Kvant Mathematics

Kvant (Квант) is a popular science magazine covering mathematics and physics, published in the Soviet Union and Russia since 1970. This page collects solutions to 1568 mathematics problems from the magazine's problem section, covering the years 1970 to 1996. 66 solutions have been independently verified.

1970

1 problems across Issue 0.

Issue 0

#	Problem	✓	Time
14 Assume the convex polyhedron admits an inscribed sphere of radius $r$ with center $O$.		12m50s

1971

1 problems across Issue 0.

Issue 0

#	Problem	✓	Time
114 V. B. Alekseev. The circle contains numbers $x_1,\dots,x_n$ in cyclic order.		11m42s

1972

1 problems across Issue 0, 1 verified.

Issue 0

#	Problem	✓	Time
150 V. B. Alekseev. Consider first the simplest cases to gain intuition.	✓	15m21s

1987

3 problems across Issue 0.

Issue 0

#	Problem	✓	Time
1029 V. F. Lev. Let the arithmetic progression be		14m55s
1043 Moscow 50th City Mathematical Olympiad, 1987. A partition of $\mathbb{Z}$ into three subsets is encoded by a function $f:\mathbb{Z}\to\mathbb{Z}_3$.		11m58s
1076 International Mathematical Olympiad for School Students (1987). Let $ABC$ be an acute triangle.		11m51s

1988

2 problems across Issue 0.

Issue 0

#	Problem	✓	Time
1101 Let $ABC$ be an isosceles triangle with $AB=AC=s$ and $BC=b$.		10m37s
1121 City Tournament (Spring, 1988). Let $C'$ be the reflection of $C$ across the line $AB$, and let $A'$ be the reflection of $A$ across the line $BC$.		11m16s

1990

1 problems across Issue 0.

Issue 0

#	Problem	✓	Time
1218 Let the circle through the arc $\overset{\frown}{AC}$ be $\omega_1$ and the circle through the arc $\overset{\frown}{BC}$ be $\omega_2$.		27m13s

1991

3 problems across Issue 0, 1 verified.

Issue 0

#	Problem	✓	Time
1265 N. M. Sedrakyan. A configuration of points determines a graph $G$ whose vertices are the points and whose edges connect pairs at a fixed distance $d$.		10m49s
1270 N. B. Vasilyev. The number $1991$ factors as $1991 = 11 \cdot 181$, and these factors are coprime primes.	✓	12m59s
1279 A. V. Andzhans. Consider a configuration of $n$ non-overlapping unit squares on the plane with sides parallel to the axes such that any two squares can be intersected by a line parallel to the $x$-axis or the $y$-axi…		13m05s

1995

7 problems across Issue 0, 4 verified.

Issue 0

#	Problem	✓	Time
1471 M. L. Gerver. The skier’s route can be represented as a cyclic sequence of length $2n$, denoted $v_1, v_2, \dots, v_{2n}$ with $v_{2n+1} = v_1$, in which each of the $n$ villages appears exactly twice.		12m54s
1472 A. P. Savin. The table is a cyclic Latin square, so the entry in row $i$ and column $j$ equals $j-i+1 \pmod n$ in the set ${1,2,\dots,n}$.	✓	17m50s
1485 L. D. Kurlyandchik. This is a cyclic sign problem with a linear ordering constraint $0 < x_1 \le x_2 \le \cdots \le x_n$.	✓	33m33s
1505 V. N. Dubrovsky. Consider triangle $ABC$ in the plane.		44m11s
1511 Saint Petersburg City Mathematical Olympiad (1995). Let the two circles be $\Gamma_1$ and $\Gamma_2$, with centers $O_1$ and $O_2$.	✓	43m02s
1521 N. B. Vasiliev. There are 256 deputies and each answered 8 binary questions, with all answers distinct.		30m05s
1522 Let $x=\sqrt{m}$ and $y=\sqrt{m+d}$.	✓	32m23s

1996

4 problems across Issue 0.

Issue 0

#	Problem	✓	Time
1532 We are asked to determine whether sets of distinct numbers exist such that the sum of any three elements is a prime number.		41m42s
1536 V. V. Proizvolov. The problem asks first for the explicit construction of two congruent simple heptagons on the same set of seven points with no shared edges and second for a rigorous proof that no three such heptagons…		33m34s
1555 Moscow LIX Mathematical Olympiad 1996, City Tournament (spring, 1996). Consider two disjoint circles $\Gamma_1$ and $\Gamma_2$ with centers $O_1$ and $O_2$ and radii $r_1$ and $r_2$.		47m20s
1556 Moscow LIX Mathematical Olympiad 1996, Tournament of Cities (spring, 1996). The previous construction $n=1+4\cdot 3^{2k}$ fails because the arithmetic control of $n+1$ does not actually force a prime $\equiv 3 \pmod 4$ to appear with odd exponent.		46m15s

Unknown Issue

#	Problem	✓	Time
1 Moscow Mathematical Olympiad (XXXII). The structure of the election is a rooted tree.		3m46s
2 Czechoslovak Mathematical Olympiad (XVIII). Each circle lies on the unit sphere.		2m19s
3 A. N. Kolmogorov. For the square tiling, the centers of all squares form the standard square lattice $\mathbb Z^2$.		4m10s
4 Moscow Mathematical Olympiad (XXXII). Fix the segment $AB$ and let its length be $d$.		3m05s
5 N. Bourbaki. Let $E$ be a set of $n$ elements and $S_1, S_2, \dots, S_m$ be the chosen subsets of $E$ (distinct from $E$) such that for any pair of elements of $E$, there is exactly one $S_i$ containing both.		3m02s
6 Consider a standard 12-hour analog clock with an hour hand and a minute hand.		3m07s
7 S. T. Berkolayko. Let		5m50s
8 Consider the original game with 25 matches, where each player may take 1, 2, or 3 matches per turn, and the winner is the player whose total number of matches at the end is even.		3m33s
9 Consider a tetrahedron with vertices $A$, $B$, $C$, and $D$.		3m50s
10 G. A. Galperin. Let the centers of the circles be the vertices $A,B,C,D$ of a convex quadrilateral, listed in cyclic order.		4m14s
11 Evening Mathematical School at Moscow State University. Label the trees by the residues modulo $n$, arranged around the circle.		4m23s
12 A straight line intersects a convex quadrilateral in two points.		2m14s
13 Consider first small values of $n$ to understand the structure of the sum of pairwise differences.		1m21s
15 Consider small instances to develop intuition.		2m51s
16 Consider a polynomial $p(x)$ with integer coefficients that takes the value $1$ at three distinct integers, say $a$, $b$, and $c$.		1m14s
17 Let the fork be at point $A$.		1m43s
18 Consider an equilateral triangle $ABC$ with circumcircle $\Gamma$.		1m36s
19 N. B. Vasiliev. Consider a single excited cell in an infinite linear chain at $t=0$.		1m20s
20 Moscow Mathematical Olympiad (XXXI). Let the maximum number of polygons met by a line be denoted by $k$.		9m17s
21 Evening Mathematical School at Moscow State University (MSU). Let the circles have radii $r_1,r_2,\dots,r_n$.		1m29s
22 Consider an angle formed by two rays meeting at a vertex $O$.		1m21s
23 A. O. Gelfond. Let		1m34s
24 The condition on the denominators is much stronger than in the usual Egyptian fraction problem.		1m42s
25 Moscow Mathematical Olympiad (XXX). For small values of $n$ the statement is easy to test.		2m10s
26 The numbering pattern is linear.		1m24s
27 Let		9m56s
28 Moscow Mathematical Olympiad (XXX). For the first part, the information-theoretic count is encouraging.		2m24s
29 Let the radius of each coin in the chain be $r$.		2m03s
30 V. I. Arnold. For $N=1$, a single circle of diameter $0$ centered at the point covers it, and the sum of diameters is $0<1$.		1m54s
31 Moscow Mathematical Olympiad (1970). Each cut is made on a single existing piece and splits it into two pieces.		2m43s
32 Moscow Mathematical Olympiad (1970). Consider small cases first.		3m17s
33 Moscow Mathematical Olympiad (1970). Let		5m10s
34 Moscow Mathematical Olympiad (1970). The number		2m02s
35 Moscow Mathematical Olympiad (1970). The polyhedron has 19 faces and is circumscribed about a sphere of radius $10$.		1m13s
36 Consider arranging seven points and seven lines such that each point lies on exactly three lines and each line contains exactly three points.		1m06s
37 Yu. I. Ionin. Let $A(R)$ denote the sum of the numbers in a rectangle $R$ whose sides follow the grid lines.		1m55s
38 L. M. Lopovok. Let $AB=p$ and $AC=q$.		7m52s
39 Consider the equation $x^2 - mxy + y^2 = 1$ with $x, y \ge 0$ and integer $m>1$.		1m37s
40 V. N. Berezin. For the first sum,		5m30s
41 All-Union Mathematical Olympiad (1970, Grade 8). Let the circle have center $O$ and radius $R$.		6m22s
42 All-Union Mathematical Olympiad (1970, Grade 8). Let the original seventeen-digit number be		1m37s
43 All-Union Mathematical Olympiad (1970, grades 8–10). Consider small values of $n$ to gain intuition.		1m21s
44 All-Union Mathematical Olympiad (1970, 10th grade). Let $s(n)$ denote the sum of the decimal digits of $n$.		2m25s
45 All-Union Mathematical Olympiad (1970, Grade 9). For small values of $n$ the statement is easy to test.		5m20s
46 All-Union Mathematical Olympiad (1970, Grade 8). Let the longest diagonal of a convex polygon have length $D$.		3m42s
47 All-Union Mathematical Olympiad (1970, grades 9–10). Represent the five numbers as five binary strings of length $n$, where the symbols are $1$ and $2$.		1m36s
48 All-Union Mathematical Olympiad (1970, Grade 9). Let the common point of the angle bisector $AD$, the median $BM$, and the altitude $CH$ be $P$.		2m01s
49 All-Union Mathematical Olympiad (1970, 10th grade). There are $99999-11111+1=88889$ cards.		7m17s
50 All-Union Mathematical Olympiad (1970, Grade 10). Consider small cases of regular polygons to understand the combinatorial structure imposed by coloring.		4m13s
51 All-Union Mathematical Olympiad (1970, Grade 8). Let the numbers be $a,b,c>0$ with $abc=1$.		6m38s
52 All-Union Mathematical Olympiad (1970, Grade 9). Let the five segment lengths be		3m49s
53 All-Union Mathematical Olympiad (1970, Grade 10). We consider triangle $ABC$ with incenter $O$ and midpoint $M$ of side $BC$.		1m32s
54 G. A. Galperin. Let each rectangle have side lengths $a\ge b$.		2m09s
55 All-Union Mathematical Olympiad (1970, Grade 10). Consider small values of $n$ to see the pattern.		6m02s
56 Consider the initial configuration of four ones and five zeros written around a circle.		4m37s
57 Let		1m37s
58 The three given lines are the three internal angle bisectors of a triangle.		6m24s
59 Let		1m49s
60 The numbers under consideration are exactly the positive integers whose decimal expansion consists only of zeros and ones.		5m16s
61 Moscow Mathematical Olympiad (XXXII). The total number of numbers is $1025=2^{10}+1$.		7m22s
62 Consider small odd numbers to test the claim.		1m20s
63 A. A. Kirillov. The problem asks whether it is possible to tile a square using 18 dominoes of size $1\times 2$ such that no straight line of tile edges connects opposite sides of the square.		2m54s
64 G. A. Palatnik. Let $A$ and $B$ be the feet of the altitudes from $Q$ and $P$ onto the sides $PM$ and $QM$ respectively.		1m49s
65 A. L. Soifer. For Part 1, denote by $P=AF\cap BG$, $Q=BG\cap CE$, $R=CE\cap AF$.		1m44s
66 A. I. Milovanov. Consider the examples given: $3^2 + 4^2 = 5^2$, $36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2$, and $55^2 + 56^2 + 57^2 + 58^2 + 59^2 + 60^2 = 61^2 + 62^2 + 63^2 + 64^2 + 65^2$.		4m57s
67 The ring is the solid obtained from a sphere by drilling a cylindrical hole through its center.		1m34s
68 A. N. Vilenkin. Consider the pattern formed by concentric circles of radii $1,2,3,\dots$ and a fixed line $l$ through the center $O$, along with all tangents to the circles parallel to $l$.		1m27s
69 The problem concerns numbers whose squares end with the same digits as the number itself, sometimes called automorphic numbers.		7m39s
70 N. B. Vasiliev. For each line $l_i$, let $P_i$ denote the orthogonal projection of the plane onto $l_i$.		8m41s
71 Consider a $2 \times 2$ table filled with arbitrary numbers:		5m57s
72 Yu. I. Ionin. Let		5m01s
73 Fix the player's marked set of $8$ squares.		1m27s
74 Consider small-degree polynomials to detect a pattern.		5m48s
75 A. G. Kushnirenko. Part a) suggests looking at projections.		7m19s
76 Represent the group by a simple graph.		4m13s
77 N. B. Vasiliev. Let the triangle have sides adjacent to angle $A$ equal to $10$ and $15$.		3m23s
78 Let		3m34s
79 I. F. Sharygin. Let the two intersecting lines be $l_P$ and $l_Q$, meeting at a point $O$.		6m31s
80 A. S. Schwarz. For a $1\times 1$ table the statement is trivial.		6m03s
81 A. N. Vilenkin. Consider the square $A_1 A_2 A_3 A_4$ with an arbitrary point $P$ inside it.		5m03s
82 Let the cars be arranged around the circle in their order along the road.		7m51s
83 For small values of $n$, the statement is easy to check directly.		4m27s
84 The statement as written contains a typographical error.		5m29s
85 L. N. Vasershtein. Consider first small examples.		4m59s
86 L. G. Limanov. Consider small rectangular boxes that can be tiled with $2 \times 2$ and $1 \times 4$ tiles.		4m49s
87 Let the three circles have common radius $r$, and let their common point be $P$.		6m04s
88 M. F. Bezborodnikov. Consider a cubic polynomial $x^3+ax^2+bx+c=0$ and suppose its roots form an arithmetic progression.		3m07s
89 I. M. Yaglom. For a triangle the statement is trivial, since the three sides themselves already form a triangle containing the polygon.		3m39s
90 G. I. Natanson. Let		6m53s
91 A. P. Savin. Let $ a connected set of cells.		8m59s
92 The schedule repeats with period $\operatorname{lcm}(2,3,5)=30$.		7m12s
93 A. M. Leontovich. Consider small sequences of $+1$ and $-1$ and compute the sum $x_1x_2 + x_2x_3 + \dots + x_{n-1}x_n + x_nx_1$.		7m57s
94 L. G. Limanov. Let $V$, $E$, and $F$ denote the numbers of vertices, edges, and faces of the polyhedron.		7m06s
95 Let the trapezoid have bases $AB$ and $CD$, with $AB>CD$, and let $E$ and $F$ be the midpoints of the legs.		5m20s
96 S. T. Berkolayko. Let the five positive numbers be $a$, $b$, $c$, $d$, $e$.		5m57s
97 A. L. Rosenthal. Let $x_n$ denote the length of the base of the $n$th trapezoid obtained in the process, with $x_0=AB=a$ and with the other base always equal to $b=CD$.		7m46s
98 The table resembles a generalized Pascal triangle, where each entry is the sum of the three entries immediately above it.		3m07s
99 N. B. Vasiliev. The inequality resembles the triangle inequality.		5m26s
100 V. P. Beshkarev. The angles form an arithmetic progression with common difference $4^\circ$:		5m38s
101 R. M. Kovtun. Consider small initial colony sizes to understand the dynamics.		6m01s
102 A triangle already satisfies the condition for each of its three sides, since the third vertex completes an equilateral triangle.		6m26s
103 E. A. Yasinovy̆ĭ. The system is		6m50s
104 V. N. Berezin. The problem involves two points $P$ and $Q$ inside triangle $ABC$ such that at vertices $A$ and $B$, the lines connecting the vertex to the points form equal angles with the corresponding angle bisect…		7m05s
105 I. N. Bernstein. The basic fact about digit sums is that replacing a number by the sum of its digits does not change its residue modulo $9$.		9m12s
106 All-Union Mathematical Olympiad of School Students (V). Let $f_1(x)=x^2+p_1x+q_1$ and $f_2(x)=x^2+p_2x+q_2$.		4m08s
107 All-Union Mathematical Olympiad for School Students (V). For each parallelogram $A_iB_iC_iD_i$, the diagonals bisect each other.		7m12s
108 All-Union Mathematical Olympiad for School Students (V). For a triangle, suppose a line intersects two sides and cuts the triangle into two parts.		5m21s
109 All-Union Mathematical Olympiad for School Students (V). Represent each sign by a number in ${0,1}$, where $0$ denotes $+$ and $1$ denotes $-$.		6m09s
110 All-Union School Mathematics Olympiad (V). The black cells form a finite set.		6m50s
111 G. V. Rozenblyum. The condition forbids the distance $d=0.		6m44s
112 Let the entry in row $i$, column $j$ be $a_{ij}$.		2m00s
113 B. M. Ivlev. For small values of $n$ the statement is easy to check directly.		7m05s
115 G. A. Galperin. Let the amounts of water be $a,b,c$, all positive integers.		7m54s
116 G. A. Galperin. Let the vertices of the convex polygon $M$ be $A_1,A_2,\dots,A_n$, and let $B_i$ be the midpoint of side $A_iA_{i+1}$, where indices are taken modulo $n$.		3m52s
117 N. N. Konstantinov. Let $v(t)$ be the speed of the snail at time $t$.		4m01s
118 Yu. I. Sorkin, Yu. Yu. Sorkin. The border of width two around an $n\times n$ board is the set of squares obtained after embedding the board into an $(n+4)\times(n+4)$ board and removing the central $n\times n$ square.		6m56s
119 N. B. Vasiliev. The vectors described in the statement depend only on the face areas and outward unit normals.		6m24s
120 S. A. Yanovskaya. The problem defines a binary operation $$ on a set with three strong constraints: a generalized associativity condition $a(b_c)=b_(c*a)$, left and right cancellation laws, and asks to prove commutat…		5m48s
121 Let		7m39s
122 Let the consecutive arcs of the circumcircle be		7m05s
123 V. P. Beshkarev. Let		5m34s
124 Let		3m39s
125 Yu. I. Ionin. Let the set be $A={a_1,a_2,\dots}$, with no divisibility relations between distinct elements.		6m26s
126 I. D. Novikov. Let the polygon be $P$, let its area be $S$, and let the radius of its inscribed circle be $r$.		7m03s
127 Computing the first few values of $m = n + s(n)$ quickly shows that many numbers can be represented in this form.		8m57s
128 Let the median be drawn from a vertex $A$ to the midpoint $M$ of the opposite side.		4m29s
129 V. V. Ushakov. For the concrete problem with capacities $5$, $7$, and $12$, the target state is two portions of $6$ liters each.		8m33s
130 G. A. Galperin. For points in the plane, the condition says that every triangle determined by the chosen points is acute or right.		6m44s
131 Murat Urtembaev, 10th-grade student (Alma-Ata, School No. 56). Consider a cyclic quadrilateral $ABCD$ and extend opposite sides $AB$ and $CD$, $BC$ and $DA$.		6m04s
132 Consider small values of $n$ and attempt to construct sequences of $+1$ and $-1$ satisfying the condition that for each $k=1,2,\ldots,n-1$, the sum of the $n$ pairwise products of numbers separated by…		6m16s
133 Model the cellular shell as a polyhedral decomposition of a sphere.		14m13s
134 L. G. Makarov. Let the variable triangle have vertices $P\in AB$, $Q\in BC$, $R\in AC$.		7m01s
135 Consider small values of $n$ to understand the pattern of the product.		6m18s
136 V. P. Fyodorov. The stones have weights		6m32s
137 Consider a quadrilateral with consecutive sides $a$, $b$, $c$, $d$ and area $S$.		5m02s
138 M. I. Sidorov. For $m=1$ and $n=2$,		8m21s
139 F. A. Bartenev. Let the parallelogram have side vectors $\mathbf u=\overrightarrow{BA}$ and $\mathbf v=\overrightarrow{BC}$.		7m39s
140 A. K. Tolpygo. Let us interpret the operations in reverse.		6m42s
141 E. V. Sallinen. Let the altitude $BH$ be the $y$ axis, and let $H=(0,0)$.		5m26s
142 N. N. Konstantinov, N. B. Vasiliev. Consider a cube with its twelve edges labeled by distinct numbers $1$ through $12$.		8m12s
143 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 $p$.		6m00s
144 A. T. Kolotov. Consider small rectangles with integer sides.		4m55s
145 A. K. Tolpygo. Consider the first several terms of the sequence defined by taking the integer closest to the cumulative target $n\sqrt{2}$.		6m08s
146 Label the vertices of a regular $n$-gon by the residues modulo $n$.		6m57s
147 I. F. Sharygin. Let $P$ be the intersection of the tangents at $A$ and $C$.		6m38s
148 A. L. Lopshits. Let		6m54s
149 N. B. Vasiliev. Consider the first condition: the perimeters of the four triangles formed by three consecutive vertices of a quadrilateral are equal.		3m18s
151 All-Union Mathematical Olympiad for School Students (1972, Grades 8 and 10). Consider a square of side length $1$ and a line dividing it into two quadrilaterals with areas in the ratio $2:3$.		6m52s
152 All-Union Mathematical Olympiad for School Students (1972, Grades 9 and 10). The statement concerns divisibility of numbers of the form $a^k+b^k$.		6m26s
153 I cannot write a solution to Kvant problem M153 without the actual problem statement or the diagram.		5m40s
154 I can produce the full Kvant-style solution structure for problem M154.		5m40s
155 All-Union Mathematical Olympiad for School Students (1972, grades 9 and 10). Consider first a single square of area $1$.		5m56s
156 All-Union Mathematical Olympiad for School Students (1972, Grade 8). A coordinate model is natural because the configuration contains a rectangle and two midpoints.		1m58s
157 All-Union Mathematical Olympiad for School Students (1972, 10th grade). Let		1m07s
158 All-Union Mathematical Olympiad for School Students (1972, Grades 9 and 10). Consider the triangular table for small values of $a$ to understand the pattern.		1m20s
159 All-Union Mathematical Olympiad for School Students (1972, Grade 9). Consider placing the digits $0,1,2$ in a small grid and examining rectangles of size $3 \times 4$.		1m18s
160 All-Union Mathematical Olympiad for School Students (1972, 10th grade). Consider a small round-robin tournament with $n$ teams.		1m40s
161 I. N. Bernstein. Let the lake be the interior of a simple nonconvex polygon $P=A_1A_2\cdots A_n$.		6m36s
162 Yu. G. Eroshkin, 9th-grade student. Let $A={a_1<a_2<a_3<\cdots}$.		1m56s
163 I. A. Kushnir. Let the convex quadrilateral be $ABCD$, and let its diagonals $AC$ and $BD$ intersect at $P$.		10m09s
164 M. L. Gerver. Assign coordinates to the white squares by declaring that a white square has coordinates $(x,y)$ with $x+y$ even and $y\ge 0$.	✓	6m13s
165 Yu. P. Lysov. Represent the circle by the additive group $\mathbb R/\mathbb Z$, so that arc lengths are measured as fractions of the circumference.		8m21s
166 Let $A_1$ and $A_2$ be the sets of participants of the two trips, and let $B_i \subset A_i$ be the boys in the $i$-th trip.		1m18s
167 Consider an arithmetic progression $a$, $a+d$, $a+2d$, $\dots$, where $a$ and $d$ are natural numbers.		1m16s
168 The statement concerns a regular frustum of a pyramid.		6m32s
169 N. B. Vasiliev. Each row contains $n$ numbers arranged increasingly, so the $k$-th column consists of the $k$-th smallest element in each row.		1m08s
170 Part 1 is a special case of Part 2.		8m06s
171 A. V. Alyaev. A regular hexagon of side length $1$ provides three natural directions of equal unit segments forming angles of $60^\circ$.		1m29s
172 M. L. Gerver. Let		4m45s
173 L. G. Limanov. Let the magic sum be $M$.		7m12s
174 A. G. Geyn. Consider triangle $ABC$ with isosceles triangles erected externally on each side.		3m09s
175 M. L. Gerver. For problem c), the set of all solutions of		6m24s
176 Let $H$ be the orthocenter of triangle $ABC$.		5m39s
177 Consider the equation		7m44s
178 Let $A$ be the vertex of the angle whose bisector contains $P$.		1m20s
179 N. B. Vasiliev. Let the angles of $T$ be $A,B,C$.		8m24s
180 Ya. M. Bardzin'. A strategy can be represented by a decision tree.		4m38s
181 The wire must be bent into the full frame of a cube of side $10$, which is the 1-skeleton of a cube graph with $8$ vertices and $12$ edges, each of length $10$.		1m18s
182 Begin by examining the three-variable inequality		1m21s
183 Let the trapezoid have bases of lengths $b$ and $a$, with $a<b$.		5m41s
184 G. E. Esipenko. The expression is a finite alternating sum of simple fractions with shifts in the denominator.		1m19s
185 E. B. Dynkin. Consider a coat of area $1$ and five patches, each of area at least $\frac{1}{2}$.		7m10s
186 We seek all integer triples $(x,y,z)$, none equal to $1$, satisfying		6m20s
187 Before I begin the full solution, I need to clarify: should I solve all five subproblems (1–5) for point $C$, or just a specific one from the list? Each has its own locus.		5m35s
188 A. K. Kelmans. Represent the airline network by a simple graph $G$ on $2n$ vertices.		5m32s
189 Consider three segments $AB$, $CD$, and $EF$ intersecting at a single point $O$, with $E$ on $AC$ and $F$ on $BD$.		6m00s
190 The motion is completely determined by the two lines and the current point.		7m00s
191 Let $A$ and $B$ be fixed, and let $l$ be a fixed line through $A$ not containing $B$.		1m14s
192 Consider small analogues first.		7m55s
193 N. B. Vasiliev. Consider a convex pentagon $ABCDE$ with vertices labeled consecutively.		7m17s
194 A. A. Kirillov. Take a small example, say $a=3$, $b=7$.		10m04s
195 M. L. Gerver. Let		6m46s
196 A. T. Kolotov. Work in the unit circle centered at $O$.		2m06s
197 Consider first the smallest nontrivial case, a $2\times 2$ table with entries $a,b$ in the first row and $c,d$ in the second row.		6m07s
198 V. L. Gutenmacher. The conditions place $H$ on the line $AB$ and $K$ on the line $BC$.		7m51s
199 D. A. Fridkin. We begin by examining the first sum for small values of $n$.		6m06s
200 A. N. Kolmogorov. For part (a), the six points are the intersection points of four lines in general position.		5m24s
201 Let triangle $ABC$ have sides $a=BC$, $b=CA$, $c=AB$.		2m02s
202 N. B. Vasilyev. Let the arithmetic progression be		7m17s
203 Let $ABCD$ be a cyclic quadrilateral with diagonals $AC$ and $BD$ intersecting at $P$.		1m13s
204 G. A. Gurevich. Consider the total number of $n$-digit numbers, which is $9 \cdot 10^{n-1}$.		3m23s
205 Consider the matrix of size $24 \times 25$ with entries $0$ and $1$, where $1$ indicates that a student solved a problem.		8m34s
206 V. A. Ufnarovsky. Let the digits of the infinite sequence be $a_1,a_2,a_3,\dots$, where each $a_i \in {0,1,\dots,9}$.		1m21s
207 N. D. Nagaev. Consider the problem geometrically by placing triangle $A_1 A_2 A_3$ in the plane and attempting to construct a triangle $M_1 M_2 M_3$ similar to a given triangle $B_1 B_2 B_3$ with the given side-ver…		6m06s
208 V. B. Peller. Let		7m05s
209 M. L. Gerver. Let		3m57s
210 G. A. Gurevich. The operation does not act on individual digits.		6m21s
211 All-Union Mathematical Olympiad for School Students (VII, 8th grade). The problem asks for an orientation of all edges between $n$ points.		3m33s
212 All-Union Mathematical Olympiad for School Students (VII, Grades 8 and 9). We are asked whether an expert can convince the court, using only three weighings on a balance scale, that exactly seven out of fourteen coins are counterfeit.		5m03s
213 All-Union Mathematical Olympiad for School Students (VII, Grades 9 and 10). Let the circle have center $I$.		6m31s
214 All-Union Mathematical Olympiad of School Students (VII, 10th grade). Let $f(x)=ax^{2}+bx+c$ and assume the equation $f(x)=x$ has no real roots.		1m04s
215 All-Union Mathematical Olympiad for School Students (VII, Grade 10). Consider a small patch of the grid with just one black cell at $(0,0)$.		6m16s
216 All-Union Mathematical Olympiad for School Students (VII, 8th grade). We interpret the situation as a simple undirected graph on $N$ vertices, where each vertex represents a person and each edge represents a mutual acquaintance.		1m27s
217 All-Union Mathematical Olympiad for School Students (VII, Grade 9). Consider first a triangle, the simplest convex polygon.		5m54s
218 All-Union Mathematical Olympiad for School Students (VII, Grade 10). We are asked to compare the square of a sum of five positive numbers with four times a sum of specific pairwise products taken cyclically.		4m56s
219 All-Union Mathematical Olympiad of School Students (VII, 10th grade). Let the four points be $A,B,C,D$ in space, not lying in one plane.		1m22s
220 All-Union School Mathematics Olympiad (VII, 10th grade). The problem concerns a king moving on an $8\times 8$ chessboard, visiting every square exactly once, and returning to the starting square.		4m04s
221 Consider an arbitrary compact planar blot.		8m51s
222 Consider small convex polyhedra such as the tetrahedron, cube, and octahedron.		6m59s
223 Perfect numbers are rare and highly structured.		4m03s
224 E. G. Gotman. Consider a trihedral angle, that is, three planes meeting at a common vertex, forming three plane angles $\alpha$, $\beta$, and $\gamma$ at the vertex.		1m41s
225 G. A. Galperin. Consider a die with faces numbered so that opposite faces sum to $7$.		6m12s
226 Yu. I. Ionin. Place the square in the coordinate plane with vertices		6m11s
227 E. V. Sallinen. Let the parallelogram be mapped by an affine transformation to the unit square, since affine maps preserve parallelism, ratios of areas, and the condition of a point lying on a segment.		1m16s
228 Consider small values of $n$ first.		4m35s
229 Let the square have side length $a$.		7m04s
230 S. V. Konyagin. Let the side length of the equilateral pentagon be $1$, and let its consecutive vertices be $A_1,A_2,A_3,A_4,A_5$.		6m41s
231 Consider the equation $n^x + n^y = n^z$ in natural numbers.		7m59s
232 P. S. Pankov. For a triple of points $A,B,C$, the condition that the triangle is obtuse means that one of the three angles exceeds $90^\circ$, equivalently one of the three opposite-side inequalities of the form		1m27s
233 G. A. Halperin. Consider a small case to understand the process.		4m45s
234 Alice moves on the integer lattice starting at $(0,0)$, and her motion is periodic, determined by a string of $n$ moves repeated indefinitely.	✓	4m24s
235 I. N. Bernstein. Consider a lion moving along a polygonal path inside a circular arena of radius $R = 10$ meters.		6m02s
236 A. Yu. Soifer, S. G. Slobodnik. For the first part, the numbers involved are all two-digit numbers, so each number can be represented as an ordered pair $(a,b)$ with $a,b \in {1,2,\dots,9,0}$, $a\neq 0$.		7m00s
237 B. D. Ginzburg. Consider an acute-angled triangle with vertices $A$, $B$, and $C$ and corresponding angles $\alpha$, $\beta$, and $\gamma$, and sides $a = BC$, $b = AC$, $c = AB$.		5m06s
238 F. G. Shleifer. Let		4m39s
239 Let points $A$ and $B$ be fixed on the plane, and let $C$ lie on the perpendicular bisector of segment $AB$, since it must satisfy $/AC/ = /BC/$.		6m12s
240 E. G. Belaga. The examples suggest that divisions should be used together with repeated squaring.	✓	13m56s
241 S. I. Meyzus. The exponent $1974$ is large, so direct computation is impossible.		1m57s
242 Denote the sides opposite $A_1,A_2,A_3$ by		4m55s
243 A. M. Lopshits. Let the two given lines be denoted $l_1$ and $l_2$.		2m21s
244 The desired inequality can be rewritten as		6m36s
245 M. L. Gerver. Consider the task of placing $N$ points in the plane such that the distance between any two points $M_i$ and $M_j$ is a given number $r_{ij}$.		6m13s
246 Yu. A. Gryaznov. Let the triangle be $ABC$ with circumcenter $O$.		1m17s
247 A $6 \times 6$ square contains $36$ unit squares.		6m15s
248 I. A. Kushnir. Let $S$ denote the area of the polygon $A_1A_2\cdots A_n$.		7m54s
249 I. F. Sharygin. Consider a cube $ABCDA'B'C'D'$ with an inscribed sphere, whose center coincides with the cube's center and whose radius is half the cube's edge length.		6m06s
250 V. N. Vaguten. Represent friendship by a graph $G$ whose vertices are the knights, with an edge joining two friends.		7m48s
251 F. G. Shleifer. The condition says no color appears more than $\frac{n}{2}$ times.		1m16s
252 G. A. Galperin. Consider a regular octagon with side length $a$ placed on a plane.		7m14s
253 B. V. Martynov. Let the given points be $O$, $I$, and $I_a$, where $O$ is the circumcenter, $I$ the incenter, and $I_a$ one of the excenters of triangle $ABC$.		1m36s
254 A. A. Egorov. Consider small cases of numbers of the form $0.		7m16s
255 I. F. Sharygin. Let the centers of the spheres be $O_1$ and $O_2$, with radii $R_1$ and $R_2$.		1m27s
256 A. N. Chernyshyov. Consider first a simple case: a triangle circumscribed around a circle, with the incircle touching the sides at points $A'$, $B'$, and $C'$, forming the inscribed triangle.		6m13s
257 The inequality can be written as		4m06s
258 A. P. Savin. Consider a convex quadrilateral with vertices $A$, $B$, $C$, $D$ in order, and let $K$, $L$, $N$ be the midpoints of three of its sides.		5m42s
259 Consider a simple case of a triangle circumscribed around a circle, where the inscribed circle is tangent to its sides at points $A', B', C'$.		3m28s
260 G. A. Gurevich. Label the $n$ equal elementary arcs by the colors of the segments		6m30s
261 S. G. Gindikin. Consider a hoop of radius $R$ placed over a fixed circle of radius $r < R$.		1m48s
262 The problem asks for the maximal number of rooks or queens on an $8 \times 8$ chessboard such that each piece is attacked by at most one other piece.		8m52s
263 E. G. Gotman. Let the rectangle have coordinates		6m48s
264 The graph described by Fig.		3m40s
265 M. L. Gerver. Consider a rectangular parallelepiped with edges of length $a$, $b$, and $c$.		6m15s
266 A. V. Karzanov. Consider the circle through three consecutive vertices $A_{i-1},A_i,A_{i+1}$.		1m35s
267 F. A. Bartenev. Let the $n$th triple be $(a_n,b_n,c_n)$, with		6m37s
268 The game is played on the graph of an $n\times n$ chessboard, where vertices are squares and edges correspond to standard knight moves $(\pm2,\pm1)$ and $(\pm1,\pm2)$.		1m16s
269 E. A. Yasinovyi. The quantity $T_k(n)$ is the $k$-th elementary symmetric polynomial in the numbers $1,2,\dots,n$:		5m15s
270 I. F. Sharygin, A. I. Yanovsky. The conditions mean that $KA \perp AB$, $KC \perp CD$, $HB \perp AB$, and $HD \perp CD$.		7m37s
271 All-Union Mathematical Olympiad of School Students (1974, grades 9 and 10). For small values, direct checking clarifies the constraint.		1m09s
272 All-Union Mathematical Olympiad for School Students (1974, 9th grade). Consider two circles of radii $R$ and $r$ that are externally tangent.		7m25s
273 All-Union Mathematical Olympiad for School Students (1974, 10th grade). The condition		5m24s
274 All-Union Mathematical Olympiad for School Students (1974, Grades 8 and 9). We seek the smallest positive value attained by the given differences.		4m04s
275 All-Union Mathematical Olympiad for school students (1974, grades 9 and 10). Let the vectors be represented by points on the unit circle.		6m44s
276 All-Union Mathematical Olympiad for School Students (1974, 10th grade). A direct synthetic approach would require tracking the foot of a perpendicular from $B$ to the line $PC$, which suggests that a coordinate representation or vector projection will likely reduce the co…		9m33s
277 All-Union Mathematical Olympiad for School Students (1974, Grade 10). Let $E$ be the number of segments whose endpoints have different colors.		7m19s
278 All-Union Mathematical Olympiad of School Students (1974, 8th grade). The configuration involves a convex hexagon with side lengths bounded below or above and three “long” diagonals connecting every second vertex.		1m24s
279 All-Union Mathematical Olympiad for School Students (1974, Grade 9). Let the numbers on the cards be $a_1,\dots,a_n$, where each $a_i\in{\pm1}$.		3m59s
280 All-Union Mathematical Olympiad for School Students (1974, Grade 10). Consider a triangle $ABC$ of area $1$ with midpoints $A_1$, $B_1$, and $C_1$ of the sides $BC$, $AC$, and $AB$ respectively.		3m21s
281 Moscow Mathematical Olympiad (1974). Consider small convex polygons whose diagonals are defined as segments joining non-adjacent vertices.		1m10s
282 Moscow Mathematical Olympiad (1974). Consider a small table, for instance $2 \times 2$, with entries		5m56s
283 Moscow Mathematical Olympiad (1974). Consider small examples of convex polygons, starting with triangles and quadrilaterals, and examine what happens when each side is shifted outward by a fixed distance.		6m07s
284 Moscow Mathematical Olympiad (1974). Consider smaller analogues of the problem to understand its structure.		3m31s
285 Moscow Mathematical Olympiad (1974). Consider small examples to understand the claim.		6m02s
286 Let $m(N)$ denote the minimum possible number of distinct marked points.		5m42s
287 Consider a sequence of natural numbers $a_1 < a_2 < a_3 < \dots$ such that every natural number $n$ can be represented uniquely as $a_j - a_i$ with $j > i$.		7m10s
288 S. V. Konyagin. Model the congress by a simple graph.		7m36s
289 S. V. Konyagin. Let the total weight be $S$, and suppose the $N$ weights are partitioned into $K$ piles each of sum $T$, so $S = KT$.		1m19s
290 S. V. Konyagin. Let the closed non-self-intersecting broken line have vertices		8m41s
291 V. M. Fishman. Let the triangle have vertices $A_1,A_2,A_3$.		1m18s
292 F. G. Shleifer. Consider a small example with numbers $1, 2, 3$.		8m18s
293 Let $\gamma_n = \angle C_{n+1} C_n O$.		1m30s
294 V. P. Fedotov. The inequality is homogeneous and symmetric in a suggestive way.		7m45s
295 N. B. Vasiliev. The problem involves a convex polyhedron intersected by three parallel planes $p_0$, $p_1$, $p_2$, with $p_1$ equidistant between the outer planes.		5m06s
296 Let the rows be numbered from top to bottom by $1,\dots,n$, and let $a_{ij}$ be the entry in row $i$, column $j$.		4m57s
297 L. P. Kuptsov. The problem involves four squares arranged on a plane with shared vertices, forming a chain: the second vertex of the first square coincides with a vertex of the second square, and so on, closing back…		6m12s
298 N. B. Vasilyev. For $m=5$ the consecutive fractions		6m30s
299 N. B. Vasilyev. Consider a ruled sheet of paper with parallel lines spaced a fixed distance apart, and suppose a regular $n$-gon is drawn so that all vertices lie on these lines.		6m16s
300 A. M. Styopin. We are given a finite or otherwise fixed collection of forbidden words over the alphabet ${a,b,c}$, each forbidden word having length at least $2$, and all forbidden words having pairwise distinct len…		1m11s
301 For $n=1$ the statement is immediate.		7m43s
302 Let $O = AC \cap BD$ in the trapezoid $ABCD$ with $AB \parallel CD$.		1m14s
303 A. V. Sherstyuk. Consider placing a small number of identical weights on the vertices of a $1 \times 1$ grid.		6m10s
304 A. A. Grigoryan. The axioms resemble the algebraic properties of the bitwise exclusive-or operation.		7m53s
305 A. I. Shirshov. The concurrency of $AA'$, $BB'$, $CC'$ at $P$ together with products $/AP/\cdot/A'P/=t$ suggests a fixed-power relation, which is characteristic of inversion centered at $P$.		2m08s
306 V. P. Fedotov. Let the removed corner be the unit square with vertices $(0,0)$, $(1,0)$, $(1,1)$, $(0,1)$.		6m49s
307 V. P. Golubyatnikov. Consider a single vertex where three hexagonal walls meet.		7m01s
308 Yu. I. Ionin. 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 $x$.		3m30s
309 V. G. Shleifer. For the first question, divisibility by $x^2+x+1$ suggests evaluating the polynomial at the nonreal cube roots of unity.		7m32s
310 G. A. Gurevich. An $n$-digit number is a sequence of digits $d_1d_2\ldots d_n$ where $d_1 \in {1,\dots,9}$ and $d_i \in {0,\dots,9}$ for $i \ge 2$.		1m00s
311 Consider the growth process for small numbers.		6m06s
312 A. A. Grigoryan. The outer parallelogram $P_1$ admits an affine normalization to a unit square without changing incidence relations such as “lying on a side” and “being parallel to fixed directions.		1m13s
313 I. N. Bronshtein. Consider an angle with vertex $O$ and denote its sides by rays $OA$ and $OB$.		6m10s
314 A. G. Leiderman. Consider the difference between a number and the product of its digits, denoted $N - P(N)$, where $N$ is a 9-digit number with digits $d_1, d_2, \dots, d_9$ in ${1,2,\dots,9}$ and $P(N) = d_1 d_2 \dot…		8m35s
315 A. M. Zubkov. Each edge of the convex polyhedron is oriented, so the 1-skeleton becomes an orientation of a connected planar graph embedded on the sphere.		1m25s
316 E. G. Gotman. Consider the sum of squares of $k$ consecutive natural numbers beginning at $n$, expressed as		3m37s
317 G. V. Egorov. Consider a small graph representing countries, where vertices are countries and edges connect neighboring countries.		4m08s
318 A. P. Savin. Let		6m36s
319 I cannot write a solution to Kvant problem M319 because the actual problem statement is not present in your message.		5m08s
320 The statement asks for a classification.		7m16s
321 A. V. Brailov. The table may be taken to be the unit square $[0,1]\times[0,1]$.		7m27s
322 S. V. Fomin. Each circle contributes boundary pieces only where it is the lowest among the $N$ radii in some direction, since the intersection of disks can be described as the set of points satisfying $d(x,O_i)\le…		1m25s
323 V. A. Sergeev. Consider a function $f:\mathbb{R}\to\mathbb{R}$.		6m29s
324 S. V. Fomin. Consider a single pile with a small number of stones.		4m20s
325 Let $x_{n,m}$ be the entry in row $n$ and position $m$, where $0\le m\le n$.	✓	14m26s
326 E. G. Gotman. Let the circle have radius $R$ and let a chord $AB$ be at a distance $h$ from the center $O$.		6m28s
327 Consider small groups to understand how the structure of \		5m58s
328 Let the tetrahedron have edge length $1$.		7m46s
329 Let the vertices of the convex $n$-gon be $P_1,P_2,\dots,P_n$ in cyclic order.		6m36s
330 N. B. Vasiliev. Let $M_0$ and $M_1$ be convex polygons.		4m41s
331 All-Union Mathematical Olympiad for School Students (1975, grades 8–10). Let the rotation about the circumcenter $O$ have angle $\varphi$, where $0<\varphi<180^\circ$.		8m14s
332 All-Union Mathematical Olympiad for School Students (1975, 9th grade). Consider small values of $k$ first.		2m46s
333 All-Union Mathematical Olympiad for School Students (1975, 8th grade). Let the flies be at positions $P(t),Q(t),R(t)$ on the sides of triangle $ABC$.		2m20s
334 All-Union School Mathematics Olympiad (1975, 9th grade). Consider small polynomials such as $P(x) = x$, $P(x) = x+1$, or $P(x) = x^2$.		4m41s
335 All-Union Mathematical Olympiad for School Students (1975, grades 8–10). Represent the marked cell centers by a $0$-$1$ matrix.		10m27s
336 All-Union Mathematical Olympiad for School Students (1974, Grade 10). For two polygons the statement is immediate.		1m39s
337 Please provide the textual version of Kvant problem M337.		4m04s
338 All-Union Mathematical Olympiad for School Students (1975, Grades 8 and 10). Consider the operation on a small set of digits.		1m27s
339 All-Union Mathematical Olympiad for School Students (1975, Grades 8–9). For each of the given lines, orient it upward.		3m59s
340 All-Union Mathematical Olympiad for School Students (1974, Grade 10). Consider first a $2 \times 2$ table:		3m16s
341 Yu. A. Shneider. Consider first small cases to understand the tension between a European team dominating the European Championship yet performing worst in the World Championship.		4m54s
342 S. V. Fomin. The problem is a binary coding problem.		6m39s
343 S. L. Eliseev. Model the country by a connected graph.		6m21s
344 A. N. Pechkovsky. The 64 marked points are the centers of the squares of an $8\times 8$ grid, hence they can be identified as		7m32s
345 G. A. Gurevich. Let $a_n$ be the sequence, with		6m10s
346 Yu. G. Bogaturov, 10th grade student (Kutaisi). Place the square in a coordinate system with algebraic convenience so that perpendicularity can be tested by a dot product condition.		5m29s
347 A. A. Grigoryan. Let the chosen pair be an unknown $2$ element subset of ${1,\dots,25}$.		6m17s
348 S. M. Ageev. The numbers in the table are		4m51s
349 A. P. Savin. Let the given triangle have sides $a,b,c$ opposite angles $A,B,C$.		5m49s
350 E. Ya. Gik, A. B. Zhornitsky. Consider a small $n\times m$ chessboard, for instance $4\times 5$.		7m26s
351 M. M. Imerishvili, 9th-grade student (Tbilisi). Let the unknown triangle be $ABC$, and suppose that $H$ is the foot of the altitude from $A$ onto $BC$.		10m16s
352 D. K. Faddeev. Write		6m05s
353 Before proceeding, I need the precise textual statement of Kvant problem M353.		5m51s
354 Let the number of sides be		3m34s
355 I cannot write a solution to Kvant problem M355 because the actual problem statement is not present in your message.		2m48s
356 The operation that produces $A_{k+1}B_{k+1}C_{k+1}$ from $A_kB_kC_k$ is the pedal construction with respect to the fixed point $M$.		2m37s
357 Let		2m56s
358 I can produce a complete, rigorous Kvant-style solution, but I need the text of problem M358 to proceed.		6m09s
359 The statement of problem M359 is incomplete.		4m43s
360 V. P. Golubyatnikov. Let		7m19s
361 Please provide the full text of Kvant problem M361 so I can write the complete, rigorous solution in the requested six-section format.		4m29s
362 The statement is affine in nature.		3m39s
363 For two parabolas with parallel axes, it is natural to choose coordinates so that the common direction of the axes is vertical.		2m40s
364 The requirement that every training session consists of 4 disjoint crews of 4 cosmonauts means that each session partitions the 16 cosmonauts into 4-element subsets.		1m45s
365 Consider first a simple case of two numbers summing to $1$.		6m31s
366 V. E. Kolosov. Assume such a configuration exists and consider the finite set of triangles.		8m46s
367 For three consecutive natural numbers $n, n+1, n+2$, the key structural feature is that any two of them are coprime.		2m14s
368 S. V. Fomin. Choose coordinates so that the three cylinder axes are parallel to the coordinate axes.		6m32s
369 The circle $\gamma$ is centered at the orthocenter $H$ and lies inside the acute triangle $ABC$.		7m04s
371 S. V. Fomin. Consider the problem on a $2\times 2$ chessboard first.		2m45s
372 Consider the triangle $ABC$ and the inequality $/AP/ + /BP/ + /CP/ \ge /AC/ + /BC/$ for an arbitrary point $P$ in the plane.		7m43s
373 An infinite decimal expansion determines an infinite sequence of digits, hence an infinite word over the alphabet ${0,1,\dots,9}$.		7m06s
374 The expression on the left contains two square roots.		2m05s
375 I cannot write a solution to Kvant problem M375 from the information provided, because the actual problem statement is missing.		5m09s
376 Let the black piece start at cell $1$ and the white piece at cell $N$.		6m47s
377 The required condition is		12m03s
378 For part (a), the classical obstruction for sums of three cubes is modulo $9$.		3m10s
379 I can proceed with this framework, but I need the text of Kvant problem M379 in order to produce a rigorous solution.		4m28s
381 I cannot write a solution to Kvant problem M381 because the actual problem statement is not present in your message.		6m08s
382 The hypothesis concerns the values of $f(x)$ modulo $p+1$ at the $p+1$ consecutive integers		2m36s
383 M. L. Gerver. Consider small examples to understand the problem.		7m06s
384 The two squares share the vertex $O$ and have the same orientation.		2m22s
385 The statement asks only for part (a), namely that the number of lattice points in the dilated polygon $nF$ is a polynomial in $n$.		2m31s
386 I can proceed with that, but I need the text of Kvant Problem M386 to provide the complete solution.		6m12s
387 Let the natural number be $n$, and let it have $k$ decimal digits.		2m53s
388 I cannot write a rigorous solution to Kvant problem M388 because the problem statement itself is not provided in the conversation.		7m50s
389 A line running along a grid line is either vertical or horizontal.		2m25s
390 Moscow Mathematical Olympiad (1976). Let $s(m)$ denote the sum of decimal digits of $m$.		2m01s
391 I cannot write a solution to Kvant problem M391 from the information provided, because the actual problem statement is missing.		5m45s
392 All-Union Mathematical Olympiad of School Students (1976, 10th grade). Let the positions of the three pedestrians at time $t$ be represented by vectors $A(t), B(t), C(t)$ in the plane.		6m44s
393 Let		6m39s
394 All-Union Mathematical Olympiad for School Students (1976, Grades 9 and 10). Let		15m09s
395 All-Union Mathematical Olympiad for School Students (1976, Grade 10). Label the vertices of a regular $n$-gon by $0,1,\dots,n-1$ in cyclic order.		2m05s
396 All-Union Mathematical Olympiad for School Students (1976, grades 8 and 9). The side length of the equilateral triangle is much larger than the lower bound $1$ imposed on the sides of the desired triangles.		6m44s
397 Each marked arc belongs to a circle of the same radius.		2m52s
398 I cannot write a rigorous solution to Kvant problem M398 from the information provided, because the problem statement itself is missing.		3m24s
399 A. P. Savin. For a set of points on a segment, the condition “there exist two points at distance $m$” is equivalent to requiring that the difference set of all chosen coordinates contains every integer $1,2,\dots,…		2m07s
400 All-Union Mathematical Olympiad for School Students (1976, Grade 9). Consider small values of $N$ to understand the structure of universal sequences.		7m15s
401 Let $A,B,C$ be the angles of $\triangle ABC$.		7m13s
402 Yu. I. Ionin. From the condition with $n=m=1$ we obtain		2m37s
403 A convex polyhedron has the property that every vertex has even degree.		8m44s
404 A. P. Savin. Start with small $n$.		2m00s
405 All-Union Mathematical Olympiad for School Students (X, 8th grade). The condition concerns preservation of adjacency.		7m21s
406 Let the circle have center $O$ and radius $R$.		2m58s
407 S. V. Konyagin. Consider small examples to understand the structure of the problem.		3m58s
408 Let the original rectangles have side lengths $a$ and $b$, and let their ratio be $r = \frac{a}{b} > 0$.		9m21s
409 The transformation replaces each entry in a row by the frequency of that value in the same row.		9m54s
410 Consider the sphere of radius $1$ centered at the origin in $\mathbb{R}^3$, and let the equatorial plane be the $xy$-plane.		4m02s
411 Consider a triangle with sides $a$, $b$, $c$, and a point inside it through which three segments pass, each parallel to a side and all of equal length $x$.		2m09s
412 Model the city as a finite directed graph $G=(V,E)$ in which vertices are squares and directed edges are one-way streets.		9m18s
413 I. M. Yaglom. Consider the equation $f(x+a)-f(x)=0$ for a function $f$ continuous on $[0,1]$ with $f(0)=f(1)=0$.		9m58s
414 Let the convex pentagon be $A_1A_2A_3A_4A_5$, with indices taken modulo $5$.		26m19s
415 The problem asks for the maximum number of mutually non-attacking kings on an $n\times n$ toroidal board.		2m23s
416 Interpret the drawn segments as the edges of a graph whose vertices are the given points.		6m34s
417 V. V. Proizvolov. The object is a closed polygonal line drawn on the surface of a unit cube, with the condition that every face of the cube contains at least one entire segment of the polygonal line.		7m06s
418 For small values of $n$,		6m34s
419 Let $D_{16}$ denote the closed disk of radius $16$ centered at the origin, and suppose $650$ points $P_1, \dots, P_{650}$ are placed in $D_{16}$.		27m06s
420 G. A. Gurevich, B. Makarevich. The allowed operations on a fraction $\frac{a}{b}$ replace the integer pair $(a,b)$ by one of $(a-b,b)$, $(a+b,b)$, or $(b,a)$.		2m11s
421 Let the cells of the infinite graph paper be indexed by integer coordinates $(x,y)$, where each cell corresponds to one pair of integers.		6m42s
422 Let $\triangle ABC$ be arbitrary.		24m02s
423 The left-hand side contains the three quantities		3m37s
424 Let $ABCD$ be a tetrahedron.		6m51s
425 Suppose such an $N$ exists.		6m34s
426 The figure describes a standard cyclic filling of an $n\times n$ table with the numbers $1,2,\dots,n$ in such a way that each row is a cyclic shift of the previous one.		7m12s
427 Let		7m23s
428 The problem is naturally translated into graph theory.		1m43s
429 Write $x=n+t$ with $n=[x]\in\mathbb{Z}$ and $t={x}\in[0,1)$.		6m48s
430 For the planar statement, the number $2$ strongly suggests a relation between the area of a convex figure and the area of a rectangle determined by two orthogonal widths.		3m53s
431 The trees are vertical cylinders.	✓	6m37s
432 Consider the sum of the digits of perfect squares.		7m31s
433 The configuration imposes five independent parallelism relations between each side of a convex pentagon and a diagonal.		7m01s
434 D. K. Faddeev. Consider the sum		4m58s
435 S. V. Konyagin. Let $A=(a_{ij})$ be an $m\times n$ matrix.		1m53s
436 S. T. Berkolayko. We are asked to partition all pairwise sums of two sets of ten numbers each into ten groups of ten, each with the same total.		4m49s
437 Let the odd number be		9m38s
438 The configuration is a fixed circular segment determined by a chord $AB$ of a circle with center $O$.		1m58s
439 A. G. Kushnirenko. For part 1, write		7m08s
441 G. A. Gurevich. Let the vertices of the convex $2n$-gon be $A_1,A_2,\dots,A_{2n}$ in cyclic order.		3m50s
442 For small primes the structure is very rigid.		7m02s
443 Before I begin writing the complete solution, I need the full textual statement of Kvant problem M443.		5m56s
445 I cannot write a solution to Kvant problem M445 from the information provided, because the actual problem statement is missing.		5m40s
447 I cannot write a solution to Kvant problem M447 because the actual problem statement is not present in your message.		4m39s
448 Let the quadrilateral have diagonals intersecting at a point $O$.		2m10s
449 I cannot write a rigorous solution to Kvant problem M449 without the actual problem statement or the diagram.		5m39s
450 Normalize the width of the bottom rectangle to $1$, and let the common height of all rectangles be fixed.		12m12s
451 All-Union Mathematical Olympiad for School Students (XI, 1977, Grade 8). Consider first the simplest nontrivial configuration of points, namely three points not lying on a line.		7m17s
452 All-Union Mathematical Olympiad for School Students (XI, 1977, grades 8–10). Let $ABC$ be the triangle $T_1$ inscribed in a circle with center $O$.		1m55s
453 All-Union Mathematical Olympiad for School Students (XI, 1977, 8th grade). Let $S$ be a subset of ${a_1,\dots,a_n}$ and write $s(S)$ for its sum.		7m05s
454 All-Union Mathematical Olympiad for School Students (XI, 1977, Grade 8). Let the dwarfs act in order $1,2,\dots,7$ around the table.		6m50s
456 All-Union Mathematical Olympiad of School Students (XI, 1977, 10th grade). At each vertex of the polyhedron, exactly three edges meet, so the vertex figure is a trihedral angle.		9m15s
457 All-Union Mathematical Olympiad for School Students (XI, 1977, Grade 8). Let the vertices of the simple closed polygonal line be $A_1,A_2,\dots,A_n$ in cyclic order, and let $e_i=A_iA_{i+1}$, with indices taken modulo $n$.		3m44s
458 All-Union School Mathematics Olympiad (XI, 1977, 10th grade). Consider the polynomial $x^{10}+a_9x^9+\dots+a_1x+1$ with all coefficients initially unspecified except for the leading and constant terms, which are $1$.		7m07s
459 All-Union Mathematical Olympiad for School Students (XI, 1977, Grade 9). Let the route produced by the minimum-greedy algorithm be		24m48s
460 All-Union Mathematical Olympiad of School Students (XI, 1977, 8th and 10th grades). We begin with small values of $n$ to understand the structure.		7m13s
461 All-Union Mathematical Olympiad for School Students (XI, 1977, 9th grade). Consider a small number of weights, for instance $n=2$ or $n=3$, each with distinct masses $w_1<w_2<w_3$.		6m02s
462 Let the apex of the regular square pyramid be $S$, and let the base square be $ABCD$ with center $O$.		1m46s
463 All-Union Mathematical Olympiad for School Students (XI, 1977, Grade 9). Consider small examples to understand the problem concretely.		6m14s
464 All-Union Mathematical Olympiad for School Students (XI, 1977, Grade 9). Let each square correspond to its center.	✓	9m26s
465 All-Union Mathematical Olympiad of School Students (XI, 1977, grades 8 and 10). A ticket is a length-$k$ word over the alphabet ${0,1,\dots,9}$.		4m28s
466 S. V. Fomin. Consider first a smaller version of the problem.		6m04s
467 The triangle is equilateral, so placing it in a coordinate system with one side horizontal allows algebraic control over all distances and angles.		2m06s
468 Yu. I. Ionin. Consider four points $A$, $B$, $C$, $D$ in the plane, and the scalar products $\overrightarrow{MA} \cdot \overrightarrow{MB}$ and $\overrightarrow{MC} \cdot \overrightarrow{MD}$ for a variable point $…		4m06s
469 V. V. Vavilov. Let $P(x)=x^4+ax^3+bx+c$ have four distinct real roots $r_1<r_2<r_3<r_4$.		2m04s
470 L. D. Kurlandchik, A. D. Lisitsky. We begin by examining the two sums for small values of $n$ to detect patterns.		5m13s
471 S. V. Fomin. Two intersecting circles partition the plane into exactly three bounded regions: the common lens $R_0$, the two asymmetric caps $R_1$ and $R_2$ lying respectively in the first and second circle but ou…		7m20s
472 V. V. Prasolov. Consider a cube of side length $1$ for simplicity.		4m36s
473 Let the weights in the first group be	✓	17m39s
474 We begin by examining the properties of perfect numbers modulo small integers.		6m15s
475 Assume that an equilateral triangle has all three vertices at lattice points.	✓	30m48s
476 For the planar statement, the condition that no lattice points lie on the boundary except the vertices means that every side joins two lattice points with relatively prime coordinate differences.		7m08s
477 S. V. Konyagin. The sequence is defined by iteration of an integer polynomial $P$ satisfying $P(x)>x$ for all natural $x$.		10m04s
478 Moscow Mathematical Olympiad (XL, 1977, grades 8–10). Represent the tournament by a directed complete graph.		16m46s
479 Moscow Mathematical Olympiad (XL, 1977, grades 9 and 11). Consider a set of distinct natural numbers ${a_1, a_2, \dots, a_n}$ with the property that for any two elements $a_i$ and $a_j$, the sum $a_i + a_j$ is divisible by their difference $a_i - a_j$.		5m05s
480 The recurrence is		2m04s
481 Let		6m26s
482 V. V. Proizvolov, A. P. Savin. Let the regular tetrahedron be $ABCD$, with edge length $a$.		33m51s
483 Consider a right triangle with legs $a$ and $b$ and hypotenuse $c$, where $c^2 = a^2 + b^2$.		6m19s
484 S. N. Mironov. A dissection of a convex polygon into regular polygons means that every piece is an equilateral triangle, a square, or a regular polygon of higher order, all glued edge-to-edge without overlap.		7m06s
485 T. Martynenko, R. P. Ushakov. The interval is		8m14s
487 Let $O_1$ and $O_2$ be the centers of circles $\gamma_1$ and $\gamma_2$, with radii $R_1$ and $R_2$.		1m58s
488 The recurrence		6m00s
489 The transformation replaces each term by the average of the other two.		1m39s
490 S. V. Fomin. Let the given integers be $a_1,\dots,a_{p-1}$, none divisible by $p$.		9m24s
491 Let three consecutive terms be $a,ar,ar^2$, where all terms are integers.		3m43s
492 Consider triangle $ABC$ and points $A_1$, $B_1$, $C_1$ on sides $BC$, $CA$, and $AB$, respectively, with cevians $AA_1$, $BB_1$, and $CC_1$ concurrent at $P$.		6m22s
493 The expression		2m49s
494 Let the square be partitioned into a regular grid of $n \times n$ congruent squares, each of side length $1/n$.		9m50s
495 G. A. Halperin. Each satellite moves along a circular orbit centered at $O$ with constant angular velocity.		6m11s
496 S. V. Fomin. A six-digit number can be written as the product of two three-digit numbers if and only if it belongs to the set	✓	9m01s
497 V. L. Gutenmakher. Consider triangle $ABC$ with arbitrary points $A_1$ on $BC$, $B_1$ on $CA$, and $C_1$ on $AB$.		3m44s
498 N. B. Vasiliev, A. A. Egorov. Consider small values of $n$.		2m18s
499 G. A. Gurevich. Consider what it means for a number to be balanced.		6m14s
500 Label the people $1,2,\dots,N$ from left to right.		9m26s
501 A power $3^k$ begins with the digit $9$ precisely when there exists an integer $m$ such that		3m48s
502 The three segments $AA_1$, $BB_1$, $CC_1$ are parallel and not coplanar.		8m45s
503 The condition		3m46s
504 Let $F$ be the set of free squares.		6m54s
505 P. Blecher, G. A. Halperin, M. Kelbert. For the one-dimensional problem, the state of the process is not the point $O_k$ itself but the set of material points lying in the interval of length $2r$ centered at $O_k$.		9m24s
506 Let $x=a^2,\; y=b^2,\; z=c^2,\; w=d^2$.		7m09s
507 We are asked to consider sequences of $n$ distinct natural numbers $a_1 < a_2 < \dots < a_n < 2n$ with $n \ge 6$, and to find bounds for the minimum of their least common multiples and the maximum of…		6m36s
508 I. F. Sharygin. The three semicircles with diameters $AB$, $BC$, $AC$ lie on the same line $AB$, with centers at the midpoints of $AB$, $BC$, and $AC$.		2m00s
509 Equation (1), $2^x + 1 = 3^y$, suggests searching for powers of 2 that are one less than a power of 3.		3m09s
510 I. Birger, R. P. Ushakov. Let		6m43s
511 All-Union Mathematical Olympiad for School Students (XII, 1978, grades 8–9). ABMD is a parallelogram, so the vertices satisfy the affine relation $a+m=b+d$, hence $m=a+d-b$.		2m08s
512 All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10). Compute the first few values of $f$ for small natural numbers greater than $1$.		6m26s
513 All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10). Consider a square inscribed in the graph of $y = A \sin x$.		8m01s
514 All-Union 12th School Mathematics Olympiad, 1978, grades 9–10. We seek an infinite bounded sequence $(x_n)$ such that every two distinct terms satisfy		6m35s
515 All-Union Mathematical Olympiad for School Students (XII, 1978, Grades 8 and 10). Starting with two points $A$ and $B$ at distance 1, reflecting one about the other generates points along the line $AB$.		1m20s
516 All-Union Mathematical Olympiad for School Students (XII, 1978, grades 8–10). The three machines modify cards in distinct ways.		2m36s
517 All-Union Mathematical Olympiad for School Students (XII, 1978, Grades 8–9). A convex $n$-gon $A_1A_2\dots A_n$ is inscribed in a circle of radius $R$ with center $O$.		42m23s
518 All-Union School Mathematics Olympiad (XII, 1978, 9th grade). Consider the inequality		4m19s
519 All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 9). Let $W$ denote a winning position for the player to move and $L$ a losing position.		6m20s
520 All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10). Consider the sequence $x_n=(1+\sqrt{2}+\sqrt{3})^n$.		5m27s
521 All-Union Mathematical Olympiad for School Students (XII, 1978, Grades 8–9). Consider the first few values of $a_n$.		6m44s
522 All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 8). For one segment the answer is trivial.		8m01s
523 All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 8). Consider small $n \times n$ boards and simulate the game.		9m54s
524 All-Union Mathematical Olympiad for School Students (XII, 1978, 8th grade). Consider the numbers $1978^m - 1$ and $1000^m - 1$ for small values of $m$.		6m40s
525 All-Union Mathematical Olympiad for School Students (XII, 1978, Grade 10). The area of the orthogonal projection of a polyhedron onto a plane depends on the direction of projection.		4m59s
526 Label the convex quadrilateral $ABCD$ with consecutive sides $AB = a$, $BC = b$, $CD = c$, and $DA = d$.		1m32s
527 Denote		5m33s
528 Consider the $8 \times 8$ chessboard with one chip on each square.		4m41s
529 A homothety with ratio $k<0$ reverses directions through its center.		2m17s
530 Let each cell be represented by a variable in $\mathbb F_2$, where $1$ means black and $0$ means white.		6m59s
531 N. B. Vasiliev. Consider two points $A$ and $B$ on a line and a motorist starting from $A$ and a cyclist starting from $B$, both moving toward each other at constant speeds $v_m$ and $v_c$.		4m39s
532 For small values,		3m44s
533 V. G. Boltyanskii. A heptagon has $7$ vertices and $14$ diagonals.		3m00s
534 Consider triangle $ABC$ with a point $P$ inside it, through which three lines are drawn, each parallel to one side of the triangle.		7m18s
535 The defining condition of a trigram is		7m00s
536 For the first part, the condition means that every domino of the upper layer must cross the boundary between two dominoes of the lower layer.		6m55s
537 International Mathematical Olympiad (XX, 1978). Let $O$ be the center of the circumcircle of the isosceles triangle $ABC$, and let $M$ be the midpoint of $PQ$.		6m27s
538 International Mathematical Olympiad for School Students (XX, 1978). Begin by examining small values to understand the recursive structure imposed by $g(n)=f(f(n))+1$.		3m23s
539 International Mathematical Olympiad for School Students (XX, 1978). Let the given sphere have center $O$ and radius $R$.		4m13s
540 International Mathematical Olympiad for School Students (XX, 1978). Let the members of each country form a set of integers contained in ${1,2,\dots,1978}$.		7m02s
541 Consider a small social network where each person has exactly three friends.		7m08s
542 Let us denote the initial right triangle as $A_0A_1A_2$, with right angle at $A_2$, and legs $/A_0A_2/=a$ and $/A_1A_2/=b$.		7m06s
543 The expression		8m21s
544 S. N. Bychkov. Consider first the case $n=4$.		3m11s
545 V. M. Galperin, G. A. Galperin. Consider three points on the plane.		11m13s
546 Let the rectangle be centered at the origin, with sides parallel to the coordinate axes.		4m53s
547 The equation is		10m01s
548 For four points on a circle, label them by position vectors $a,b,c,d$ on a circle with center $O$, taken as the origin.		4m47s
549 V. E. Matizen. Let $N$ be a natural number and let its divisors be $d \mid N$.		8m56s
550 S. S. Krotov. Let the optimal finishing time be $T$.		9m40s
551 Moscow Mathematical Olympiad. Consider the case of a triangle first.		6m05s
552 Let the roots of		4m35s
553 Consider a triangle $ABC$ with sides $BC < AC < AB$.		3m37s
554 USA Mathematical Olympiad (1978). Consider small examples of natural numbers and attempt to write them as sums of numbers whose reciprocals add to one.		6m02s
555 Consider first the intersection of two cylinders of equal radius $r$ with axes perpendicular.		3m01s
556 A. A. Egorov. The answer is yes.		31m46s
557 A. T. Kolotov. Suppose, contrary to the statement, that none of the given numbers is prime.		3m02s
558 V. V. Proizvolov. Let the black sectors have angular lengths $\alpha_1,\dots,\alpha_k$, where each		2m40s
559 Let		2m11s
560 For a fixed position of the cover, let $C$ be the convex cover and let $H$ be the hole.		7m15s
561 L. P. Kuptsov. The condition says that corresponding sides are parallel, but the directions are reversed.	✓	11m32s
562 Our systems have detected unusual activity coming from your system.		27m20s
563 S. V. Fomin. Assume, seeking a contradiction, that	✓	8m47s
564 Let $BC=a$, and place the triangle in coordinates		7m57s
565 For each $k$, the quantity $b_k$ is the average of all products of $k$ distinct elements from $a_1,\ldots,a_n$.		2m06s
566 All-Union 13th Olympiad of School Students, grades 8–9. Let the fixed isosceles right triangle be placed as a unit right isosceles triangle with vertices $A(0,0)$, $B(1,0)$, $C(0,1)$.		2m46s
567 13th All-Union School Students' Olympiad, Grade 9. Take a small example, say $p=2$, $q=3$.		9m47s
568 13th All-Union School Olympiad, Grade 10. Let		3m34s
569 13th All-Union School Students' Olympiad, Grades 9 and 10. Starting from $0$ and $1$, the first new number that can be obtained is $\frac12$, since the mean of $0$ and $1$ is $\frac12$.		7m25s
570 All-Union 13th School Olympiad, Grade 8. Let the squares have side lengths $a_1, a_2, \dots, a_n$, so that		2m18s
571 All-Union Mathematical Olympiad of School Students (1979, 10th grade). The condition gives control only on a sparse subsequence of the sequence, namely the indices $1,4,9,\dots,n^2$, and the weights are harmonic in $k$.		9m59s
572 All-Union Mathematical Olympiad for School Students (1979, 8th grade). The kangaroo moves in the integer lattice of the first quadrant with vectors $v_1=(1,-1)$ and $v_2=(-5,7)$, always staying in $x\ge 0$, $y\ge 0$.		2m26s
573 All-Union Mathematical Olympiad for School Students (1979, Grade 9). Let the lines through $O$ be $l_1,\dots,l_{1979}$.		7m24s
574 All-Union Mathematical Olympiad for School Students (1979, Grade 9). Let	✓	13m05s
575 All-Union Mathematical Olympiad of School Students (1979, 10th grade). Let $A_0A_1,\dots,A_{n-1}A_n$ be consecutive segments on a line with each length at most $1$.		9m49s
576 All-Union Mathematical Olympiad for School Students (1979, Grade 8). Represent each chosen vector by an oriented edge of a directed graph whose vertices are the given points.	✓	10m41s
577 All-Union Mathematical Olympiad for School Students (1979, Grades 8 and 10). Place coordinates on the board by identifying each square with the pair $(i,j)$, where $1\le i,j\le n$.		3m25s
578 All-Union Mathematical Olympiad for School Students (1979, Grades 8 and 10). Introduce		9m12s
579 All-Union Mathematical Olympiad for School Students (1979, Grade 9). For $n=1$ the inequality becomes	✓	15m32s
580 All-Union Mathematical Olympiad of School Students (1979, grades 8–10). Let $G$ be the graph whose vertices are parliamentarians and edges represent mutual enmity.		2m10s
581 The first question asks for a three-digit integer $x$ such that $x^3$ ends in $777$, equivalently		2m02s
582 A. V. Kelarev. Let the cyclic quadrilateral be $ABCD$, and let its diagonals $AC$ and $BD$ intersect at $P$.		7m06s
583 Let the stone masses be $x_1,\dots,x_n$ with $0<x_i\le 2$ and $\sum_{i=1}^n x_i=50$.		27m17s
584 F. V. Vainshtein. Suppose such a family of lines exists.		7m29s
585 S. V. Konyagin, P. Blecher. The majority are chemists, and chemists are perfectly reliable.		6m59s
586 All-Russian Mathematical Olympiad for School Students (1979, 8th grade). Let $B=60^\circ$ and let $O$ be the incenter of triangle $ABC$.		2m15s
587 All-Russian School Mathematics Olympiad (1979, 9th grade). The operation replaces two numbers $x,y$ by		3m28s
588 All-Russian Mathematical Olympiad for School Students (1979, Grade 9). For the planar analogue, take a triangle $ABC$ and a point $P$ inside it.	✓	8m21s
589 All-Russian School Mathematics Olympiad (1979, 9th grade). Let the given vectors be $v_1,\dots,v_n$.		6m10s
590 All-Russian Mathematical Olympiad for School Students (1979, Grade 10). Consider first the expression $/\cos x/ + /\cos 2x/$.		6m31s
591 International Mathematical Olympiad for School Students (XXI, 1979). Let		1m40s
592 Consider a triangle $ABC$ with circumcircle $\Gamma$.		5m06s
593 F. Kabdykarov, V. V. Proizvolov. Consider first the simplest case, $n=2$, with two circles inside a larger circle $\mathit\Gamma$.		6m58s
594 International Mathematical Olympiad (XXI, 1979). Let		5m38s
595 International Mathematical Olympiad for School Students (XXI, 1979). Label the vertices of the regular octagon cyclically by		3m55s
596 International Mathematical Olympiad (XXI, 1979). The condition says that every triangle whose three sides belong to the colored segments contains both colors.		9m33s
597 The sequence $x_n=1+\frac12+\dots+\frac1n$ is the $n$-th harmonic number.		3m24s
598 I can proceed, but I need the text of problem M598 first.		5m37s
599 Let $A=4^{5^6}+6^{5^4}$.		2m09s
600 N. B. Vasiliev, I. F. Sharygin. Let the circles intersect at points $A$ and $B$.		2m02s
601 Let $H$ be the orthocenter of triangle $ABC$, let $M$ be the midpoint of $BC$, and let $D$ be the point on the circumcircle diametrically opposite $A$.		1m22s
602 Let the three consecutive entries in row $n$ be		1m50s
603 L. P. Kuptsov. The denominators suggest introducing		1m39s
604 I cannot write a solution to Kvant problem M604 from the information provided, because the actual problem statement is missing.		58s
605 A reflection with respect to a point $A$ is the central symmetry $x\mapsto 2A-x$.		2m23s
606 The recurrence		1m48s
607 V. F. Lev. An isosceles trapezoid includes rectangles as a special case, since a rectangle has a pair of parallel sides and equal legs.		2m03s
608 M. L. Kontsevich, 10th-grade student. The polygon is rectilinear: every side lies on a grid line, hence every side is horizontal or vertical.		1m46s
609 For the planar statement, choose coordinates so that the two given perpendicular directions are the coordinate axes.		2m05s
610 A. K. Tolpygo. For part 1 it is natural to reinterpret a nondecreasing tuple		2m04s
611 The statement involves two circles.		1m36s
612 Assume that the infinite digit string obtained by concatenating		1m39s
613 L. P. Kuptsov. The data of the problem are naturally encoded by a similarity.		1m59s
614 Let $s(n)$ denote the sum of the digits of the single number $n$.		2m04s
615 V. A. Senderov. A triangular pyramid is a tetrahedron.		2m11s
616 S. T. Berkolayko. For the numbers $1,2,\dots,30$, the total sum is		1m44s
617 Let the triangle be $ABC$.		1m49s
618 Testing small values of $n$ shows that the divisibility condition $n^2+1 \mid n!$ is rarely satisfied for small integers, as $n^2+1$ grows faster than $n$.		3m22s
619 I. F. Sharygin. Let the bisectors of $\angle A$ and $\angle B$ meet at a point $P$.		6m58s
620 Consider small values of $n$ first.		9m40s
621 Let the circle have center $O$ and radius $r$.		5m31s
622 Consider the two Diophantine equations		1m15s
623 V. A. Senderov. A cube is highly symmetric, so the number of axes of symmetry should be larger than in simpler polyhedra.		1m11s
624 Compute the first few terms of the sequence $(a_n)$ directly from the recursive formula.		1m25s
625 The operations are purely projective.		1m43s
626 V. V. Proizvolov. The quadrilateral is cut by two families of lines.		1m47s
627 A. K. Tolpygo. For part 1, suppose every natural number appears exactly once.		6m57s
628 Consider a spherical triangle with one side of length $120^\circ$.		3m21s
629 For the first statement, computing small cases is instructive.		1m31s
630 I. F. Sharygin. The point $M$ is defined from the circle through $P,Q,K$.	✓	22m44s
631 All-Union Mathematical Olympiad (XIV, 1980, Grade 8). Let		1m45s
632 All-Union Mathematical Olympiad (14th, 1980, Grades 8 and 9). The problem involves packing 18 tons of cargo into at least 35 containers, with seven spacecraft available, each capable of carrying 3 tons, and the assertion that any selection of 35 containers can b…		1m13s
633 All-Union Mathematical Olympiad (XIV, 1980, 9th grade). Let the circle have center $O$ and radius $R$.		1m51s
634 All-Union Mathematical Olympiad (XIV, 1980, Grade 8). Define		1m44s
635 All-Union Mathematical Olympiad (XIV, 1980, Grade 8). Let $S_t$ be the set of sick Mites on day $t$.		8m36s
636 All-Union Mathematical Olympiad (XIV, 1980, Grade 10). Consider small examples to understand how the set $A$ might grow.		4m39s
637 All-Union Mathematical Olympiad (XIV, 1980, 9th grade). Consider an equilateral triangle $ABC$ with side length normalized to $1$ for convenience.		8m29s
638 All-Union Mathematical Olympiad (XIV, 1980, 8th grade). Consider small examples first.		7m32s
639 All-Union Mathematical Olympiad (XIV, 1980, Grade 10). Let		6m42s
640 All-Union Mathematical Olympiad (XIV, 1980, Grade 10). Let the decimal expansion of $x_k$ be		7m04s
641 E. G. Gotman. Place the regular hexagon in the coordinate plane with center		29m30s
642 I. K. Zhuk. The coefficients are restricted to the set ${-1,0,1}$, and two neighboring coefficients cannot both be nonzero.		5m12s
643 A shuffle takes an initial segment of the deck and inserts it somewhere later, preserving the internal order of the removed block and of the remaining cards.		5m44s
644 G. A. Gurevich. A convex equiangular $n$-gon has exterior angle $2\pi/n$ at every vertex.		7m12s
645 Normalize the speeds so that Warnicke moves with speed $1$ and the criminal with speed $\frac12$.		39m32s
646 Consider the problem for small values of $n$ to understand the geometric constraints.		7m13s
647 S. V. Fomin. The inequality is symmetric in $a$ and $b$.		5m42s
648 I. F. Sharygin. Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ intersect at $P$ and satisfy $AC \perp BD$.		7m48s
649 S. L. Manukyan. Let		7m02s
650 A. A. Razborov. We are asked about sequences of numbers (natural numbers or integers) such that every element in a certain target set (all naturals, all integers, or subsets thereof) can be represented uniquely as a…		6m26s
651 Let the sofa, suitcase, valise, picture, basket, cardboard box, and dog have weights $S, U, V, P, B, C, D$ respectively.		1m56s
652 Consider the set of faces of a convex polyhedron.		3m57s
653 The ruler has two fixed marks.		5m06s
654 Consider small examples of six natural numbers and examine the divisibility patterns.		7m45s
655 Consider small cases by simulating the procedure described.		4m59s
656 A. K. Tolpygo. The statement concerns only the directions of the vectors, since scaling a nonzero vector does not change its angle with another vector.		7m02s
657 A. V. Anzhan. Let the rows be $R_1,\dots,R_n$, each a vector of length $n$.		5m26s
658 A. V. Andzhan. Consider a square of side length $1$ with a collection of horizontal and vertical segments inside it.		7m26s
659 A. Odessky, 10th-grade student, D. B. Fuchs. For the Fibonacci sequence $f_1=1$, $f_2=2$, $f_{k+1}=f_k+f_{k-1}$, the first terms are		2m07s
660 Consider the dynamics of the allowed operations on the circle.		8m22s
661 N. B. Vasilyev. Let the speeds of the motorboat and rowboat be constant, equal to $v_M$ and $v_R$.		5m40s
662 A. G. Kushnirenko. The statement concerns a piggy bank containing coins whose total value is $4$ rubles.		2m01s
663 Consider small prime numbers to understand the behavior of the expression $2^p + p^2$.		3m24s
664 Consider a convex quadrilateral $ABCD$ with area $S$.		4m04s
665 I cannot write a rigorous solution to Kvant problem M665 because the actual problem statement is missing.		3m21s
666 The problem considers a network of $n$ vertices connected by $m$ undirected edges with positive integer weights.		12m58s
667 N. B. Vasilyev. Consider a triangle $ABC$ with the smallest angle $\widehat A$ and suppose the differences $d = /AB/ - /BC/$ and $e = /AC/ - /BC/$ are given.		7m48s
668 Computing the first few terms of the sequence $(x_i)$ modulo small integers may reveal patterns.		4m45s
669 Consider a cyclic quadrilateral $ABCD$ with circumcircle $\Gamma$.		4m21s
670 Let each vertex be a point, and let its color at time $t$ be represented by a sign $s_v(t)\in{+1,-1}$.		10m14s
671 Let the cyclic quadrilateral be $ABCD$ with diagonals $AC$ and $BD$.		6m20s
672 Let $f(n)=2^n-1$.		9m33s
673 A. A. Razborov. Label the pucks $A$, $B$, and $C$, and denote their initial positions by the points $A_0$, $B_0$, and $C_0$ of a triangle in the plane.		5m30s
674 Consider an acute triangle $ABC$ with arbitrary points $A_1$ on $BC$, $B_1$ on $AC$, and $C_1$ on $AB$.		7m17s
675 G. A. Gurevich, A. T. Kolotov. The condition states that all subset sums of the chosen set are distinct.		9m38s
676 A. V. Savkin, 9th-grade student. We begin by computing small powers of $1981$ and observing the sums of their digits.		8m16s
677 Consider a triangle $ABC$ and a point $M$ which is simultaneously the centroid, incenter, and orthocenter.		6m51s
678 A. F. Sidorenko. Consider small examples first.		6m23s
679 V. V. Proizvolov. Let the circles be $\omega_1,\omega_2,\omega_3,\omega_4$ with consecutive tangency points $A,B,C,D$.		9m23s
680 A. A. Razborov. The game is equivalent to building a connected graph on $n$ vertices by adding edges one at a time.		7m10s
681 Let		7m41s
682 Consider an arbitrary acute-angled triangle $\triangle ABC$.		6m59s
683 Represent each circle by a vertex, and join two vertices when the corresponding circles touch.		5m11s
684 Each ship occupies an entire row or an entire column of an $n\times n$ board, and different ships are disjoint, so all ships are either rows or columns exclusively.		2m13s
685 Let the partition be		4m56s
686 V. V. Prasolov. Let		3m17s
687 We model the configuration as follows.		2m18s
688 Let		3m16s
689 Each tile is an isosceles trapezoid with bases $3$ and $1$ and height $1$.		2m25s
690 A. V. Kelarev. For a convex polygon, the quantity $\dfrac{2S}{P}$ has a geometric meaning.		5m51s
691 All-Union Mathematical Olympiad of School Students (1981, 8th grade). Let $P(n,k)=n(n+1)\cdots(n+k-1)$ for $n\ge 2$.		2m13s
692 All-Union Mathematical Olympiad for School Students (1981, Grade 8). Consider triangle $ABC$ with arbitrary side lengths $AB=c$, $BC=a$, $CA=b$.		3m13s
693 All-Union School Mathematics Olympiad (1981, Grade 9). Consider small-scale analogues of the village communication problem.		4m54s
694 All-Union Mathematical Olympiad for School Students (1981, 9th grade). The operation affects the two endpoints of an edge.		6m59s
695 All-Union School Olympiad, Grade 10, 1981. Let the table have $m$ rows and $n$ columns.		7m13s
696 Consider the problem for small $k \times k$ squares.		2m58s
697 S. V. Fomin. Let the square have side length $1$.		9m51s
698 Let the cyclic quadrilateral be $ABCD$, with side lengths		7m57s
699 Working		30m36s
700 Consider the set of all terminating decimal fractions.		7m34s
701 Let the sides of the acute triangle $LMN$ be		8m41s
702 I. K. Zhuk. For the first values,	✓	8m45s
703 The first equation links three expressions of the form $t + \frac{1}{t}$ multiplied by constants 3, 4, and 5.		5m04s
704 N. B. Vasiliev. Consider a square $ABCD$ and a parallelogram $PQRS$ that circumscribes it, with each vertex of the square touching a different side of the parallelogram.		3m07s
705 G. A. Galperin, V. V. Proizvolov. Represent each cell of the sheet by a vertex.		4m45s
706 A. P. Savin. Consider two circles with centers $O_1$ and $O_2$ and radii $R_1$ and $R_2$.		4m32s
707 A. F. Sidorenko. Let the clubs be represented by sets of students.		8m20s
708 Consider a convex quadrilateral $ABCD$ and construct squares externally on its sides $AB$, $BC$, $CD$, and $DA$, with centers $P$, $Q$, $R$, $T$ respectively.		10m06s
709 The rhombus tiles are the unit lozenges of the triangular lattice.		6m51s
710 S. V. Konyagin. The requirement that no term is equal to the sum of several others is ensured by a stronger property: for every $n$, if		31m15s
711 Consider a convex quadrilateral $ABCD$ inscribed in a circle with diagonals $AC$ and $BD$ perpendicular at some point $P$.		6m28s
712 We seek to represent an arbitrary positive real number as a sum of nine numbers whose decimal expansions use only digits $0$ and $7$.		2m05s
713 V. V. Prasolov. Let the finite set be $M$.		3m49s
714 A. V. Anzhan. Consider small values of $N$ to build intuition.		5m55s
715 M. L. Kontsevich. The rule replaces one chip by two chips placed one step north and one step east, provided those target cells are empty.		6m50s
716 International Mathematical Olympiad for School Students (XXII, 1981). Let		6m32s
717 International Mathematical Olympiad for School Students (XXII, 1981). Consider small values of $n$ and $r$ to detect a pattern.		5m07s
718 International Mathematical Olympiad for School Students (XXII, 1981). The equation is		7m19s
719 International Mathematical Olympiad for School Students (XXII, 1981). Consider small values of $n$ to understand the property.		4m19s
720 International Mathematical Olympiad for School Students (XXII, 1981). The function $f$ is defined recursively on nonnegative integers.		6m07s
721 A. Zolotykh, 10th-grade student (Moscow, Specialized Physics and Mathematics School No. 18 at Moscow State University). Let $ABC$ be the given triangle.		41m44s
722 A. A. Razborov. Consider the simplest nontrivial cases first.		6m09s
723 We seek an infinite set $S \subset \mathbb{N}$ such that no element of $S$ and no finite sum of distinct elements of $S$ is a perfect power $a^k$ with $k \ge 2$.		2m05s
724 V. V. Prasolov. Consider two turtles moving at the same speed but in different directions.		3m04s
725 N. B. Vasilyev. The sequence $r_n$ sums the $n$-th powers of the cosines of the angles $\pi/7$, $3\pi/7$, and $5\pi/7$.		6m17s
726 V. V. Prasolov. Let the regular $2n$-gon have vertices $A_1,A_2,\dots,A_{2n}$ in cyclic order, and let $P$ be a point inside the polygon.	✓	23m55s
727 Since the perimeter is $2$, we have		6m08s
728 I. F. Sharygin. Consider a parallelepiped with vertex $P$ at the origin, edges along vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ leading to adjacent vertices $A = P + \vec{a}$, $B = P + \vec{b}$, $C = P + \vec{c}$.		7m13s
729 N. B. Vasiliev. The statement asks for a natural number with a specified property, not for all such numbers.		5m39s
730 V. S. Shevelev. The sequence $(a_n)$ is defined recursively by $a_1=0$ and $a_{2n}=a_{2n+1}=n-a_n$.		6m11s
731 V. G. Boltyansky. The game begins with the first player choosing an initial value $P_0 \in {2,3,4,5,6,7,8,9}$.		30m23s
732 I cannot write a rigorous solution to Kvant problem M732 because the problem statement itself is not present in your message.		5m55s
733 V. V. Prasolov. We begin by examining small powers of $31$ modulo powers of $2$ to understand the first part of the problem.		4m43s
734 Let $AB=c$, $AC=b$, and let $\angle A = \alpha$.		9m30s
735 Consider first the case of covering a circle of diameter $1$ with strips of paper.		6m15s
736 The statement involves a median and an angle bisector meeting at a point.		9m36s
737 A. V. Zelevinsky. Let the houses have populations $h_1,h_2,\ldots,h_n$, arranged in nonincreasing order.		6m25s
738 V. V. Prasolov. Consider a polygon in the plane, labeled $A_1 A_2 \dots A_n$.		5m03s
740 Consider a right circular cylindrical pot with radius $R$ and height $H$.		6m53s
741 Let		9m44s
742 A. Mikhailovsky, V. V. Prasolov. Let the points be represented by vectors $x_1,\dots,x_n$ from the center of the circle or sphere.		6m31s
743 L. D. Kurlyandchik, S. Okhitin. Part (1) is the classical two-color complete graph statement.		7m41s
744 The configuration consists of two similar triangles $ABC$ and $A_1B_1C_1$, with $A_1 \in BC$, $B_1 \in CA$, $C_1 \in AB$.		2m09s
745 For the first problem, let		5m41s
746 Let the square have side length $2$ and center $O$.	✓	30m44s
747 V. V. Prasolov. For the first part, write the numbers as $x_1,\dots,x_n$, let $M=\max x_i$ and $m=\min x_i$.		5m39s
748 Consider first the planar problem with parabolas.		6m09s
749 Part (1) concerns the cyclic expression		23m09s
750 S. N. Bespamyatnykh. The first statement is a classical rectangle theorem.		3m46s
751 All-Union Mathematical Olympiad for School Students (1982, Grade 9). Begin by examining small examples.		8m19s
752 All-Union Mathematical Olympiad for School Students (1982, Grade 8). Let the entries of the table be integers, and neighboring cells differ by at most $1$.		7m14s
753 All-Union School Mathematics Olympiad (1982, 10th grade). The problem gives three numbers $a$, $b$, $c$ in the interval $(0, \frac{\pi}{2})$ satisfying		6m21s
754 All-Union Mathematical Olympiad for School Students (1982, 10th grade). The expression		2m14s
755 All-Union School Mathematics Olympiad (1982, 9th grade). Consider a tetrahedron with vertices $A$, $B$, $C$, and $D$, and a point $M$ inside it.		4m52s
756 Model the airline network as a connected undirected graph.		5m22s
757 G. A. Halperin. Let an arithmetic progression of reciprocals be		10m56s
758 All-Union Mathematical Olympiad for School Students (1982, 9th grade). Let $R$ be the set of remaining integers, and let $A=R\setminus{1}$.		3m21s
759 All-Union Mathematical Olympiad for School Students (1982, Grade 10). Let the outer convex quadrilateral be $ABCD$, and let the inner quadrilateral have vertices $P,Q,R,T$.		2m23s
760 All-Union Mathematical Olympiad for Schoolchildren (1982, Grade 9). Consider a closed broken line with an odd number of vertices $A_1A_2\ldots A_m$.		9m58s
761 E. G. Gotman. The statement is affine in nature.		7m13s
762 The two inequalities are		6m21s
763 V. N. Dubrovsky. Let	✓	15m51s
764 O. V. Mazurov. The task is purely existential.		6m28s
766 Yu. I. Ionin. Consider three consecutive integers $n-1$, $n$, $n+1$ and compute the sum of their squares.		1m57s
767 V. V. Prasolov. For the first statement, choose coordinates so that the bisecting line $l$ is the vertical line $x=0$.		7m26s
768 V. P. Grinberg. Consider small values of $n$ first.		5m37s
769 R. A. Mazov. Let $I=L$ be the incenter.		5m49s
770 V. A. Senderov. Let the common value of the three face angles be $\alpha$:		2m43s
771 Let $O$ be the common point which is simultaneously the circumcenter of $\triangle ABC$ and the incenter of $\triangle ABK$.		4m38s
772 We are asked to train 8 workers on 5 machines so that any 3 absences still allow operation of all machines.		6m51s
773 P. B. Gusyatnikov, S. V. Reznichenko. Let the side lengths be		9m58s
774 P. B. Gusyatnikov. Condition (2) resembles a midpoint convexity-type inequality, but in the reversed direction: usually convexity gives $f\left(\frac{x+y}{2}\right)\le \frac{f(x)+f(y)}{2}$, whereas here we have $f\left(…		9m49s
775 A. V. Anzhants. Consider small values of $n$ to detect patterns.		8m29s
776 International Mathematical Olympiad (XXIII, 1982). Consider a regular hexagon $ABCDEF$ with vertices labeled consecutively.		7m02s
777 International Mathematical Olympiad for School Students (XXIII, 1982). The given equation is cubic in two variables, $x$ and $y$:		4m00s
778 International Mathematical Olympiad for School Students (XXIII, 1982). Let the triangle be isosceles with $A_2A_3=a_1$ as the base and $A_1A_2=A_1A_3$.		8m06s
779 International Mathematical Olympiad for School Students (XXIII, 1982). We are asked to study sequences of positive numbers $x_0, x_1, x_2, \dots$ with $x_0 = 1$ and $x_0 \ge x_1 \ge x_2 \ge \dots$.		9m50s
780 International Mathematical Olympiad for School Students (XXIII, 1982). The condition says that every point of the boundary of the square lies within distance $\tfrac12$ of the polygonal line $L$.		12m19s
781 L. V. Kim. Consider triangle $ABC$ and a line parallel to $AC$ intersecting $AB$ at $D$ and $BC$ at $E$ such that $/AD/ = /BE/$.		5m27s
782 S. V. Fomin. Let $a+b=30030$ with $a,b\in \mathbb{N}$.		2m05s
783 Consider the first system of inequalities: $1 < x < 2$, $2 < x^2 < 3$, $3 < x^3 < 4$, and so on up to $n < x^n < n+1$.		6m14s
784 A. P. Savin. The length of daylight at latitude $\varphi$ depends on the declination $\delta$ of the star.		10m41s
785 M. U. Gafurov. The hypothesis states that for every integer $k>1$ there exists a constant $b_k$ such that		9m37s
786 A. N. Kozachenko. Consider small values of $n$ and $k$ to identify a pattern.		7m18s
787 B. S. Pitskel. Let the right triangle have legs of lengths $a$ and $b$, with hypotenuse $c = \sqrt{a^2 + b^2}$.		4m08s
788 E. D. Moskalensky, Yu. V. Chinyaev. Let	✓	9m34s
789 V. V. Proizvolov. Consider first a small number of points on a circle, for example, four points dividing the circle into four equal arcs.		7m02s
790 The hypothesis is that a map $F:\mathbb{R}^2\to\mathbb{R}^2$ preserves unit distance, meaning every pair of points at distance $1$ is mapped to a pair of points at distance $1$.		2m14s
791 S. B. Gashkov. Petya's calculator allows addition, subtraction, increment by one, and reciprocal operations.		6m54s
792 S. N. Bychkov, V. V. Prasolov, L. D. Kurlandchik. For the first two equations, small natural numbers $x$ and $y$ can be tested directly.		4m09s
793 V. N. Dubrovsky. Let		8m31s
794 Alla Ivchenko, 9th-grade student (Mohyliv-Podilskyi, School No. 1). Let the first circle be $\omega_1$ and the second circle be $\omega_2$.	✓	24m15s
795 V. F. Lev. Consider small values of $n$ and their divisor sums $\sigma(n)$.		3m19s
796 L. D. Kurlandchik. Consider a square $ABCD$ with a point $P$ inside it.		10m09s
797 D. B. Fuchs. We interpret the problem as asking whether, for every fixed block of $n$ decimal digits $A=a_1a_2\ldots a_n$, there exists an integer $x$ such that the last $n+1$ digits of $x^2$ have the form $A b$,…		2m21s
798 S. V. Fomin. Consider first small values of $k$.		7m25s
799 S. S. Vallander. Consider the first equation, $3^{x+1} + 100 = 7^{x-1}$.		10m02s
800 A. B. Goncharov. Consider the square lattice $\mathbb{Z}^2$ with distinguished origin $O=(0,0)$.		3m55s
801 V. V. Kisil'. Compute several small cases to see the pattern.		4m45s
802 L. P. Kuptsov. The data involve two right triangles erected externally on sides $AB$ and $BC$.		2m14s
803 R. A. Mazov, A. P. Savin. Let $x,y \in \mathbb{Q}\setminus{0}$ satisfy		8m00s
804 I. K. Zhuk. Place a right circular cylinder vertically with axis along the $z$-axis and center at the origin, so that $O=(0,0,0)$ is the midpoint of the axis.		7m34s
805 R. P. Ushakov. Consider the planar case first.		10m42s
806 Let		13m10s
807 V. V. Prasolov. Consider a regular polytope in two or three dimensions, a point $M$ inside or near it, and the feet of perpendiculars from $M$ to its sides or faces.		13m59s
808 D. G. Azov. Begin by examining small values of $k$.		4m14s
809 V. V. Proizvolov. Compute the first few terms of the series to detect a pattern.		10m33s
810 F. V. Vainshtein. Let $M$ be a convex polygon with area $S$.		6m59s
811 D. M. Milošević. The quantities involving medians and altitudes suggest rewriting everything in terms of the sides.		10m38s
812 S. I. Maizus. The sum is	✓	13m31s
813 V. V. Prasolov. Consider three segments $OA$, $OB$, and $OC$ of equal length $R$, with $B$ inside the angle $AOC$.		1m43s
814 L. D. Kurlyandschik. A natural number is representable as a sum of two squares if and only if every prime congruent to $3 \pmod 4$ appears with even exponent in its prime factorization.		9m33s
815 A. A. Razborov. Place the $4k$ points on a circle in their cyclic order and denote their labels by $a_1,a_2,\ldots,a_{4k}$.		32m37s
816 A. D. Lisitsky. Let $s(n)$ denote the sum of the decimal digits of $n$.		6m31s
817 A. L. Toom. Let		7m55s
818 M. L. Kontsevich. Consider small $n$-gons to understand what a uniform set of blue vertices looks like.		16m10s
819 L. M. Kogan. We are given the complete graph on $n$ vertices with each edge oriented in exactly one direction.		7m22s
820 V. V. Proizvolov. A parallelogram has two pairs of parallel sides.		3m59s
821 Yu. I. Ionin. The equation is		2m16s
822 S. B. Shlosman. The initial stack is the periodic sequence of length $4n$ with pattern $G,R,B,Y$ repeating every four cards, so every consecutive block of four cards contains all four colors exactly once.		5m06s
823 Before I begin the full solution, I need clarification: does the photograph provide any perspective cues such as vanishing points, the relative positions of the corners, or angles of the visible wall?		7m19s
824 I need the diagram from Figure 3 to solve the problem rigorously.		4m20s
825 All-Union Mathematical Olympiad for School Students (1983, Grade 10). Let the segments be	✓	17m20s
826 All-Union School Mathematics Olympiad (1983, 8th grade). The operation described on the board replaces one number with the sum of the other two numbers minus one.		7m57s
827 All-Union Mathematical Olympiad for School Students (1983, Grade 8). The figure consists of a triangle subdivided into smaller regions, four of which are blue triangles of equal area.		7m27s
828 All-Union Mathematical Olympiad for School Students (1983, 8th grade). Consider a function $a_{i,j}$ on the integer lattice.		1m58s
829 All-Union Mathematical Olympiad for School Students (1983, Grade 9). Consider small values of $m$ to detect a pattern.		14m07s
830 All-Union Mathematical Olympiad for School Students (1983, 10th grade). Let the first quadratic be $x^2 + p_1 x + q_1 = 0$ with two distinct real roots $r_1 \le s_1$.		9m39s
831 V. V. Prasolov. Introduce position vectors for the vertices $A,B,C,D$ in the plane.		7m40s
832 V. A. Li. Consider first the case of dividing a square into smaller squares.		4m08s
833 V. E. Matizen. Compute the first few terms to detect patterns.		7m17s
834 N. B. Vasilyev. This is a two-part geometric covering problem.		34m45s
835 L. D. Meniches. Represent the previous encounters by a bipartite graph.		7m31s
836 International Mathematical Olympiad for School Students (XXIV, 1983). The coordinate setup in the proposed solution is correct and can be carried through to completion.		38m54s
837 International Mathematical Olympiad for School Students (XXIV, 1983). Consider the simpler case where two of the numbers are coprime.		9m11s
838 International Mathematical Olympiad for School Students (XXIV, 1983). Consider an equilateral triangle $ABC$.		10m41s
839 International Mathematical Olympiad for School Students (XXIV, 1983). For a set of integers with no three-term arithmetic progression, the classical example is obtained by writing numbers in base $3$ and allowing only digits $0$ and $1$.		8m47s
840 International Mathematical Olympiad for School Students (XXIV, 1983). The first expression can be expanded into a difference of two homogeneous cyclic sums:		7m52s
841 Consider a right triangle $ABC$ with right angle at $C$.		1m17s
842 L. D. Kurlyandchik. Consider the first identity $\sin\alpha + \sin\beta + \sin\gamma$ under the constraint $\alpha + \beta + \gamma = 0$.		7m11s
843 A. A. Yagubyants. Let the plane of the triangle be $z=0$.	✓	10m37s
844 V. E. Kolosov. For the first representation, the coefficients are constrained by $0\le a_k\le k$.		4m37s
845 V. G. Belov. We consider the problem of forming a centrally symmetric polygon using two types of tiles: a “corner” formed by four $1\times1$ squares arranged in an L-shape, and $4\times1$ rectangles.		1m39s
846 V. F. Lev. Let the regular polygon have $n$ sides and circumradius $R$.		1m57s
847 I. V. Vetrov, A. G. Kogan. The game is played on the edge set of the $n\times n$ square grid graph.		2m14s
848 P. G. Satyanov. The function $f_0(x) = //x-1/-2//x/-3//$ involves nested absolute values.		1m46s
849 ```		15m44s
850 V. N. Dubrovsky. Let $A,B,C$ be a nondegenerate triangle with side lengths $BC=a$, $CA=b$, $AB=c$.		10m08s
851 A. B. Khodulyev. Place the square in a coordinate system so that computations can be expressed in terms of two parameters.		9m53s
852 ```		9m12s
853 A square $ABCD$ rotates about its fixed center $O$, while a fixed line $l$ is taken as the $x$-axis.	✓	29m33s
854 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.		3m10s
855 ```		3m03s
856 ```		3m01s
857 A number is marked if it can be written as a sum of exactly five powers of two, repetitions allowed.		9m36s
858 P. B. Gusyatnikov. Let		2m09s
859 V. P. Pikulin. Let		55s
860 Consider the triangle $ABC$ with circumcircle $(O)$ and incircle $(Z)$.		1m40s
861 Consider small values of $n$ to understand the behavior of the sums modulo $1$.		58s
862 E. A. Yasinovy̆ĭ. Consider first the equilateral triangle case.		4m09s
863 Consider a small board, $n=3$.		1m32s
864 A. V. Savkin. Consider first a right triangle.		7m10s
865 B. M. Ivlev. For consecutive pairs $a_{i-1}<a_i$, the expression		12m45s
866 Cities Tournament (spring, 1984). Represent the cells by lattice points $(i,j)$, where $1\le i\le m$ and $1\le j\le n$, the coordinates being the centers of the cells.		7m01s
867 City Tournament (Spring, 1984). Let the boys' heights be $b_1,\dots,b_{17}$ and the girls' heights be $g_1,\dots,g_{17}$.		7m22s
868 Tournament of the Cities (spring, 1984). Let the tetrahedron be $ABCD$, where $ABC$ is the base.		4m25s
869 Tournament of Towns (spring, 1984). A number is powerful when every prime divisor appears with exponent at least $2$, so each such number has the form		8m56s
870 Cities Tournament (Spring, 1984). Let the occupied rooms be represented by the multiset of integer positions of all pianists.		7m15s
871 All-Union Mathematical Olympiad for School Students (1984). We encode each entry $x_{i,j}\in{\pm1}$ by $a_{i,j}\in\mathbb{F}2$ via $x{i,j}=(-1)^{a_{i,j}}$.		2m20s
872 All-Union School Mathematics Olympiad (1984). Let $O_1,O_2,O_3$ be the centers of the circles $C_1,C_2,C_3$.		8m18s
873 All-Union Mathematical Olympiad for School Students (1984). The process can be encoded in terms of the coefficients $a$ and $b$ of the quadratic $x^2+ax+b$, starting from $(a,b)=(10,20)$ and ending at $(20,10)$.		8m18s
874 All-Union School Mathematical Olympiad (1984). We begin by testing small integer values to see whether the equation $(5+3\sqrt{2})^m = (3+5\sqrt{2})^n$ admits any obvious solutions.		11m31s
875 All-Union Mathematical Olympiad for School Students (1984). Let		37m39s
876 Leningrad City Mathematical Olympiad (50th, 1984). Consider the circle inscribed in an angle with vertex $O$ and the two diametrically opposite points $A$ and $B$.		5m51s
877 Leningrad City Mathematical Olympiad (Problem 50, 1984). Consider a smaller version of the problem first.		3m25s
878 Leningrad City Mathematical Olympiad (50, 1984). Consider a pyramid with apex $A$ and base $B_1B_2\dots B_n$.		10m23s
879 Leningrad City Mathematical Olympiad (50, 1984). Work is carried out in the residue field $\mathbb{F}_p$, where $p$ is odd, so $2$ is invertible.		10m55s
880 Leningrad City Mathematical Olympiad (50, 1984). The sequence begins as $1, 0, 1, 0, 1, 0$ and each subsequent term is defined as the last digit of the sum of the preceding six terms.		4m49s
881 Our systems have detected unusual activity coming from your system.		26m56s
882 Leningrad City Mathematical Olympiad (50th, 1984). Consider small integer triples $(a,b,c)$ satisfying $a+b+c=0$.		4m53s
883 Working		14m25s
884 The reviewer correctly identified a critical error.		32m13s
885 Tournament of Towns (Spring 1984). The quantity $q(n)$ equals the sum, over all partitions $\lambda$ of $n$, of the number of distinct values appearing among the parts of $\lambda$.	✓	24m27s
886 All-Russian Mathematical Olympiad for School Students (Grade 10, 1984). Label the cells on the boundary of the $n\times n$ square cyclically by		1m53s
887 All-Russian Mathematical Olympiad for School Students (X, 1984). Consider a circle $\Gamma_1$ with tangents $CA$ and $CB$ meeting at $C$, so $A$ and $B$ are points of tangency.		7m39s
888 All-Russian Mathematical Olympiad for School Students (10th, 1984). Let $ab=cd$.		4m11s
889 All-Russian Mathematical Olympiad for School Students (Grade 10, 1984). Consider the problem of choosing three points $A$, $B$, $C$ in the plane such that every point $P$ has at least one segment $PA$, $PB$, or $PC$ of irrational length.		7m03s
890 All-Russian Mathematical Olympiad for School Students (Grade 10, 1984). The problem concerns connecting 51 cities in a square-shaped country of side 1000 km with 11,000 km of highways.		8m30s
891 A. A. Muratov. Let $ABC$ be a triangle with a circle tangent to two of its sides and to two of its medians.		26m28s
892 A. I. Zaichik. Write		8m30s
893 Moscow Mathematical Olympiad (1984). The complete graph on $n$ vertices is $K_n$.		12m10s
894 The previous proof contains critical errors in both parts.	✓	36m39s
895 Moscow Mathematical Olympiad (1984). Place the cube of side $2$ in coordinates with center at the origin, so its vertices are $(\pm1,\pm1,\pm1)$ and its inscribed sphere is $x^2+y^2+z^2=1$.		2m16s
896 The condition that the circle with diameter $AB$ is tangent to the line $CD$ has a simple metric interpretation.		4m18s
897 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 $7$.		3m02s
898 Consider the given odd natural numbers $a<b<c<d$ satisfying $ad=bc$, $a+d=2^k$, and $b+c=2^m$.		8m48s
899 A. P. Savin. Each number $x$ can be written uniquely as $x = \lfloor x \rfloor + t$, where $t \in (0,1)$.		7m15s
900 M. D. Kovalev. A convex polyhedron with six faces is a cube or a more general hexahedron.		6m12s
901 Consider triangle $ABC$ with bisectors $AK$ and $BM$ intersecting at $O$.		2m59s
902 A. V. Kelarev. Suppose the partition consists of finitely many arithmetic progressions		9m23s
903 A. A. Dorogovtsev. A plane section of a convex polyhedron changes combinatorially only when the plane passes through a vertex.		2m56s
904 I cannot write a solution because the actual problem statement is missing.		1m16s
905 Consider the equation		10m22s
906 M. V. Slavinskyi. Consider the equation		27m41s
907 T. A. Dzhortmenadze. Let $A=\widehat A$, $B=\widehat B$, $C=\widehat C$.		6m49s
908 Before I begin writing the full Kvant-style solution, I need the text of problem M908.		2m49s
909 R. N. Azizyan, V. Tolstykh, A. V. Alyaev. We seek an arithmetic progression		10m21s
910 Let the regular hexagon be $P_1P_2P_3P_4P_5P_6$, and let the points of the problem be chosen on its sides so that $A_i\in P_iP_{i+1}$, indices modulo $6$.		1m36s
911 M. V. Stark. Place quadrilateral $ABCD$ in the plane and select points $E$ on $AB$ and $F$ on $CD$.		3m06s
912 V. P. Pikulin. For a polynomial to be monotonically increasing on the whole real line, it is enough that its derivative be nonnegative everywhere.		6m29s
913 Consider triangle $ABC$ with circumcircle $\Gamma$.		6m05s
914 V. G. Ilyichyov. Let $(g,b,c)$ denote the numbers of gray, brown, and crimson chameleons.	✓	12m15s
915 L. D. Kurlandchik. The inequality is cyclic rather than symmetric:		7m26s
916 A. A. Azamov. Let the acute triangle be $ABC$.		8m00s
917 Consider six-digit numbers from $000000$ to $999999$.		8m32s
918 V. V. Prasolov. Let the triangle have sides $a,b,c$ and semiperimeter $s=\frac{a+b+c}{2}$.		9m39s
919 Yu. I. Ionin. For the first integral equality, the two integrals involve complementary functions: the tangent function on $[0,\pi/4]$ and the arctangent function on $[0,1]$.		6m07s
920 R. A. Mazov. The equation is		5m39s
921 The problem involves a convex quadrilateral $ABCD$ with two given angles, $\angle A = \alpha$ and $\angle B = \beta$, and a special relation between its sides and area: the doubled area satisfies $2S…		9m17s
922 A. M. Sedletskii. Let		3m30s
923 Consider a unit cube in three-dimensional space with edges parallel to the axes.		4m54s
924 I. I. Tsalenchuk, 10th grade student. Each pair of points is connected by a directed edge, so the structure is a tournament.		14m51s
925 A. L. Toom. Consider a small blue region, for example, a disk of radius $r<1$.		3m40s
926 S. V. Duzhin. The given conditions state that the vectors $(x,y)$ and $(u,v)$ both lie on the unit circle and are orthogonal with respect to the standard dot product since $xu+yv=0$.		7m06s
927 V. E. Kolosov. The problem asks whether a sequence of allowed replacements can form a nontrivial cycle.		4m09s
928 I. B. Alekseev-Astafyev. Consider small values of $N$ to understand the dynamics of the seat-shifting process.		2m12s
929 V. D. Yakovlev. Consider the equation $a^4 + b^4 + c^4 + d^4 = e^4$ modulo small primes to understand divisibility constraints.		4m27s
930 A. D. Valiev, 9th grade student. We are asked to prove that in any partition of the integers from $1$ to $1985$ into six classes, one class must contain either a triple $a,b,c$ with $a+b=c$ or a pair $a,2a$.		2m09s
931 Moscow Mathematical Olympiad. Consider triangle $ABC$ with an incircle touching sides $AB$, $BC$, and $CA$ at points $C_1$, $A_1$, and $B_1$ respectively.		1m34s
932 Moscow Mathematical Olympiad. The anaconda is an arbitrary polygonal line of total length $10$ contained in the unit square.		3m48s
933 Moscow Mathematical Olympiad. Let the clans be represented by labels.		5m06s
934 Moscow Mathematical Olympiad. Interpret the $2n$ points as vertices of a graph $G$ with $2n$ vertices and $n^2+1$ edges.		5m24s
935 Moscow Mathematical Olympiad. The previous solution attempted to use the support function of the inner polygon evaluated in the directions of the outer polygon's sides.		27m03s
936 City Tournament (Spring, 1985). Consider the simplest nontrivial case $n=1$.		1m02s
937 Tournament of Towns (Spring 1985). The question asks whether there is a set $F$ which by itself cannot contain any semicircle of radius $1$, while two congruent copies of $F$ can together contain the whole unit circle.		4m34s
938 Tournament of Towns (Spring 1985). Let the angular speed be $\dfrac{360^\circ}{n}$ per second.		1m22s
939 City Tournament (Spring, 1985). The problem has two parts.		1m46s
940 City Tournament (spring, 1985). For the planar statement, the natural idea is to look at one fixed side of the square, say the left side.		5m50s
941 All-Union Mathematical Olympiad (1985). Consider first the case $k=2$, which corresponds to a regular decagon.		9m09s
942 All-Union Mathematical Olympiad (1985). For $n=1$, the partition is ${1}$ and ${2}$, hence		3m25s
943 All-Union Mathematical Olympiad (1985). Let us compute the first terms.		4m47s
944 All-Union Mathematical Olympiad (1985). The figure is the regular hexagon of side length $2$ subdivided into $24$ congruent equilateral triangles.	✓	11m35s
945 All-Union Mathematical Olympiad (1985). Let		3m07s
946 All-Russian Mathematical Olympiad for School Students (XI). Position two parabolas in the plane with perpendicular axes.		1m31s
947 All-Russian School Mathematics Olympiad (XI). Consider first small cases.		10m44s
948 All-Russian Mathematical Olympiad for School Students (XI). The problem is a combinatorial-geometric proof.		33m48s
949 All-Russian Mathematical Olympiad for School Students (XI). Consider small cases first.		5m00s
950 All-Russian School Mathematics Olympiad (XI). The $25$ plots form the $5\times5$ grid graph.		10m39s
951 Let the hexagon be $ABCDEF$ in convex order with $AB=BC=CD=DE=EF=FA=1$.		2m25s
952 Write		1m28s
953 V. V. Prasolov. Each special point is an intersection of three lines determined by pairs of the six given points.		9m48s
954 V. N. Dubrovsky. Consider first the case of a rectangle inscribed in a triangle.		8m23s
955 V. B. Alekseev. Consider first small numbers of participants.		4m55s
956 Let the four circles have the same radius $r$.		6m47s
957 International Mathematical Olympiad (XXVI) for School Students. Each allowed integer has the form		9m14s
958 International Mathematical Olympiad (XXVI). Let		3m21s
959 A. A. Razborov. Consider first small examples.		7m47s
960 Something went wrong.		29m02s
961 A. P. Savin, N. A. Paravyan. Let the side length of the square be $6$.		7m12s
962 Let $P(x)\in\mathbb{Z}[x]$ and suppose there exist distinct integers $x_1,\dots,x_n$ with $n\ge 3$ such that		2m05s
963 Let the hexagon be $A B C D E F$ in cyclic order.		3m50s
964 A. A. Stolin. The sequence $(a_n)$ consists of distinct positive integers with the growth constraint $a_n < 100n$.		7m44s
965 N. B. Vasiliev. Let	✓	13m19s
966 L. D. Kurlandchik. 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.		7m38s
967 V. F. Lev. For small values,		3m20s
968 Solution to Kvant M968		34m21s
969 Unusual activity has been detected from your device.		7m01s
970 S. L. Eliseev. Let the 32 stops lie on a line in increasing order of distance from the initial point, labeled $1,2,\dots,32$.		10m54s
971 A. T. Ukrainets. Consider a tournament of $8$ volleyball teams where each team plays every other team exactly once.		7m37s
972 A. V. Andzhans. The sequence $(x_n)$ begins with $x_1 = \frac12$ and satisfies the recurrence $x_{n+1} = x_n^2 + x_n$.		1m56s
973 I. F. Sharygin. Let		3m04s
974 S. V. Fomin. Suppose both players start with equal time and make alternating moves.		7m40s
975 A. K. Tolpygo. Consider first a simplified scenario: a small $n\times n$ board, say $n=5$, with just a few hypothetical pieces each attacking a limited number of squares.		1m33s
976 E. G. Gotman. Place the square in coordinates:		1m22s
977 The problem asks whether $x$ can be expressed using only addition, subtraction, and multiplication from given polynomials.		1m35s
978 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 $\frac1{\sqrt2}$.		1m22s
979 Consider the definition of an exceptional set of $k$ numbers $a_1, a_2, \dots, a_k$, all strictly between 0 and 1.		1m32s
980 V. G. Boltyanskyi. Consider first a convex polygon in the plane with vertices $A_1, A_2, \dots, A_n$ and a point $O$ inside it.		1m17s
981 L. D. Kurlandchik. Consider small repunit numbers of the form $R_n = 11\ldots1$ with $n$ ones.		1m37s
982 Construct triangle $ABC$ on paper and build the external squares $ABB_1A_2$, $BCB_1C_2$, $CAA_1C_2$.		1m14s
983 K. P. Kokhasy̆a. The tournament is a complete directed graph on $16$ vertices.		2m08s
984 V. N. Dubrovsky. Consider a square $ABCD$ and an arbitrary point $K$ inside it.		1m28s
985 A. B. Goncharov. We are asked to count configurations of three lines through a point in space with prescribed pairwise angles, up to congruence.		3m01s
986 The inequality is		1m42s
987 M. Bona, high school student (Hungary). Consider small instances to gain intuition.		1m39s
988 Consider small values of $n$ and $k$ to build intuition.		1m25s
989 I begin by examining small natural numbers $a$ to see which of them satisfy the given conditions.		1m51s
990 V. N. Dubrovsky. Consider three lines in space, each pair of which is skew, and they are not all parallel to the same plane.		1m22s
991 Consider triangle $ABC$ with an altitude $CH$ and median $CK$.		1m21s
992 Consider small examples of social networks where each person has at least 10 friends.		1m14s
993 Let $x$ be the smallest of $n$ consecutive natural numbers.		1m13s
994 Let		1m39s
995 Let		1m52s
996 V. V. Proizvolov. The octagon is the intersection of two congruent squares.		1m17s
997 D. A. Mitkin. Let		2m43s
998 I. F. Sharygin. Consider a tetrahedron $AXBY$ circumscribed about a sphere with fixed points $A$ and $B$.		1m45s
999 L. D. Kurlyandchik, A. S. Merkuryev. Let		2m59s
1000 Archimedes (Syracuse). Let $O$ be the center of the circle containing the arc $AB$, and let $\angle AOB=2\alpha$.		1m37s
1001 Leningrad City Mathematics Olympiad (1986). Let $S(n)$ denote the total sum of all recorded products when a pile of $n$ stones is repeatedly split until all piles contain one stone.		1m40s
1002 Moscow Mathematical Olympiad. The door opens as soon as some block of three consecutive pressed digits coincides with the code.		1m42s
1003 Leningrad City Mathematical Olympiad (1986). The quantities involve segments cut off by the feet of the altitudes.		1m36s
1004 All-Union Mathematical Olympiad (1986). Let a line through $A$ be fixed.		1m40s
1005 All-Union Mathematical Olympiad (1986). Consider a $3 \times 3$ table first.		1m37s
1006 G. A. Halperin, A. P. Savin. Let the triangle be $ABC$.		1m40s
1007 V. P. Chichin. The equality		7m12s
1008 S. L. Eliseev. Number the steps from $1$ at the bottom to $2n+1$ at the top.		6m17s
1009 I. F. Sharygin. Let the parallelogram be represented by vectors.		9m50s
1010 O. T. Izhboldin, L. D. Kurylandchik. Consider the sequence defined by $r_1=2$ and $r_{n+1}=r_1 r_2 \cdots r_n + 1$.		3m07s
1011 L. D. Kuryandchik. For the first inequality,		3m30s
1012 D. V. Fomin. Consider arrangements of circles in the plane where each circle touches several others.		2m50s
1013 V. V. Rozhdestvensky. Consider triangle $ABC$ with points $M$ on $AB$ and $N$ on $BC$.		7m05s
1014 V. F. Lev. Consider small examples of pairwise coprime numbers, such as $a_1=2$, $a_2=3$, $a_3=5$.		7m24s
1015 S. L. Manukyan. The polynomial is		5m37s
1016 I. Z. Weinstein. For a polygon circumscribed about a circle of radius $r$, let the sides be $s_1,\dots,s_n$, with corresponding side lengths $\ell_1,\dots,\ell_n$.		7m39s
1017 International Mathematical Olympiad for School Students (XXVII). Consider assigning integers to the vertices of a regular pentagon and performing the prescribed operation whenever a vertex carries a negative number.		5m14s
1018 International Mathematical Olympiad for School Students (XXVII). Consider a regular $n$-gon $A_1 A_2 \dots A_n$ with center $O$.		9m33s
1019 International Mathematical Olympiad for School Students (XXVII). Consider first a small example on a $3 \times 3$ portion of the grid.		7m20s
1020 V. V. Prasolov, G. A. Halperin. Consider a sphere of radius $1$ with a curve drawn on it, either open of length less than $\pi$ or closed of length less than $2\pi$.		7m06s
1021 Consider how the mountaineer’s progress depends on the day’s starting point.		7m06s
1022 M. I. Shterenberg. For numbers $1,2,\dots,2n$, suppose they are arranged in two rows and $n$ columns.		1m50s
1023 G. A. Gurevich. For small numbers of triangles the statement is false.		4m55s
1024 R. P. Ushakov. Consider two triangles with angles $\alpha, \beta, \gamma$ and $\alpha_1, \beta_1, \gamma_1$.		5m21s
1025 I. F. Sharygin. Consider a convex quadrilateral $ABCD$ with extensions of opposite sides $AB$ and $CD$, and $AD$ and $BC$, intersecting at points $P$ and $Q$ respectively.		1m48s
1026 A. V. Shvetsov. Let the common measure of each arc be $x$.		9m02s
1027 V. V. Proizvolov. The number $1987$ is prime, since it is not divisible by any prime not exceeding $\sqrt{1987}<45$.		5m24s
1028 M. A. Volchkevich, 10th-grade student (Moscow). Begin by considering the configuration of two intersecting lines and points $D$ and $E$ on them.		10m13s
1030 A. B. Goncharov. Consider simple convex polyhedra such as nested cubes, tetrahedra, or pyramids.		6m09s
1031 L. D. Kurlyandchik. Reflecting on the problem, the point $M$ is chosen on the line $\ell$ to minimize the sum $MA + MB$.		8m29s
1032 A. V. Andzhans. Begin with small values of $n$ to detect a pattern.		1m51s
1033 V. V. Proizvolov. Let the square have vertices $A,B,C,D$ in cyclic order.		6m40s
1034 S. V. Fomin. Consider small chocolate bars first.		10m32s
1035 V. S. Grinberg. Consider marking points on $[0,1]$ sequentially.		3m26s
1036 S. M. Khosid. Consider a pentagon and imagine cutting it into two smaller pentagons of equal area and shape.		26m51s
1037 A. I. Zaychik. Consider the equation $x^y - y^x = x + y$ with $x, y \in \mathbb{N}$.		4m14s
1038 M. Khovanov, 9th-grade student, A. P. Savin. The rectangle contains $mn$ cells.		1m45s
1039 V. E. Matizen. Label the tetrahedron vertices as $A$, $B$, $C$, $D$.		16m36s
1040 V. B. Alekseev, S. Savchev (Bulgaria). For $n=1$ the three groups are ${1},{2},{3}$, and $3=1+2$.		8m36s
1041 Moscow 50th City Mathematical Olympiad, 1987. A regular pentagon is determined up to congruence by any three consecutive vertices.		7m51s
1042 50th Moscow City Mathematical Olympiad, 1987. Let the class contain $n$ students.		17m08s
1044 50th Moscow City Mathematical Olympiad, 1987.. Let		2m48s
1045 Moscow 50th City Mathematical Olympiad, 1987. Consider the geometry of the kingdom, which is a square of side $2$ km.		32m23s
1046 City Tournament (Spring, 1987). Consider an acute-angled triangle $ABC$ with $\angle A = 60^\circ$.		2m37s
1047 City Tournament (Spring, 1987). Consider a small round-robin tournament with $n$ players.		1m32s
1048 Tournament of Towns (Spring 1987). Consider the simplest nontrivial cases of the knight’s tour game.		1m49s
1049 Consider a cylinder $\text{Ц}_1$ with radius $R_1$ and height $H_1$, and define its diameter-to-height ratio $k = \frac{2R_1}{H_1}$.		1m33s
1050 L. D. Kurlandchik. Let the chosen points be		2m50s
1051 Cities Tournament (Spring, 1987). Consider a $3\times3$ cluster of pieces on an $8\times8$ chessboard.		16m32s
1052 City Tournament (Spring, 1987). Consider a convex $n$-gon with vertices labeled cyclically as $A_1, A_2, \dots, A_n$.		17m05s
1053 The first few Fibonacci numbers with at least four digits are		16m33s
1054 Yu. K. Koba. Consider four spheres in three-dimensional space.		23m46s
1055 City Tournament (Spring, 1987). Consider a circle with a small number of points to understand the behavior of arcs subtending at most $120^\circ$.		1m23s
1056 A. S. Merkuryev. Consider small cases first.		1m12s
1057 D. V. Fomin. A move consists of writing a number that is not a divisor of any previously written number.		1m17s
1058 D. G. Flaas. Consider a finite subset of $\mathbb{Z}^2$ as a candidate for the marked points and examine what happens when we translate each by all vectors from the given finite set.		1m12s
1059 A. V. Klyushin. Let		1m21s
1060 A. Serdyukov, D. G. Flaass. Consider two closed polygonal chains in the plane, each with an odd number of sides.		1m22s
1061 V. E. Kolosov. Interpret the cities and roads as a graph.		1m19s
1062 T. A. Dzhortmenadze, E. Ya. Gleibman. The first part of the problem deals with a triangle $ABC$ with points $D$ on $AC$ and $E$ on $AB$, forming the intersecting lines $BD$ and $CE$ at $M$.		1m49s
1063 G. O. Elsting. Let the digits of the $n$-digit number $a$ be		1m37s
1064 D. B. Fuchs. Let the closed broken line have vertices $V_1,\dots,V_n$ and segments $e_i=V_iV_{i+1}$, where indices are taken modulo $n$.		1m36s
1065 F. V. Weinstein. We are asked to study vectors $(x;y)$ with non-negative integer coordinates and to decide when they can be written as sums of generating vectors, i.		1m36s
1066 S. G. Salnikov. Consider six points in the plane with all pairwise distances at most $1$.		2m00s
1067 V. E. Matizen. Let		1m36s
1068 R. O. Burdin. Consider an angle $AOB$ with points $A$ on one side and $B$ on the other.		1m30s
1069 Tournament of Towns (Spring, 1987). Consider a small number of families, say three or four, each in a distinct apartment.		1m39s
1070 V. N. Dubrovsky. Let the tetrahedron have vertices $A,B,C,D$.		1m34s
1071 Leningrad City Mathematical Olympiad (1987). Consider smaller versions of the game to understand the parity dynamics.		2m23s
1072 Leningrad City Mathematical Olympiad (1987). The expression $989 \cdot 1001 \cdot 1007 + 320$ appears to involve three numbers spaced by six units: $989$, $1001$, $1007$.		1m50s
1073 Leningrad City Mathematical Olympiad (1987). Consider a hexagon $A_1A_2A_3A_4A_5A_6$ with a point $O$ from which all sides are seen under an angle of $60^\circ$.		1m42s
1074 Leningrad City Mathematical Olympiad (1987). Let $m=2n+1$.		11m28s
1075 Leningrad City Mathematical Olympiad (1987). Consider the problem in terms of digit patterns.		2m01s
1077 International Mathematical Olympiad for School Students (1987). Let $X(\sigma)$ denote the number of fixed points of a permutation $\sigma$ of an $n$ element set.		1m44s
1078 International Mathematical Olympiad for School Students (1987). Assume that a function $f:\mathbb N_0\to\mathbb N_0$ satisfies		1m42s
1079 International Mathematical Olympiad for School Students (1987). For $n=3$ the problem asks for a single triangle whose three side lengths are irrational and whose area is a nonzero rational number.		2m25s
1080 International Mathematical Olympiad for School Students (1987). Let		2m22s
1081 V. I. Plachko. Compute a few values:		1m43s
1082 A. P. Savin. The given equality resembles the identity for the sum of squares of the sides of a quadrilateral.		2m09s
1083 L. G. Khanin. Consider small values of $n$ to understand the inequality.		2m22s
1084 Let the two given circles be $\omega_1$ and $\omega_2$, intersecting at $A$ and $B$.		1m55s
1085 S. L. Tabachnikov. Consider the problem geometrically.		2m05s
1086 M. V. Sapir. Consider the problem of reaching a target number from $0$ using only two operations: doubling the current number or adding $1$.		1m51s
1087 Let $h_a,h_b,h_c$ be the altitudes of triangle $ABC$.		1m50s
1088 The condition is		1m41s
1089 D. Yu. Burago, F. L. Nazarov. Let the inradius of triangle $AOB$ be $r_1$, of $BOC$ be $r_2$, of $COD$ be $r_3$, and of $DOA$ be $r_4$.		2m01s
1090 Yu. V. Deikalo. Testing small values helps build intuition about the inequality.		1m43s
1091 N. I. Zilberberg. A positive integer is called lucky when its digits can be split into two disjoint groups with equal sum.		37m33s
1092 S. V. Kazakov. Consider a single fold of a convex polygon and a subsequent straight cut.		7m15s
1093 Represent the configuration by numbers $a_1,\dots,a_n\in{0,1,2}$ arranged cyclically.		8m08s
1094 V. A. Senderov. The two inequalities are		7m30s
1095 R. O. Burdin. The problem involves constructing a chord $MN$ of a circle with center $O$ seen from $A$ under a given angle $\alpha$, with additional geometric constraints.		1m15s
1096 Let the circle have radius $R=\dfrac d2$.		1m21s
1097 V. V. Proizvolov. Consider small examples of isosceles triangles whose vertices have integer coordinates.		1m28s
1098 V. G. Chvanov. Consider the game for small values of $n$.		5m16s
1099 N. N. Silkin, M. V. Volkov. Consider small examples to gain intuition.		1m09s
1100 V. G. Ilyichyov. Consider a finite set of logs lying on a straight riverbank, each forming an angle less than $45^\circ$ with the bank.		7m10s
1102 L. D. Kurlyandchik. For $n=3$ it is natural to search among classical identities involving sums of three cubes.		3m56s
1103 Begin with the first part of the problem, which concerns tiling an infinite plane with $1\times 2$ dominoes after some non-overlapping dominoes are already placed.		10m39s
1104 V. N. Dubrovsky. Let		1m14s
1105 N. P. Dolbilin, M. I. Shtogrin. The problem concerns unfolding a convex polyhedron along straight-line cuts so that its surface lies flat as a single polygon, with specified identifications of points on the boundary.		1m20s
1106 V. V. Proizvolov. Consider a convex hexagon $ABCDEF$.		7m29s
1107 L. D. Kurlandchik. The inequality is homogeneous in the ratios of the sides.		1m37s
1108 Consider small cases first.		1m56s
1109 V. S. Shevelyov. Let the vertices of an inscribed equilateral triangle be		17m21s
1110 V. F. Lev. Consider the first few natural numbers and compute the greatest common divisors of all distinct pairs.		2m21s
1111 B. I. Chinik. Consider triangle $ABC$ with acute angles and its circumcircle $\Gamma$.		2m16s
1112 A. A. Berzinsh, V. G. Ilyichev. Starting with the numbers $1$ and $2$ on the board, the rule allows us to produce $ab + a + b$ whenever $a$ and $b$ are present.		8m02s
1113 D. V. Fomin. Model the situation as a graph on $21$ vertices, the cities.		7m11s
1114 I. F. Sharygin. Consider a tetrahedron with vertices $A$, $B$, $C$, $D$ and let $a = AB$ and $b = CD$ be two skew edges.		4m45s
1115 A. V. Andzhans. Consider the first problem.		5m56s
1116 Consider a rectangle drawn on a square grid where the unit squares are the cells.		9m26s
1117 A. A. Gorbachev. Let the sides of the given triangle $ABC$ be		10m40s
1118 S. G. Mamikonyan. Expanding the left-hand side gives		8m06s
1119 Consider first small values of $k$.		8m52s
1120 V. F. Lev. Consider the sequence defined by $a_0 = 0$ and $a_n = P(a_{n-1})$ for $n \ge 1$, where $P(x)$ is a polynomial with integer coefficients and $P(x) > 0$ for $x \ge 0$.		3m13s
1122 City Tournament (Spring, 1988). Let		9m02s
1123 Tournament of Towns (Spring 1988). Label the cells by coordinates $(i,j)$, where $i,j\in\mathbb N$ and $i,j\ge 1$.		10m09s
1124 City Tournament (Spring, 1988). Consider a trapezoid $ABCD$ with bases $AB$ and $CD$, where $AB$ is the shorter base.		8m33s
1125 I cannot write a solution to Kvant problem M1125 from the information provided, because the actual problem statement is missing.		2m53s
1126 Leningrad City Mathematical Olympiad (1988). The statement resembles a converse of a familiar fact about equal angles subtending the same segment.		8m51s
1127 I notice that the problem statement itself is not yet provided.		9m18s
1128 Leningrad City Mathematical Olympiad (1988). Consider first the case of a $2 \times 2$ chessboard with two pieces.		8m33s
1129 I cannot write a solution to Kvant problem M1129 because the actual problem statement is not present in the conversation.		7m17s
1130 Leningrad City Mathematical Olympiad (1988). Consider first a simple convex polygon, such as a triangle or a square.		6m10s
1131 International Mathematical Olympiad for School Students (XXIX). Consider the case $n=1$ first.		4m04s
1132 International Mathematical Olympiad for School Students (XXIX). Let		7m47s
1133 International Mathematical Olympiad for School Students (XXIX). Consider the sum		3m22s
1134 International Mathematical Olympiad for School Students (XIX). Consider a right triangle $ABC$ with right angle at $A$ and altitude $AD$.		3m33s
1135 International Mathematical Olympiad for School Students (XXIX). Let		3m16s
1136 D. P. Mavlo. Testing small integer values for $A$, $M$, and $S$ helps to gain intuition about the inequality.		2m28s
1137 K. P. Kokhas. Consider first small polygons.		2m58s
1138 L. D. Kurlyandchik. The expression $n^2+n+3\sqrt n$ is not always an integer.		3m57s
1139 V. E. Matisen. For a convex polyhedron whose faces are all squares, every face angle equals $90^\circ$.		8m09s
1140 S. L. Tabachnikov. Each intersection point is a crossing of two branches.		7m26s
1141 N. M. Sedrakyan. Consider a trapezoid $ABCD$ with $AB$ and $CD$ as the bases, $AB \parallel CD$, and a circle inscribed within it.		7m59s
1142 Consider small tables first.		4m02s
1143 V. V. Proizvolov. Consider first small circular arrangements of weights with integer masses and total mass divisible into parts.		3m39s
1144 A. I. Shekhorsky. Let		7m12s
1145 Consider a circle with a point $P$ outside it and two tangents $PB$ and $PC$, forming an angle $\angle BPC > 90^\circ$.		2m51s
1146 E. G. Gotman. Place the equilateral triangle $ABC$ in the plane with convenient coordinates.		3m55s
1147 The condition says that every closed path contains an even number of red edges.		3m08s
1148 R. B. Alekseev, L. D. Kurlyandchik. Consider small values of $a$ and $n$ to understand the pattern.		7m30s
1149 V. V. Shabunin. Consider two rays $p$ and $q$ with vertices $P$ and $Q$, respectively.		1m40s
1150 E. G. Moiseev, 10th-grade student. For $n=3$ the inequality becomes		2m09s
1151 I cannot write a solution to Kvant problem M1151 from the information provided, because the actual problem statement is missing.		5m34s
1152 I do not have the statement of Kvant problem M1152, and the prompt says that only the graphical version is currently available.		9m36s
1153 I can do that.		3m45s
1154 I cannot write a solution to Kvant problem M1154 because the actual problem statement is not present in your message.		6m43s
1155 A complete solution cannot be written from the information provided.		5m30s
1156 There are eight teams, each playing once against every other team, so each team plays $7$ games.		7m23s
1157 Let the three triangles be $T_W,T_R,T_G$, and let $M$ be a point lying in the interior of each of them.		9m57s
1158 We are asked to minimize $(x+y)(x+z)$ under the constraint $xyz(x+y+z)=1$, with $x$, $y$, $z$ positive.		7m27s
1159 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.		6m52s
1160 S. L. Eliseev. Consider the situation with only two kangaroos first.		7m10s
1161 N. P. Dolbilin. Consider first the configuration of ten identical billiard balls arranged snugly in a triangular container.		7m09s
1162 D. V. Fomin. Consider the Diophantine equation		4m11s
1163 Let the position of the first turtle at time $t$ be $P(t)$ and the position of the second turtle be $Q(t)$.		7m06s
1164 V. V. Shabunin. Let $\sigma(n)$ denote the sum of all positive divisors of $n$.		4m11s
1165 D. J. Newman, B. D. Kotlyar. Consider a square of side length $n$ on a standard graph paper with $1\times1$ cells.		3m06s
1166 Tournament of Towns (Autumn, 1988). The inequality involves the side lengths $a$, $b$, $c$ of a triangle and three numbers $p$, $q$, $r$ summing to zero.		3m53s
1167 Tournament of Cities (Autumn, 1988). Let $p(i)$ denote the position of $i$ in the permutation.		1m55s
1168 Tournament of Towns (Autumn, 1988). Model the country by a graph with $1989$ vertices and $4000$ edges.		1m41s
1169 City Tournament (Autumn, 1988). Consider a rectangle $ABCD$ with sides $AB = CD = a$ and $BC = AD = b$.		1m34s
1170 City Tournament (autumn, 1988). A triangulation of a convex $n$-gon contains exactly $n-3$ diagonals.		2m05s
1171 L. D. Kurlandchik. Let		1m31s
1172 S. N. Bychkov. Let		6m17s
1173 G. N. Zaitsev. Let the three lines through the interior point $P$ meet the sides of triangle $ABC$ in such a way that they cut off three corner triangles of areas $S_1,S_2,S_3$.		6m14s
1174 Compute the first few values of the sequence to look for patterns.		7m37s
1175 D. A. Tereshin. Consider the problem for small values of $n$ first.		7m25s
1176 S. A. Stolyarov. The expression to be proved resembles the area formula for a quadrilateral written in terms of the lengths of two diagonals of some auxiliary figure.		4m34s
1177 K. P. Kokhas, V. M. Televka. Consider the inequality for small values of $n$ to understand its structure.		7m37s
1178 Z. A. Skopets. The inequality involves the three classical quantities		4m41s
1179 B. A. Wertheim. Consider the first recurrence, $a_{n+1} = \frac{n}{n+1}(a_n+1)$ with $a_1 = 0$.		7m53s
1180 All-Union Mathematical Olympiad (XXIII, 1989). Consider two spheres intersecting along a circle.		7m41s
1181 All-Union Mathematical Olympiad (XXIII, 1989). Consider small instances of the chessboard and examine the parity of black squares occupied by pieces when each row and each column contains exactly one piece.		7m55s
1182 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.		8m26s
1183 I cannot write a solution to Kvant problem M1183 because the actual problem statement is not present in your message.		5m49s
1184 I do not have access to the published graphical version of Kvant M1184, and you mentioned the textual version is being prepared.		10m11s
1185 I do not yet have the full text of Kvant problem M1185.		5m49s
1186 Solution to Kvant math problem 1186.		6m09s
1187 City Tournament (Spring, 1989). Let the required order be $a_1,a_2,\dots,a_{m-1}$.		8m53s
1188 I cannot write a solution to Kvant problem M1188 from the information provided because the actual problem statement is missing.		7m23s
1189 Tournament of Towns (Spring 1989). For $n=1$, a single line divides the plane into two regions.		12m01s
1190 Tournament of Towns (Spring 1989). For the first part, reinterpret the table as a bipartite graph.		4m52s
1191 I. F. Akulich. Let $S_A$ denote the central symmetry with center $A$.		8m15s
1192 V. A. Senderov. The polyhedron has all edges of equal length and every edge is tangent to a sphere.		8m00s
1193 Let		7m10s
1194 A. A. Azamov. Let the rectangle have vertices		7m39s
1195 O. T. Izhboldin. Consider the given inequality		4m05s
1196 Leningrad City Mathematical Olympiad (1989). Denote the transformation applied to a chosen pair $(a,b)$ by		7m13s
1197 Leningrad City Mathematical Olympiad (1989). Let		7m33s
1198 Leningrad City Mathematical Olympiad (1989). The problem asks for the largest set of 10-digit binary words such that no two words can be obtained from each other by reversing a contiguous subsequence of even sum.		7m19s
1199 Leningrad City Mathematical Olympiad (1989). Consider the quartic polynomial $P(x) = ax^4 + bx^3 + cx^2 + dx + e$ and the quadratic polynomial $Q(x) = ax^2 + (c-b)x + (e-d)$.		7m16s
1200 Leningrad City Mathematical Olympiad (1989). Consider a small number of arcs on a circle.		10m06s
1201 N. B. Vasiliev. Let the numbers of voters for parties $A,B,C$ in a district be $a,b,c$, expressed as fractions of the district electorate.		7m17s
1202 Let the two rays from $A$ be $r_1$ and $r_2$.		7m18s
1203 S. L. Eliseev. Let the side of the large square be $1000$ m.		4m23s
1204 Consider three circles centered at points $A$, $B$, and $C$, each expanding at the same constant rate.		1m22s
1205 I cannot write a solution to Kvant problem M1205 because the actual problem statement is not present in the conversation.		1m01s
1206 I see that the problem statement for Kvant M1206 is not yet provided.		1m01s
1207 I see the problem statement itself is not fully provided yet.		1m06s
1208 I need the textual statement of Kvant problem M1208 in order to produce a rigorous solution according to your strict rules.		51s
1209 I do not yet have the full textual version of Kvant problem M1209.		1m00s
1210 K. P. Kokhas. The position of the game is completely determined by the current number of matches in the pile and the set of numbers already written on the sheet.		1m57s
1211 City Tournament (Autumn, 1989). A plane intersects a tetrahedron along a triangle and a sphere along a circle.		1m01s
1212 Tournament of Towns (Autumn, 1989). Consider first partitioning the integers into arithmetic progressions with positive integer differences.		1m23s
1213 Tournament of Towns (Autumn, 1989). Consider a convex hexagon that can be subdivided into $N$ parallelograms of equal area.		1m13s
1214 Cities Tournament (Autumn, 1989). Consider small tables first.		1m09s
1215 Tournament of Towns (Autumn, 1989). To understand $k(n)$, consider $n=15$, where the problem gives an example: three triples $(1,6,8)$, $(2,4,9)$, $(3,5,7)$, all sums equal to $15$, and all nine numbers distinct.		2m23s
1216 Consider an acute-angled triangle $ABC$ with angle bisector $AD$ from vertex $A$.		1m32s
1217 Yu. M. Burman. Compute the sum for small values of $n$ to gain intuition.		3m20s
1219 Begin by testing the inequality for small values of $n$ to develop intuition.		2m13s
1220 D. V. Fomin. The recurrence is		3m22s
1221 V. P. Chichin. Consider a triangle $ABC$ with sides $AB$ and $AC$ given.		1m12s
1222 V. F. Lev. Let the given integers be considered modulo $m$.		4m46s
1223 A. A. Razborov. Consider small cases first.		3m10s
1224 V. A. Senderov. Consider a triangle $ABC$ with incenter $I$ and an arbitrary point $D$ on side $BC$.		3m07s
1225 Let		2m30s
1226 A. P. Savin. Let the convex quadrilateral be $ABCD$.		3m12s
1227 Consider small tournaments first.		5m06s
1228 D. V. Fomin. The expression is		2m53s
1229 For the first expression, compute a few values:		3m57s
1230 Please provide the full textual statement of Kvant problem M1230.		5m05s
1231 I notice the problem statement itself is not yet included.		4m34s
1232 I cannot write a solution to Kvant problem M1232 from the information provided, because the actual problem statement is missing.		2m53s
1233 I can do that.		6m52s
1234 I can start preparing the full Kvant-style solution, but I need the actual problem statement for M1234 to proceed.		3m34s
1235 I cannot write a solution to Kvant problem M1235 because the actual problem statement is not included in your message.		5m20s
1236 I see the framework and instructions, but I do not yet have the actual text of Kvant problem M1236.		5m44s
1237 I cannot write a solution to Kvant problem M1237 because the actual problem statement is not included in your message, and I do not have access to the graphical version from the published issue.		12m04s
1238 I can follow your requested six-section format and rigorous style.		1m42s
1239 The problem statement is not actually present in your message.		6m20s
1240 D. V. Fomin. A broken line from $A$ to $C$ of length $2n$ along the grid lines must consist of exactly $n$ horizontal and $n$ vertical unit steps.		6m43s
1241 All-Union Mathematical Olympiad (XXIV, 1990). For a small instance with piles $1,2,3$, one move cannot remove all stones, because the three pile sizes are different.		8m06s
1242 All-Union Mathematical Olympiad (XXIV, 1990). Consider a small regular polygon, for instance a regular hexagon or octagon, to visualize the positions of points $K$ on $AB$ and $N$ on $BC$.		5m54s
1243 All-Union Mathematical Olympiad (XXIV, 1990). For the quadratic game, the first player provides three numbers $a$, $b$, $c$, and the second player can assign them arbitrarily to the coefficients of $x^2$, $x$, and the constant term.		15m20s
1244 All-Union Mathematical Olympiad (XXIV, 1990). Label the 30 senators as vertices of a graph where edges represent friendship.		1m16s
1245 All-Union Mathematical Olympiad (XXIV, 1990). Begin by considering small cases, such as $n = 2$ or $n = 3$, to develop intuition.		4m18s
1246 Leningrad City Mathematical Olympiad (1990). Let the arithmetic progression be		7m04s
1247 Leningrad City Mathematical Olympiad (1990). Consider tiling the plane with squares of side lengths $1, 2, 4, 8, \dots$ under the two constraints: using each size at most ten times or using each size once.		7m33s
1248 Leningrad City Mathematical Olympiad (1990). For the first statement, it is natural to place the original segment on the real line as $[0,L]$.		3m58s
1249 Leningrad City Mathematical Olympiad (1990). The road system is a tournament: for every pair of cities exactly one directed edge joins them.		4m58s
1250 with full Olympiad-level rigor.. I cannot write a rigorous solution to Kvant problem M1250 from the information provided, because the actual problem statement is missing.		6m44s
1251 I notice that the problem statement for Kvant M1251 is not fully given in text form.		4m50s
1252 I cannot write a solution to Kvant problem M1252 because the actual problem statement is not present.		6m20s
1253 A complete solution cannot be written because the problem statement is not available.		4m50s
1254 Before I write the full solution, I need the precise text of Kvant problem M1254.		8m01s
1255 I cannot write a rigorous solution to Kvant problem M1255 because the actual problem statement is not present in your message.		7m38s
1256 V. A. Senderov. The circles are equal and tangent to each other.		9m48s
1257 Before I start the full solution, I need the precise statement of Kvant problem M1257.		3m03s
1258 I see that you have provided the framework for solving Kvant problem M1258, but I do not yet have the text of the problem itself.		6m15s
1259 Before proceeding, I need the full textual statement of Kvant problem M1259.		5m10s
1260 International Mathematical Olympiad (1990). Examine small integers $n>1$ to see when $\frac{2^n+1}{n^2}$ is an integer.		6m11s
1261 Consider small examples with three colors and a uniform degree graph.		6m07s
1262 L. D. Kurlandchik. Let the side lengths be $a,b,c$.		5m22s
1263 Let the outer circle have center $O$ and radius $R$.		7m54s
1264 Consider a small portion of the infinite grid and attempt to construct a $2\times 2$ black square using only $3\times 3$ and $4\times 4$ flip operations.		6m03s
1266 K. P. Kokhas. Consider small cases with integer points in a bounded region.		4m24s
1267 L. D. Kurlandchik. Consider small values of $n$ to understand the behavior of the remainders $r_k$.		6m07s
1268 Consider triangle $ABC$ with an interior point $X$, and let $AX$, $BX$, $CX$ intersect the opposite sides at $A_1$, $B_1$, $C_1$ respectively.		7m06s
1269 Let $a=AB$ and $c=AC$.		6m28s
1271 V. A. Senderov. Let the semicircle have radius $R$ and center $O$.		3m49s
1272 L. D. Kurylandchik. Begin with small values of $n$ to gain intuition.		3m20s
1273 Before I can write the full solution, I need the full text of Kvant problem M1273 in order to proceed.		6m09s
1274 G. A. Halperin. Denote		7m15s
1275 Let		7m47s
1276 Consider a fixed chord $MN$ of a circle and a variable diameter $AB$ that does not intersect $MN$.		6m09s
1277 L. D. Kurylandchik. Begin by testing small values of $n$.		6m03s
1278 For $n=2$, the conditions give $x_2=-x_1$ and $2x_1^2=1$, hence $x_1x_2=-\frac12\le-\frac12=-\frac1n$.		6m53s
1280 The decimal expansion of $\dfrac1{3^{100}}$ is purely periodic after a finite preperiod, because $3^{100}$ is coprime to $10$.		6m42s
1281 Tournament of Towns (Autumn 1990). Consider two positive numbers $x$ and $y$ such that $xy > x + y$.		5m04s
1282 Tournament of Towns (Autumn, 1990). Consider two trapezoids $ABCD$ and $A'B'C'D'$ such that the legs of each are equal to the bases of the other.		9m22s
1283 D. V. Fomin. The figure shows three allowed tile types.		7m21s
1284 Cities Tournament (Autumn, 1990). Consider an isosceles triangle $ACB$ with base $AB$ and vertex $C$.		6m03s
1285 Tournament of Towns (Autumn, 1990). Let a permutation of the deck be written as a sequence.		6m42s
1286 Consider a small case to understand the structure.		7m51s
1287 Consider a parallelogram $ABCD$ with $AC > BD$ and a point $M$ on $AC$ such that $BCDM$ is cyclic.		6m07s
1288 The number is		5m17s
1289 Let		8m30s
1290 The folding is performed only along the grid lines.		8m51s
1291 The problem statement is missing.		20m13s
1292 I. N. Sergeev. Begin by considering the simplest nontrivial cases.		2m45s
1293 I. F. Sharygin. Consider an angle with vertex $A$ and two rays forming the sides.		2m49s
1294 A. V. Spivak. Consider a smaller cube, for instance $2\times2\times2$ or $4\times4\times4$, with a checkerboard coloring.		11m15s
1295 Consider small examples of the problem to understand the dynamics.		4m33s
1296 All-Union Mathematical Olympiad (25th, 1991). Consider the operation described in the problem: a polygon is cut along a line segment into two pieces, one piece is flipped, and the pieces are reattached along the cut line.		3m12s
1297 All-Union Mathematical Olympiad (XXV, 1991). The equations for $\alpha$ and $\beta$ are cubic but not immediately factorable in integer roots.		7m21s
1298 I can prepare the solution, but I need the text of the Kvant M1298 problem to proceed.		3m21s
1299 All-Union Mathematical Olympiad (XXV, 1991). Consider small values of $n$ to gain insight.		6m34s
1300 I do not yet have the full textual statement of Kvant problem M1300.		1m43s
1301 I cannot write a solution to Kvant problem M1301 from the information provided, because the problem statement itself is missing.		15m37s
1302 I cannot write a solution to Kvant problem M1302 because the actual problem statement is not available in your message.		1m33s
1303 I can begin writing a full solution, but I need the text of problem M1303 from Kvant before proceeding.		1m29s
1304 I cannot write a solution to Kvant problem M1304 from the information currently available.		1m32s
1305 I do not have the statement of Kvant problem M1305, and the prompt indicates that the textual version is not available here.		1m31s
1306 I do not have the full text of Kvant Problem M1306 from your input.		1m32s
1307 Solution to Kvant math problem 1307.		1m30s
1308 Please provide the text version of the Kvant M1308 problem so I can write the rigorous solution in the six-section format you requested.		1m31s
1309 I can do that.		1m33s
1310 Solution to Kvant math problem 1310.		1m31s
1311 Leningrad City Mathematical Olympiad (1991). Consider a triangle with integer side lengths $x$, $y$, $z$ and let the altitudes corresponding to these sides be $h_x$, $h_y$, $h_z$, where $h_x$ is the altitude to side $x$, $h_y$ to $y$, and $h_z$…		1m44s
1312 I cannot write a rigorous solution to problem M1312 from the information provided, because the problem statement itself is missing.		1m31s
1313 I can do that.		1m32s
1314 I need the full text of Kvant problem M1314 in order to write the solution.		1m33s
1315 I can do that.		1m33s
1316 I can prepare the full six-section solution, but I need the text of Kvant problem M1316.		1m33s
1317 Before I begin, I need the exact text of Kvant problem M1317 in order to produce a fully rigorous solution according to the framework you specified.		1m33s
1318 I cannot write a rigorous solution to Kvant problem M1318 without the actual problem statement or the diagram.		1m32s
1319 Before I can write the complete solution, I need the full text of Kvant problem M1319.		1m34s
1320 I can follow your structure exactly, but I need the full text of Kvant problem M1320 to write the solution.		1m33s
1321 For small boards the bound looks plausible.		1m34s
1322 I cannot write a solution to Kvant problem M1322 without the actual problem statement or a description of the diagram.		1m33s
1323 I need the full text of Kvant problem M1323 in order to produce a complete, rigorous solution.		1m30s
1324 Before I start the solution, I need the full text of Kvant problem M1324.		1m29s
1325 I cannot write a rigorous solution to Kvant problem M1325 because the actual problem statement is missing.		1m30s
1326 M. N. Vyalyi. The recurrence can be rewritten in a much more suggestive form:		1m34s
1327 I can do that.		1m32s
1328 Before proceeding with a full solution, I need the full text of Kvant problem M1328 in order to construct the rigorous, step-by-step proof according to your specifications.		1m31s
1329 Solution to Kvant math problem 1329.		1m31s
1330 I do not have the statement of Kvant problem M1330.		1m33s
1331 Consider a unit square $ABCD$ with points $K$, $M$, $N$, $L$ on sides $BC$, $CD$, $DA$, $AB$ respectively, such that lines $AK$, $BM$, $CN$, $DL$ create four corner triangles of areas $S_1$, $S_2$, $S…		7m43s
1332 A regular tetrahedron has four triangular faces and six edges.		12m23s
1333 Before I begin, I need the text of Kvant problem M1333.		4m16s
1334 I cannot write a rigorous solution to Kvant M1334 without the actual problem statement.		7m20s
1335 Solution to Kvant math problem 1335.		11m39s
1336 I cannot write a solution to Kvant problem M1336 because the actual problem statement is not present in your message.		7m24s
1337 I can do that.		4m20s
1338 I notice that the statement of Kvant problem M1338 has not been provided.		7m15s
1339 I cannot write a solution to Kvant problem M1339 from the information provided, because the actual problem statement is missing.		7m19s
1340 I can start preparing the complete Kvant M1340 solution in your requested format.		10m18s
1341 L. D. Kurlandchik, V. A. Senderov. Denote by		4m36s
1342 I need the full text of Kvant problem M1342 to write a complete, rigorous solution.		3m22s
1343 I cannot write a rigorous solution to problem M1343 because the actual problem statement is not present in your message.		7m18s
1344 I notice that the actual textual statement of Kvant problem M1344 is not included.		5m38s
1345 Before I begin the full solution, I need to clarify the precise statement of Kvant problem M1345.		7m18s
1346 I cannot write a rigorous solution to problem M1346 without the actual problem statement or diagram.		6m48s
1347 Tournament of Towns. Consider a smaller version of the problem to gain insight.		7m07s
1348 I can provide a full solution in the structure you specified, but I need the exact problem statement for Kvant M1348.		7m25s
1349 Tournament of Towns. Let $a_i$ be the number of counters in sector $i$, with indices taken modulo $n$.		10m23s
1350 Tournament of Towns. Let us understand exactly what is being counted.		7m30s
1351 Tournament of Towns. Let the right angle of triangle $ABC$ be at $A$.		7m31s
1352 City Tournament. Consider the definition of \		5m49s
1353 I can prepare the complete Kvant-style solution once you provide the full textual problem statement for M1353.		7m25s
1354 I cannot write a rigorous solution to Kvant problem M1354 from the information provided, because the actual problem statement is missing and the published version is graphical only.		7m19s
1356 We are given a triangle with sides $a$, $b$, $c$ and the relation		35m03s
1357 I can follow that framework precisely.		8m49s
1358 I can write the complete solution in the format you specify, but I need the text of Kvant problem M1358.		6m42s
1359 I can proceed with that, but I need the text of Kvant problem M1359 to write a complete solution.		7m02s
1360 I cannot write a solution to Kvant problem M1360 because the actual problem statement is missing.		6m10s
1361 I can proceed, but I need the text of problem M1361 from Kvant.		8m44s
1362 The problem statement for Kvant M1362 is not included in your message, and the web-accessible archive currently only indicates that the textual version is still being prepared rather than providing th…		7m24s
1363 I do not have the statement of Kvant problem M1363, and the prompt indicates that only the graphical version is available.		11m44s
1364 I do not have the textual version of Kvant problem M1364.		7m26s
1365 I cannot write a solution to Kvant problem M1365 because the actual problem statement is not present in your message.		7m10s
1366 I cannot write a solution to Kvant problem M1366 because the actual problem statement is not present in your message.		7m16s
1367 I see that the problem statement for Kvant M1367 has not been provided in text.		7m23s
1368 I can prepare the full solution in the requested format.		4m27s
1369 I do not have the textual version of Kvant Problem M1369.		10m19s
1370 I do not have the statement of Kvant problem M1370.		4m19s
1371 Interrepublican Mathematical Olympiad 1992. Place points $A$ and $B$ on a circle and pick a point $P$ on the smaller arc $AB$.		7m35s
1372 I can prepare a complete solution in that format, but I need the text version of Kvant problem M1372 to proceed.		7m20s
1373 Interrepublican Mathematical Olympiad 1992. Let the sphere have radius $R$, and let the given plane be $\pi$.		7m44s
1374 I cannot write a rigorous solution to Kvant problem M1374 without the actual problem statement or diagram.		3m23s
1375 I see that the problem statement for Kvant M1375 is not yet fully provided.		7m24s
1376 I cannot write a valid solution to Kvant problem M1376 from the information provided, because the actual problem statement is missing.		12m01s
1377 I can produce a full solution in your requested format, but I need the text of Kvant problem M1377.		1m30s
1378 I cannot write a rigorous solution to Kvant problem M1378 because the actual problem statement is not present in the conversation.		7m22s
1379 I do not have access to the graphical version of Kvant problem M1379, and the textual statement is not included in your message.		8m36s
1380 International Mathematical Olympiad for School Students (XXXIII, 1992, problem from the jury materials). Consider the expression		8m54s
1381 I cannot write a solution to Kvant problem M1381 without the actual problem statement or the diagram.		6m50s
1382 Before I begin the full solution, I need the text of Kvant problem M1382.		7m14s
1383 I cannot write a solution to Kvant problem M1383 because the problem statement itself is not present in your message.		7m09s
1384 I can follow your requested format precisely, but I need the full text of Kvant problem M1384 to produce the complete solution.		5m49s
1385 I can do that.		6m50s
1386 I do not have access to the graphical version of Kvant problem M1386, and the text of the problem is not included in your message.		6m38s
1387 I cannot write a solution to Kvant M1387 because the problem statement itself is not included in your message.		6m37s
1388 St. Petersburg City Mathematical Olympiad. Let $f(x)=x^2+bx+c$ and $g(x)=x^2+px+q$, since both quadratics have leading coefficient $1$.		9m06s
1389 I cannot write a rigorous solution to Kvant problem M1389 because the actual problem statement is not present in your message.		6m44s
1390 I can follow your requested format rigorously, but I need the text of Kvant problem M1390 to proceed.		52s
1391 Solution to Kvant math problem 1391.		6m49s
1392 M. L. Kontsevich. The quadrilateral $ABCD$ has three consecutive sides equal, $AB = BC = CD = 1$, and points $B$ and $C$ are fixed.		7m28s
1393 I cannot write a solution to Kvant problem M1393 because the actual problem statement is not present in your message.		6m42s
1394 I see that you have provided the full template and instructions for solving Kvant problem M1394, but the actual problem statement has not been included yet.		8m35s
1395 F. L. Nazarov. Consider a small social network where each person has a certain number of acquaintances.		7m00s
1396 I do not have the full text of Kvant problem M1396, so I cannot write a complete, rigorous solution yet.		2m47s
1397 Consider first a simple convex polyhedron, such as a tetrahedron.		7m14s
1398 The statement of Kvant problem M1398 is not included in your message.		6m57s
1399 I cannot write a rigorous solution to Kvant problem M1399 because the actual problem statement is not present in your message.		9m41s
1400 I. F. Sharygin. A shortest closed route that visits all four faces can be replaced by a polygonal route whose vertices lie on the faces.		8m52s
1401 Consider triangle $ABC$ with circumcircle $\Gamma$ and a point $K$ chosen on the arc $BC$ that does not contain $A$.		4m20s
1402 L. D. Kurlyandchik, A. Meltzer. Consider the inequality for small values of $n$ to understand its behavior.		7m09s
1403 E. A. Yasinovy̆. Consider a convex $n$-gon $A_1A_2\ldots A_n$ and construct points $B_k$ on each side $A_kA_{k+1}$ such that $A_{k+1}B_k = A_kA_{k+1}$.		9m13s
1404 We are asked to maximize the expression		11m55s
1405 V. A. Senderov. Consider a small example, such as a pyramid with a square base.		4m28s
1406 I. S. Rubanov. Consider small examples.		7m35s
1407 S. V. Konyagin. Consider a small example with two central men.		8m52s
1408 Consider small circles of people and the statements they can make.		7m17s
1409 Consider the equilateral triangle of side $n$ subdivided into $n^2$ unit equilateral triangles.		6m22s
1410 Consider first the two-dimensional question.		7m20s
1411 I can follow your requested format, but I need the full textual statement of Kvant problem M1411 to produce the solution.		3m01s
1412 I cannot write a rigorous solution to Kvant problem M1412 because the actual problem statement is missing.		6m53s
1413 Let $N$ be the number of inhabitants, $T$ the number of truth-tellers, and $L=N-T$ the number of liars.		6m05s
1414 I do not yet have the full text of Kvant problem M1414.		8m26s
1415 Consider two regular decagons, each with 10 vertices, and numbers $a_1,\dots,a_{10}$ on the first decagon and $b_1,\dots,b_{10}$ on the second, all natural numbers, summing to 99 on each decagon.		4m47s
1416 V. A. Ufnarovsky. Consider a small finite set of gangsters.		8m59s
1417 I see that the problem statement for Kvant M1417 is not yet fully provided in text form.		9m54s
1418 I cannot write a solution to Kvant problem M1418 because the problem statement itself is not present in the conversation.		4m06s
1419 I cannot write a solution to Kvant problem M1419 because the actual problem statement is not present in your message.		6m48s
1420 Before I begin writing the solution, I need the text of Kvant problem M1420.		7m19s
1421 I cannot write a solution to Kvant M1421 because the problem statement itself is not present in your message.		7m19s
1422 Before proceeding, I need the full textual statement of Kvant M1422.		7m23s
1423 Consider three players $A$, $B$, and $C$, each playing the same number of games against one another.		7m21s
1424 S. I. Tokarev. For a sequence $a_1,\dots,a_n$, the transformation replaces each entry by the number of strictly larger elements to its right.		12m07s
1425 V. V. Proizvolov. Consider a quadrilateral with three interior angles of $45^\circ$.		7m31s
1426 M. L. Gerver. Consider the function $f(n) = n + S(n)$, where $S(n)$ is the sum of the digits of $n$.		3m43s
1427 I notice that the textual statement of Kvant problem M1427 has not been provided.		10m14s
1428 Before I start, I need the full text of Kvant problem M1428.		4m20s
1429 I see that you want a full Kvant-style solution in six sections, but the actual textual statement of problem M1429 is missing.		7m24s
1430 V. A. Senderov. The sequence ${a_n}$ is monotonic and integer-valued.		10m20s
1431 B. D. Ginzburg. Let the operation be $f(10k+d)=k+4d$ with $0\le d\le 9$.		4m53s
1432 L. D. Kurylandchik. Consider a sequence of positive numbers $a_1, a_2, \dots$.		7m27s
1433 I can do this, but I need the textual statement of Kvant problem M1433 to proceed.		7m24s
1434 Consider a simple convex polyhedron, such as a cube or a tetrahedron, and imagine casting a shadow onto a plane with a point light source.		10m12s
1435 Before writing the full solution, I need the exact textual statement of Kvant Problem M1435.		4m23s
1436 The statement of Kvant problem M1436 is not included, and the note indicates only a graphical version exists in the issue.		7m24s
1437 The problem statement is not provided in textual or graphical form.		10m17s
1438 The textual statement of Kvant M1438 is not included, and the graphical version is not accessible here.		4m19s
1439 Before I can begin the full solution in your requested six-section format, I need the precise text of Kvant problem M1439.		8m04s
1440 I can write the solution as requested, but I need the full text of Kvant problem M1440 in order to proceed.		7m26s
1441 The textual statement of Kvant problem M1441 is not included, and the graphical version is referenced but not provided here.		4m24s
1442 Solution to Kvant math problem 1442.		6m47s
1443 The problem statement for Kvant M1443 is not included, and the graphical version is not available here.		6m54s
1444 Moscow Mathematical Olympiad (LVII, 1994). Consider polynomials with a negative coefficient, such as $P(x) = -x + 2$ or $P(x) = x^2 - x + 1$.		7m19s
1445 Moscow Mathematical Olympiad (LVII, 1994). Let a natural number $N$ satisfy the condition that after deleting one digit that is not the first digit, the resulting number $M$ divides $N$ with an integer quotient.		7m13s
1446 Let $P$ be a convex polyhedron in $\mathbb{R}^3$ with vertices $V(P)={A,V_1,\dots,V_8}$.		3m14s
1447 No mathematical exploration is possible because the statement of Kvant problem M1447 is not provided.		7m11s
1451 Russian Mathematical Olympiad (XX). Let $d=\gcd(a,b)$ and write $a=dx$, $b=dy$ with $\gcd(x,y)=1$.		9m15s
1453 Russian Mathematical Olympiad (XX). Let $n_k$ be the repunit of length $k$, so		7m51s
1460 Russian Mathematical Olympiad (XX). Let $A$ and $B$ be the two figures.	✓	15m00s
1461 Represent the $n$ definitions by vertices $1,\dots,n$.	✓	20m43s
1466 V. K. Kovaldzhi. Two artists play a sequential map-coloring game.		30m17s
1467 International Mathematical Olympiad for School Students (XXXV). Let		9m49s
1468 International Mathematical Olympiad for School Students (XXXV). Place the isosceles triangle in coordinates as		38m52s
1469 International Mathematical Olympiad for School Students (XXXV). Each integer $n$ with exactly three ones in binary has the form		9m52s
1470 International Mathematical Olympiad for School Students (XXXV). We need a set $A$ of positive integers such that every infinite set $S$ of primes contains, among the squarefree numbers formed from distinct primes of $S$, two numbers with the same number $k\ge2$ of…		9m45s
1474 B. D. Ginzburg. Let the direction of the line used in the $i$-th projection be denoted by $\alpha_i$.	✓	22m16s
1475 A. H. Shen. Let $a_n$ denote the number of admissible fillings.	✓	17m24s
1479 L. D. Kurlyandchik. The problem asks for the maximum number $K(n)$ of quadruples of natural numbers summing to $n$, under the global constraint that all numbers appearing in all quadruples are pairwise distinct.		9m15s
1481 Let $a=BC$, $b=CA$, $c=AB$.		6m30s
1482 Let	✓	7m58s
1483 B. D. Ginzburg. Each vector lies in the first quadrant and has length $1$.		10m28s
1484 N. B. Vasilyev. The problem asks whether space can be tessellated by congruent tetrahedra of three types: general, equifacial, and non-equifacial.		31m11s
1486 S. I. Tokarev. Consider sequences $(a_1, a_2, a_3, \dots)$ satisfying the recurrence $a_k = a_{k-1} - a_{k-2}$ for $k \ge 3$, with terms chosen from the set $1, \frac12, \frac13, \dots$.		32m39s
1487 V. A. Senderov. Place the circumcenter $O$ as a reference point and represent the triangle on its circumcircle.		7m47s
1488 Let the squares in the sequence be		38m09s
1489 A. I. Galochkin. Work over the vector space $\mathbb F_2^{mn}$ of all $0$-$1$ configurations on the $m\times n$ rectangle.	✓	14m54s
1490 V. A. Ufnarovsky. The statement as written can only be meaningful if the second triangle has side lengths $\sin x$, $\sin y$, $\sin z$.	✓	26m37s
1500 City Tournament (Spring, 1995). Represent the group by a simple graph $G$ with $50$ vertices.	✓	29m05s
1501 V. A. Senderov. For small $x$, the functions admit linear approximations $\sin(kx)\sim kx$ and $\sin x\sim x$.		9m09s
1502 V. V. Proizvolov. Let the regular $2n$-gon have vertices labeled $A=A_0, A_1, \dots, A_{2n-1}$ in cyclic order.		41m25s
1503 Let $B$ be the set of black numbers and $W$ the set of white numbers.		9m53s
1506 G. V. Kondakov. For a fixed interval $[a,b]$, the condition that the sums of integrals over white and black subintervals are equal for every polynomial in a given family can be rewritten as the vanishing of a signed…		9m47s
1512 Saint Petersburg City Mathematical Olympiad (1995). Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$ with $a_n\ne 0$, and define for a natural number $k$		33m09s
1515 Russian Mathematical Olympiad. Let		3m42s
1518 Russian Mathematical Olympiad. Let the common intersection point of the altitudes be $H$.		23m19s
1520 Russian Mathematical Olympiad. Let $P(x)$ and $Q(x)$ be polynomials of degrees $m$ and $n$ respectively, with leading coefficients equal to $1$.		14m15s
1523 L. D. Kurlandchik. Let	✓	29m20s
1524 Let the incenters of the triangles $ABP$, $BCP$, $CDP$, $DAP$ be $X,Y,Z,W$ respectively.		12m20s
1525 International Mathematical Olympiad for School Students (XXXVI). Place the four collinear points on the $x$ axis as	✓	9m04s
1526 International Mathematical Olympiad for School Students (XXXVI). The condition $abc=1$ suggests eliminating one variable from denominators by rewriting powers of $a,b,c$ in terms of the other two variables.		9m34s
1527 International Mathematical Olympiad for School Students (XXXVI). We shall prove that the only integer $n>3$ for which a configuration of points $A_1,\dots,A_n$ and real numbers $r_1,\dots,r_n$ satisfying the problem conditions exists is		27m53s
1528 International Mathematical Olympiad for School Students (XXXVI). The relation		10m47s
1529 International Mathematical Olympiad for School Students (XXXVI). The equalities		8m28s
1530 International Mathematical Olympiad of School Students (XXXVI). Numbers $1,2,\dots,2p$ split into residue classes modulo $p$ as		9m31s
1531 Each question consists of choosing a line and learning in which of the three sets determined by that line the point $P$ lies: one open half-plane, the other open half-plane, or the line itself.		10m23s
1533 N. B. Vasilyev. Let $\ell$ be a line through $C$.		28m24s
1551 Moscow LIX Mathematical Olympiad, 1996, Tournament of Towns (Spring 1996). A closed broken line with six segments is a hexagonal cycle whose six vertices lie on one circle.	✓	9m50s
1552 59th Moscow Mathematical Olympiad, 1996, Tournament of Towns (Spring 1996). The polynomial $P_n(x)=1+x+\cdots+x^{n-1}$ satisfies	✓	22m50s
1553 I cannot produce a mathematical solution to Kvant M1553 because the problem statement itself is missing.	✓	21m42s
1554 Moscow 59th Mathematical Olympiad, 1996, Tournament of the Cities (spring, 1996). Consider a triangle $ABC$ with outward squares $ABMN$, $BCKL$, and $ACPQ$ constructed on its sides.	✓	21m00s
1557 Moscow LIX Mathematical Olympiad 1996, Tournament of Towns (Spring, 1996). Let $O$ be the center of the given circle.		9m57s
1558 Consider the game on small chessboards to observe patterns.		8m44s
1559 59th Moscow Mathematical Olympiad (1996), Tournament of Towns (Spring 1996). Let the given plane be $\Pi$.		9m11s
1560 59th Moscow Mathematical Olympiad, 1996, Tournament of Towns (Spring 1996). Consider the population as a finite set of $N$ individuals arranged along a circle.		24m56s
1561 All-Russian Mathematical Olympiad for School Students (1996). Let the convex polygon be $A_1A_2\ldots A_n$, indexed cyclically.		9m22s
1562 All-Russian Mathematical Olympiad for School Students (1996). We are asked whether a $5\times7$ rectangle can be covered by L-trominoes in several layers so that each cell of the rectangle is covered by the same number of cells from the trominoes.		27m44s
1563 All-Russian Mathematical Olympiad for School Students (1996). For $n=0$, the condition is	✓	8m28s
1564 All-Russian School Mathematics Olympiad (1996). Let	✓	15m45s
1565 All-Russian Mathematical Olympiad for School Students (1996). A query chooses 50 of the 100 elements and reveals their induced linear order, which determines all pairwise comparisons inside the chosen set.		18m55s
1566 All-Russian Mathematical Olympiad for School Students (1996). Each committee has $80$ members, and there are $16000$ committees.		8m01s
1568 All-Russian Mathematical Olympiad for School Students (1996). This is a Type B problem.		27m35s
1569 Working		24m01s
1570 Soros Mathematical Olympiad. Consider three pairs of diametrically opposite points on a sphere, denoted $A, A'$, $B, B'$, $C, C'$, where $O$ is the center of the sphere.		20m58s
1580 Tournament of Towns (Autumn 1996). The problem asks whether a circle can be dissected into finitely many pieces whose boundaries consist of line segments and circular arcs, and then reassembled into a square of the same area.		12m23s
1591 Tournament of the Towns (spring, 1997). Let $BL$ be the internal angle bisector at $B$ meeting $AC$ at $L$, and $AK$ be the internal angle bisector at $A$ meeting $BC$ at $K$.	✓	17m12s
1592 Tournament of Cities (spring, 1997). Let integers $a \le b$ be such that the required sum is		3m46s
1593 Tournament of Towns (Spring 1997). Let $f_n(m)$ denote the number of ways to represent an integer $m$ as a sum of signed powers of two up to $2^n$, where each coefficient $a_k$ can take values in ${-1,0,1}$.		16m22s
1594 The functional equation is		10m08s
1595 City Tournament (spring, 1997). Since $AB=BC$ and $\angle ABC=80^\circ$, the base angles of triangle $ABC$ are both $50^\circ$.	✓	23m46s
2226 Let $A(k,n)$ denote the number of integer $k$-tuples $(x_1,\dots,x_k)$ satisfying the inequality		24m33s
2445 International Mathematical Olympiad (57th). Let $A_{k+1} = A_1$, and let $q = p^t$ be an arbitrary prime power dividing the odd integer $n$.		33m36s
2496 Consider a finite set of $n$ cities in the plane, with $n$ odd, such that all pairwise distances between cities are distinct.		26m37s
2576 A.P. Savin Tournament (25th). The problem concerns tilings of an $8\times8$ board with dominoes of size $1\times2$ (horizontal) or $2\times1$ (vertical).		28m32s
2632 Working		32m22s
2709 Caucasian Mathematical Olympiad. The problem can be formulated in graph-theoretic terms.		32m35s
2738 All-Russian Olympiad in Mathematics (XLIX, regional stage). Let		12m06s
2755 Caucasian Mathematical Olympiad (VIII). Consider the $3 \times 101$ board with rows labeled $1,2,3$ and columns labeled $1,\dots,101$, with the central cell $(2,51)$ initially crossed out.		50m26s
2768 Let $P(x) = \prod_{i=1}^n (x + a_i)$ and let $A = \prod_{i=1}^n a_i$.		34m08s
2786 All-Russian Mathematics Olympiad (L, Regional Round). Let the value of a completed coloring be the number of edges of the 100-cycle whose endpoints have different colors.		10m22s
2802 Caucasian Mathematical Olympiad (IX). Let		10m19s
2806 We construct a closed 20-segment polygonal chain satisfying the required intersection properties explicitly.		37m28s
2810 Ural Tournament of Young Mathematicians (62nd). The problem requires counting the number of ways to color the cells of an $n\times n$ square with four colors so that any two cells sharing a side or a vertex receive different colors.	✓	26m12s
2825 Sirius Mathematical Olympiad. Let $ABC$ be an acute-angled, non-isosceles triangle with altitudes $AD$, $BE$, $CF$ meeting at the orthocenter $H$.		28m55s
2826 Place the square in coordinates with $A(0,0)$, $B(1,0)$, $C(1,1)$, $D(0,1)$.	✓	40m44s
2827 Our systems have detected unusual activity coming from your system.		22m35s
2830 Let $S$ be the set of amounts obtainable with at most $1000$ coins of denominations $a$ and $b$.		9m59s
2834 All-Russian Olympiad in Mathematics (LI, regional stage). Let the original integers be $a_1, a_2, \dots, a_n$ and let $S = a_1 + a_2 + \cdots + a_n$.	✓	19m28s
2842 Euler Olympiad (XVII, regional stage). Let $a, a+1, \dots, a+n-1$ be $n>1$ consecutive natural numbers.		38m17s
2843 Introduce Cartesian coordinates adapted to the symmetry of the equilateral triangle.		20m27s
2844 We place square $ABCD$ in the coordinate plane with $A=(0,0)$, $B=(1,0)$, $C=(1,1)$, and $D=(0,1)$.		33m26s
2845 Tournament of Cities (XLIV). A cube in three-dimensional space with vertices in $\mathbb{Z}^3$ can be described by a vertex and three vectors $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{Z}^3$ that are pairwise orthogonal and…	✓	21m22s
2846 All-Russian Mathematics Olympiad for School Students (LI, Final Round). The word “ПЕТЯ” has five letters.		10m11s
2849 All-Russian School Mathematics Olympiad (LI, final stage). This is a C: optimization problem.		25m51s
2850 Label the marked points by the elements of $\mathbb Z_{2n}$ in cyclic order.	✓	21m45s
2854 Let $n=\prod p_i^{\alpha_i}$ with all $p_i<100$.		11m39s
2855 I. F. Sharygin Geometry Olympiad (XXI, correspondence round). Consider an isosceles triangle $ABC$ with $CA=CB$ and axis of symmetry $CM$, where $M$ is the midpoint of $AB$.		19m25s
2856 For a query polynomial $Q$, the answer is the set of real roots of		15m16s
2857 Olympiad of the Presidential Physics and Mathematics Lyceum No. 239 of St. Petersburg. Let the hats be $x_1,x_2,x_3,x_4$, placed on sages $1,2,3,4$ respectively, with all values distinct in ${1,\dots,100}$.		2m18s
2858 Consider a tree with $n$ vertices, each initially containing a single chip.	✓	20m14s
2859 Assume that a function $f:\mathbb R\to\mathbb R$ satisfies	✓	11m34s
2860 Consider the problem of determining whether both sums	✓	30m15s
2861 The configuration contains two coupled constructions on the base line $BC$ and two circumcenter loci attached to triangles $NEC$ and $MFB$.		23m40s
2862 Consider the equation		28m26s
2863 The configuration is a complete quadrilateral formed by the four lines $AB, BC, CD, DA$.		10m17s
2864 Based on an Iranian Olympiad problem.. The arrangement of the lines is a simple arrangement: every pair of lines intersects and no three are concurrent.		11m44s
2865 The operation in every part is a conditional transposition: two entries are swapped exactly when the larger stands to the left.		2m20s
2866 South Russian Mathematical Olympiad “Assara” (IV). The problem asks whether the integers from $1$ through $127$ can be divided into two groups such that the sum of the products of the numbers in each group ends with exactly ten zeros or exactly thirty…	✓	30m37s
2867 Tournament of the Towns (XLVII). Label the people around the circle as a cyclic sequence of 60 vertices, each colored either $B$ (boy) or $G$ (girl), with exactly 30 of each color.		7m56s
2868 Tournament of Cities (XLVII). We are asked whether Vasya can determine the total sum of 60 real numbers written on cards if he is allowed to ask about sums of 17-card subsets.		34m43s
2869 Tournament of Towns (XLVII). Let $ABCD$ be a convex quadrilateral with diagonals $AC$ and $BD$.		26m33s
2870 City Tournament (XLVII). We are asked to prove that if a snail moves along a closed, non-self-intersecting polygonal line in the plane using only three directions—up, to the right, and down-left at an angle of $45^\circ$ to t…		32m17s
2871 Southern Mathematical Tournament (XX). Let $p>2$ be a prime number and $k$ an integer with $0<k<p-1$.		33m36s
2872 South-Russian Mathematical Olympiad “Assara” (IV). Two distinct natural numbers $x$ and $y$ form a beautiful pair when	✓	27m30s
2873 Something went wrong.		39m12s
2874 Let $M$ be the midpoint of $AC$.		2m07s
2875 Working		13m42s
2876 Southern Mathematical Tournament (XX). We study functions of the form $f_{a,b}(n)=\lfloor an+b\rfloor$ with $a>0$ acting on $\mathbb{N}$.		10m12s
2877 Southern Mathematical Tournament (XX). The flaw in the previous solution is the assumption that Wolf must win on every graph, and that showing a single losing graph is sufficient to refute his strategy.		29m45s
2878 Southern Mathematical Tournament (XX). This is a Type B (prove) problem.		36m33s
2879 Sirius Mathematical Olympiad. Consider small natural numbers $n$ and compute $S(n)$, $S(7n)$, and $S(9n)$.		11m32s
2880 Sirius Mathematical Olympiad. This is a Type B (prove) problem.		22m30s
2881 Southern Mathematical Tournament (XX). Consider first the simplest nontrivial convex polygon, a triangle $A_1A_2A_3$, with a point $O$ inside.		2m40s
2882 All-Russian Olympiad in Mathematics (52nd, regional stage). The game begins with $1000$ heaps containing $1,2,3,\dots,1000$ matches respectively.		34m08s
2883 All-Russian Mathematics Olympiad (LII, regional stage). The previous solution attempted to prove nonexistence by separating cases according to the prime factorization of $n$.	✓	24m39s
2884 All-Russian Mathematics Olympiad (LII, Regional Round). The task is to determine the smallest integer $k$ such that, after adaptive subset queries with a binary oracle that may err but never on two consecutive days, the Wizard can always guarantee that at…		12m31s
2885 All-Russian Olympiad of School Students in Mathematics (LII, regional stage). Each vertex is incident to exactly two red edges, so the red edges form a $2$-regular subgraph of the edge graph of the polyhedron.		9m35s