Navvye/Olympiad-Problems-CSV huggingface dataset:
1992 All Soviet Union Mathematical Olympiad
A country contains $n$ cities and some towns. There is at most one road between each pair of towns and at most one road between each town and each city, but all the towns and cities are connected, directly or indirectly. We call a route between a city and a town a gold route if there is no other route between them which passes through fewer towns. Show that we can divide the towns and cities between $n$ republics, so that each belongs to just one republic, each republic has just one city, and each republic contains all the towns on at least one of the gold routes between each of its towns and its city.
1992 All Soviet Union Mathematical Olympiad
Given an infinite sheet of square ruled paper. Some of the squares contain a piece. A move consists of a piece jumping over a piece on a neighbouring square (which shares a side) onto an empty square and removing the piece jumped over. Initially, there are no pieces except in an $m x n$ rectangle ($m, n > 1$) which has a piece on each square. What is the smallest number of pieces that can be left after a series of moves?
1992 All Soviet Union Mathematical Olympiad
Does there exist a $4$-digit integer which cannot be changed into a multiple of $1992$ by changing $3$ of its digits?
1992 All Soviet Union Mathematical Olympiad
$A$ and $B$ lie on a circle. $P$ lies on the minor arc $AB$. $Q$ and $R$ (distinct from $P$) also lie on the circle, so that $P$ and $Q$ are equidistant from $A$, and $P$ and $R$ are equidistant from $B$. Show that the intersection of $AR$ and $BQ$ is the reflection of $P$ in $AB$.
1992 All Soviet Union Mathematical Olympiad
Find all real $x, y$ such that  $\begin{cases}(1 + x)(1 + x^2)(1 + x^4) = 1+ y^7 \\
 (1 + y)(1 + y^2)(1 + y^4) = 1+ x^7 \end{cases}$ .
1992 All Soviet Union Mathematical Olympiad
An $m \times n$ rectangle is divided into mn unit squares by lines parallel to its sides. A gnomon is the figure of three unit squares formed by deleting one unit square from a $2 \times 2$ square. For what $m, n$ can we divide the rectangle into gnomons so that no two gnomons form a rectangle and no vertex is in four gnomons?
1992 All Soviet Union Mathematical Olympiad
Show that for any real numbers $x, y > 1$, we have $\frac{x^2}{y - 1}+ \frac{y^2}{x - 1} \ge  8$
1992 All Soviet Union Mathematical Olympiad
Show that if $15$ numbers lie between $2$ and $1992$ and each pair is coprime, then at least one is prime.
1992 All Soviet Union Mathematical Olympiad
A cinema has its seats arranged in $n$ rows $\times m$ columns. It sold mn tickets but sold some seats more than once. The usher managed to allocate seats so that every ticket holder was in the correct row or column. Show that he could have allocated seats so that every ticket holder was in the correct row or column and at least one person was in the correct seat. What is the maximum $k$ such that he could have always put every ticket holder in the correct row or column and at least $k$ people in the correct seat?
1992 All Soviet Union Mathematical Olympiad
Circles $C$ and $C'$ intersect at $O$ and $X$. A circle center $O$ meets $C$ at $Q$ and $R$ and meets $C'$ at $P$ and $S$. $PR$ and $Q$ meet at $Y$ distinct from $X$. Show that $\angle YXO = 90^o$.
1992 All Soviet Union Mathematical Olympiad
Define the sequence $a_1 = 1, a_2, a_3, ...$ by $a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$.

Show that $1$ is the only square in the sequence.
1992 All Soviet Union Mathematical Olympiad
$ABCD$ is a parallelogram. The excircle of $ABC$ opposite $A$ has center $E$ and touches the line $AB$ at $X$. The excircle of $ADC$ opposite $A$ has center $F$ and touches the line $AD$ at $Y$. The line $FC$ meets the line$ AB$ at $W$, and the line $EC$ meets the line $AD$ at $Z$. Show that $WX = YZ$.
1992 All Soviet Union Mathematical Olympiad
Half the cells of a $2m \times n$ board are colored black and the other half are colored white. The cells at the opposite ends of the main diagonal are different colors. The center of each black cell is connected to the center of every other black cell by a straight line segment, and similarly for the white cells. Show that we can place an arrow on each segment so that it becomes a vector and the vectors sum to zero.
1992 All Soviet Union Mathematical Olympiad
A graph has $17$ points and each point has $4$ edges.

Show that there are two points which are not joined and which are not both joined to the same point.
1992 All Soviet Union Mathematical Olympiad
Let $f(x) = a \cos(x + 1) + b \cos(x + 2) + c \cos(x + 3)$, where $a, b, c$ are real.

Given that $f(x)$ has at least two zeros in the interval $(0, \pi)$, find all its real zeros.
1992 All Soviet Union Mathematical Olympiad
A plane intersects a sphere in a circle $C$. The points $A$ and $B$ lie on the sphere on opposite sides of the plane. The line joining $A$ to the center of the sphere is normal to the plane. Another plane $p$ intersects the segment $AB$ and meets $C$ at $P$ and $Q$. Show that $BP\cdot BQ$ is independent of the choice of $p$.
1992 All Soviet Union Mathematical Olympiad
If you have an algorithm for finding all the real zeros of any cubic polynomial, how do you find the real solutions to $x = p(y), y = p(x)$, where $p$ is a cubic polynomial?
1992 All Soviet Union Mathematical Olympiad
Find all integers $k > 1$ such that for some distinct positive integers $a, b$, the number $k^a + 1$ can be obtained from $k^b + 1$ by reversing the order of its (decimal) digits.
1992 All Soviet Union Mathematical Olympiad
An equilateral triangle side $10$ is divided into $100$ equilateral triangles of side $1$ by lines parallel to its sides. There are m equilateral tiles of $4$ unit triangles and $25 - m$ straight tiles of $4$ unit triangles (as shown below). For which values of $m$ can they be used to tile the original triangle. [The straight tiles may be turned over.]
1992 All Soviet Union Mathematical Olympiad
$1992$ vectors are given in the plane. Two players pick unpicked vectors alternately. The winner is the one whose vectors sum to a vector with larger magnitude (or they draw if the magnitudes are the same). Can the first player always avoid losing?
1992 All Soviet Union Mathematical Olympiad
If $a > b > c > d > 0$ are integers such that $ad = bc$, show that $(a - d)^2 \ge 4d + 8$.
1991 All Soviet Union Mathematical Olympiad
Find all integers $a, b, c, d$ such that $ab - 2cd = 3, ac + bd = 1$.
1991 All Soviet Union Mathematical Olympiad
$n$ numbers are written on a blackboard. Someone then repeatedly erases two numbers and writes half their arithmetic mean instead, until only a single number remains. If all the original numbers were $1$, show that the final number is not less than $\frac{1}{n}$.
1991 All Soviet Union Mathematical Olympiad
Four lines in the plane intersect in six points. Each line is thus divided into two segments and two rays. Is it possible for the eight segments to have lengths $1, 2, 3, ... , 8$? Can the lengths of the eight segments be eight distinct integers?
1991 All Soviet Union Mathematical Olympiad
A lottery ticket has $50$ cells into which one must put a permutation of $1, 2, 3, ... , 50$. Any ticket with at least one cell matching the winning permutation wins a prize. How many tickets are needed to be sure of winning a prize?
1991 All Soviet Union Mathematical Olympiad
Find unequal integers $m, n$ such that $mn + n$ and $mn + m$ are both squares.

Can you find such integers between $988$ and $1991$?
1991 All Soviet Union Mathematical Olympiad
$ABCD$ is a rectangle. Points $K, L, M, N$ are chosen on $AB, BC, CD, DA$ respectively so that $KL$ is parallel to $MN$, and $KM$ is perpendicular to $LN$. Show that the intersection of $KM$ and $LN$ lies on $BD$.
1991 All Soviet Union Mathematical Olympiad
An investigator works out that he needs to ask at most $91$ questions on the basis that all the answers will be yes or no and all will be true. The questions may depend upon the earlier answers. Show that he can make do with $105$ questions if at most one answer could be a lie.
1991 All Soviet Union Mathematical Olympiad
A minus sign is placed on one square of a $5 \times  5$ board and plus signs are placed on the remaining squares. A move is to select a $2  \times  2, 3  \times  3, 4  \times  4$ or $5  \times  5$ square and change all the signs in it. Which initial positions allow a series of moves to change all the signs to plus?
1991 All Soviet Union Mathematical Olympiad
Show that $\frac{(x + y + z)^2}{3} \ge x\sqrt{yz} + y\sqrt{zx} + z\sqrt{xy}$ for all non-negative reals $x, y, z$.
1991 All Soviet Union Mathematical Olympiad
Does there exist a triangle in which two sides are integer multiples of the median to that side?

Does there exist a triangle in which every side is an integer multiple of the median to that side?
1991 All Soviet Union Mathematical Olympiad
The numbers $1, 2, 3, ... , n$ are written on a blackboard (where $n \ge 3$). A move is to replace two numbers by their sum and non-negative difference. A series of moves makes all the numbers equal $k$. Find all possible $k$
1991 All Soviet Union Mathematical Olympiad
The figure below is cut along the lines into polygons (which need not be convex). No polygon contains a $2 \times 2$ square. What is the smallest possible number of polygons?



[missing figure]
1991 All Soviet Union Mathematical Olympiad
$ABC$ is an acute-angled triangle with circumcenter $O$. The circumcircle of $ABO$ intersects$ AC$ and $BC$ at $M$ and $N$. Show that the circumradii of $ABO$ and $MNC$ are the same.
1991 All Soviet Union Mathematical Olympiad
A polygon can be transformed into a new polygon by making a straight cut, which creates two new pieces each with a new edge. One piece is then turned over and the two new edges are reattached. Can repeated transformations of this type turn a square into a triangle?
1991 All Soviet Union Mathematical Olympiad
An $h \times k$ minor of an $n \times n$ table is the $hk$ cells which lie in $h$ rows and $k$ columns. The semiperimeter of the minor is $h + k$. A number of minors each with semiperimeter at least $n$ together include all the cells on the main diagonal. Show that they include at least half the cells in the table.
1991 All Soviet Union Mathematical Olympiad
a) $r_1, r_2, ... , r_{100}, c_1, c_2, ... , c_{100}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $100 \times 100$ array. The product of the numbers in each column is $1$. Show that the product of the numbers in each row is $-1$.

b) $r_1, r_2, ... , r_{2n}, c_1, c_2, ... , c_{2n}$ are distinct reals. The number $r_i + c_j$ is written in position $i, j$ of a $2n \times 2n$ array. The product of the numbers in each column is the same. Show that the product of the numbers in each row is also the same.
1991 All Soviet Union Mathematical Olympiad
A sequence of positive integers is constructed as follows. If the last digit of $a_n$ is greater than $5$, then $a_{n+1}$ is $9a_n$. If the last digit of $a_n$ is $5$ or less and an has more than one digit, then $a_{n+1}$ is obtained from $a_n$ by deleting the last digit. If $a_n$ has only one digit, which is $5$ or less, then the sequence terminates. Can we choose the first member of the sequence so that it does not terminate?
1991 All Soviet Union Mathematical Olympiad
$p(x)$ is the cubic $x^3 - 3x^2 + 5x$. If $h$ is a real root of $p(x) = 1$ and $k$ is a real root of $p(x) = 5$, find $h + k$.
1991 All Soviet Union Mathematical Olympiad
The chords $AB$ and $CD$ of a sphere intersect at $X. A, C$ and $X$ are equidistant from a point $Y$ on the sphere. Show that $BD$ and $XY$ are perpendicular.
1991 All Soviet Union Mathematical Olympiad
Do there exist $4$ vectors in the plane so that none is a multiple of another, but the sum of each pair is perpendicular to the sum of the other two? Do there exist $91$ non-zero vectors in the plane such that the sum of any $19$ is perpendicular to the sum of the others?
1991 All Soviet Union Mathematical Olympiad
$ABCD$ is a square. The points $X$ on the side $AB$ and $Y$ on the side $AD$ are such that $AX\cdot AY = 2 BX\cdot DY$. The lines $CX$ and $CY$ meet the diagonal $BD$ in two points. Show that these points lie on the circumcircle of $AXY$.
1991 All Soviet Union Mathematical Olympiad
$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members?
1991 All Soviet Union Mathematical Olympiad
The real numbers $x_1, x_2, ... , x_{1991}$ satisfy $|x_1 - x_2| + |x_2 - x_3| + ... + |x_{1990} - x_{1991}| = 1991$. What is the maximum possible value of $|s_1 - s_2| + |s_2 - s_3| + ... + |s_{1990} - s_{1991}|$, where $s_n = \frac{x_1 + x_2 + ... + x_n}{n}$?
1990 All Soviet Union Mathematical Olympiad
Show that $x^4 > x  - \frac12$ for all real $x$.
1990 All Soviet Union Mathematical Olympiad
The line joining the midpoints of two opposite sides of a convex quadrilateral makes equal angles with the diagonals. Show that the diagonals are equal.
1990 All Soviet Union Mathematical Olympiad
A graph has $30$ points and each point has $6$ edges. Find the total number of triples such that each pair of points is joined or each pair of points is not joined.
1990 All Soviet Union Mathematical Olympiad
Does there exist a rectangle which can be dissected into $15$ congruent polygons which are not rectangles?

Can a square be dissected into $15$ congruent polygons which are not rectangles?
1990 All Soviet Union Mathematical Olympiad
The point $P$ lies inside the triangle $ABC$. A line is drawn through $P$ parallel to each side of the triangle. The lines divide $AB$ into three parts length $c, c', c"$ (in that order), and $BC$ into three parts length $a, a', a"$ (in that order), and $CA$ into three parts length $b, b', b"$ (in that order). Show that $abc = a'b'c' = a"b"c"$.
1990 All Soviet Union Mathematical Olympiad
Find three non-zero reals such that all quadratics with those numbers as coefficients have two distinct rational roots.
1990 All Soviet Union Mathematical Olympiad
What is the largest possible value of $|...| |a_1 - a_2| - a_3| - ... - a_{1990}|$, where $a_1, a_2, ... , a_{1990}$ is a permutation of $1, 2, 3, ... , 1990$?
1990 All Soviet Union Mathematical Olympiad
An equilateral triangle of side $n$ is divided into $n^2$ equilateral triangles of side $1$. A path is drawn along the sides of the triangles which passes through each vertex just once. Prove that the path makes an acute angle at at least $n$ vertices.
1990 All Soviet Union Mathematical Olympiad
Can the squares of a $1990 \times 1990$ chessboard be colored black or white so that half the squares in each row and column are black and cells symmetric with respect to the center are of opposite color?
1990 All Soviet Union Mathematical Olympiad
Let $x_1, x_2, ..., x_n$ be positive reals with sum $1$. Show that $\frac{x_1^2}{x_1 + x_2}+ \frac{x_2^2}{x_2 + x_3} +... + \frac{x_{n-1}^2}{x_{n-1} + x_n} + \frac{x_n^2}{x_n + x_1} \ge \frac12$.
1990 All Soviet Union Mathematical Olympiad
$ABCD$ is a convex quadrilateral. $X$ is a point on the side $AB. AC$ and $DX$ intersect at $Y$. Show that the circumcircles of $ABC, CDY$ and $BDX$ have a common point.
1990 All Soviet Union Mathematical Olympiad
Two grasshoppers sit at opposite ends of the interval $[0, 1]$. A finite number of points (greater than zero) in the interval are marked. A move is for a grasshopper to select a marked point and jump over it to the equidistant point the other side. This point must lie in the interval for the move to be allowed, but it does not have to be marked. What is the smallest $n$ such that if each grasshopper makes $n$ moves or less, then they end up with no marked points between them?
1990 All Soviet Union Mathematical Olympiad
Find all integers $n$ such that $\left[\frac{n}{1!}\right] + \left[\frac{n}{2!}\right] + ... + \left[\frac{n}{10!}\right]  = 1001$.
1990 All Soviet Union Mathematical Olympiad
$A, B, C$ are adjacent vertices of a regular $2n$-gon and $D$ is the vertex opposite to $B$ (so that $BD$ passes through the center of the $2n$-gon). $X$ is a point on the side $AB$ and $Y$ is a point on the side $BC$ so that angle $XDY = \frac{\pi}{2n}$. Show that DY bisects angle $\angle XYC$.
1990 All Soviet Union Mathematical Olympiad
A graph has $n$ points and $\frac{n(n-1)}{2}$ edges. Each edge is colored with one of $k$ colors so that there are no closed monochrome paths. What is the largest possible value of $n$ (given $k$)?
1990 All Soviet Union Mathematical Olympiad
Given a point $X$ and $n$ vectors $\overrightarrow{x_i}$ with sum zero in the plane. For each permutation of the vectors we form a set of $n$ points, by starting at $X$ and adding the vectors in order. For example, with the original ordering we get $X_1$ such that $XX_1 = \overrightarrow{x_1}, X_2$ such that $X_1X_2 = \overrightarrow{x_2}$ and so on. Show that for some permutation we can find two points $Y, Z$ with angle $\angle YXZ = 60^o $, so that all the points lie inside or on the triangle $XYZ$.
1990 All Soviet Union Mathematical Olympiad
Two unequal circles intersect at $X$ and $Y$. Their common tangents intersect at $Z$. One of the tangents touches the circles at $P$ and $Q$. Show that $ZX$ is tangent to the circumcircle of $PXQ$.
1990 All Soviet Union Mathematical Olympiad
Given $1990$ piles of stones, containing $1, 2, 3, ... , 1990$ stones. A move is to take an equal number of stones from one or more piles. How many moves are needed to take all the stones?
1990 All Soviet Union Mathematical Olympiad
A quadratic polynomial $p(x)$ has positive real coefficients with sum $1$. Show that given any positive real numbers with product $1$, the product of their values under $p$ is at least $1$.
1990 All Soviet Union Mathematical Olympiad
A cube side $100$ is divided into a million unit cubes with faces parallel to the large cube. The edges form a lattice. A prong is any three unit edges with a common vertex. Can we decompose the lattice into prongs with no common edges?
1990 All Soviet Union Mathematical Olympiad
For which positive integers $n$ is $3^{2n+1} - 2^{2n+1} - 6^n$ composite?
1990 All Soviet Union Mathematical Olympiad
If every altitude of a tetrahedron is at least $1$, show that the shortest distance between each pair of opposite edges is more than $2$.
1990 All Soviet Union Mathematical Olympiad
A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?
1990 All Soviet Union Mathematical Olympiad
Given $2n$ genuine coins and $2n$ fake coins. The fake coins look the same as genuine coins but weigh less (but all fake coins have the same weight). Show how to identify each coin as genuine or fake using a balance at most $3n$ times.
1989 All Soviet Union Mathematical Olympiad
$7$ boys each went to a shop $3$ times. Each pair met at the shop. Show that $3$ must have been in the shop at the same time.
1989 All Soviet Union Mathematical Olympiad
Can $77$ blocks each $3 \times 3 \times1$ be assembled to form a $7 \times 9 \times 11$ block?
1989 All Soviet Union Mathematical Olympiad
The incircle of $ABC$ touches $AB$ at $M$. $N$ is any point on the segment $BC$. Show that the incircles of $AMN, BMN, ACN$ have a common tangent.
1989 All Soviet Union Mathematical Olympiad
A positive integer $n$ has exactly $12$ positive divisors $1 = d_1 < d_2 < d_3 < ... < d_{12} = n$. Let $m = d_4 - 1$. We have $d_m = (d_1 + d_2 + d_4) d_8$. Find $n$.
1989 All Soviet Union Mathematical Olympiad
Eight pawns are placed on a chessboard, so that there is one in each row and column. Show that an even number of the pawns are on black squares.
1989 All Soviet Union Mathematical Olympiad
$ABC$ is a triangle. $A' , B' , C'$ are points on the segments $BC, CA, AB$ respectively. $\angle B' A' C' = \angle A$ , $\frac{AC'}{C'B} = \frac{BA' }{A' C} = \frac{CB'}{B'A}$. Show that $ABC$ and $A'B'C'$ are similar.
1989 All Soviet Union Mathematical Olympiad
One bird lives in each of $n$ bird-nests in a forest. The birds change nests, so that after the change there is again one bird in each nest. Also for any birds $A, B, C, D$ (not necessarily distinct), if the distance $AB < CD$ before the change, then $AB > CD$ after the change. Find all possible values of $n$.
1989 All Soviet Union Mathematical Olympiad
Show that the $120$ five digit numbers which are permutations of $12345$ can be divided into two sets with each set having the same sum of squares.
1989 All Soviet Union Mathematical Olympiad
We are given $1998$ normal coins, $1$ heavy coin and $1$ light coin, which all look the same. We wish to determine whether the average weight of the two abnormal coins is less than, equal to, or greater than the weight of a normal coin. Show how to do this using a balance $4$ times or less.
1989 All Soviet Union Mathematical Olympiad
A triangle with perimeter $1$ has side lengths $a, b, c$. Show that $a^2 + b^2 + c^2 + 4abc <\frac 12$.
1989 All Soviet Union Mathematical Olympiad
$ABCD$ is a convex quadrilateral. $X$ lies on the segment $AB$ with $\frac{AX}{XB} = \frac{m}{n}$. $Y$ lies on the segment $CD$ with $\frac{CY}{YD} = \frac{m}{n}$. $AY$ and $DX$ intersect at $P$, and $BY$ and $CX$ intersect at $Q$. Show that $\frac{S_{XQYP}}{S_{ABCD}} < \frac{mn}{m^2 + mn + n^2}$.
1989 All Soviet Union Mathematical Olympiad
A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares?
1989 All Soviet Union Mathematical Olympiad
Do there exist two reals whose sum is rational, but the sum of their $n$ th powers is irrational for all $n > 1$?

Do there exist two reals whose sum is irrational, but the sum of whose $n$ th powers is rational for all $n > 1$?
1989 All Soviet Union Mathematical Olympiad
An insect is on a square ceiling side $1$. The insect can jump to the midpoint of the segment joining it to any of the four corners of the ceiling. Show that in $8$ jumps it can get to within $1/100$ of any chosen point on the ceiling
1989 All Soviet Union Mathematical Olympiad
$ABCD$ has $AB = CD$, but $AB$ not parallel to $CD$, and $AD$ parallel to $BC$. The triangle is $ABC$ is rotated about $C$ to $A'B'C$. Show that the midpoints of $BC, B'C$ and $A'D$ are collinear.
1989 All Soviet Union Mathematical Olympiad
Show that for each integer $n > 0$, there is a polygon with vertices at lattice points and all sides parallel to the axes, which can be dissected into $1 \times 2$ (and / or $2  \times 1$) rectangles in exactly $n$ ways.
1989 All Soviet Union Mathematical Olympiad
Find the smallest positive integer $n$ for which we can find an integer $m$ such that $\left[\frac{10^n}{m}\right] = 1989$.
1989 All Soviet Union Mathematical Olympiad
$ABC$ is a triangle. Points $D, E, F$ are chosen on $BC, CA, AB$ such that $B$ is equidistant from $D$ and $F$, and $C$ is equidistant from $D$ and $E$. Show that the circumcenter of $AEF$ lies on the bisector of $EDF$.
1989 All Soviet Union Mathematical Olympiad
$S$ and $S'$ are two intersecting spheres. The line $BXB'$ is parallel to the line of centers, where $B$ is a point on $S, B'$ is a point on $S'$ and $X$ lies on both spheres. $A$ is another point on $S$, and $A'$ is another point on S' such that the line $AA'$ has a point on both spheres. Show that the segments $AB$ and $A'B'$ have equal projections on the line $AA'$.
1989 All Soviet Union Mathematical Olympiad
Two walkers are at the same altitude in a range of mountains. The path joining them is piecewise linear with all its vertices above the two walkers. Can they each walk along the path until they have changed places, so that at all times their altitudes are equal?
1989 All Soviet Union Mathematical Olympiad
Find the least possible value of $(x + y)(y + z)$ for positive reals satisfying $(x + y + z) xyz = 1$.
1989 All Soviet Union Mathematical Olympiad
A polyhedron has an even number of edges. Show that we can place an arrow on each edge so that each vertex has an even number of arrows pointing towards it (on adjacent edges).