vertex $A_i$ of an acute-angled triangle $A_1A_2A_3$ ($i=1$, 2, 3). Prove that one of the three products $|H_1M_1|\cdot|A_2A_3|$, $|H_2M_2|\cdot|A_3A_1|$, $|H_3M_3|\cdot|A_1A_2|$ equals the sum of the other two. Is this statement also true for right-angled and obtuse-angled triangles? S. Salnikov, 10th-grade student (Mary) Exploration Denote the sides...

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Kvant Math Problem 239

Verified: no Verdicts: SKIP + SKIP Solve time: 6m12s Source on kvant.digital Problem On the plane, two points $A$ and $B$ are given. Let $C$ be a point equidistant from $A$ and $B$. Construct a sequence of points $C_1=C$, $C_2$, $C_3$, $\ldots$, $C_n$, $C_{n+1}$, $\ldots$ where $C_{n+1}$ is the circumcenter of triangle $AC_nB$. For which position of the point $C$ does the point $C_n$ fall at the midpoint of segment...

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Kvant Math Problem 238

Verified: no Verdicts: SKIP + SKIP Solve time: 4m39s Source on kvant.digital Problem Prove that $$C_n^1+C_n^3\cdot1973+C_n^5\cdot1973^2+C_n^7\cdot1973^3+\ldots$$ is divisible by $2^{n-1}$. (Here $C_n^k$ are the coefficients of the polynomial $$(a+b)^n=\textstyle\sum\limits_{k=0}^n C_n^k a^k b^{n-k}.\Big)$$ F. G. Shleifer Exploration Let $$S=C_n^1+C_n^3\cdot1973+C_n^5\cdot1973^2+\cdots.$$ The exponent of $1973$ increases by $1$ whenever the upper index of the binomial coefficient increases by $2$. Thus $$S=\sum_{m\ge0} C_n^{2m+1}1973^m.$$ The appearance of only odd binomial coefficients suggests separating the odd...

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Kvant Math Problem 236

Verified: no Verdicts: SKIP + SKIP Solve time: 7m00s Source on kvant.digital Problem There are 51 two-digit numbers. Prove that from these numbers one can select at least 6 numbers such that no two of the selected numbers have the same digit in any position. Natural numbers $k$ and $n$, $1\lt k\lt n$ are given. For what smallest $m$ is the following statement true: for any arrangement of $m$ rooks...

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Kvant Math Problem 235

Verified: no Verdicts: SKIP + SKIP Solve time: 6m02s Source on kvant.digital Problem A lion runs around the arena of a circular circus of radius 10 m. Moving along a polygonal path, it has run 30 km. Prove that the sum of all the angles through which it turned (Fig. 3) is at least 2998 radians. Fig. 3 I. N. Bernstein Exploration Consider a lion moving along a polygonal path...

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Kvant Math Problem 233

Verified: no Verdicts: SKIP + SKIP Solve time: 4m45s Source on kvant.digital Problem At the ends of a segment, two ones are written. In the middle between them, their sum — the number 2 — is written. Then, in the middle between each pair of neighboring written numbers, their sum is written again, and so on — 1973 times (Fig. 1). How many times will the number 1973 be written?...

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Kvant Math Problem 231

Verified: no Verdicts: SKIP + SKIP Solve time: 7m59s Source on kvant.digital Problem Find all solutions in natural numbers of the equation $$n^x+n^y=n^z.$$ R. Egoryan, 9th-grade student Exploration Consider the equation $n^x + n^y = n^z$ in natural numbers. First, observe symmetry in $x$ and $y$, allowing us to assume without loss of generality that $x \le y$. Test small values of $n$. For $n=2$, one sees that $2^1 +...

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Kvant Math Problem 230

Verified: no Verdicts: SKIP + SKIP Solve time: 6m41s Source on kvant.digital Problem Prove that from any convex equilateral (but not necessarily regular) pentagon one can cut out an equilateral triangle, one of whose sides coincides with a side of the pentagon (Fig. 2). Figure 2 S. V. Konyagin Exploration Let the side length of the equilateral pentagon be $1$, and let its consecutive vertices be $A_1,A_2,A_3,A_4,A_5$. The statement asks...

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Kvant Math Problem 228

Verified: no Verdicts: SKIP + SKIP Solve time: 4m35s Source on kvant.digital Problem A sheet of squared paper of size $n\times n$ cells was colored using $n$ colors (each cell was either colored with one of these colors or left uncolored). A coloring is called proper if no two cells of the same color occur in any row or in any column. Is it always possible to complete the coloring...

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Kvant Math Problem 226

Verified: no Verdicts: SKIP + SKIP Solve time: 6m11s Source on kvant.digital Problem Three grasshoppers are located at three vertices of a square. They are playing leapfrog. If grasshopper $A$ jumps over grasshopper $B$, then after the jump it ends up at the same distance from $B$ (but, naturally, on the other side and on the same line; Fig. 1). Can one of the grasshoppers reach the fourth vertex of...

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Kvant Math Problem 225

Verified: no Verdicts: SKIP + SKIP Solve time: 6m12s Source on kvant.digital Problem The faces of a die are numbered 1, 2, 3, 4, 5, 6 so that the numbers on opposite faces sum to 7. The die is rolled from the lower left corner to the upper right corner of a chessboard of size $50\times50$ squares (each square of the board is equal in size to a face of...

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Kvant Math Problem 223

Verified: no Verdicts: SKIP + SKIP Solve time: 4m03s Source on kvant.digital Problem A natural number is called perfect if it is equal to the sum of all its divisors except the number itself. (For example, 28 is a perfect number: $28=1+2+4+7+14$.) Prove that a perfect number cannot be a perfect square. A. Makarichev, 9th-grade student Exploration Perfect numbers are rare and highly structured. The first few examples are $6$,...

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Kvant Math Problem 221

Verified: no Verdicts: SKIP + SKIP Solve time: 8m51s Source on kvant.digital Problem A blot was placed on a sheet of paper. For each point of the blot, the smallest distance to the boundary of the blot and the largest distance to the boundary of the blot were determined (the figure shows the largest and smallest distances to the boundary for the point $K$). Among all the smallest distances, the...

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Kvant Math Problem 220

Verified: no Verdicts: SKIP + SKIP Solve time: 4m04s Source on kvant.digital Problem A king traversed a chessboard $8\times8$, visiting each square exactly once and returning to the starting square on the final move. (The king moves according to the usual rules: in one move, it can go to any adjacent square horizontally, vertically, or diagonally.) When its path was drawn by consecutively connecting the centers of the squares it...

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Kvant Math Problem 218

Verified: no Verdicts: SKIP + SKIP Solve time: 4m56s Source on kvant.digital Problem Prove that if $x_1$, $x_2$, $x_3$, $x_4$, $x_5$ are positive numbers, then $$(x_1 + x_2 + x_3 + x_4 + x_5)^2 \ge 4(x_1x_2 + x_2x_3 + x_3x_4 + x_4x_5 + x_5x_1).$$ B. D. Ginzburg All-Union Mathematical Olympiad for School Students (VII, Grade 10) Exploration We are asked to compare the square of a sum of five positive...

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Kvant Math Problem 217

Verified: no Verdicts: SKIP + SKIP Solve time: 5m54s Source on kvant.digital Problem A convex $n$-gon with pairwise nonparallel sides and a point inside it are given. Prove that no more than $n$ lines can be drawn through this point, each of which divides the area of the $n$-gon into two equal parts. E. V. Sallinen All-Union Mathematical Olympiad for School Students (VII, Grade 9) Exploration Consider first a triangle,...

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Kvant Math Problem 215

Verified: no Verdicts: SKIP + SKIP Solve time: 6m16s Source on kvant.digital Problem On an infinite square-grid sheet of white paper, $n$ cells are colored black. At times $t = 1$, 2, $\ldots$, all cells of the sheet are recolored simultaneously according to the following rule: each cell $k$ acquires the color that the majority of the following three cells had at the previous moment: the cell $k$ itself and...

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Kvant Math Problem 213

Verified: no Verdicts: SKIP + SKIP Solve time: 6m31s Source on kvant.digital Problem Given an angle with vertex $O$ and a circle tangent to its sides at points $A$ and $B$. From point $A$, a ray is drawn parallel to $OB$, intersecting the circle at point $C$. Segment $OC$ intersects the circle at point $E$, and the lines $AE$ and $OB$ intersect at point $K$. Prove that $OK = KB$....

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Kvant Math Problem 212

Verified: no Verdicts: SKIP + SKIP Solve time: 5m03s Source on kvant.digital Problem At a trial, 14 coins are presented as physical evidence. An expert has determined that seven of them are counterfeit and the remaining seven are genuine, and moreover has identified exactly which coins are counterfeit and which are genuine. The court knows only that all counterfeit coins have the same weight, all genuine coins have the same...

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Kvant Math Problem 210

Verified: no Verdicts: SKIP + SKIP Solve time: 6m21s Source on kvant.digital Problem Consider sequences consisting of 3000 digits, each either 1 or 2. In such a sequence, it is allowed to swap any two adjacent triples of digits. Two sequences are called equivalent if one can be transformed into the other by a finite number of such swaps. How many inequivalent sequences exist in total? G. A. Gurevich Exploration...

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Kvant Math Problem 208

Verified: no Verdicts: SKIP + SKIP Solve time: 7m05s Source on kvant.digital Problem It is known that the difference between the largest and the smallest of the real numbers $$x_1, \ x_2, \ x_3, \ \ldots, \ x_{10}$$ is equal to 1. What can be the greatest, least possible difference between the largest and the smallest of the 10 numbers $$x_1, \ \frac{x_1 + x_2}{2}, \ \frac{x_1 + x_2 +...

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Kvant Math Problem 207

Verified: no Verdicts: SKIP + SKIP Solve time: 6m06s Source on kvant.digital Problem Given two triangles $A_1 A_2 A_3$ and $B_1 B_2 B_3$. Construct around triangle $A_1 A_2 A_3$ a triangle $M_1 M_2 M_3$ of maximum area similar to triangle $B_1 B_2 B_3$ (with vertex $A_1$ lying on line $M_2 M_3$, vertex $A_2$ on line $M_3 M_1$, and vertex $A_3$ on line $M_1 M_2$). N. D. Nagaev Exploration Consider the...

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Kvant Math Problem 205

Verified: no Verdicts: SKIP + SKIP Solve time: 8m34s Source on kvant.digital Problem 24 students solved 25 problems. The instructor has a table $24 \times 25$ recording which problems each student solved. It turned out that every problem was solved by at least one student. Prove that it is possible to mark some problems with a “check” so that each student solved an even number (possibly zero) of the marked...

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Kvant Math Problem 204

Verified: no Verdicts: SKIP + SKIP Solve time: 3m23s Source on kvant.digital Problem Let us call a positive integer good if the digits 1973 occur consecutively in its decimal representation, and bad otherwise. (For example, the number $197 \ 639 \ 917$ is bad, while $116 \ 519 \ 732$ is good.) Prove that there exists a positive integer $n$ such that among all $n$-digit numbers (from $10^{n-1}$ to $10^{n}-1$),...

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Kvant Math Problem 202

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem Prove that from the sequence $$a,\ a + d,\ a + 2d,\ \ldots,\ a + nd,\ \ldots,$$ which is an infinite arithmetic progression $(d \neq 0)$, one can select a subsequence forming an infinite geometric progression if and only if the ratio $a/d$ is rational. N. B. Vasilyev Exploration Let the arithmetic progression be $$a,a+d,a+2d,\ldots ,$$ with...

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Kvant Math Problem 200

Verified: no Verdicts: SKIP + SKIP Solve time: 5m24s Source on kvant.digital Problem a) Figure 3 shows six points, with three points lying on each of four lines. Prove that there are 24 different ways to map this set of six points onto itself so that every triple of points lying on one line is mapped to a triple of points that also lie on one line. Figure 3 b)...

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Kvant Math Problem 199

Verified: no Verdicts: SKIP + SKIP Solve time: 6m06s Source on kvant.digital Problem Prove that the sum $$C_n^0-C_{n-1}^1\cdot\dfrac14+C_{n-2}^2\cdot\dfrac1{4^2}-\ldots+(-1)^i C_{n-i}^i\cdot\frac1{4^i}+\ldots$$ $\Big($the sum is taken over all integers $i$, $0\le i\le\dfrac n2\Big)$ is equal to $\dfrac{n+1}{2^n}$. Prove that if $p$ and $q$ are distinct numbers and $p+q=1$, then the sum $$C_n^0-C_{n-1}^1pq+C_{n-2}^2p^2q^2-\ldots+(-1)^i C_{n-i}^i p^iq^i+\ldots,$$ analogous to the previous one is equal to $\dfrac{p^{n+1}-q^{n+1}}{p-q}$ for arbitrary $n$. Here $C_n^k$ are binomial coefficients, i.e., $C_n^0=1$...

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Kvant Math Problem 197

Verified: no Verdicts: SKIP + SKIP Solve time: 6m07s Source on kvant.digital Problem In a rectangular table with $m$ rows and $n$ columns, $mn$ arbitrary positive numbers are written. Let us compute the product of the numbers in each column and then the sum $S$ of all $n$ such products. Prove that if the numbers in each row are rearranged in ascending order, the sum $S$ of the new table...

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Kvant Math Problem 195

Verified: no Verdicts: SKIP + SKIP Solve time: 6m46s Source on kvant.digital Problem Given a triangle $ABC$. How many points $D$ exist such that the perimeters of the quadrilaterals $ADBC$, $ABDC$, and $ABCD$ are equal? M. L. Gerver Exploration Let $$P_1=\operatorname{per}(ADBC),\qquad P_2=\operatorname{per}(ABDC),\qquad P_3=\operatorname{per}(ABCD).$$ Writing the perimeters explicitly, $$P_1=AD+DB+BC+CA,$$ $$P_2=AB+BD+DC+CA,$$ $$P_3=AB+BC+CD+DA.$$ The condition is $$P_1=P_2=P_3.$$ Subtracting pairwise gives linear relations among the distances from $D$ to the vertices. From $P_1=P_2$, $$AD+BC=AB+DC.$$...

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Kvant Math Problem 193

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem Prove that the sum of the areas of the five triangles formed by pairs of sides and diagonals of a convex pentagon (see Fig. 1) is greater than the area of the entire pentagon. Figure 1 N. B. Vasiliev Exploration Consider a convex pentagon $ABCDE$ with vertices labeled consecutively. The problem involves the five triangles formed by...

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Kvant Math Problem 192

Verified: no Verdicts: SKIP + SKIP Solve time: 7m55s Source on kvant.digital Problem The numbers 1, 2, 3, ..., 1000 are given. Find the greatest $m$ with the following property: no matter which $m$ of the given numbers are crossed out, among the remaining $1000-m$ numbers there will be two such that one is divisible by the other. Exploration Consider small analogues first. For numbers $1$ through $8$, we want...

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Kvant Math Problem 190

Verified: no Verdicts: SKIP + SKIP Solve time: 7m00s Source on kvant.digital Problem On the plane, two lines $a$ and $b$ are given. At point $A_1$, located on line $a$ at a distance less than 1 from line $b$, a flea sits. Then the flea sequentially jumps to points $B_1$, $A_2$, $B_2$, $A_3$, $B_3$, $\ldots$, following these rules (Fig. 2): points $A_1$, $A_2$, $A_3$, $\ldots$ lie on line $a$, points...

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Kvant Math Problem 188

Verified: no Verdicts: SKIP + SKIP Solve time: 5m32s Source on kvant.digital Problem Air service has been established between some of $2n$ cities, and each city is connected to at least $n$ others (by non-stop flights). Prove that even if any $n-1$ flights are canceled, it will still be possible to travel by air (with transfers) from any city to any other. Determine all cases in which this “connectivity” is...

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Kvant Math Problem 186

Verified: no Verdicts: SKIP + SKIP Solve time: 6m20s Source on kvant.digital Problem Find all integer solutions, different from 1, of the equation $$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 1$$. V. Slepchenko, 10th-grade student Exploration We seek all integer triples $(x,y,z)$, none equal to $1$, satisfying $$\frac1x+\frac1y+\frac1z=1.$$ Since the equation is symmetric, we may assume $$|x|\le |y|\le |z|.$$ First consider whether any variable can be negative. If two are negative...

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Kvant Math Problem 185

Verified: no Verdicts: SKIP + SKIP Solve time: 7m10s Source on kvant.digital Problem On a coat of area 1, there are 5 patches, each of area at least $\dfrac{1}{2}$. Prove that there exist two patches whose overlap has area at least $\dfrac{1}{5}$. E. B. Dynkin Exploration Consider a coat of area $1$ and five patches, each of area at least $\frac{1}{2}$. The sum of the areas of all patches is...

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Kvant Math Problem 183

Verified: no Verdicts: SKIP + SKIP Solve time: 5m41s Source on kvant.digital Problem Find the height of a trapezoid whose bases are $a$ and $b$ ($a \lt b$), the angle between its diagonals is $90^{\circ}$, and the angle between the extensions of its lateral sides is $45^{\circ}$. Exploration Let the trapezoid have bases of lengths $b$ and $a$, with $a<b$. Put the longer base on the $x$-axis. Let the lower...

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Kvant Math Problem 182

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem Prove that if $a >0$, $b>0$, and $c>0$, then $\dfrac{a}{b+c} + \dfrac{b}{c+a} + \dfrac{c}{a+b} \ge \dfrac{3}{2}$. 2. $a >0$, $b>0$, $c>0$, and $d>0$, then $\dfrac{a}{b+c+d} + \dfrac{b}{a+c+d} + \dfrac{c}{a+b+d} + \dfrac{d}{a+b+c} \ge \dfrac{4}{3}$. 3. $a_1$, $a_2$, $a_3$, ..., $a_n$ are positive numbers ($n \ge 2$), then $\dfrac{a_1}{a_2+a_3+...+a_n} + \dfrac{a_2}{a_1+a_3+...+a_n} + ... + \dfrac{a_n}{a_1+a_2+...+a_{n-1}} \ge \dfrac{n}{n-1}$. Exploration Begin...

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Kvant Math Problem 180

Verified: no Verdicts: SKIP + SKIP Solve time: 4m38s Source on kvant.digital Problem Two people play the following game. One thinks of a positive integer $n$, and the other asks questions of the form “is it true that $n \ge x$?” (he may choose $x$ at his discretion) and receives the answers “yes” or “no.” To each possible strategy $T$ of the second player, associate a function $f_T(n)$ equal to...

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Kvant Math Problem 620

Verified: no Verdicts: SKIP + SKIP Solve time: 9m40s Source on kvant.digital Problem Let $x_1$, $x_2$, $\ldots$, $x_n$ be real numbers such that $x_1^2+x_2^2+\ldots+x_n^2=1$. Prove that the sum of the absolute values of $2^n$ numbers $$\pm x_1 \pm x_2 \pm \ldots \pm x_n$$ (with all possible combinations of the signs “$+$” and “$-$”) does not exceed $2^n$. I. Kasakovskikh Exploration Consider small values of $n$ first. For $n=1$, the numbers...

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Kvant Math Problem 621

Verified: no Verdicts: SKIP + SKIP Solve time: 5m31s Source on kvant.digital Problem A $n$-gon is circumscribed about a circle. An arbitrary point $P$ inside the circle is joined to all its vertices and points of tangency. The resulting $2n$ triangles are colored alternately red and blue. Prove that the product of the areas of the red triangles is equal to the product of the areas of the blue triangles....

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Kvant Math Problem 179

Verified: no Verdicts: SKIP + SKIP Solve time: 8m24s Source on kvant.digital Problem For each non-right triangle $T$, let $T_1=H(T)$ denote the triangle whose vertices are the feet of the altitudes of triangle $T_1$; let $T_2=H(T_1)$ denote the triangle whose vertices are the feet of the altitudes of triangle $T_1$; further, let $T_3=H(T_2)$; $T_4=H(T_3)$, $\ldots$ What must the angles of triangle $T$ be so that the triangle $H(T)$ is acute?...

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Kvant Math Problem 619

Verified: no Verdicts: SKIP + SKIP Solve time: 6m58s Source on kvant.digital Problem Prove that if, for the cyclic quadrilateral $ABCD$, the equality $|CD|=|AD|+|BC|$ holds, then the bisectors of its angles $A$ and $B$ intersect on the side $CD$. I. F. Sharygin Exploration Let the bisectors of $\angle A$ and $\angle B$ meet at a point $P$. Since the quadrilateral is cyclic, opposite angles satisfy $$\angle A+\angle C=180^\circ,\qquad \angle B+\angle...

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Kvant Math Problem 618

Verified: no Verdicts: SKIP + SKIP Solve time: 3m22s Source on kvant.digital Problem Prove that there exist infinitely many natural numbers $n$ such that $n!$ is divisible by $n^2+1$. Prove that for any $\alpha \gt 0$ there exist infinitely many natural numbers $n$ such that $[\alpha n]!$ is divisible by $n^2+1$. (Here $m!=1\cdot 2\cdot \ldots \cdot m$; $[x]$ is the integer part of the number $x$.) A. Sivatsky, 10th grade...

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Kvant Math Problem 554

Verified: no Verdicts: SKIP + SKIP Solve time: 6m02s Source on kvant.digital Problem Call a natural number $n$ good if there exist natural numbers $a_1$, $a_2$, $\ldots$, $a_k$ (not necessarily distinct) such that $$a_1+a_2+\ldots+a_k=n$$ and $$\dfrac1{a_1}+\dfrac1{a_2}+\ldots+\dfrac1{a_k}=1.$$ It is known that all numbers between 33 and 73 are good. Prove that all numbers greater than 73 are also good. USA Mathematical Olympiad (1978) Exploration Consider small examples of natural numbers and...

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Kvant Math Problem 555

Verified: no Verdicts: SKIP + SKIP Solve time: 3m01s Source on kvant.digital Problem Consider the intersection of two; three cylinders of equal radius $r$ whose axes are mutually perpendicular and pass through a single point. How many planes of symmetry does this intersection have? What is its volume? S. Pukhov Exploration Consider first the intersection of two cylinders of equal radius $r$ with axes perpendicular. Label the axes $x$ and...

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Kvant Math Problem 550

Verified: no Verdicts: SKIP + SKIP Solve time: 9m40s Source on kvant.digital Problem From point $A$ to point $B$, which are $d$ km apart, $n$ cyclists must travel. They have $m$ bicycles available. Each person can walk at a speed of $u~\text{км/ч}$ or ride a bicycle at a speed of $v~\text{км/ч}$. What is the minimum time in which all $n$ cyclists can get from $A$ to $B$? (The time is...

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Kvant Math Problem 177

Verified: no Verdicts: SKIP + SKIP Solve time: 7m44s Source on kvant.digital Problem Find all solutions of the equation $$\sqrt[\scriptstyle n~]{x^n-a^n}+\sqrt[\scriptstyle n~]{2a^n-x^n}=a,$$, where $a$ is a given real number, and $n$ is a natural number greater than one. T. Temirov Exploration Consider the equation $$\sqrt[n]{x^n - a^n} + \sqrt[n]{2a^n - x^n} = a,$$ with $a \in \mathbb{R}$ and $n > 1$ a natural number. The structure suggests symmetry around $x...

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Kvant Math Problem 551

Verified: no Verdicts: SKIP + SKIP Solve time: 6m05s Source on kvant.digital Problem What is the smallest number of points that is sufficient to mark inside a convex pentagon so that every triangle whose vertices are vertices of the pentagon contains at least one marked point? The same question for a convex $n$-gon. Moscow Mathematical Olympiad Exploration Consider the case of a triangle first. No points are needed inside a...

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Kvant Math Problem 552

Verified: no Verdicts: SKIP + SKIP Solve time: 4m35s Source on kvant.digital Problem Find at least one pair $(p, q)$ of nonzero integers for which the trinomials $x^2+px+q$ and $x^2+qx+p$ have integer roots. Find all such pairs $(p,q)$. E. Turkevich Exploration Let the roots of $$x^2+px+q$$ be integers $a,b$. Then $$a+b=-p,\qquad ab=q.$$ Likewise, if the roots of $$x^2+qx+p$$ are integers $c,d$, then $$c+d=-q,\qquad cd=p.$$ Substituting $p=cd$ and $q=ab$ into the...

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Kvant Math Problem 547

Verified: no Verdicts: SKIP + SKIP Solve time: 10m01s Source on kvant.digital Problem For the equation $\dfrac1x-\dfrac1y=\dfrac1n$, where $n$ is a positive integer, to have a unique solution in positive integers $x$, $y$, it is necessary and sufficient that $n$ be prime. Prove this. A. Danielyan Exploration The equation is $$\frac1x-\frac1y=\frac1n,$$ with $x,y,n\in\mathbb Z_{>0}$. First rewrite it: $$\frac{y-x}{xy}=\frac1n,$$ hence $$n(y-x)=xy.$$ Moving all terms to one side gives $$xy+nx-ny=0.$$ Adding and...

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Kvant Math Problem 175

Verified: no Verdicts: SKIP + SKIP Solve time: 6m24s Source on kvant.digital Problem Each side of an equilateral triangle is divided into $m$ equal parts, and through the division points lines are drawn parallel to the sides, cutting the triangle into $m^2$ small triangles. Among the vertices of the resulting triangles, one must select $N$ vertices so that for no two selected vertices $A$ and $B$ is the segment $AB$...

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Kvant Math Problem 545

Verified: no Verdicts: SKIP + SKIP Solve time: 11m13s Source on kvant.digital Problem On the plane, $n$ points are given. It is required to place $n$ spotlights at these points, each illuminating an angle of $360^\circ /n$, so as to illuminate the entire plane. Prove that this is possible for any arrangement of the given points if a) $n=3$; b) $n=4$; c) $n$ — any natural number. d) Now let...

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Kvant Math Problem 548

Verified: no Verdicts: SKIP + SKIP Solve time: 4m47s Source on kvant.digital Problem Four points are placed on a circle. Through the midpoint of the chord joining two of them, draw a line perpendicular to the chord joining the other two points. (Such a line is drawn for each pair of points.) Prove that all six constructed lines pass through a single point. Five points are placed on a circle....

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Kvant Math Problem 546

Verified: no Verdicts: SKIP + SKIP Solve time: 4m53s Source on kvant.digital Problem From an arbitrary point $M$ on the circle circumscribed about a rectangle, perpendiculars $MP$ and $MQ$ were dropped to two of its opposite sides, and perpendiculars $MR$ and $MT$ were dropped to the extensions of the other two sides. Prove that the lines $PR$ and $QT$ are perpendicular to each other, and that their point of intersection...

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Kvant Math Problem 173

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem In the square array $4 \times 4$, the numbers 1, 2, 3, $\ldots$, 16 are arranged so that the sum of the four numbers in each row, in each column, and on each of the two diagonals is the same, with 1 and 16 placed in opposite corners of the array. Prove that in this “magic square”...

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Kvant Math Problem 544

Verified: no Verdicts: SKIP + SKIP Solve time: 3m11s Source on kvant.digital Problem What is the largest number of vertices of a nonconvex $n$-gon from which no diagonal (lying entirely inside the polygon) can be drawn? First solve this problem for $n=4$, 5, 6, and 7. S. N. Bychkov Exploration Consider first the case $n=4$. A quadrilateral has two possible diagonals. If the quadrilateral is convex, both diagonals lie entirely...

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Kvant Math Problem 542

Verified: no Verdicts: SKIP + SKIP Solve time: 7m06s Source on kvant.digital Problem A right triangle $A_0A_1A_2$ is given with legs $|A_0A_2|=a$ and $|A_1A_2|=b$. An ant crawls along an infinite broken line $A_2A_3A_4A_5\ldots$, where $A_nA_{n+1}$ is the altitude of triangle $A_{n-2}A_{n-1}A_n$. Find the length of its path (consisting of an infinite number of segments). Construct the limit point $L$ toward which the ant approaches. At what distances from the legs...

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Kvant Math Problem 541

Verified: no Verdicts: SKIP + SKIP Solve time: 7m08s Source on kvant.digital Problem In a company of $N$ people, each person has exactly three friends. Prove that $N$ is even. Is it always possible to divide such a company into $N/2$ pairs so that the people in each pair are friends? D. Bernstein Exploration Consider a small social network where each person has exactly three friends. For $N=4$, each person...

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Kvant Math Problem 540

Verified: no Verdicts: SKIP + SKIP Solve time: 7m02s Source on kvant.digital Problem The international society consists of representatives of six different countries. The membership list of the society contains 1978 surnames, numbered 1, 2, $\ldots$, 1978. Prove that there exists at least one member of the society whose number is equal either to the sum of the numbers of two members from his country or to twice the number...

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Kvant Math Problem 172

Verified: no Verdicts: SKIP + SKIP Solve time: 4m45s Source on kvant.digital Problem Prove that for every prime $p$, the number $$\underbrace{11 \ldots 1} {p}\underbrace{22 \ldots 2} {p}\underbrace{33 \ldots 3} {p} \ldots \underbrace{99 \ldots 9} {p} - 123456789$$ is divisible by $p$. M. L. Gerver Exploration Let $$N=\underbrace{11\ldots1} {p}\underbrace{22\ldots2} {p}\underbrace{33\ldots3} {p}\cdots \underbrace{99\ldots9} {p}.$$ The statement is that $N-123456789$ is divisible by the prime $p$. Write $$R=\underbrace{11\ldots1}_{p}=1+10+\cdots+10^{p-1}.$$ Then each block of...

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Kvant Math Problem 539

Verified: no Verdicts: SKIP + SKIP Solve time: 4m13s Source on kvant.digital Problem Let $P$ be a given point inside a given sphere, and let $A$, $B$, $C$ be any three points on this sphere such that the segments $PA$, $PB$, $PC$ are mutually perpendicular. Let $Q$ be the vertex of the parallelepiped defined by the segments $PA$, $PB$, and $PC$, diagonally opposite to $P$. Determine the locus of points...

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Kvant Math Problem 170

Verified: no Verdicts: SKIP + SKIP Solve time: 8m06s Source on kvant.digital Problem Let $M$ and $N$ be the points of tangency of the incircle of triangle $ABC$ with sides $AB$ and $AC$, and let $P$ be the intersection point of line $MN$ with the bisector of angle $B$. Prove that angle $BPC$ is right. Prove a more general fact: if a point $O$ inside triangle $ABC$ is such that...

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Kvant Math Problem 538

Verified: no Verdicts: SKIP + SKIP Solve time: 3m23s Source on kvant.digital Problem The set of all positive integers is the union of two disjoint subsets ${f(1),f(2),\ldots,f(n),\ldots}$, ${g(1),g(2),\ldots,g(n),\ldots}$, where $f(1)\lt f(2)\lt\ldots\lt f(n)\lt\ldots$, $g(1)\lt g(2)\lt\ldots\lt g(n)\lt\ldots$ and $g(n)=f(f(n))+1$ for all $n\ge1$. Determine $f(240)$. International Mathematical Olympiad for School Students (XX, 1978) Exploration Begin by examining small values to understand the recursive structure imposed by $g(n)=f(f(n))+1$. Denote the sequences ${f(n)}$ and ${g(n)}$...

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Kvant Math Problem 536

Verified: no Verdicts: SKIP + SKIP Solve time: 6m55s Source on kvant.digital Problem Prove that any rectangle of size $m \times 2n$ cells can be tiled with two layers of dominoes (tiles of size $1 \times 2$ cells) in such a way that each tile in the upper layer rests on two different tiles in the lower layer. A rectangle of size $2k \times 2n$ cells has already been tiled...

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Kvant Math Problem 535

Verified: no Verdicts: SKIP + SKIP Solve time: 7m00s Source on kvant.digital Problem Fig. 2 Suppose that on the plane there is given a system of three doubly infinite sequences of points $A_k$, $B_l$, $C_m$ (the indices $k$, $l$, and $m$ range over all integers). We shall call such a system a trigram if, for any $k$ and $m$, the line $A_kC_m$ contains one more point $B_{k+m}$ of this system....

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Kvant Math Problem 534

Verified: no Verdicts: SKIP + SKIP Solve time: 7m18s Source on kvant.digital Problem Fig. 1 Three lines, parallel to the sides of triangle $ABC$ and passing through a single point, cut trapezoids from $\triangle ABC$. The three diagonals of these trapezoids, having no common endpoints, divide the triangle into seven parts, four of which are triangles (Fig. 1). Prove that the sum of the areas of three of these triangles,...

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Kvant Math Problem 168

Verified: no Verdicts: SKIP + SKIP Solve time: 6m32s Source on kvant.digital Problem In a regular frustum of a pyramid (Fig. 1), the point $K$ is the midpoint of some side $AB$ of the upper base, and $L$ is the midpoint of some side $CD$ of the lower base. Prove that the projections of the segments $AB$ and $CD$ onto the line $KL$ are equal in length. Figure 1 G....

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Kvant Math Problem 530

Verified: no Verdicts: SKIP + SKIP Solve time: 6m59s Source on kvant.digital Problem On a rectangular sheet of graph paper, some cells are colored black. Then all cells of the sheet are recolored simultaneously according to the following rule: a cell that had an even number of black neighbors becomes white, while a cell that had an odd number of black neighbors becomes black. (Neighbors are cells that share a...

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Kvant Math Problem 531

Verified: no Verdicts: SKIP + SKIP Solve time: 4m39s Source on kvant.digital Problem A motorist and a cyclist simultaneously set out from points $A$ and $B$ toward each other. Having met at point $C$, they immediately turned around and headed back (at the same speeds as before). Upon reaching their respective points $A$ and $B$, they turned around again and met for the second time at point $D$; there they...

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Kvant Math Problem 532

Verified: no Verdicts: SKIP + SKIP Solve time: 3m44s Source on kvant.digital Problem Let $a_n=\sqrt{n+1}+\sqrt{n\vphantom2}$ and $b_n=\sqrt{4n+2}$. Prove that for every positive integer $n$: $[a_n]=[b_n]$, where $[x]$ denotes the integer part of the number $x$; $0\lt b_n-a_n\lt\dfrac1{16n\sqrt n}$. Exploration For small values, $$a_1=\sqrt2+1\approx2.414,\qquad b_1=\sqrt6\approx2.449,$$ so $[a_1]=[b_1]=2$ and $$b_1-a_1\approx0.035.$$ For $n=2$, $$a_2=\sqrt3+\sqrt2\approx3.146,\qquad b_2=\sqrt{10}\approx3.162,$$ again the integer parts coincide. To compare $a_n$ and $b_n$, compute their squares: $$a_n^2=(\sqrt{n+1}+\sqrt n)^2=2n+1+2\sqrt{n(n+1)},$$ $$b_n^2=4n+2.$$ The difference...

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Kvant Math Problem 167

Verified: no Verdicts: SKIP + SKIP Solve time: 1m16s Source on kvant.digital Problem Prove that in any arithmetic progression $a$, $a+d$, $a+2d$, $\ldots$, $a+nd$, $\ldots$ consisting of natural numbers, there exist infinitely many terms whose prime factorization involves exactly the same prime numbers. G. Pólya Exploration Consider an arithmetic progression $a$, $a+d$, $a+2d$, $\dots$, where $a$ and $d$ are natural numbers. Each term has a unique prime factorization. The claim...

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Kvant Math Problem 527

Verified: no Verdicts: SKIP + SKIP Solve time: 5m33s Source on kvant.digital Problem Let $x_1$, $x_2$, $\ldots$, $x_n$ be real numbers, $0\le x_i\le1$. Prove that the quantity $$x_1+x_2+\ldots+x_{n-1}+x_n-x_1x_2-x_2x_3-\ldots-x_{n-1}x_n-x_nx_1$$ does not exceed 1 when $n=3$; 2 when $n=4$; $\left[\dfrac n2\right]$ for any $n\ge3$. V. Batyrev, 10th-grade student Exploration Denote $$S=\sum_{i=1}^n x_i-\sum_{i=1}^n x_i x_{i+1},$$ where indices are taken modulo $n$. For $n=3$, $$S=x_1+x_2+x_3-x_1x_2-x_2x_3-x_3x_1.$$ If $x_1=x_2=x_3=\frac12$, then $S=\frac34$. If $x_1=1,x_2=0,x_3=0$, then $S=1$. Thus...

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Kvant Math Problem 528

Verified: no Verdicts: SKIP + SKIP Solve time: 4m41s Source on kvant.digital Problem On each square of a chessboard there is a chip (Fig. 1). The chips must be rearranged so that the distance between every pair of chips does not decrease compared to their distance in the initial arrangement. In how many ways can this be done? (The distance between chips is defined as the distance between the centers...

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Kvant Math Problem 529

Verified: no Verdicts: SKIP + SKIP Solve time: 2m17s Source on kvant.digital Problem The polygon $M'$ is the image of the convex polygon $M$ under a homothety with ratio $k=-\dfrac12$. Prove that there exists a translation $T$ such that $T(M')\subset M$. For which homothety ratios $k\lt0$ does an analogous statement hold for a convex polyhedron $M$ in space? N. Yu. Netsvetaev Exploration A homothety with ratio $k<0$ reverses directions through...

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Kvant Math Problem 165

Verified: no Verdicts: SKIP + SKIP Solve time: 8m21s Source on kvant.digital Problem On a circle there is a set $F$ consisting of 100 arcs. It is known that for any rotation $R$ of the circle, the set $R(F)$ has a common point with $F$. (In other words, for any $\alpha$ from $0^\circ$ to $180^\circ$, one can indicate in the set $F$ two points whose distance apart along the circle...

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Kvant Math Problem 523

Verified: no Verdicts: SKIP + SKIP Solve time: 9m54s Source on kvant.digital Problem A checker stands in a corner of an $n \times n$ chessboard. The two players take turns moving it to an adjacent square (sharing a side with the square on which the checker stands). It is forbidden to move to a square that the checker has already visited. The player who has no legal move loses. Prove...

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Kvant Math Problem 526

Verified: no Verdicts: SKIP + SKIP Solve time: 1m32s Source on kvant.digital Problem Prove that the area of a convex quadrilateral is equal to $$S=\dfrac{\tg\phi,|a^2-b^2+c^2-d^2|}4,$$ where $a$, $b$, $c$, and $d$ are the lengths of the consecutive sides of the quadrilateral, $\phi$ is the measure of the angle between its diagonals, and $\phi\ne90^\circ$ Is it possible to determine the area of $S$, knowing $a$, $b$, $c$, and $d$, if $\phi=90^\circ$?...

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Kvant Math Problem 525

Verified: no Verdicts: SKIP + SKIP Solve time: 4m59s Source on kvant.digital Problem Prove that for any tetrahedron there exist two planes such that the ratio of the areas of the projections of the tetrahedron onto these planes is at least $\sqrt{2}$. A. A. Berzinsh All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10) Exploration The area of the orthogonal projection of a polyhedron onto a plane depends on...

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Kvant Math Problem 522

Verified: no Verdicts: SKIP + SKIP Solve time: 8m01s Source on kvant.digital Problem On the plane, several non-intersecting line segments are given, no two of which lie on the same straight line. We want to draw additional segments connecting the endpoints of the given segments so that all the segments together form a single non-self-intersecting broken line. Is this always possible? V. V. Proizvolov All-Union Mathematical Olympiad for School Students...

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Kvant Math Problem 163

Verified: no Verdicts: SKIP + SKIP Solve time: 10m09s Source on kvant.digital Problem Prove that if the diagonals of a convex quadrilateral are mutually perpendicular, then the projections of their intersection point onto all four sides (or their extensions) lie on a single circle. I. A. Kushnir Exploration Let the convex quadrilateral be $ABCD$, and let its diagonals $AC$ and $BD$ intersect at $P$. Assume $AC \perp BD$. Let $E,F,G,H$...

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Kvant Math Problem 521

Verified: no Verdicts: SKIP + SKIP Solve time: 6m44s Source on kvant.digital Problem Let $a_n$ denote the integer nearest to $\sqrt{n}$. Find the sum $$\dfrac1{a_1}+\dfrac1{a_2}+\dfrac1{a_3}+\ldots+\dfrac1{a_{1980}}.$$ G. A. Gurevich All-Union Mathematical Olympiad for School Students (XII, 1978, Grades 8–9) Exploration Consider the first few values of $a_n$. For $n = 1$, $\sqrt{1} = 1$, so $a_1 = 1$. For $n = 2$, $\sqrt{2} \approx 1.414$, rounding to the nearest integer gives...

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Kvant Math Problem 520

Verified: no Verdicts: SKIP + SKIP Solve time: 5m27s Source on kvant.digital Problem Consider the sequence of numbers $x_n=(1+\sqrt{2}+\sqrt{3})^n$. Each of them can be represented in the form $$x_n=q_n+r_n\sqrt{2}+s_n\sqrt{3}+t_n\sqrt{6},$$, where $q_n$, $r_n$, $s_n$, $t_n$ are integers. Find the limits $$\lim_{n\to\infty}\dfrac{r_n}{q_n}, \quad \lim_{n\to\infty}\dfrac{s_n}{q_n}, \quad \lim_{n\to\infty}\dfrac{t_n}{q_n}.$$ I. N. Bernstein All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10) Exploration Consider the sequence $x_n=(1+\sqrt{2}+\sqrt{3})^n$. Expanding this using the binomial or multinomial theorem...

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Kvant Math Problem 519

Verified: no Verdicts: SKIP + SKIP Solve time: 6m20s Source on kvant.digital Problem Two piles of matches are given. Initially one pile contains $m$ matches, the other contains $n$ matches, with $m \gt n$. Two players take turns removing matches from a pile. In a single move, a player removes from one pile any nonzero number of matches that is a multiple of the number of matches in the other...

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Kvant Math Problem 518

Verified: no Verdicts: SKIP + SKIP Solve time: 4m19s Source on kvant.digital Problem The numbers $x_1$, $x_2$, $\ldots$, $x_n$ belong to the interval $[a;b]$, where $0\lt a\lt b$. Prove the inequality $$(x_1+x_2+\ldots+x_n)\left(\dfrac1{x_1}+\dfrac1{x_2}+\ldots+\dfrac1{x_n}\right)\le \dfrac{(a+b)^2}{4ab},n^2.$$. S. V. Fomin All-Union School Mathematics Olympiad (XII, 1978, 9th grade) Exploration Consider the inequality $$(x_1+\dots+x_n)\left(\frac{1}{x_1}+\dots+\frac{1}{x_n}\right)\le \frac{(a+b)^2}{4ab} n^2$$ for numbers $x_i\in[a,b]$, with $0<a<b$. For $n=2$, we have $(x_1+x_2)\left(\frac1{x_1}+\frac1{x_2}\right)=2+\frac{x_1}{x_2}+\frac{x_2}{x_1}$. The expression $\frac{x_1}{x_2}+\frac{x_2}{x_1}$ is maximal when $x_1=a$ and $x_2=b$,...

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Kvant Math Problem 514

Verified: no Verdicts: SKIP + SKIP Solve time: 6m35s Source on kvant.digital Problem Prove that there exists an infinite bounded sequence ${x_n}$ such that for any distinct $m$ and $k$ the inequality $$|x_m-x_k|\ge|m-k|^{-1}.$$ holds. S. V. Konyagin All-Union 12th School Mathematics Olympiad, 1978, grades 9–10 Exploration We seek an infinite bounded sequence $(x_n)$ such that every two distinct terms satisfy $$|x_m-x_k|\ge \frac1{|m-k|}.$$ The condition depends only on the difference of...

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Kvant Math Problem 161

Verified: no Verdicts: SKIP + SKIP Solve time: 6m36s Source on kvant.digital Problem A lake has the shape of a non-convex $n$-gon. Prove that the set of points in the lake from which all its shores are visible is either empty or fills the interior of a convex $m$-gon, where $m\le n$. I. N. Bernstein Exploration Let the lake be the interior of a simple nonconvex polygon $P=A_1A_2\cdots A_n$. A...

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Kvant Math Problem 516

Verified: no Verdicts: SKIP + SKIP Solve time: 2m36s Source on kvant.digital Problem Three machines print pairs of positive integers on cards. The machines operate as follows. The first machine, after reading a card $(a; b)$, produces a new card $(a+1; b+1)$; the second, after reading a card $(a; b)$, produces a card $\left(\dfrac a2; \dfrac b2\right)$ (it operates only when $a$ and $b$ are even); the third, given two...

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Kvant Math Problem 513

Verified: no Verdicts: SKIP + SKIP Solve time: 8m01s Source on kvant.digital Problem Prove that there exists a number $A$ such that at least 1978 pairwise noncongruent squares can be inscribed in the graph of the function $y = A \sin x$. (A square is called inscribed if all its vertices belong to the graph.) V. Fedotov All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10) Exploration Consider a...

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Kvant Math Problem 515

Verified: no Verdicts: SKIP + SKIP Solve time: 1m20s Source on kvant.digital Problem A finite set $K_0$ is given. To it are added all points that can be obtained by reflecting one point of this set centrally about another. The resulting set is denoted by $K_1$. Similarly, $K_2$ is obtained from the set $K_1$, $K_3$ from $K_2$, and so on. Let the set $K_0$ consist of two points $A$ and...

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Kvant Math Problem 510

Verified: no Verdicts: SKIP + SKIP Solve time: 6m43s Source on kvant.digital Problem In the book Hungarian Mathematical Olympiads the following problem (No. 148) is given: “Prove that for all positive $\alpha\lt\pi$, the inequality $$\sin\alpha+\dfrac12\sin 2\alpha+\dfrac13\sin3\alpha\gt0,\text{»}.$$ holds. Prove the following generalization of this inequality: for all $0\lt\alpha\lt\pi$ and every positive integer $n$, $$\sin\alpha+\dfrac12\sin2\alpha+\dfrac13\sin3\alpha+\ldots+\dfrac1n\sin n\alpha\gt0.$$ I. Birger, R. P. Ushakov Exploration Let $$S_n(\alpha)=\sum_{k=1}^{n}\frac{\sin k\alpha}{k}.$$ The statement claims that $S_n(\alpha)>0$ for every...

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Kvant Math Problem 160

Verified: no Verdicts: SKIP + SKIP Solve time: 1m40s Source on kvant.digital Problem When a round-robin hockey tournament ended, it turned out that for any group of teams, one could find a team (possibly from the same group) that scored an odd number of points in games against the teams in that group. Prove that the tournament involved an even number of teams. (Loss — 0 points, draw — 1...

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Kvant Math Problem 159

Verified: no Verdicts: SKIP + SKIP Solve time: 1m18s Source on kvant.digital Problem Is it possible to place the digits 0, 1, 2 in the cells of a grid sheet of size $100 \times 100$ so that in every rectangle of size $3 \times 4$, whose sides run along the grid lines, there are three zeros, four ones, and five twos? V. E. Lapitsky All-Union Mathematical Olympiad for School Students...

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Kvant Math Problem 158

Verified: no Verdicts: SKIP + SKIP Solve time: 1m20s Source on kvant.digital Problem The triangular table is constructed according to the following rule: the natural number $a \gt 1$ is written in the top row, and thereafter, under each number $k$, the number $k^2$ is written on the left and the number $k+1$ on the right. For example, when $a=2$, the following table is obtained: Add a picture to the...

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Kvant Math Problem 156

Verified: no Verdicts: SKIP + SKIP Solve time: 1m58s Source on kvant.digital Problem In the rectangle $ABCD$, the point $M$ is the midpoint of the side $AD$, and $N$ is the midpoint of the side $BC$. On the extension of the segment $DC$ beyond the point $D$, the point $P$ is taken. Let the point of intersection of the lines $PM$ and $AC$ be denoted by $Q$. Prove that $\angle...

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Kvant Math Problem 112

Verified: no Verdicts: SKIP + SKIP Solve time: 2m00s Source on kvant.digital Problem In the table $m\times n$, numbers are arranged so that for any two rows and any two columns, the sum of the numbers at two opposite corners of the rectangle they form equals the sum of the numbers at the other two corners. Some numbers have been erased, but from the remaining ones it is possible to...

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Kvant Math Problem 509

Verified: no Verdicts: SKIP + SKIP Solve time: 3m09s Source on kvant.digital Problem Solve the equations in positive integers: $$2^x+1=3^y;$$ $$z^x+1=(z+1)^2;$$ $$z^x+1=(z+1)^y.$$ D. Fleishman Exploration Equation (1), $2^x + 1 = 3^y$, suggests searching for powers of 2 that are one less than a power of 3. Small integer experimentation shows $x=1$, $y=1$ works: $2^1 + 1 = 3$. Next, $x=3$ gives $2^3 + 1 = 9 = 3^2$, which...

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Kvant Math Problem 505

Verified: no Verdicts: SKIP + SKIP Solve time: 9m24s Source on kvant.digital Problem Let $n$ material points of equal mass be placed on a line. Consider an arbitrary segment of length $2r$ containing at least one of these points, and find the center of mass $O_1$ of all points within it. Now consider a segment of length $2r$ with midpoint $O_1$ and find the center of mass $O_2$ of all...

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Kvant Math Problem 504

Verified: no Verdicts: SKIP + SKIP Solve time: 6m54s Source on kvant.digital Problem A chessboard of $n \times n$ squares is covered with $k$ tiles of size $2 \times 1$, arranged so that it is impossible to place the $(k+1)$-th tile without moving the already placed tiles. Prove that the number of remaining free squares does not exceed $\dfrac{n^2+n+1}3$; $\dfrac{n^2+2}3$; $\dfrac{n^2}3$. Is it possible, for some $n$, to obtain an...

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Kvant Math Problem 502

Verified: no Verdicts: SKIP + SKIP Solve time: 8m45s Source on kvant.digital Problem Given three parallel segments $AA_1$, $BB_1$, $CC_1$ that do not lie in a single plane. Let $M$ be the point of intersection of the planes $ABC_1$, $BCA_1$, and $CAB_1$, and let $M_1$ be the point of intersection of the planes $A_1B_1C$, $B_1C_1A$, and $C_1A_1B$. Prove that the segment $MM_1$ is parallel to the three original segments. Z....

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Kvant Math Problem 501

Verified: no Verdicts: SKIP + SKIP Solve time: 3m48s Source on kvant.digital Problem Choose from the sequence of powers of three, $$~3,~9,~27,~81,~243,~729,~2187, 6561, \ldots$$, all numbers that begin with the digit 9; let these numbers (in order) be $3^{~f(1)}$, $3^{~f(2)}$, $3^{~f(3)}$, $\ldots$ (in particular, $f(1)=2$, since the first such number is $3^2=9$). Find $f(2)$ and $f(3)$, the indices of the second and third such numbers. Prove that there are infinitely...

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Kvant Math Problem 155

Verified: no Verdicts: SKIP + SKIP Solve time: 5m56s Source on kvant.digital Problem Several squares are given whose total area is equal to 1. Prove that they can be placed without overlap inside a square of area 2. G. A. Galperin All-Union Mathematical Olympiad for School Students (1972, grades 9 and 10) Exploration Consider first a single square of area $1$. Its side length is $1$, so it can obviously...

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Kvant Math Problem 963

Verified: no Verdicts: SKIP + SKIP Solve time: 3m50s Source on kvant.digital Problem Three pairs of opposite sides of a hexagon are parallel. Prove that the segments joining their midpoints intersect at a single point. T. Gazaryan, 10th-grade student Exploration Let the hexagon be $A B C D E F$ in cyclic order. The condition says that $A B \parallel D E$, $B C \parallel E F$, and $C D...

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Kvant Math Problem 153

Verified: no Verdicts: SKIP + SKIP Solve time: 5m40s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be found in the published issue. I cannot write a solution to Kvant problem M153 without the actual problem statement or the diagram. The prompt says that the text version is unavailable and that the problem appears only in graphical form. Since...

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Kvant Math Problem 154

Verified: no Verdicts: SKIP + SKIP Solve time: 5m40s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I can produce the full Kvant-style solution structure for problem M154. Before I start, I need the text version of the problem statement , because the graphical version alone is insufficient to construct a rigorous, self-contained...

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Kvant Math Problem 152

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem Let $a$, $b$, $m$, $n$ be positive integers, with $a$ relatively prime to $b$ and $a\ge1$. Prove that if $a^m + b^m$ is divisible by $a^n + b^n$, then $m$ is divisible by $n$. D. Yu. Grigoriev All-Union Mathematical Olympiad for School Students (1972, Grades 9 and 10) Exploration The statement concerns divisibility of numbers of the...

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Kvant Math Problem 151

Verified: no Verdicts: SKIP + SKIP Solve time: 6m52s Source on kvant.digital Problem Each of the nine lines divides the square into two quadrilaterals whose areas are in the ratio $2:3$. Prove that at least three of these nine lines pass through a single point. B. M. Ivlev All-Union Mathematical Olympiad for School Students (1972, Grades 8 and 10) Exploration Consider a square of side length $1$ and a line...

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Kvant Math Problem 956

Verified: no Verdicts: SKIP + SKIP Solve time: 6m47s Source on kvant.digital Problem Four circles of equal radius are drawn in the plane so that three of them pass through the point $A$ and three pass through the point $B$ (Fig. 1). Prove that the four points of their pairwise intersections, distinct from $A$ and $B$, are the vertices of a parallelogram. Fig. 1 V. Kapovich, 10th-grade student, I. Kapovich,...

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Kvant Math Problem 150

Verified: no Verdicts: SKIP + SKIP Solve time: 5m57s Source on kvant.digital Problem From the numbers 1, 2, $\ldots$, $k$ all possible $(a_1,a_2,\ldots,a_n)$-tuples of length $n$ are formed (it is easy to see that there are $k^n$ of them). Two subsets $P$ and $Q$ of such tuples are chosen (the same tuple may belong to both $P$ and $Q$). It is known that if we take an arbitrary tuple $(p_1,p_2,\ldots,p_n)$...

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Kvant Math Problem 148

Verified: no Verdicts: SKIP + SKIP Solve time: 6m54s Source on kvant.digital Problem The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined by the following conditions: $x_0=1$, $x_1=\lambda$, for any $n\gt1$ $$(\alpha+\beta)^n x_n=\alpha^n x_n x_0+\alpha^{n-1}\beta x_{n-1} x_1+\alpha^{n-2}\beta^2 x_{n-2} x_2+\ldots+\beta^n x_0 x_n.$$ Here $\lambda$, $\alpha$, $\beta$ are given positive numbers. Find $x_n$ and determine for which $n$ the quantity $x_n$ attains its greatest value. A. L. Lopshits Exploration Let $$F_n=(\alpha+\beta)^n x_n .$$...

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Kvant Math Problem 147

Verified: no Verdicts: SKIP + SKIP Solve time: 6m38s Source on kvant.digital Problem Prove that if the cyclic quadrilateral $ABCD$ is such that the tangents to the circle at the points $A$ and $C$ intersect on the extension of the diagonal $BD$, then the tangents at the points $B$ and $D$ intersect on the extension of the diagonal $AC$; the bisectors of the interior angles $A$ and $C$ of the...

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Kvant Math Problem 146

Verified: no Verdicts: SKIP + SKIP Solve time: 6m57s Source on kvant.digital Problem Black and white counters are placed at the vertices of a regular heptagon. Prove that there exist three counters of the same color located at the vertices of an isosceles triangle. Is the analogous statement true for an octagon? Determine for which regular $n$-gons the analogous statement is true and for which it is not. A. Romanov...

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Kvant Math Problem 149

Verified: no Verdicts: SKIP + SKIP Solve time: 3m18s Source on kvant.digital Problem Let $O$ be the point of intersection of the diagonals of quadrilateral $ABCD$. Prove that if the perimeters of triangles $ABC$, $BCD$, $CDA$, and $DAB$ are equal, then $ABCD$ is a rectangle; if the perimeters of triangles $ABO$, $BCO$, $CDO$, and $DAO$ are equal, then $ABCD$ is a rhombus. N. B. Vasiliev Exploration Consider the first condition:...

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Kvant Math Problem 949

Verified: no Verdicts: SKIP + SKIP Solve time: 5m00s Source on kvant.digital Problem Given 1985 weights with masses 1 g, 2 g, 3 g, $\ldots$, 1984 g, 1985 g. Is it possible to divide them into five groups so that both the number of weights and their total mass are the same in all five groups? E. P. Eroshenkov All-Russian Mathematical Olympiad for School Students (XI) Exploration Consider small cases...

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Kvant Math Problem 145

Verified: no Verdicts: SKIP + SKIP Solve time: 6m08s Source on kvant.digital Problem $A$ promises to pay $B$ an average of $\sqrt{2}$ rubles per day. They agreed that on the $n$-th day $A$ would receive an integer number $a_n$ of rubles ($a_n$ equals 1 or 2), chosen so that the total amount received during the first $n$ days ($a_1+a_2+\ldots+a_n$) is as close as possible to the number $n\sqrt{2}$. For example,...

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Kvant Math Problem 943

Verified: no Verdicts: SKIP + SKIP Solve time: 4m47s Source on kvant.digital Problem The sequence $a_1$, $a_2$, $\ldots$ is defined by the rules: $a_{2n}=a_n$ for $n\ge 1$ and $a_{4n+1}=1$, $a_{4n+3}=0$ for $n\ge 0$. Prove that this sequence is not periodic. Yu. V. Nesterenko All-Union Mathematical Olympiad (1985) Exploration Let us compute the first terms. Since $a_{2n}=a_n$, every term is eventually reduced by repeatedly dividing its index by $2$ until an...

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Kvant Math Problem 142

Verified: no Verdicts: SKIP + SKIP Solve time: 8m12s Source on kvant.digital Problem a) Prove that it is impossible to label the edges of a cube with the numbers $1$, $2$, $\ldots$, $12$ so that for every vertex, the sum of the labels of the three edges incident to it is the same. b)* Is it possible to delete one of the numbers $1$, $2$, $\ldots$, $12$, $13$ and use...

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Kvant Math Problem 144

Verified: no Verdicts: SKIP + SKIP Solve time: 4m55s Source on kvant.digital Problem Find the necessary and sufficient conditions that the numbers $a$, $b$, $\alpha$, $\beta$ must satisfy in order for the rectangle $a\times b$ to be partitioned into several rectangles $\alpha\times\beta$. For example, can the rectangle $50\times 60$ be partitioned into rectangles of the following dimensions: $20\times 15$; $5\times 8$; $6{,}25\times 15$; $(2-\sqrt{2})\times (2+\sqrt{2})$. A. T. Kolotov Exploration Consider...

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Kvant Math Problem 143

Verified: no Verdicts: SKIP + SKIP Solve time: 6m00s Source on kvant.digital Problem Find the smallest positive integer $n$ for which the following condition holds: if $n$ is divisible by $p-1$ and $p$ is prime, then $n$ is divisible by $p$. Exploration Consider small positive integers $n$ and examine the condition that if $n$ is divisible by $p-1$ for some prime $p$, then $n$ must also be divisible by $p$....

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Kvant Math Problem 937

Verified: no Verdicts: SKIP + SKIP Solve time: 4m34s Source on kvant.digital Problem Does there exist a figure $F$ such that it cannot cover a semicircle of radius $1$, but two copies of it can cover a circle of radius $1$, if $F$ is: an arbitrary figure; a convex figure? N. B. Vasiliev, A. G. Samosvat Tournament of Towns (Spring 1985) Exploration The question asks whether there is a set...

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Kvant Math Problem 140

Verified: no Verdicts: SKIP + SKIP Solve time: 6m42s Source on kvant.digital Problem The following operations may be performed on a natural number (written in decimal notation): append the digit $4$ to the end; append the digit $0$ to the end; divide by $2$ (if the number is even). For example, if the operations B, B, A, and B are performed successively on the number $4$, the result is the...

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Kvant Math Problem 141

Verified: no Verdicts: SKIP + SKIP Solve time: 5m26s Source on kvant.digital Problem Choose an arbitrary point $P$ on the altitude $BH$ of triangle $ABC$. Let $K$ be the intersection point of the lines $AP$ and $BC$, and $L$ the intersection point of the lines $CP$ and $AB$. Prove that the segments $KH$ and $LH$ make equal angles with the altitude $BH$. E. V. Sallinen Exploration Let the altitude $BH$...

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Kvant Math Problem 139

Verified: no Verdicts: SKIP + SKIP Solve time: 7m39s Source on kvant.digital Problem From vertex $B$ of the parallelogram $ABCD$, its altitudes $BK$ and $BH$ are drawn. The segments $KH=a$ and $BD=b$ are known. Find the distance from point $B$ to the intersection point of the altitudes of triangle $BKH$. F. A. Bartenev Exploration Let the parallelogram have side vectors $\mathbf u=\overrightarrow{BA}$ and $\mathbf v=\overrightarrow{BC}$. Since $BK$ and $BH$ are...

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Kvant Math Problem 138

Verified: no Verdicts: SKIP + SKIP Solve time: 8m21s Source on kvant.digital Problem Prove that if $m$ and $n$ are integers and $1 \le m \lt n$, then $$\sum_{k=1}^n (-1)^k k^m C_n^k=0,$$ where $C_n^k$ are the binomial coefficients, that is, the coefficients of the polynomial $$(1+x)^m=\sum_{k=0}^n C_n^k x^k.$$ For example, if $n=4$, then $C_4^0=1$, $C_4^1=4$, $C_4^2=6$, $C_4^3=4$, $C_4^4=1$, and the equalities $$\begin{gather*} 1 \cdot 4+2 \cdot 6-3 \cdot 4+4 \cdot...

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Kvant Math Problem 932

Verified: no Verdicts: SKIP + SKIP Solve time: 3m48s Source on kvant.digital Problem In a square cell with a side of 1 m, there is an anaconda 10 m long. Baron Munchausen claims that at any moment he can shoot the anaconda in 6 places with a single shot. Is the Baron exaggerating? (The anaconda can be considered an arbitrary broken line of length 10, located inside the square $1\times1$.)...

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Kvant Math Problem 136

Verified: no Verdicts: SKIP + SKIP Solve time: 6m32s Source on kvant.digital Problem Is it possible to transport $50$ stones from a quarry, whose weights are $370~\text{кг}$, $372~\text{кг}$, $374~\text{кг}$, $\ldots$, $468~\text{кг}$ (the weights form an arithmetic progression with difference $2~\text{кг}$), using seven three-ton trucks? V. P. Fyodorov Exploration The stones have weights $370,372,374,\ldots,468,$ so there are $\frac{468-370}{2}+1=50$ stones. Each truck can carry at most $3000$ kg. The first question is...

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Kvant Math Problem 137

Verified: no Verdicts: SKIP + SKIP Solve time: 5m02s Source on kvant.digital Problem Let $a$, $b$, $c$, $d$ be the lengths of four consecutive sides of a quadrilateral, and $S$ its area. Prove that $2S \le ab+cd$. Prove that $2S \le ac+bd$. Prove that if equality holds in at least one of these inequalities, the quadrilateral can be inscribed in a circle. Exploration Consider a quadrilateral with consecutive sides $a$,...

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Kvant Math Problem 133

Verified: no Verdicts: SKIP + SKIP Solve time: 14m13s Source on kvant.digital Problem One of the simplest multicellular organisms—the alga Volvox —consists of a spherical shell made up primarily of heptagonal, hexagonal, and pentagonal cells (that is, cells having seven, six, or five neighbors), with three cells meeting at each “vertex” (Fig. 3). There are specimens that also contain quadrilateral and octagonal cells, but biologists have observed that if there...

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Kvant Math Problem 927

Verified: no Verdicts: SKIP + SKIP Solve time: 4m09s Source on kvant.digital Problem A finite set of points is given in the plane, with no three of them lying on the same line. Several segments have been drawn with endpoints at the given points. These segments may be modified: if two of them, $AC$ and $BD$, intersect, they may be erased and replaced by: the segments $AB$ and $CD$, the...

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Kvant Math Problem 135

Verified: no Verdicts: SKIP + SKIP Solve time: 6m18s Source on kvant.digital Problem Prove that for every positive integer $n \gt 1$ the identity $$\sin x \cdot \sin \left(x+\frac{\pi}{n}\right) \cdot \sin \left(x+\frac{2 \pi}{n}\right) \cdot \ldots \cdot \sin \left(x+\frac{(n-1) \pi}{n}\right)= c_n \sin nx,$$ holds, where $c_n$ is some number (depending on $n$), and find $c_n$. V. Maresin Exploration Consider small values of $n$ to understand the pattern of the product. For...

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Kvant Math Problem 920

Verified: no Verdicts: SKIP + SKIP Solve time: 5m39s Source on kvant.digital Problem Find at least one solution of the equation $$x^3+y^3+z^3=x^2y^2z^2$$ in natural numbers. Prove that the equation $$x^3+y^3+z^3=nx^2y^2z^2$$ has a solution in natural numbers only for $n=1$ and $n=3$, and find all such solutions. R. A. Mazov Exploration The equation is $$x^3+y^3+z^3=nx^2y^2z^2,$$ with positive integers $x,y,z$. Since the equation is symmetric, arrange $$x\le y\le z.$$ The first task...

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Kvant Math Problem 132

Verified: no Verdicts: SKIP + SKIP Solve time: 6m16s Source on kvant.digital Problem Insert illustration Suppose that $n$ numbers $x_1$, $x_2$, ..., $x_n$ are written around a circle, each of which is equal to $(+1)$ or $(—1)$. The sum of the $n$ pairwise products of neighboring numbers is equal to $0$ (as in Problem M93 , p. 42), and, more generally, for each $k=1$, $2$, $\ldots,$ $n—1$, the sum of...

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Kvant Math Problem 134

Verified: no Verdicts: SKIP + SKIP Solve time: 7m01s Source on kvant.digital Problem What set of points is traced out by the centroids of triangles whose three vertices lie respectively on the three sides $AB$, $BC$, and $AC$ of a given triangle $ABC$? L. G. Makarov Exploration Let the variable triangle have vertices $P\in AB$, $Q\in BC$, $R\in AC$. We seek the locus of the centroid $$G=\frac{P+Q+R}{3}.$$ The condition that...

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Kvant Math Problem 129

Verified: no Verdicts: SKIP + SKIP Solve time: 8m33s Source on kvant.digital Problem A bucket contains 12 L of milk. Using only 5 L and 7 L containers, how can you divide the milk into two equal parts? Solve the general problem: for which $a$ and $b$ is it possible to divide $(a+b)$ L of milk in half using only $a$ L, $b$ L, and $(a+b)$ L containers? V. V....

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Kvant Math Problem 912

Verified: no Verdicts: SKIP + SKIP Solve time: 6m29s Source on kvant.digital Problem Prove that the polynomial $x^2$, any polynomial can be represented as the difference of two polynomials, each of which is a monotonically increasing function. V. P. Pikulin Exploration For a polynomial to be monotonically increasing on the whole real line, it is enough that its derivative be nonnegative everywhere. Thus the problem becomes a question about derivatives....

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Kvant Math Problem 131

Verified: no Verdicts: SKIP + SKIP Solve time: 6m04s Source on kvant.digital Problem Insert illustration Prove that the four points at which the bisectors of the angles between the extensions of opposite sides of a cyclic quadrilateral intersect its sides are the vertices of a rhombus (Fig. 1). Murat Urtembaev, 10th-grade student (Alma-Ata, School No. 56) Exploration Consider a cyclic quadrilateral $ABCD$ and extend opposite sides $AB$ and $CD$, $BC$...

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Kvant Math Problem 130

Verified: no Verdicts: SKIP + SKIP Solve time: 6m44s Source on kvant.digital Problem What is the greatest number of points that can be placed in the plane; in space, so that none of the triangles with vertices at these points is obtuse? G. A. Galperin Exploration For points in the plane, the condition says that every triangle determined by the chosen points is acute or right. A first guess is...

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Kvant Math Problem 127

Verified: no Verdicts: SKIP + SKIP Solve time: 8m57s Source on kvant.digital Problem For each natural number $n$, let $s(n)$ denote the sum of its digits (in decimal notation). Call a natural number $m$ special if it cannot be represented in the form $m=n+s(n)$, where $n$ is some natural number. (For example, the number $117$ is not special, since $117= 108+s(108)=108+9$, while the number $121$ is, as is easy to...

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Kvant Math Problem 908

Verified: no Verdicts: SKIP + SKIP Solve time: 2m49s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. Before I begin writing the full Kvant-style solution, I need the text of problem M908. Could you provide the problem statement in text form? If you only have a graphical version, I can help reconstruct it...

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Kvant Math Problem 126

Verified: no Verdicts: SKIP + SKIP Solve time: 7m03s Source on kvant.digital Problem A polygon circumscribed about a circle of radius $r$ has been partitioned in some manner into triangles. Prove that the sum of the radii of the incircles of these triangles is greater than $r$. I. D. Novikov Exploration Let the polygon be $P$, let its area be $S$, and let the radius of its inscribed circle be...

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Kvant Math Problem 125

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem a) Does there exist an infinite sequence of natural numbers with the following property: None of these numbers divides another, yet among any three numbers one can choose two whose sum is divisible by the third? b) If not, then how many numbers can there be in a set with this property? c) Solve the same problem...

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Kvant Math Problem 128

Verified: no Verdicts: SKIP + SKIP Solve time: 4m29s Source on kvant.digital Problem Find the ratio of the sides of a triangle, one of whose medians is divided by the incircle into three equal parts. Exploration Let the median be drawn from a vertex $A$ to the midpoint $M$ of the opposite side. If the incircle cuts this median into three equal parts, then the two intersection points of the...

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Kvant Math Problem 900

Verified: no Verdicts: SKIP + SKIP Solve time: 6m12s Source on kvant.digital Problem Can the projection of a convex polyhedron with 6 faces onto a plane be an octagon? a nonagon? What is the greatest number of sides that the projection of a convex polyhedron with $n$ faces can have? M. D. Kovalev Exploration A convex polyhedron with six faces is a cube or a more general hexahedron. The projection...

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Kvant Math Problem 123

Verified: no Verdicts: SKIP + SKIP Solve time: 5m34s Source on kvant.digital Problem Find all natural numbers $m$ for which $$1!\cdot3!\cdot5!\cdot\ldots\cdot(2m-1)!=\left( \frac{m(m+1)}{2} \right)!$$ (here $n!$ denotes the product $1\cdot2\cdot\ldots\cdot n$). V. P. Beshkarev Exploration Let $$P_m=1!\cdot3!\cdot5!\cdots(2m-1)!$$ and $$N=\frac{m(m+1)}2.$$ The equation is $$P_m=N!.$$ Checking small values: $$m=1:\quad 1!=1!=1,$$ so $m=1$ works. $$m=2:\quad 1!\cdot3!=6,\qquad 3!=6,$$ so $m=2$ works. $$m=3:\quad 1!\cdot3!\cdot5!=720,\qquad 6!=720,$$ so $m=3$ works. $$m=4:\quad 1!\cdot3!\cdot5!\cdot7!=3628800,\qquad 10!=3628800,$$ so $m=4$ works. $$m=5:\quad P_5=P_4\cdot9!=10!\cdot9!,$$...

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Kvant Math Problem 121

Verified: no Verdicts: SKIP + SKIP Solve time: 7m39s Source on kvant.digital Problem Prove that for any $n$ real numbers $a_1$, $a_2$, ..., $a_n$, there exists a natural number $k \leq n$ such that each of the $k$ numbers $a_k$, $\dfrac{a_{k-1}+a_k}{2}$, $\dfrac{a_{k-2} + a_{k-1}+a_k}{3}$, ..., $\dfrac{a_1 + a_2+\ldots +a_k}{k}$ does not exceed $\dfrac{a_1 + a_2+\ldots +a_n}{n}$. Exploration Let $$A=\frac{a_1+\cdots+a_n}{n}$$ be the average of all given numbers. The statement asks for...

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Kvant Math Problem 122

Verified: no Verdicts: SKIP + SKIP Solve time: 7m05s Source on kvant.digital Problem The pentagon $ABCDE$ is inscribed in a circle. It is known that the distances from the point $E$ to the lines $AB$, $BC$, and $CD$ are equal to $p$, $q$, and $r$, respectively. Find the distance from the point $E$ to the line $AD$. Exploration Let the consecutive arcs of the circumcircle be $$\widehat{AB}=2x,\quad \widehat{BC}=2y,\quad \widehat{CD}=2z,\quad \widehat{DE}=2w,\quad...

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Kvant Math Problem 124

Verified: no Verdicts: SKIP + SKIP Solve time: 3m39s Source on kvant.digital Problem Given a triangle $ABC$. Find a point $O$ inside it with the following property: for any line passing through the point $O$ and intersecting the sides of the triangle $AB$ at point $K$ and $BC$ at point $L$, the equality $$\dfrac{AK}{KB}+\dfrac{CL}{LB}=1.$$ holds. More generally, prove that if $p$ and $q$ are arbitrary positive numbers, then inside the...

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Kvant Math Problem 897

Verified: no Verdicts: SKIP + SKIP Solve time: 3m02s Source on kvant.digital Problem Find at least one pair of integers $(x,y)$ such that the number $$(x+y)^7-x^7-y^7$$ is divisible by $7^7$, while the number $(x+y)xy$ is not divisible by 7. Exploration The problem asks for integers $(x,y)$ such that $(x+y)^7 - x^7 - y^7$ is divisible by $7^7$, while $(x+y)xy$ is not divisible by $7$. Testing small integers modulo 7 is...

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Kvant Math Problem 889

Verified: no Verdicts: SKIP + SKIP Solve time: 7m03s Source on kvant.digital Problem Do there exist three points $A$, $B$, $C$ in the plane such that for every point of the plane $P$, at least one of the segments $PA$, $PB$, and $PC$ has irrational length? A. M. Slinko All-Russian Mathematical Olympiad for School Students (Grade 10, 1984) Exploration Consider the problem of choosing three points $A$, $B$, $C$ in...

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Kvant Math Problem 120

Verified: no Verdicts: SKIP + SKIP Solve time: 5m48s Source on kvant.digital Problem An operation $*$ is defined on a certain set, assigning to every two elements $a$ and $b$ of this set an element $a * b$ of the same set. It is known that for any three elements $a$, $b$, and $c$, $$a*(b c)=b (c*a);$$ if $a b = a c$, then $b=c$; if $a * c =...

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Kvant Math Problem 118

Verified: no Verdicts: SKIP + SKIP Solve time: 6m56s Source on kvant.digital Problem A border of width two squares has been constructed around all four sides of a $n\times n$ chessboard. Prove that a knight can traverse this border, visiting each square exactly once, if and only if $n-1$ is divisible by 4. Yu. I. Sorkin, Yu. Yu. Sorkin Exploration The border of width two around an $n\times n$ board...

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Kvant Math Problem 119

Verified: no Verdicts: SKIP + SKIP Solve time: 6m24s Source on kvant.digital Problem Prove that if, on each face of a convex polyhedron, one chooses a point and draws from that point a vector perpendicular to the corresponding face, directed outward, whose length is equal to the area of that face, then the sum of all such vectors is equal to zero. N. B. Vasiliev Exploration The vectors described in...

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Kvant Math Problem 115

Verified: no Verdicts: SKIP + SKIP Solve time: 7m54s Source on kvant.digital Problem Three vessels are filled with an integer number of liters of water. It is allowed to pour into any vessel an amount of water equal to the quantity it already contains, taken from any other vessel. Prove that, by performing several such pourings, it is possible to empty one of the vessels. (The vessels are sufficiently large:...

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Kvant Math Problem 117

Verified: no Verdicts: SKIP + SKIP Solve time: 4m01s Source on kvant.digital Problem Several people observed a snail for $t$ minutes. Each person observed it for exactly 1 minute and noticed that during that minute the snail crawled exactly 1 meter. At no moment was the snail left unobserved. What is the smallest and what is the largest distance it could have crawled during those $t$ minutes? Try to solve...

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Kvant Math Problem 116

Verified: no Verdicts: SKIP + SKIP Solve time: 3m52s Source on kvant.digital Problem Insert illustration Prove that if one joins the midpoints of consecutive sides of a convex $n$-gon $M$ (Fig. 1), then the resulting polygon has a) perimeter not less than half the perimeter of $M$ ($n\geq3$); b) area not less than half the area of $M$ ($n\geq4$). G. A. Galperin Exploration Let the vertices of the convex polygon...

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Kvant Math Problem 882

Verified: no Verdicts: SKIP + SKIP Solve time: 4m53s Source on kvant.digital Problem The sum of three integers $a$, $b$, and $c$ is equal to 0. Prove that the number $2a^4+2b^4+2c^4$ is the square of an integer. L. D. Kurlyandchik, A. S. Merkuryev, S. V. Fomin Leningrad City Mathematical Olympiad (50th, 1984) Exploration Consider small integer triples $(a,b,c)$ satisfying $a+b+c=0$. If $a=1$, $b=1$, then $c=-2$, and $2a^4+2b^4+2c^4 = 2+2+32 =...

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Kvant Math Problem 114

Verified: no Verdicts: SKIP + SKIP Solve time: 7m56s Source on kvant.digital Problem Several numbers are written around a circle. If for some four consecutive numbers $a$, $b$, $c$, $d$ it turns out that $(a-b)(c-d)\lt0$, then the numbers $b$ and $c$ may be interchanged. Prove that this operation can be performed only finitely many times. V. B. Alekseev Exploration Let the numbers around the circle be $x_1,x_2,\dots,x_n$, with indices taken...

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Kvant Math Problem 113

Verified: no Verdicts: SKIP + SKIP Solve time: 7m05s Source on kvant.digital Problem Prove that for every positive integer $n$ there exists a number composed only of the digits 1 and 2 that is divisible by $2^n$. B. M. Ivlev Exploration For small values of $n$ the statement is easy to check directly. For $n=1$, the number $2$ works. For $n=2$, the number $12$ works since $12$ is divisible by...

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Kvant Math Problem 876

Verified: no Verdicts: SKIP + SKIP Solve time: 5m51s Source on kvant.digital Problem On the circle tangent to the sides of an angle with vertex $O$, two diametrically opposite points $A$ and $B$ are chosen (distinct from the points of tangency). The tangent to the circle at point $B$ intersects the sides of the angle at points $C$ and $D$, and the line $OA$ at point $E$ (Fig. 1). Prove...

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Kvant Math Problem 110

Verified: no Verdicts: SKIP + SKIP Solve time: 6m50s Source on kvant.digital Problem On an infinite sheet of graph paper, $N$ cells are colored black. Prove that it is possible to cut out a finite number of squares from the sheet such that the following two conditions are satisfied: all black cells lie within the cut-out squares; in any cut-out square $K$, the area of black cells is at least...

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Kvant Math Problem 111

Verified: no Verdicts: SKIP + SKIP Solve time: 6m44s Source on kvant.digital Problem In a square of side length 1, there is a figure such that the distance between any two of its points is not equal to $0{,}001$. Prove that the area of this figure does not exceed $0{,}34$. (You may assume that the boundary of the figure mentioned in the statement consists of line segments and circular arcs.)...

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Kvant Math Problem 109

Verified: no Verdicts: SKIP + SKIP Solve time: 6m09s Source on kvant.digital Problem At vertex $A_1$ of a regular 12-gon $A_1A_2A_3\ldots A_{12}$ there is a minus sign, and at all the other vertices there are plus signs. One is allowed simultaneously to change each sign to the opposite one at any six consecutive vertices of the polygon. Prove that after several such operations it is impossible to obtain a configuration...

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Kvant Math Problem 866

Verified: no Verdicts: SKIP + SKIP Solve time: 7m01s Source on kvant.digital Problem Each cell of the square $20 \times 20$ contains one soldier. For the largest possible $d$, how can the soldiers be rearranged into other cells so that each moves a distance of at least $d$? (Distance is measured as a straight line between the centers of the old and new cells; the side of a cell is...

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Kvant Math Problem 108

Verified: no Verdicts: SKIP + SKIP Solve time: 5m21s Source on kvant.digital Problem a) Prove that the line dividing a given triangle into two polygons of equal area and equal perimeter passes through the center of the circle inscribed in the triangle. b) Prove the analogous statement for an arbitrary polygon into which a circle can be inscribed. Yu. I. Ionin All-Union Mathematical Olympiad for School Students (V) Exploration For...

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Kvant Math Problem 107

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem Insert illustration a) Given a convex polygon $A_1A_2\ldots A_n$ (Fig. 1). On side $A_1A_2$, points $B_1$ and $D_2$ are chosen; on side $A_2A_3$, points $B_2$ and $D_3$; ..., on side $A_nA_1$, points $B_n$ and $D_1$ are chosen so that if the parallelograms $A_1B_1C_1D_1$, $A_2B_2C_2D_2$, ..., $A_nB_nC_nD_n$ are constructed, then the lines $A_1C_1$, $A_2C_2$, ..., $A_nC_n$ intersect at...

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Kvant Math Problem 105

Verified: no Verdicts: SKIP + SKIP Solve time: 9m12s Source on kvant.digital Problem The sum of the digits of a number after multiplication by 8 can decrease: $75\cdot 8=300$ — the sum of the digits was $7+5=12$, and became 3. However, it cannot decrease by more than a factor of 8. Prove that $$\dfrac{S(8M)}{S(M)}\geq\dfrac{1}{8},$$ where $M$ is a natural number, and $S(A)$ is the sum of the digits of the...

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Kvant Math Problem 862

Verified: no Verdicts: SKIP + SKIP Solve time: 4m09s Source on kvant.digital Problem Inside a given equilateral triangle, indicate the set of all points $M$ such that the distances from $M$ to its sides themselves serve as the side lengths of some triangle. Inside a given regular tetrahedron, indicate the set of all points $M$ such that the distances from $M$ to the faces of the tetrahedron serve as the...

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Kvant Math Problem 104

Verified: no Verdicts: SKIP + SKIP Solve time: 7m05s Source on kvant.digital Problem Insert illustration Inside triangle $ABC$ there are two points $P$ and $Q$ such that segments $AP$ and $AQ$ form equal angles with the bisector of angle $A$ of the triangle, and segments $BP$ and $BQ$ form equal angles with the bisector of angle $B$. Prove that segments $CP$ and $CQ$ form equal angles with the bisector of...

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Kvant Math Problem 103

Verified: no Verdicts: SKIP + SKIP Solve time: 6m50s Source on kvant.digital Problem Investigate how many solutions the system of equations $$\left{\begin{array}{l} x^2+y^2+xy=a,\ x^2-y^2=b, \end{array}\right.$$ has, where $a$ and $b$ are some real numbers. E. A. Yasinovy̆ĭ Exploration The system is $$\begin{cases} x^2+y^2+xy=a,\ x^2-y^2=b. \end{cases}$$ The first equation is a positive definite quadratic form, since $$x^2+y^2+xy=\left(x+\frac y2\right)^2+\frac34y^2\ge 0,$$ with equality only at $(0,0)$. Hence $a\ge0$ is necessary for solvability. The...

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Kvant Math Problem 102

Verified: no Verdicts: SKIP + SKIP Solve time: 6m26s Source on kvant.digital Problem A set in the plane consisting of finitely many points has the following property: for any two points $A$ and $B$ of the set, there exists a point $C$ of the set such that the triangle $ABC$ is equilateral. How many points can such a set contain? V. Gurari Exploration A triangle already satisfies the condition for...

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Kvant Math Problem 100

Verified: no Verdicts: SKIP + SKIP Solve time: 5m38s Source on kvant.digital Problem Prove that the sum of the 45 numbers $$\tg 1\deg + \tg 5\deg+\tg 9\deg+ \ldots +\tg 173\deg+\tg 177\deg$$ is equal to 45. V. P. Beshkarev Exploration The angles form an arithmetic progression with common difference $4^\circ$: $$1^\circ,5^\circ,9^\circ,\ldots,177^\circ.$$ The number of terms is $$\frac{177-1}{4}+1=45.$$ A natural idea is to pair terms whose angles add to $180^\circ$. Since $$\tan(180^\circ-\theta)=-\tan\theta,$$...

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Kvant Math Problem 101

Verified: no Verdicts: SKIP + SKIP Solve time: 6m01s Source on kvant.digital Problem A single virus enters a colony consisting of $n$ bacteria. During the first minute, it destroys one bacterium, then divides into two new viruses, and simultaneously each of the remaining bacteria also divides into two new bacteria. During the next minute, the two newly formed viruses destroy two bacteria, and then both viruses and all the remaining...

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Kvant Math Problem 97

Verified: no Verdicts: SKIP + SKIP Solve time: 7m46s Source on kvant.digital Problem In the trapezoid $ABCD$ with bases $AB=a$ and $CD=b$, a segment $A_1B_1$ is drawn connecting the midpoints of the diagonals. In the resulting trapezoid $A_1B_1CD$, a segment $A_2B_2$ is again drawn connecting the midpoints of the diagonals, and so on (Fig. 1). Can any number appear twice in the sequence of segment lengths $AB$, $A_1B_1$, $A_2B_2$, ...?...

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Kvant Math Problem 99

Verified: no Verdicts: SKIP + SKIP Solve time: 5m26s Source on kvant.digital Problem In the triangle $ABC$, the side $AC$ is the largest. Prove that for any point $M$ of the plane, $AM + CM$ is not less than $BM$. In what cases does equality occur? N. B. Vasiliev Exploration The inequality resembles the triangle inequality. Since $AC$ is the largest side of triangle $ABC$, one has $$AC \ge AB,\qquad...

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Kvant Math Problem 98

Verified: no Verdicts: SKIP + SKIP Solve time: 3m07s Source on kvant.digital Problem Prove that in the table $$\begin{array}{ccccccccc} & & & &1\ & & &1&1&1\ & &1&2&3&2&1\ &1&3&6&7&6&3&1\ .&.&.&.&.&.&.&.&., \end{array}$$, where each number is equal to the sum of the three numbers above it, there is an even number in every row (starting from the third). Does every row (except for the first two) contain a number divisible by...

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Kvant Math Problem 96

Verified: no Verdicts: SKIP + SKIP Solve time: 5m57s Source on kvant.digital Problem It is known of five positive numbers that if from the sum of any three of them one subtracts the sum of the remaining two, then the difference is positive. Prove that the product of all ten such differences does not exceed the square of the product of the given five numbers. S. T. Berkolayko Exploration Let...

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Kvant Math Problem 94

Verified: no Verdicts: SKIP + SKIP Solve time: 7m06s Source on kvant.digital Problem Prove that there does not exist a polyhedron such that at least four edges are incident to each vertex and at least four edges are incident to each face. L. G. Limanov Exploration Let $V$, $E$, and $F$ denote the numbers of vertices, edges, and faces of the polyhedron. The conditions translate into two combinatorial inequalities. Since...

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Kvant Math Problem 93

Verified: no Verdicts: SKIP + SKIP Solve time: 7m57s Source on kvant.digital Problem Each of the numbers $x_1$, ..., $x_n$ is equal to either $+1$ or $-1$. It is known that $$x_1x_2+x_2x_3+\ldots+x_{n-1}x_n+x_nx_1=0.$$ Prove that $n$ is divisible by 4. A. M. Leontovich Exploration Consider small sequences of $+1$ and $-1$ and compute the sum $x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1$. For $n = 2$, the sum is...

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Kvant Math Problem 95

Verified: no Verdicts: SKIP + SKIP Solve time: 5m20s Source on kvant.digital Problem A trapezoid was drawn on the board. Its midline $EF$ was drawn, and a perpendicular $OK$ was dropped from the point $O$ where the diagonals intersect to the longer base. Then the trapezoid was erased. How can the drawing be reconstructed from the remaining segments $EF$ and $OK$ (Fig. 2)? Insert illustration here. Exploration Let the trapezoid...

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Kvant Math Problem 92

Verified: no Verdicts: SKIP + SKIP Solve time: 7m12s Source on kvant.digital Problem Petya plans to spend all 90 days of the holidays in the village, strictly following this schedule: every second day (that is, every other day) he goes swimming in the lake, every third day he goes to the store for groceries, and every fifth day he solves math problems. (On the first day, Petya did all three...

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Kvant Math Problem 821

Verified: no Verdicts: SKIP + SKIP Solve time: 2m16s Source on kvant.digital Problem Solve the equation $x^3+x^2+x=-\dfrac{1}{3}.$ Yu. I. Ionin Exploration The equation is $$x^3+x^2+x=-\frac13.$$ Multiplying by $3$ gives $$3x^3+3x^2+3x+1=0.$$ The left-hand side resembles the expansion of $(x+1)^3$: $$(x+1)^3=x^3+3x^2+3x+1.$$ Subtracting yields $$3x^3+3x^2+3x+1 =3(x+1)^3-(3x^2+6x+2).$$ This does not immediately factor. Another possibility is to remove the fraction first: $$x^3+x^2+x+\frac13=0.$$ Multiplying by $27$ suggests testing the substitution $y=3x$: $$y^3+3y^2+9y+9=0.$$ Now $$y^3+3y^2+9y+9=(y+3)(y^2+9),$$ because $$(y+3)(y^2+9)=y^3+3y^2+9y+27.$$...

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Kvant Math Problem 91

Verified: no Verdicts: SKIP + SKIP Solve time: 8m59s Source on kvant.digital Problem Two players play “tic-tac-toe” on an infinite sheet of graph paper. The first player places a cross in any cell. On each subsequent turn, they must place a cross in any empty cell adjacent to a cell already containing a cross; a cell is considered adjacent if it shares a side or a vertex with the given...

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Kvant Math Problem 90

Verified: no Verdicts: SKIP + SKIP Solve time: 6m53s Source on kvant.digital Problem Prove that if $x_1 \lt x_2 \lt x_3 \lt \ldots$ are natural numbers, then $$\frac{\sqrt{x_2-x_1}}{x_2} + \frac{\sqrt{x_3-x_2}}{x_3}+ \ldots + \frac{\sqrt{x_n-x_{n-1}}}{x_n} \lt 1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n^2-1}+\frac{1}{n^2}.$$ G. I. Natanson Exploration Let $$d_k=x_k-x_{k-1}\qquad (k\ge 2).$$ Then $$x_k=d_2+d_3+\cdots+d_k+x_1.$$ The summand is $$\frac{\sqrt{d_k}}{x_k}.$$ Since $x_k\ge d_k$, the crude estimate $$\frac{\sqrt{d_k}}{x_k}\le \frac1{\sqrt{d_k}}$$ is far too weak, because the right-hand side need not be summable in...

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Kvant Math Problem 87

Verified: no Verdicts: SKIP + SKIP Solve time: 6m04s Source on kvant.digital Problem Prove that if three circles of equal radius pass through a single point, then the three other points of pairwise intersection of these circles lie on a circle of the same radius (Fig. 1). Insert illustration Exploration Let the three circles have common radius $r$, and let their common point be $P$. Denote the circles by $\omega_1,\omega_2,\omega_3$....

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Kvant Math Problem 86

Verified: no Verdicts: SKIP + SKIP Solve time: 4m49s Source on kvant.digital Problem The bottom of a rectangular box was tiled with tiles of sizes $2 \times 2$ and $1 \times 4$. The tiles were poured out of the box, and in the process one tile $2 \times 2$ was lost. In its place, it was possible to obtain a tile $1 \times 4$. Prove that it is now impossible...

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Kvant Math Problem 88

Verified: no Verdicts: SKIP + SKIP Solve time: 3m07s Source on kvant.digital Problem What condition must the coefficients $a$, $b$, $c$ of the equation $x^3+ax^2+bx+c=0$ satisfy for its three roots to form an arithmetic progression? M. F. Bezborodnikov Exploration Consider a cubic polynomial $x^3+ax^2+bx+c=0$ and suppose its roots form an arithmetic progression. Denote the roots by $r-d$, $r$, and $r+d$, where $r$ is the middle root and $d$ is the...

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Kvant Math Problem 89

Verified: no Verdicts: SKIP + SKIP Solve time: 3m39s Source on kvant.digital Problem Insert illustration Prove that in any convex polygon other than a parallelogram, one can choose three sides such that the extensions of their lines form a triangle containing the given polygon. (For example, in Figure 2, where the polygon is outlined in black, the three red lines satisfy the required condition, whereas the three blue lines do...

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Kvant Math Problem 800

Verified: no Verdicts: SKIP + SKIP Solve time: 3m55s Source on kvant.digital Problem On the plane, all points with integer coordinates—the nodes of a square lattice—are marked, and among them one “initial” node $O$ is distinguished. For each of the remaining nodes $P$, draw the line with respect to which the nodes $O$ and $P$ are symmetric—the perpendicular bisector of the segment $OP$. The drawn lines divide the plane into...

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Kvant Math Problem 82

Verified: no Verdicts: SKIP + SKIP Solve time: 7m51s Source on kvant.digital Problem Several identical automobiles are parked along a circular road. It is known that if all the gasoline contained in these automobiles were poured into a single car, then that car could travel all the way around the circular road and return to its starting point. Prove that at least one of the cars on the road can...

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Kvant Math Problem 84

Verified: no Verdicts: SKIP + SKIP Solve time: 5m29s Source on kvant.digital Problem Let $A$ be the foot of the perpendicular dropped from the center of the given circle onto the given line $l$. Take two more points $B$ and $C$ on this line so that $AB+AC$. Through the points $B$ and $C$ draw two arbitrary secants, one of which intersects the circle at the points $P$ and $Q$, while...

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Kvant Math Problem 85

Verified: no Verdicts: SKIP + SKIP Solve time: 4m59s Source on kvant.digital Problem Prove that if $a_1$, $a_2$, ..., $a_m$ are pairwise distinct natural numbers, none of which is divisible by the square of an integer greater than one, and $b_1$, $b_2$, ..., $b_m$ are nonzero integers, then $$b_1\sqrt{a_1}+b_2\sqrt{a_2}+\ldots+b_m\sqrt{a_m}\ne 0.$$ L. N. Vasershtein Exploration Consider first small examples. If $m=1$, the sum reduces to $b_1\sqrt{a_1}$, which is nonzero since $b_1\ne...

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Kvant Math Problem 83

Verified: no Verdicts: SKIP + SKIP Solve time: 4m27s Source on kvant.digital Problem Prove that the numbers 1, 2, ..., $n$ cannot be divided into two groups in such a way that the product of the numbers in one group equals the product of the numbers in the other, for any $n$. Exploration For small values of $n$, the statement is easy to check directly. For $n=1$, the only partition...

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Kvant Math Problem 787

Verified: no Verdicts: SKIP + SKIP Solve time: 4m08s Source on kvant.digital Problem Find the ratio of the sides of a right triangle, given that half of the hypotenuse (from the vertex to the midpoint of the hypotenuse) is seen from the incenter under a right angle. B. S. Pitskel Exploration Let the right triangle have legs of lengths $a$ and $b$, with hypotenuse $c = \sqrt{a^2 + b^2}$. Denote...

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Kvant Math Problem 81

Verified: no Verdicts: SKIP + SKIP Solve time: 5m03s Source on kvant.digital Problem Inside the square $A_1 A_2 A_3 A_4$, an arbitrary point $P$ is chosen. From vertex $A_1$ a perpendicular is dropped to the line $A_2P$; from vertex $A_2$, to $A_3P$; from $A_3$, to $A_4P$; and from $A_4$, to $A_1P$. Prove that all four perpendiculars (or their extensions) pass through a single point. A. N. Vilenkin Exploration Consider the...

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Kvant Math Problem 80

Verified: no Verdicts: SKIP + SKIP Solve time: 6m03s Source on kvant.digital Problem In the table $m\times n$, arbitrary numbers are placed. It is allowed to simultaneously change the sign of all numbers in any one column or all numbers in any one row. Prove that, by repeating this operation several times, it is possible to obtain a table in which the sum of the numbers in any column and...

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Kvant Math Problem 79

Verified: no Verdicts: SKIP + SKIP Solve time: 6m31s Source on kvant.digital Problem Two points $P$ and $Q$ move along two intersecting lines with the same constant speed $v$. Prove that there exists a fixed point $A$ in the plane such that the distances from it to the points $P$ and $Q$ are equal at any moment in time. I. F. Sharygin Exploration Let the two intersecting lines be $l_P$...

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Kvant Math Problem 78

Verified: no Verdicts: SKIP + SKIP Solve time: 3m34s Source on kvant.digital Problem Prove that every nonnegative integer can be represented in the form $$\frac{(x+y)^2+3x+y}{2},$$, where $x$ and $y$ are nonnegative integers, and that this representation is unique. Exploration Let $$N=\frac{(x+y)^2+3x+y}{2}.$$ A natural first step is to introduce $$s=x+y.$$ Then $$N=\frac{s^2+3x+y}{2} =\frac{s^2+2x+s}{2} =\frac{s(s+1)}2+x.$$ This is a much simpler expression. Since $x\ge 0$ and $y\ge 0$, we have $0\le x\le s$....

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Kvant Math Problem 75

Verified: no Verdicts: SKIP + SKIP Solve time: 7m19s Source on kvant.digital Problem a) Prove that in any convex polyhedron, the sum of the lengths of all edges is greater than three times the diameter. (The diameter of a polyhedron is defined as the greatest length among all segments with endpoints at the vertices of the polyhedron.) b) Prove that for any two vertices $A$ and $B$ of a convex...

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Kvant Math Problem 77

Verified: no Verdicts: SKIP + SKIP Solve time: 3m23s Source on kvant.digital Problem The lengths of two sides of a triangle are 10 and 15. Prove that the bisector of the angle between them is at most 12. N. B. Vasiliev Exploration Let the triangle have sides adjacent to angle $A$ equal to $10$ and $15$. Let $l$ denote the length of the bisector of angle $A$. A standard formula...

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Kvant Math Problem 74

Verified: no Verdicts: SKIP + SKIP Solve time: 5m48s Source on kvant.digital Problem The polynomial $p$ has the following property: for some number $a$, $$p(x)=p(a-x).$$ Prove that $p(x)$ can be represented as a polynomial in $$\left(x-\frac{a}{2}\right)^2.$$. For example, if $p(x)=x^5+(1-x)^5$, then obviously $p(x)=p(1-x)$ and, as is easy to verify, $$p(x)=5y^2+\frac{5}{2}y+\frac{1}{16},$$ where $$y=\left(x-\frac{1}{2}\right)^2.$$. Exploration Consider small-degree polynomials to detect a pattern. For degree zero, a constant function $p(x)=c$ trivially satisfies $p(x)=p(a-x)$...

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Kvant Math Problem 76

Verified: no Verdicts: SKIP + SKIP Solve time: 4m13s Source on kvant.digital Problem In a group of $n$ people, every pair of strangers has exactly two mutual acquaintances, and every pair of acquaintances has no more than two mutual acquaintances. Prove that in this group, each person knows the same number of people. Exploration Represent the group by a simple graph. Vertices are people, adjacent vertices are acquaintances, and nonadjacent...

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Kvant Math Problem 770

Verified: no Verdicts: SKIP + SKIP Solve time: 2m43s Source on kvant.digital Problem At the base of a triangular pyramid $PABC$ lies an equilateral triangle $ABC$. Prove that if the angles $PAB$, $PBC$, $PCA$ are congruent, then the pyramid $PABC$ is regular. V. A. Senderov Exploration Let the common value of the three face angles be $\alpha$: $$\angle PAB=\angle PBC=\angle PCA=\alpha .$$ The base $ABC$ is equilateral. A regular triangular...

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Kvant Math Problem 71

Verified: no Verdicts: SKIP + SKIP Solve time: 5m57s Source on kvant.digital Problem A rectangular table with $m$ rows and $n$ columns is filled with numbers. Rearrange the numbers in each row into increasing order. Prove that if, after that, the numbers in each column are rearranged into increasing order, then the numbers in every row will still remain in increasing order. Consider what happens if the operations are performed...

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Kvant Math Problem 69

Verified: no Verdicts: SKIP + SKIP Solve time: 7m39s Source on kvant.digital Problem The number 76 has the following curious property: the last two digits of the number $76^2=5776$ are again 76. Are there any other two-digit numbers with this property? Find all three-digit numbers $A$ such that the last three digits of the number $A^2$ form the number $A$. Does there exist an infinite sequence of digits $a_1$, $a_2$,...

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Kvant Math Problem 766

Verified: no Verdicts: SKIP + SKIP Solve time: 1m57s Source on kvant.digital Problem Prove that the sum of the squares of three consecutive integers cannot be a cube of a natural number. Yu. I. Ionin Exploration Consider three consecutive integers $n-1$, $n$, $n+1$ and compute the sum of their squares. Explicitly, $$(n-1)^2 + n^2 + (n+1)^2 = n^2 - 2n + 1 + n^2 + n^2 + 2n + 1...

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Kvant Math Problem 70

Verified: no Verdicts: SKIP + SKIP Solve time: 8m41s Source on kvant.digital Problem Let $l_1$, $l_2$, ..., $l_n$ be several lines in the plane, among which there are two that intersect. Prove that it is possible to choose uniquely a point $X_1$, $X_2$, ... on each of these lines so that the perpendicular erected at the point $X_i$ to the line $l_i$ passes through the point $X_{i+1}$ (for all $i=1$,...

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Kvant Math Problem 758

Verified: no Verdicts: SKIP + SKIP Solve time: 3m21s Source on kvant.digital Problem What is the smallest number of integers that must be crossed out from the sequence 1, 2, 3, $\dots$, 1982 so that none of the remaining integers is equal to the product of two other remaining integers? L. D. Kurlyandchik All-Union Mathematical Olympiad for School Students (1982, 9th grade) Exploration Let $R$ be the set of remaining...

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Kvant Math Problem 72

Verified: no Verdicts: SKIP + SKIP Solve time: 5m01s Source on kvant.digital Problem Solve the equation $$\sqrt[3]{1-x}+\sqrt[3]{1+x}=3,$$ where $p$ is an arbitrary real number. Yu. I. Ionin Exploration Let $$a=\sqrt[3]{1-x},\qquad b=\sqrt[3]{1+x}.$$ Then the equation becomes $$a+b=3.$$ The quantities $a$ and $b$ satisfy $$a^3+b^3=(1-x)+(1+x)=2.$$ Hence we seek real numbers $a,b$ such that $$a+b=3,\qquad a^3+b^3=2.$$ Using $$a^3+b^3=(a+b)^3-3ab(a+b),$$ we obtain $$2=27-9ab,$$ so $$ab=\frac{25}{9}.$$ Now $a$ and $b$ would have to be roots of...

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Kvant Math Problem 73

Verified: no Verdicts: SKIP + SKIP Solve time: 1m27s Source on kvant.digital Problem On a lottery ticket, one must mark 8 squares out of 64. What is the probability that after the drawing, in which 8 squares out of 64 are also selected (with all such selections assumed equally likely), exactly 4 squares will have been guessed correctly? Exactly 5 squares? … all 8 squares? Exploration Fix the player's marked...

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Kvant Math Problem 66

Verified: no Verdicts: SKIP + SKIP Solve time: 4m57s Source on kvant.digital Problem Here are a few examples where the sum of the squares of $k$ consecutive natural numbers equals the sum of the squares of the next $k-1$ natural numbers: $$\begin{gathered} 3^2+4^2=5^2,\ 36^2+37^2+38^2+39^2+40^2=41^2+42^2+43^2+44^2,\ 55^2+56^2+57^2+58^2+59^2+60^2=61^2+62^2+63^2+64^2+65^2. \end{gathered}$$ Find a general formula that covers all such cases. A. I. Milovanov Exploration Consider the examples given: $3^2 + 4^2 = 5^2$, $36^2 +...

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Kvant Math Problem 68

Verified: no Verdicts: SKIP + SKIP Solve time: 1m27s Source on kvant.digital Problem The grid of lines shown on the cover of this issue of the magazine consists of concentric circles of radii 1, 2, 3, 4, ... centered at $O$, the line $l$ passing through the point $O$, and all possible tangents to the circles that are parallel to $l$. These lines divide the entire plane into cells, which...

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Kvant Math Problem 61

Verified: no Verdicts: SKIP + SKIP Solve time: 7m22s Source on kvant.digital Problem Two sages are playing a new game consisting of the following. The numbers 0, 1, 2, ..., 1024 are written down. The first sage crosses out, at his discretion, 512 numbers; the second crosses out 256 of the remaining numbers; then the first again crosses out another 128, then the second another 64 numbers, and so on....

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Kvant Math Problem 67

Verified: no Verdicts: SKIP + SKIP Solve time: 1m34s Source on kvant.digital Problem A jeweler has been commissioned to make a gold ring of width $h$, shaped as the solid bounded by the surface of a sphere with center $O$ and the surface of a cylinder of radius $r$, whose axis passes through the point $O$ (Fig. 1). The craftsman made such a ring but chose $r$ too small. How...

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Kvant Math Problem 65

Verified: no Verdicts: SKIP + SKIP Solve time: 1m44s Source on kvant.digital Problem Let $E$, $F$, $G$ be points on the sides $AB$, $BC$, and $CA$ of triangle $ABC$ such that $$\frac{AE}{EB}=\frac{BF}{FC}=\frac{CG}{GA}=k,$$, where $0\lt k\lt 1$. Find the ratio of the area of the triangle formed by the lines $AF$, $BG$, and $CE$ to the area of triangle $ABC$ (Fig. 2). Divide a triangle by six straight lines into pieces...

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Kvant Math Problem 60

Verified: no Verdicts: SKIP + SKIP Solve time: 5m16s Source on kvant.digital Problem Consider all natural numbers whose decimal representation contains only the digits 1 and 0. Divide these numbers into two groups so that the sum of any two distinct numbers from the same group contains at least two ones in its decimal representation. E. B. Dynkin, S. A. Molchanov, A. L. Rozenthal, A. K. Tolpygo Mathematical Problems. 3rd...

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Kvant Math Problem 64

Verified: no Verdicts: SKIP + SKIP Solve time: 1m49s Source on kvant.digital Problem On the plane, a line $l$ and two points $P$ and $Q$ lying on the same side of it are given. Find a point $M$ on the line $l$ such that the distance between the feet of the altitudes of triangle $PQM$ dropped onto the sides $PM$ and $QM$ is minimized. G. A. Palatnik Exploration Let $A$...

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Kvant Math Problem 62

Verified: no Verdicts: SKIP + SKIP Solve time: 1m20s Source on kvant.digital Problem Prove that for any odd number $a$ there exists a natural number $b$ such that $2^b-1$ is divisible by $a$. Exploration Consider small odd numbers to test the claim. For $a=1$, any $b$ works because $2^b-1$ is divisible by $1$. For $a=3$, $2^2-1=3$ is divisible by $3$, so $b=2$ works. For $a=5$, $2^4-1=15$ is divisible by $5$,...

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Kvant Math Problem 63

Verified: no Verdicts: SKIP + SKIP Solve time: 2m54s Source on kvant.digital Problem Is it possible to tile a square with 18 tiles of size $1\times 2$ in such a way that there is not a single straight “seam” connecting opposite sides of the square and running along the edges of the tiles? (For example, the arrangement of tiles shown in Figure 1 is not suitable, since it contains the...

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Kvant Math Problem 59

Verified: no Verdicts: SKIP + SKIP Solve time: 1m49s Source on kvant.digital Problem There are several weights with masses 1 g, 2 g, 3 g, ..., $n$ g. They must be divided into three groups of equal total weight. For which $n$ can this be done? E. B. Dynkin, S. A. Molchanov, A. L. Rosenthal, A. K. Tolpygo Mathematical Problems. 3rd ed. — Moscow: Nauka, 1971. Exploration Let $$S=1+2+\cdots+n=\frac{n(n+1)}2.$$ If...

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Kvant Math Problem 55

Verified: no Verdicts: SKIP + SKIP Solve time: 6m02s Source on kvant.digital Problem All natural numbers whose decimal representation contains at most $n$ digits are divided into two groups. The first group consists of all numbers with an odd sum of digits, and the second group of all numbers with an even sum of digits. Prove that if $1\le k\lt n$, then the sum of the $k$th powers of all...

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Kvant Math Problem 56

Verified: no Verdicts: SKIP + SKIP Solve time: 4m37s Source on kvant.digital Problem On a circle, four ones and five zeros are written in an arbitrary order. Then, in the interval between two identical numbers, a one is written, and between two different digits, a zero is written, after which the original digits are erased. Prove that, no matter how many times this process is repeated, one can never obtain...

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Kvant Math Problem 58

Verified: no Verdicts: SKIP + SKIP Solve time: 6m24s Source on kvant.digital Problem Three lines intersecting at a single point are given in the plane. A point is marked on one of them. It is known that the lines are the angle bisectors of some triangle, and the marked point is one of its vertices. Construct this triangle. E. B. Dynkin, S. A. Molchanov, A. L. Rosenthal, A. K. Tolpygo...

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Kvant Math Problem 57

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem Find the number $k$ that is divisible by 2 and 9 and has exactly 14 divisors (including 1 and $k$). Prove that if 14 is replaced by 15, the problem has multiple solutions, and if 14 is replaced by 17, there are no solutions at all. E. B. Dynkin, S. A. Molchanov, A. L. Rozental, A. K....

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Kvant Math Problem 54

Verified: no Verdicts: SKIP + SKIP Solve time: 2m09s Source on kvant.digital Problem Two congruent rectangles are positioned so that their boundaries intersect at eight points. Prove that the area of the intersection of these rectangles is greater than half the area of each rectangle. G. A. Galperin Exploration Let each rectangle have side lengths $a\ge b$. The intersection of two convex sets is convex, so the common part is...

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Kvant Math Problem 52

Verified: no Verdicts: SKIP + SKIP Solve time: 3m49s Source on kvant.digital Problem Five segments are such that any three of them can form a triangle. Prove that at least one of these triangles is acute. M. Serov All-Union Mathematical Olympiad (1970, Grade 9) Exploration Let the five segment lengths be $$a_1\le a_2\le a_3\le a_4\le a_5.$$ The condition that any three segments form a triangle is very strong. For a...

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Kvant Math Problem 50

Verified: no Verdicts: SKIP + SKIP Solve time: 4m13s Source on kvant.digital Problem The vertices of a regular $n$-gon are colored with several colors (each vertex in exactly one color) in such a way that the points of any one color are the vertices of a regular polygon. Prove that among these polygons there are two that are congruent. N. B. Vasiliev All-Union Mathematical Olympiad (1970, Grade 10) Exploration Consider...

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Kvant Math Problem 53

Verified: no Verdicts: SKIP + SKIP Solve time: 1m32s Source on kvant.digital Problem In triangle $ABC$, through the midpoint $M$ of side $BC$ and the center $O$ of the inscribed circle of this triangle, a line $MO$ is drawn. This line intersects the altitude $AH$ at the point $E$. Prove that the segment $AE$ is equal to the radius of the inscribed circle. All-Union Mathematical Olympiad (1970, Grade 10) Exploration...

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Kvant Math Problem 51

Verified: no Verdicts: SKIP + SKIP Solve time: 6m38s Source on kvant.digital Problem Prove that if the product of three positive numbers is equal to 1, and the sum of these numbers is strictly greater than the sum of their reciprocals, then exactly one of these numbers is greater than 1. All-Union Mathematical Olympiad (1970, Grade 8) Exploration Let the numbers be $a,b,c>0$ with $abc=1$. Since $$\frac1a=bc,\qquad \frac1b=ca,\qquad \frac1c=ab,$$ the...

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Kvant Math Problem 49

Verified: no Verdicts: SKIP + SKIP Solve time: 7m17s Source on kvant.digital Problem All five-digit numbers from 11111 to 99999 inclusive are written on cards. These cards are then arranged in a single sequence in an arbitrary order. Prove that the resulting $444\ 445$-digit number cannot be a power of two. All-Union Mathematical Olympiad (1970, 10th grade) Exploration There are $99999-11111+1=88889$ cards. Concatenating all of them produces a number with...

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Kvant Math Problem 45

Verified: no Verdicts: SKIP + SKIP Solve time: 5m20s Source on kvant.digital Problem Prove that from any two hundred integers one can choose one hundred whose sum is divisible by 100. Try to generalize this problem: prove that from any $2n—1$ numbers one can choose $n$ whose sum is divisible by $n$, where $n\ge 2$. All-Union Mathematical Olympiad (1970, Grade 9) Exploration For small values of $n$ the statement is...

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Kvant Math Problem 46

Verified: no Verdicts: SKIP + SKIP Solve time: 3m42s Source on kvant.digital Problem How many sides of a convex polygon can have lengths equal to the length of its longest diagonal? G. A. Galperin All-Union Mathematical Olympiad (1970, Grade 8) Exploration Let the longest diagonal of a convex polygon have length $D$. The question asks how many sides can also have length $D$. Small cases suggest a strong restriction. In...

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Kvant Math Problem 47

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem Using the digits 1 and 2, five $n$-digit numbers were formed such that any two numbers have the same digit in exactly $m$ positions, but in no position do all five numbers have the same digit. Prove that $$\frac{2}{5}\le \frac{m}{n}\le \frac{3}{5}.$$. All-Union Mathematical Olympiad (1970, grades 9–10) Exploration Represent the five numbers as five binary strings of...

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Kvant Math Problem 705

Verified: no Verdicts: SKIP + SKIP Solve time: 4m45s Source on kvant.digital Problem On a rectangular sheet of graph paper, several rectangular cards are placed so that their sides lie along the grid lines. The cards cover the sheet in two layers, that is, each cell of the sheet is covered by exactly two cards. Suppose each card has dimensions $1\times 2$ cells. Prove that it is possible to select...

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Kvant Math Problem 44

Verified: no Verdicts: SKIP + SKIP Solve time: 2m25s Source on kvant.digital Problem Prove that for every natural number $K$, there exist infinitely many natural numbers $T$ that contain no zeros in their decimal representation and such that $T$ and $KT$ have the same sum of digits. All-Union Mathematical Olympiad (1970, 10th grade) Exploration Let $s(n)$ denote the sum of the decimal digits of $n$. The condition is $$s(T)=s(KT),$$ with...

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Kvant Math Problem 41

Verified: no Verdicts: SKIP + SKIP Solve time: 6m22s Source on kvant.digital Problem Given a circle, its diameter $AB$, and a point $C$ on this diameter. Construct on the circle two points $X$ and $Y$, symmetric with respect to the diameter $AB$, such that the line $YC$ is perpendicular to the line $XA$. All-Union Mathematical Olympiad (1970, Grade 8) Exploration Let the circle have center $O$ and radius $R$. Put...

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Kvant Math Problem 48

Verified: no Verdicts: SKIP + SKIP Solve time: 2m01s Source on kvant.digital Problem In an acute triangle $ABC$, the angle bisector $AD$, the median $BM$, and the altitude $CH$ intersect at a single point. Prove that the angle $BAC$ is greater than $45^\circ$. All-Union Mathematical Olympiad (1970, Grade 9) Exploration Let the common point of the angle bisector $AD$, the median $BM$, and the altitude $CH$ be $P$. The condition...

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Kvant Math Problem 38

Verified: no Verdicts: SKIP + SKIP Solve time: 7m52s Source on kvant.digital Problem Fig. 2 A circle constructed on the altitude $AD$ of the right triangle $ABC$ as a diameter intersects leg $AB$ at point $K$ and leg $AC$ at point $M$. The segment $KM$ intersects the altitude $AD$ at point $L$. It is known that the segments $AK$, $AL$, and $AM$ form a geometric progression, i.e., $\dfrac{AK}{AL}=\dfrac{AL}{AM}$. Find the...

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Kvant Math Problem 40

Verified: no Verdicts: SKIP + SKIP Solve time: 5m30s Source on kvant.digital Problem Find the sum $$1\cdot n + 2\cdot (n-1) + 3\cdot (n-2)+ \ldots + n\cdot 1.$$ Try to solve the following, more general problem: find the sum $$\begin{aligned} S_{n,k}={}&[1\cdot 2\cdot \ldots \cdot k] \times [n(n-1)\cdot\ldots\cdot (n-k+1)]+{}\ {}+{}& [2\cdot 3\cdot \ldots \cdot (k+1)] \times [(n-1)(n-2)\cdot\ldots\cdot (n-k)]+{}\ {}+{}& [3\cdot 4\cdot \ldots \cdot (k+2)] \times [(n-2)(n-1)\cdot\ldots\cdot (n-k-1)]+{}\ {}+{}& {\ldots},{\ldots},{\ldots},{\ldots},{\ldots},{\ldots},{\ldots},{\ldots} +{}\ {}+{}&...

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Kvant Math Problem 43

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem Fig. 3 Each side of an equilateral triangle is divided into $n$ equal parts. Through the division points, lines parallel to the sides are drawn. As a result, the triangle is partitioned into $n^2$ small triangles (Fig. 3). Let us call a “chain” a sequence of triangles in which no triangle appears twice and each subsequent triangle...

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Kvant Math Problem 42

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem The digits of a certain seventeen-digit number are written in reverse order. The resulting number is added to the original one. Prove that at least one of the digits of their sum is even. All-Union Mathematical Olympiad (1970, Grade 8) Exploration Let the original seventeen-digit number be $a_{16}a_{15}\cdots a_1a_0,$ and let its reverse be $a_0a_1\cdots a_{15}a_{16}.$ The...

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Kvant Math Problem 39

Verified: no Verdicts: SKIP + SKIP Solve time: 1m37s Source on kvant.digital Problem Prove that the nonnegative integers $x$, $y$ satisfy the equation $$x^2-mxy+y^2=1$$ (where $m$ is a given integer greater than one) if and only if $x$ and $y$ are consecutive terms of the sequence $\phi_0=0$, $\phi_1=1$, $\phi_2=m$, $\phi_3=m^2-1$, $\phi_4=m^3-2m$, $\phi_5=m^4-3m^2+1$, $\ldots$, in which $\phi_{k+1}=m\phi_k-\phi_{k-1}$ for all $k\ge 1$. For example, all solutions of the equation $$x^2-3xy+y^2=1\tag1$$ in integers...

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Kvant Math Problem 37

Verified: no Verdicts: SKIP + SKIP Solve time: 1m55s Source on kvant.digital Problem In each cell of an infinite sheet of squared paper, a number is written in such a way that the sum of the numbers in any square (whose sides run along the grid lines) has absolute value at most one. Prove that there exists a number $c$ such that the sum of the numbers in any rectangle...

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Kvant Math Problem 33

Verified: no Verdicts: SKIP + SKIP Solve time: 5m10s Source on kvant.digital Problem There is a natural number $n \gt 1000$. Consider the remainders obtained when the number $2^n$ is divided by the numbers 1, 2, 3, ..., $n$, and find the sum of all these remainders. Prove that this sum is greater than $2n$. A. G. Kushnirenko Moscow Mathematical Olympiad (1970) Exploration Let $$S=\sum_{k=1}^{n} r_k,$$ where $r_k$ is the...

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Kvant Math Problem 36

Verified: no Verdicts: SKIP + SKIP Solve time: 1m06s Source on kvant.digital Problem Prove that it is impossible to arrange seven lines and seven points in the plane so that exactly three lines pass through each point and exactly three points lie on each line. Exploration Consider arranging seven points and seven lines such that each point lies on exactly three lines and each line contains exactly three points. Each...

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Kvant Math Problem 32

Verified: no Verdicts: SKIP + SKIP Solve time: 3m17s Source on kvant.digital Problem In every cell of an $100\times 100$ table there is a plus sign. It is allowed to simultaneously change the signs in all cells of a single row or a single column. Is it possible, after performing such operations several times, to obtain a table containing exactly 1970 minus signs? A. V. Zelevinsky Moscow Mathematical Olympiad (1970)...

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Kvant Math Problem 35

Verified: no Verdicts: SKIP + SKIP Solve time: 1m13s Source on kvant.digital Problem A certain 19-faced polyhedron is circumscribed about a sphere of radius 10. Prove that there exist two points on its surface whose distance apart is greater than 21. A. G. Kushnirenko Moscow Mathematical Olympiad (1970) Exploration The polyhedron has 19 faces and is circumscribed about a sphere of radius $10$. The first instinct is to consider the...

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Kvant Math Problem 34

Verified: no Verdicts: SKIP + SKIP Solve time: 2m02s Source on kvant.digital Problem Prove that if a natural number is divisible by $10,101,010,101$, then in its decimal representation at least six digits are nonzero. A. K. Tolpygo Moscow Mathematical Olympiad (1970) Exploration The number $$10,101,010,101$$ has a very suggestive decimal form. Computing, $$10,101,010,101=10^{10}+10^8+10^6+10^4+10^2+1.$$ It is useful to factor it. Since $$10^{12}-1=(10^6-1)(10^6+1),$$ and $$10^{12}-1=(10^2-1)(10^{10}+10^8+10^6+10^4+10^2+1),$$ we get $$10,101,010,101=\frac{10^{12}-1}{99}.$$ A direct computation gives...

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Kvant Math Problem 689

Verified: no Verdicts: SKIP + SKIP Solve time: 2m25s Source on kvant.digital Problem Prove that it is impossible to form a rectangle from identical tiles shaped as isosceles trapezoids with bases 3 cm and 1 cm and height 1 cm. S. Rukshin Exploration Each tile is an isosceles trapezoid with bases $3$ and $1$ and height $1$. Its area is $$\frac{3+1}{2}\cdot 1=2.$$ The nonparallel sides have length $$\sqrt{1^2+1^2}=\sqrt2,$$ because the...

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Kvant Math Problem 31

Verified: no Verdicts: SKIP + SKIP Solve time: 2m43s Source on kvant.digital Problem A square sheet of paper is cut along a straight line into two pieces. One of the resulting pieces is then cut into two, and this process is repeated many times. What is the minimum number of cuts needed so that among the resulting pieces there are exactly one hundred 20-sided polygons? I. N. Bernstein Moscow Mathematical...

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Kvant Math Problem 30

Verified: no Verdicts: SKIP + SKIP Solve time: 1m54s Source on kvant.digital Problem Prove that $N$ points in the plane can always be covered by several pairwise disjoint circles whose total sum of diameters is less than $N$ and the distance between any two of them is greater than 1. (The distance between two circles is understood to be the distance between their nearest points.) V. I. Arnold Exploration For...

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Kvant Math Problem 20

Verified: no Verdicts: SKIP + SKIP Solve time: 9m17s Source on kvant.digital Problem Is it possible to divide an equilateral triangle into one million convex polygons so that any straight line intersects no more than forty of these polygons? (We say that a line intersects a polygon if it has at least one point in common with it.) Moscow Mathematical Olympiad (XXXI) Exploration Let the maximum number of polygons met...

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Kvant Math Problem 670

Verified: no Verdicts: SKIP + SKIP Solve time: 10m14s Source on kvant.digital Problem Several points are given, with some pairs connected by lines (the points in such pairs are called neighbors ). Each point has an odd number of neighbors. Initially, all points are colored in two colors — red and blue. Then, every minute, the points are simultaneously recolored according to the following rule: each point whose majority of...

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Kvant Math Problem 29

Verified: no Verdicts: SKIP + SKIP Solve time: 2m03s Source on kvant.digital Problem $n$ identical coins are lying on a table, forming a closed chain (see the figure). How many rotations will a coin $M$ of the same size make while rolling once along the outer side of the entire chain, as shown in the figure (coin $M = 2~\text{коп.}$)? How does the answer change if the coin $M$ has...

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Kvant Math Problem 28

Verified: no Verdicts: SKIP + SKIP Solve time: 2m24s Source on kvant.digital Problem Among 19 balls, 2 are radioactive. For any collection of balls, a single test can determine whether it contains at least one radioactive ball or not (but it cannot determine how many such balls the collection contains). Prove that both radioactive balls can be identified in 8 tests. Among 11 balls, 2 are radioactive. Prove that it...

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Kvant Math Problem 25

Verified: no Verdicts: SKIP + SKIP Solve time: 2m10s Source on kvant.digital Problem In a set consisting of $n$ elements, $2^{n-1}$ subsets have been chosen, every three of which have a common element. Prove that all these subsets have a common element. Moscow Mathematical Olympiad (XXX) Exploration For small values of $n$ the statement is easy to test. If $n=1$, the family consists of one subset, and the claim is...

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Kvant Math Problem 26

Verified: no Verdicts: SKIP + SKIP Solve time: 1m24s Source on kvant.digital Problem Suppose that each issue of the Kvant problem book contains five mathematics problems, and the journal is published monthly. Denote by $f(x,y)$ the number of the first problem in the $x$-th issue for the $y$-th year (for example, $f(6, 1970)=26$). Write a general formula for $f(x,y)$, for all $x$, $y$ ($1\le x\le12$, $y\ge 1970$). Solve the equation...

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Kvant Math Problem 24

Verified: no Verdicts: SKIP + SKIP Solve time: 1m42s Source on kvant.digital Problem Prove that any fraction $\dfrac mn$, where $0\lt\dfrac mn\lt1$, can be represented in the form $$\dfrac mn=\dfrac1{q_1}+\dfrac1{q_2}+\dfrac1{q_3}+\ldots+\dfrac1{q_r},$$, where $0 \lt q_1 \lt q_2 \lt \ldots \lt q_r$ are integers, and each $q_k$ ($k = 2$, 3, $\ldots$, $r$) is divisible by $q_{k-1}$. Exploration The condition on the denominators is much stronger than in the usual Egyptian fraction...

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Kvant Math Problem 23

Verified: no Verdicts: SKIP + SKIP Solve time: 1m34s Source on kvant.digital Problem Prove that for all natural $n>1$ $$\frac{\sqrt{2}}{2} \frac{1}{\sqrt{2n}} \lt \frac{1\cdot 3 \cdot\ldots\cdot (2n-1)}{2\cdot 4\cdot \ldots \cdot 2n} \lt \frac{\sqrt{3}}{2} \frac{1}{\sqrt{2n}}.$$ A. O. Gelfond Exploration Let $$a_n=\frac{1\cdot3\cdot\ldots\cdot(2n-1)}{2\cdot4\cdot\ldots\cdot2n}.$$ The statement becomes $$\frac1{2\sqrt n}<a_n<\frac{\sqrt3}{2\sqrt{2n}}.$$ The quantity resembles the central binomial coefficient because $$a_n=\frac{(2n)!}{2^{2n}(n!)^2}=\frac1{4^n}\binom{2n}{n}.$$ The asymptotic behavior is $$a_n\sim \frac1{\sqrt{\pi n}},$$ so the constants in the problem are plausible: $$\frac1{2}=0.5,\qquad \frac1{\sqrt\pi}\approx0.564,\qquad...

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Kvant Math Problem 22

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem Two circles are inscribed in an angle; they have a common internal tangent $T_1 T_2$ ($T_1$ and $T_2$ are the points of tangency), which intersects the sides of the angle at points $A_1$ and $A_2$. Prove that $A_1 T_1=A_2 T_2$ (or, equivalently, $A_1 T_2=A_2 T_1$). Two circles are inscribed in an angle. One of them touches the...

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Kvant Math Problem 21

Verified: no Verdicts: SKIP + SKIP Solve time: 1m29s Source on kvant.digital Problem Inside a square of side length 1, several circles are placed whose total circumference is 10. Prove that there exists a line intersecting at least four of these circles. Evening Mathematical School at Moscow State University (MSU) Exploration Let the circles have radii $r_1,r_2,\dots,r_n$. Since their total circumference is $10$, $$2\pi\sum_{i=1}^n r_i=10, \qquad \sum_{i=1}^n r_i=\frac5\pi.$$ The statement...

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Kvant Math Problem 19

Verified: no Verdicts: SKIP + SKIP Solve time: 1m20s Source on kvant.digital Problem In an infinite chain of nerve cells, each cell can be in one of two states: "rest" or "excited." If a cell becomes excited at a given moment, it sends a signal that reaches both of its neighboring cells after a unit of time (say, one millisecond). A cell becomes excited if and only if a signal...

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Kvant Math Problem 18

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem Prove that for any point $M$ on the circumcircle of the equilateral triangle $ABC$, one of the three segments $MA$, $MB$, $MC$ is equal to the sum of the other two. Three equal circles $\gamma_1$, $\gamma_2$, $\gamma_3$ are pairwise tangent to each other, and a circle $\gamma$ is circumscribed around them, touching all three: $\gamma_1$, $\gamma_2$, and...

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Kvant Math Problem 17

Verified: no Verdicts: SKIP + SKIP Solve time: 1m43s Source on kvant.digital Problem A peasant, having come to a fork where two roads diverged at an angle of $60\degree$, asked, “How do I get to the village of $NN$?” He was told: “Take the left road to the village of $N$, which is eight versts from here. There you will see a broad straight road branching off to the right...

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Kvant Math Problem 15

Verified: no Verdicts: SKIP + SKIP Solve time: 2m51s Source on kvant.digital Problem A square table of size $n\times n$ is filled with non-negative numbers such that the sum of the numbers in each row and the sum of the numbers in each column are both equal to 1. Prove that it is possible to select $n$ positive numbers from the table, no two of which are in the same...

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Kvant Math Problem 16

Verified: no Verdicts: SKIP + SKIP Solve time: 1m14s Source on kvant.digital Problem Prove that a polynomial $p(x)$ with integer coefficients, which takes the value 1 for three distinct integer values of $x$, cannot have any integer roots. Exploration Consider a polynomial $p(x)$ with integer coefficients that takes the value $1$ at three distinct integers, say $a$, $b$, and $c$. Denote $p(a) = p(b) = p(c) = 1$. A natural...

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Kvant Math Problem 664

Verified: no Verdicts: SKIP + SKIP Solve time: 4m04s Source on kvant.digital Problem Given a quadrilateral $ABCD$ with area $S$. Let the points of intersection of the altitudes of triangles $ABC$, $BCD$, $CDA$, $DAB$ be denoted by $H$, $K$, $L$, $M$ respectively. Prove that the area of the quadrilateral $HKLM$ is also equal to $S$. V. Batyrev, V. Trofimov Exploration Consider a convex quadrilateral $ABCD$ with area $S$. Denote by...

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Kvant Math Problem 13

Verified: no Verdicts: SKIP + SKIP Solve time: 1m21s Source on kvant.digital Problem Prove that if the difference between the largest and the smallest of the $n$ real numbers $a_1$, $a_2$, $\ldots$, $a_n$ is equal to $d$, and the sum of the absolute values of all $\dfrac{n(n-1)}{2}$ pairwise differences of these numbers $$\sum_{i \lt j} |a_i - a_j|$$ is equal to $s$, then $$(n-1)d \le s \le \frac{n^2}{4}d.$$ Exploration Consider...

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Kvant Math Problem 657

Verified: no Verdicts: SKIP + SKIP Solve time: 5m26s Source on kvant.digital Problem In the table $n \times n$ filled with numbers, all rows are distinct. Prove that it is possible to delete some column from the table so that all rows in the remaining table are still distinct. A. V. Anzhan Exploration Let the rows be $R_1,\dots,R_n$, each a vector of length $n$. We must show that some column...

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Kvant Math Problem 653

Verified: no Verdicts: SKIP + SKIP Solve time: 5m06s Source on kvant.digital Problem Fig. 1 There is a ruler with two markings (Fig. 1). Using this ruler, one can draw arbitrary straight lines and lay off segments of a given length. Using it, construct: any right angle; a line perpendicular to a given line. V. L. Gutenmacher Correspondence Mathematical Olympiads Exploration The ruler has two fixed marks. Let the distance...

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Kvant Math Problem 652

Verified: no Verdicts: SKIP + SKIP Solve time: 3m57s Source on kvant.digital Problem Zhenya cut a convex polyhedron made of cardboard into its faces (along the edges) and mailed this set of faces to Vitya. Using all of these faces, Vitya assembled a convex polyhedron. Can it happen that Zhenya's and Vitya's polyhedra are not congruent? N. B. Vasiliev Correspondence Mathematical Olympiads Exploration Consider the set of faces of a...

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Kvant Math Problem 648

Verified: no Verdicts: SKIP + SKIP Solve time: 7m48s Source on kvant.digital Problem Prove that if the diagonals of a cyclic quadrilateral are perpendicular, then the midpoints of its sides and the feet of the perpendiculars dropped from the intersection point of its diagonals to the sides all lie on a single circle. I. F. Sharygin Exploration Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ intersect at...

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Kvant Math Problem 647

Verified: no Verdicts: SKIP + SKIP Solve time: 5m42s Source on kvant.digital Problem Prove that for all $a\ge\dfrac12$, $b\ge\dfrac12$ the inequality $$\left(\dfrac{a^2-b^2}{2}\right)^2\ge\sqrt{\dfrac{a^2+b^2}{2}}-\dfrac{a+b}{2}.$$ holds. S. V. Fomin Exploration The inequality is symmetric in $a$ and $b$. The right-hand side contains the classical comparison between the root mean square and the arithmetic mean: $$\sqrt{\frac{a^2+b^2}{2}}\ge \frac{a+b}{2}.$$ Hence the right-hand side is nonnegative. The left-hand side equals $$\frac{(a^2-b^2)^2}{4} =\frac{(a-b)^2(a+b)^2}{4}.$$ Since $a,b\ge \tfrac12$, we have...

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Kvant Math Problem 643

Verified: no Verdicts: SKIP + SKIP Solve time: 5m44s Source on kvant.digital Problem Cards numbered 1, 2, $\ldots$, 32 are stacked in order. One may remove any number of cards from the top and insert them among the remaining cards or underneath them, preserving the order within both groups, but otherwise arbitrarily. This operation is called a shuffle . Prove that in 5 shuffles one can rearrange the cards into...

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Kvant Math Problem 639

Verified: no Verdicts: SKIP + SKIP Solve time: 6m42s Source on kvant.digital Problem In the tetrahedron $ABCD$, $(AC) \perp (BC)$ and $(AD) \perp (BD)$. Prove that the cosine of the angle between the lines $AC$ and $BD$ is less than $\dfrac{|CD|}{|AB|}$. Yu. V. Nesterenko All-Union Mathematical Olympiad (XIV, 1980, Grade 10) Exploration Let $$u=\overrightarrow{AC},\qquad v=\overrightarrow{BD}.$$ The assumptions are $$u\perp \overrightarrow{BC},\qquad \overrightarrow{AD}\perp v.$$ A vector formulation seems natural. Put $$x=\overrightarrow{AB}.$$ Then...

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Kvant Math Problem 635

Verified: no Verdicts: SKIP + SKIP Solve time: 8m36s Source on kvant.digital Problem The Mites living in Flower Town suddenly began coming down with influenza. On one particular day, several Mites caught cold and fell ill, and although no one caught cold thereafter, healthy Mites became ill when visiting their sick friends. It is known that each Mite has influenza for exactly one day; moreover, after that he has immunity...

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Kvant Math Problem 634

Verified: no Verdicts: SKIP + SKIP Solve time: 1m44s Source on kvant.digital Problem Let $S(n)$ denote the sum of all digits of a natural number $n$. Does there exist a natural number $n$ such that $n+S(n)=1980$? Prove that at least one of any two consecutive natural numbers can be represented in the form $n+S(n)$ for some third natural number $n$. S. V. Konyagin All-Union Mathematical Olympiad (XIV, 1980, Grade 8)...

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Kvant Math Problem 633

Verified: no Verdicts: SKIP + SKIP Solve time: 1m51s Source on kvant.digital Problem On a diameter $AC$ of a certain circle, a point $E$ is given. Draw through it a chord $BD$ so that the area of the quadrilateral $ABCD$ is maximized. I. F. Sharygin All-Union Mathematical Olympiad (XIV, 1980, 9th grade) Exploration Let the circle have center $O$ and radius $R$. The diameter is $AC$, and the point $E$...

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Kvant Math Problem 632

Verified: no Verdicts: SKIP + SKIP Solve time: 1m13s Source on kvant.digital Problem A cargo packed in containers must be delivered to the orbital space station Salyut. The number of containers is at least 35, and the total mass of the cargo is 18 tons. There are seven Progress transport spacecraft available, each of which can deliver 3 tons of cargo into orbit. It is known that these spacecraft can...

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Kvant Math Problem 631

Verified: no Verdicts: SKIP + SKIP Solve time: 1m45s Source on kvant.digital Problem The two-digit numbers from 19 to 80 are written consecutively. Is the resulting number 192021...7980 divisible by 1980? V. Fedotov All-Union Mathematical Olympiad (XIV, 1980, Grade 8) Exploration Let $$N=192021\cdots 7980.$$ Since $$1980=2^2\cdot 3^2\cdot 5\cdot 11,$$ checking divisibility by $1980$ amounts to checking divisibility by $4$, $9$, $5$, and $11$. The last digit of $N$ is $0$,...

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Kvant Math Problem 629

Verified: no Verdicts: SKIP + SKIP Solve time: 1m31s Source on kvant.digital Problem Prove that the number $2^{2n-1}-9n^2+21n-14$ is divisible by 27 for every positive integer $n$. Prove that if the numbers $a+b$ and $a^2+b$ are divisible by $m$, then $a^n+b$ is divisible by $m$ for every $n$ ($a$, $b$, and $m$ are some positive integers). Prove that if $f(n)=a^n+b_0+b_1n+\ldots+b_kn^k$ is divisible by $m$ for $n=1$, $n=2$, $\ldots$, $n=k+1$, $n=k+2$,...

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Kvant Math Problem 626

Verified: no Verdicts: SKIP + SKIP Solve time: 1m47s Source on kvant.digital Problem Each side of a convex quadrilateral is divided into 8 equal parts. Corresponding division points on opposite sides are connected to each other, and the resulting cells are colored in a checkerboard pattern. Prove that the sum of the areas of the black cells is equal to the sum of the areas of the white cells. V....

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Kvant Math Problem 625

Verified: no Verdicts: SKIP + SKIP Solve time: 1m43s Source on kvant.digital Problem On the coordinate plane, four points with rational coordinates are given, which do not lie at the vertices of a parallelogram, and no three of them lie on the same line. It is allowed to draw a line through any two already obtained points and mark the intersection point of any two drawn lines. Prove that the...

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Kvant Math Problem 624

Verified: no Verdicts: SKIP + SKIP Solve time: 1m25s Source on kvant.digital Problem Find the sequence $(a_n)$ defined by the conditions $a_1=1$, $$1+\textstyle\sum\limits_d{}(-1)^{n/d}a_{d}=0, \tag{*}$$, where the sum $\sum$ is taken over all divisors $d$ of the number $n$ (including $d=1$ and $d=n$). For example, if $n=p$ is a prime number, then (*) takes the form $1+(-1)^{p/1}a_1+(-1)^{p/p}a_p=0$, from which $a_p=2$ if $p=2$, and $a_p=0$ if $p\gt 2$. V. Abramovich Exploration Compute...

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Kvant Math Problem 623

Verified: no Verdicts: SKIP + SKIP Solve time: 1m11s Source on kvant.digital Problem How many axes of symmetry does a cube have? A regular triangular pyramid? Prove that if a certain polyhedron has $k$ axes of symmetry ($k\ge 1$), then $k$ is odd. V. A. Senderov Exploration A cube is highly symmetric, so the number of axes of symmetry should be larger than in simpler polyhedra. For a cube, one...

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Kvant Math Problem 622

Verified: no Verdicts: SKIP + SKIP Solve time: 1m15s Source on kvant.digital Problem Prove that the number of solutions of the equation $$x^3+y^2=z^3+t^2+1$$ in natural numbers not exceeding $10^6$ is less than the number of solutions of the equation $$x^3+y^2=z^3+t^2$$ in natural numbers not exceeding $10^6$. V. Vavilov Exploration Consider the two Diophantine equations $x^3 + y^2 = z^3 + t^2$ and $x^3 + y^2 = z^3 + t^2 +...

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Kvant Math Problem 617

Verified: no Verdicts: SKIP + SKIP Solve time: 1m49s Source on kvant.digital Problem Inside a triangle are four circles $\alpha$, $\beta$, $\gamma$, $\delta$ of equal radius, arranged so that each of the circles $\alpha$, $\beta$, $\gamma$ is tangent to two sides of the triangle and to the circle $\delta$ (Fig. 1). Prove that the center of the circle $\delta$ lies on the line passing through the centers of the inscribed...

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Kvant Math Problem 616

Verified: no Verdicts: SKIP + SKIP Solve time: 1m44s Source on kvant.digital Problem Can the numbers 1, 2, $\ldots$, 30 be partitioned into groups of five numbers each, of six numbers each, so that the sums of the numbers in all groups are equal? For which $n$ and $k$ can the numbers 1, 2, $\ldots$, $nk$ be partitioned into $n$ groups of $k$ numbers each so that the sums of...

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Kvant Math Problem 615

Verified: no Verdicts: SKIP + SKIP Solve time: 2m11s Source on kvant.digital Problem Prove that the perimeter of any cross-section of a triangular pyramid by a plane does not exceed the greatest of the perimeters of its faces. V. A. Senderov Exploration A triangular pyramid is a tetrahedron. A plane section of a tetrahedron is either a triangle or a quadrilateral. The statement compares the perimeter of the section with...

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Kvant Math Problem 614

Verified: no Verdicts: SKIP + SKIP Solve time: 2m04s Source on kvant.digital Problem For each natural number $n$, let $S(n)$ denote the sum of the digits of all natural numbers from 1 to $n$ (in decimal notation): $$\begin{gather*} S(1)=1,\quad S(2)=3,\quad S(3)=6,\quad \ldots,\quad S(9)=45,\ S(10)=46,\quad S(11)=48,\quad S(12)=51,\quad \ldots \end{gather*}$$ Find $S(100)$. Prove that $S(10^k-1)=45k\cdot10^{k-1}$ for all $k=1$, 2, $\ldots$ Prove that for any two-digit number $\overline{ab}$, $$S(\overline{ab})=5a^2+ab+41a+\dfrac{b(b+1)}{2}.$$ Find a similar formula...

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Kvant Math Problem 613

Verified: no Verdicts: SKIP + SKIP Solve time: 1m59s Source on kvant.digital Problem On the sides of triangle $ABC$, similar triangles $ADB$, $BEC$, and $CFA$ are constructed externally $\Big(\dfrac{|AD|}{|DB|}=\dfrac{|BE|}{|EC|}=\dfrac{|CF|}{|FA|}=k$; $\widehat{ADB}=\widehat{BEC}=\widehat{CFA}=\alpha\Big)$. Prove that: the midpoints of the segments $AC$, $DC$, $BC$, and $EF$ are the vertices of a parallelogram; this parallelogram has two angles equal to $\alpha$, and the ratio of the lengths of its sides is $k$. L. P. Kuptsov...

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Kvant Math Problem 612

Verified: no Verdicts: SKIP + SKIP Solve time: 1m39s Source on kvant.digital Problem An increasing sequence of natural numbers $(a_n)$ is such that $a_{n+1}\le 10a_n$. Prove that if all the numbers $a_n$ are written one after another (without spaces or commas), then the resulting sequence of digits is not periodic. A. Karagulyan Exploration Assume that the infinite digit string obtained by concatenating $$a_1,a_2,a_3,\dots$$ is periodic with period $T$. Then every...

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Kvant Math Problem 611

Verified: no Verdicts: SKIP + SKIP Solve time: 1m36s Source on kvant.digital Problem An arbitrary point $M$ is chosen on the chord $AB$ of a circle with center $O$. A circle is drawn through the points $A$, $M$, and $O$; it intersects the first circle at the points $A$ and $C$. Prove that $|MB|=|MC|$. S. Kolpakov, 10th-grade student Exploration The statement involves two circles. The first circle has center $O$,...

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Kvant Math Problem 610

Verified: no Verdicts: SKIP + SKIP Solve time: 2m04s Source on kvant.digital Problem Fix $k\in \N$. Consider the set of all tuples of integers $a_1$, $\ldots$, $a_k$ such that $0\le a_1\le a_2\le\ldots \le a_k \le k$; denote the number of such tuples by $N$. Among them, consider those for which $a_k=k$; let their number be $M$. Prove that $N=2M$. Impose an additional restriction on the tuples under consideration: the sum...

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Kvant Math Problem 609

Verified: no Verdicts: SKIP + SKIP Solve time: 2m05s Source on kvant.digital Problem The lengths of the projections of a convex polygon of area $S$ onto two mutually perpendicular lines are equal to $l_1$ and $l_2$. Prove that $S \le l_1l_2$. The lengths of the projections of a convex polyhedron of volume $V$ onto three mutually perpendicular lines are equal to $l_1$, $l_2$, and $l_3$. Prove that $V \le l_1l_2l_3$....

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Kvant Math Problem 608

Verified: no Verdicts: SKIP + SKIP Solve time: 1m46s Source on kvant.digital Problem On squared paper (with unit grid spacing), an $n$-gon is drawn, all of whose sides lie on grid lines and have odd length. Prove that $n$ is divisible by 4. Prove that when $n=100$, the area of this $n$-gon is necessarily odd. Determine the parity of the area for other values of $n$. M. L. Kontsevich, 10th-grade...

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Kvant Math Problem 607

Verified: no Verdicts: SKIP + SKIP Solve time: 2m03s Source on kvant.digital Problem Cut a square into isosceles trapezoids. Cut an isosceles right triangle into isosceles trapezoids. Prove that any polygon can be cut into isosceles trapezoids. V. F. Lev Exploration An isosceles trapezoid includes rectangles as a special case, since a rectangle has a pair of parallel sides and equal legs. Thus any rectangle is an isosceles trapezoid. For...

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Kvant Math Problem 606

Verified: no Verdicts: SKIP + SKIP Solve time: 1m48s Source on kvant.digital Problem The function $f$ is such that for all real $x$ $$f(x+1)+f(x-1)=\sqrt2,f(x).$$ Prove that $f$ is a periodic function. E. Turkevich Exploration The recurrence $$f(x+1)+f(x-1)=\sqrt2,f(x)$$ relates values of the function at arguments differing by integers. Since the coefficient is constant, the first idea is to look for solutions of the form $$f(x+n)=r^n.$$ Then $$r+r^{-1}=\sqrt2,$$ hence $$r^2-\sqrt2,r+1=0.$$ The discriminant...

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Kvant Math Problem 605

Verified: no Verdicts: SKIP + SKIP Solve time: 2m23s Source on kvant.digital Problem There are $2n+1$ distinct points marked on the plane. Number them 1, 2, $\ldots$, $2n+1$ and consider the following transformation $R$ of the plane: first, reflection with respect to the first point is performed, then with respect to the second point, and so on, up to the $(2n+1)$th point. Show that this transformation $R$ has a unique...

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Kvant Math Problem 604

Verified: no Verdicts: SKIP + SKIP Solve time: 58s Source on kvant.digital Problem The text version of the problem statement is currently being prepared. The graphical version can be viewed in the published issue. I cannot write a solution to Kvant problem M604 from the information provided, because the actual problem statement is missing. The prompt says: “The text version of the problem statement is currently being prepared. The graphical...

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Kvant Math Problem 603

Verified: no Verdicts: SKIP + SKIP Solve time: 1m39s Source on kvant.digital Problem Solve the system of equations $$x+\dfrac{3x-y}{x^2+y^2}=3,\quad y-\dfrac{x+3y}{x^2+y^2}=0.$$ L. P. Kuptsov Exploration The denominators suggest introducing $$r=x^2+y^2.$$ Since the system contains the expressions $\dfrac{3x-y}{r}$ and $\dfrac{x+3y}{r}$, multiplying by $r$ should convert the system into linear relations between $x,y$ and $r$. Multiplying both equations by $r$ gives $$xr+3x-y=3r,$$ $$yr-x-3y=0.$$ The second equation immediately yields $$x=y(r-3).$$ Substituting into the first...

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Kvant Math Problem 602

Verified: no Verdicts: SKIP + SKIP Solve time: 1m50s Source on kvant.digital Problem Fig. 1. In the $n$th row, the $k$th number is equal to $C_n^k=\dfrac{n(n-1)\ldots(n-k+1)}{1\cdot2\cdot\ldots\cdot k}$ In Pascal’s triangle, whose beginning is shown in Fig. 1 (each “interior” number is equal to the sum of the two numbers above it), the seventh row contains three consecutive numbers that form an arithmetic progression. In which subsequent row will the same...

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Kvant Math Problem 601

Verified: no Verdicts: SKIP + SKIP Solve time: 1m22s Source on kvant.digital Problem Prove that in any triangle $ABC$, the midpoint of side $BC$ lies on the segment joining the intersection point of the altitudes of triangle $ABC$ to the point on the circumcircle of this triangle that is diametrically opposite vertex $A$, and bisects this segment. G. Groshev, 10th-grade student Exploration Let $H$ be the orthocenter of triangle $ABC$,...

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Kvant Math Problem 600

Verified: no Verdicts: SKIP + SKIP Solve time: 2m02s Source on kvant.digital Problem Two cyclists ride along two intersecting circles. Each cyclist travels on his own circle at a constant speed. Starting simultaneously from one of the intersection points of the circles and completing one revolution each, the cyclists meet again at that point. Prove that in the plane containing the circles there exists a fixed point such that its...

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Kvant Math Problem 27

Verified: no Verdicts: SKIP + SKIP Solve time: 9m56s Source on kvant.digital Problem Prove that if $$\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}=0,$$, then $$\frac{a}{(b-c)^2}+\frac{b}{(c-a)^2}+\frac{c}{(a-b)^2}=0.$$. Exploration Let $$ S=\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b}. $$ The denominators suggest introducing the differences $$ x=b-c,\qquad y=c-a. $$ Then $$ a=c-y,\qquad b=c+x,\qquad a-b=-(x+y). $$ Substituting into $S$ gives $$ S=\frac{c-y}{x}+\frac{c+x}{y}-\frac{c}{x+y}. $$ After bringing the terms to the common denominator $xy(x+y)$, the numerator becomes $$ c(x^2+xy+y^2)+x^3+x^2y-xy^2-y^3. $$ The cubic part factors as $$ x^3+x^2y-xy^2-y^3=(x-y)(x+y)^2. $$...

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Kvant Math Problem 14

Verified: no Verdicts: SKIP + SKIP Solve time: 3m28s Source on kvant.digital Problem Fig. 2 A convex white polyhedron has some of its faces painted black in such a way that no two black faces share an edge (Fig. 2). Prove that if at least one of the following conditions holds: the number of black faces is greater than half the total number of faces; the total area of the...

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Kvant Math Problem 11

Verified: no Verdicts: SKIP + SKIP Solve time: 4m23s Source on kvant.digital Problem Forty-four merry siskins were perched on 44 trees arranged in a circle (one siskin on each tree). From time to time, two siskins simultaneously fly to neighboring trees in opposite directions (one clockwise, the other counterclockwise; see Fig. 1). Prove that the siskins will never gather on a single tree. What if the number of siskins and...

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Kvant Math Problem 1

Verified: no Verdicts: SKIP + SKIP Solve time: 3m46s Source on kvant.digital Problem In the country of Anchuria, ruled by President Miraflores, the time for a new presidential election is approaching. The country has exactly 20 million voters, of whom only one percent (the regular army of Anchuria) support Miraflores. Naturally, Miraflores wants to be elected, but at the same time, he wants the election to appear democratic. By "democratic...

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Kvant Math Problem 10

Verified: no Verdicts: SKIP + SKIP Solve time: 4m14s Source on kvant.digital Problem Four circles whose centers are the vertices of a convex quadrilateral completely cover that quadrilateral. Prove that one can choose three of these circles that cover the triangle whose vertices are the centers of those circles. G. A. Galperin G. A. Galperin Exploration Let the centers of the circles be the vertices $A,B,C,D$ of a convex quadrilateral,...

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Kvant Math Problem 9

Verified: no Verdicts: SKIP + SKIP Solve time: 3m50s Source on kvant.digital Problem Consider the following properties of a tetrahedron (by a tetrahedron we mean an arbitrary triangular pyramid): all faces have equal area; each edge is equal to its opposite edge; all faces are congruent; the centers of the circumscribed and inscribed spheres coincide; the sums of the plane angles at each vertex of the tetrahedron are equal. Prove...

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Kvant Math Problem 8

Verified: no Verdicts: SKIP + SKIP Solve time: 3m33s Source on kvant.digital Problem Two people play the following game. From a pile containing 25 matches, they take turns taking 1, 2, or 3 matches. The winner is the one who, at the end of the game—after all the matches have been taken—has an even number of matches. Who wins with correct play—the first player or the second? How should the...

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Kvant Math Problem 7

Verified: no Verdicts: SKIP + SKIP Solve time: 5m50s Source on kvant.digital Problem $a$, $b$, $c$ are the sides of a triangle. Prove that $$\frac{a}{b+c-a} + \frac{b}{c+a-b} + \frac{c}{a+b-c} \geq 3.$$ S. T. Berkolayko Exploration Let $$S=\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c}.$$ Since $a,b,c$ are sides of a triangle, each denominator is positive by the triangle inequalities. A natural test is the equilateral triangle. For $a=b=c$, $$S=1+1+1=3.$$ This suggests that $3$ may be the minimum....

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Kvant Math Problem 6

Verified: no Verdicts: SKIP + SKIP Solve time: 3m07s Source on kvant.digital Problem You are given a clock. How many positions of the hands exist for which it is impossible to determine the time without knowing which hand is the hour hand and which is the minute hand? ( It is assumed that the position of each hand can be determined exactly, but observing their movement is not allowed. )...

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Kvant Math Problem 5

Verified: no Verdicts: SKIP + SKIP Solve time: 3m02s Source on kvant.digital Problem In the set $E$, consisting of $n$ elements, $m$ distinct subsets (different from $E$ itself) have been selected so that for every two elements of $E$ there exists exactly one of the given subsets containing both of these elements. Prove that $m \geq n$. In what cases can equality occur? N. Bourbaki Exploration Let $E$ be a...

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Kvant Math Problem 4

Verified: no Verdicts: SKIP + SKIP Solve time: 3m05s Source on kvant.digital Problem Given a segment $AB$. Find the set of points $C$ in the plane such that in the triangle $ABC$ the median drawn from vertex $A$ is equal to the altitude drawn from vertex $B$. Moscow Mathematical Olympiad (XXXII) Exploration Fix the segment $AB$ and let its length be $d$. The condition says that the median from $A$...

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Kvant Math Problem 2

Verified: no Verdicts: SKIP + SKIP Solve time: 2m19s Source on kvant.digital Problem A sphere of radius 1 is given. On it are placed equal circles $\gamma_0$, $\gamma_1$, $\ldots$, $\gamma_n$ of radius $r$ ($n\ge3$). The circle $\gamma_0$ is tangent to all circles $\gamma_1$, $\ldots$, $\gamma_n$; moreover, the circles $\gamma_1$ and $\gamma_2$; $\gamma_2$ and $\gamma_3$; $\ldots$; $\gamma_n$ and $\gamma_1$ are tangent to each other. For which $n$ is this possible? Calculate...

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Kvant Math Problem 3

Verified: no Verdicts: SKIP + SKIP Solve time: 4m10s Source on kvant.digital Problem In Figure 1, the plane is tiled by squares of five colors. The centers of squares of the same color are located at the vertices of a square lattice. For what numbers of colors is a similar tiling of the plane possible? Fig. 1 Fig. 2 In Figure 2, the plane is tiled by hexagons of seven...

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Kvant Math Problem 241

Verified: no Verdicts: SKIP + SKIP Solve time: 1m57s Source on kvant.digital Problem Prove that $3^{1974}+5^{1974}$ is divisible by 13. S. I. Meyzus Exploration The exponent $1974$ is large, so direct computation is impossible. Since the statement concerns divisibility by $13$, the natural step is to work modulo $13$. Compute small powers. Since $$3^3=27\equiv 1 \pmod{13},$$ the powers of $3$ repeat with period $3$. Also $$5^2=25\equiv -1 \pmod{13},$$ hence $$5^4\equiv...

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Mathematics

Mathematics Worked solutions to several mathematical problem collections. Each collection has its own index with the full list of problems. The International Mathematical Olympiad sets the hardest problems given to high-school students anywhere, six a year since 1959. Around the contest sit its two feeder collections: the IMO Shortlist , the pool from which each year's jury picks the final six, and the IMO Longlist , the wider set of...

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Kvant Mathematics

Kvant Mathematics Solutions to mathematics problems from Kvant magazine (925 solved, 0 verified). # Status Time 1 solved 3m46s 2 solved 2m19s 3 solved 4m10s 4 solved 3m05s 5 solved 3m02s 6 solved 3m07s 7 solved 5m50s 8 solved 3m33s 9 solved 3m50s 10 solved 4m14s 11 solved 4m23s 13 solved 1m21s 14 solved 3m28s 15 solved 2m51s 16 solved 1m14s 17 solved 1m43s 18 solved 1m36s 19 solved 1m20s...

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