Navvye/Olympiad-Problems-CSV huggingface dataset:
2004 All-Russian Olympiad
In a country there are several cities; some of these cities are connected by airlines, so that an airline connects exactly two cities in each case and both flight directions are possible. Each airline belongs to one of $k$ flight companies; two airlines of the same flight company have always a common final point. Show that one can partition all cities in $k+2$ groups in such a way that two cities from exactly the same group are never connected by an airline with each other.
2004 All-Russian Olympiad
A parallelepiped is cut by a plane along a 6-gon. Supposed this 6-gon can be put into a certain rectangle $ \pi$ (which means one can put the rectangle $ \pi$ on the parallelepiped's plane such that the 6-gon is completely covered by the rectangle). Show that one also can put one of the parallelepiped' faces into the rectangle $ \pi.$
2003 All-Russian Olympiad
Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b \in M$, the number $a^2 +b\sqrt 2$ is rational. Prove that $a\sqrt 2$ is rational for all $a \in M.$
2003 All-Russian Olympiad
Two circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$ respectively intersect at $A$ and $B$. The tangents at $A$ to $S_1$ and $S_2$ meet segments $BO_2$ and $BO_1$ at $K$ and $L$ respectively. Show that $KL \parallel O_1O_2.$
2003 All-Russian Olympiad
On a line are given $2k -1$ white segments and $2k -1$ black ones. Assume that each white segment intersects at least $k$ black segments, and each black segment intersects at least $k$ white ones. Prove that there are a black segment intersecting all the white ones, and a white segment intersecting all the black ones.
2003 All-Russian Olympiad
A sequence $(a_n)$ is defined as follows: $a_1 = p$ is a prime number with exactly $300$ nonzero digits, and for each $n \geq 1, a_{n+1}$ is the decimal period of $1/a_n$ multiplies by $2$. Determine $a_{2003}.$
2003 All-Russian Olympiad
There are $N$ cities in a country. Any two of them are connected either by a road or by an airway. A tourist wants to visit every city exactly once and return to the city at which he started the trip. Prove that he can choose a starting city and make a path, changing means of transportation at most once.
2003 All-Russian Olympiad
Let $a, b, c$ be positive numbers with the sum $1$. Prove the inequality

$$\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c} \geq \frac{2}{1+a}+\frac{2}{1+b}+\frac{2}{1+c}.$$
2003 All-Russian Olympiad
Is it possible to write a natural number in every cell of an infinite chessboard in such a manner that for all integers $m, n > 100$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n \ ?$
2003 All-Russian Olympiad
Let $B$ and $C$ be arbitrary points on sides $AP$ and $PD$ respectively of an acute triangle $APD$. The diagonals of the quadrilateral $ABCD$ meet at $Q$, and $H_1,H_2$ are the orthocenters of triangles $APD$ and $BPC$, respectively. Prove that if the line $H_1H_2$ passes through the intersection point $X \ (X \neq Q)$ of the circumcircles of triangles $ABQ$ and $CDQ$, then it also passes through the intersection point $Y \  (Y \neq Q)$ of the circumcircles of triangles $BCQ$ and $ADQ.$
2003 All-Russian Olympiad
Suppose that $M$ is a set of $2003$ numbers such that, for any distinct $a, b, c \in M$, the number $a^2 + bc$ is rational. Prove that there is a positive integer $n$ such that $a\sqrt n$ is rational for all $a \in M.$
2003 All-Russian Olympiad
The diagonals of a cyclic quadrilateral $ABCD$ meet at $O$. Let $S_1, S_2$ be the circumcircles of triangles $ABO$ and $CDO$ respectively, and $O,K$ their intersection points. The lines through $O$ parallel to $AB$ and $CD$ meet $S_1$ and $S_2$ again at $L$ and $M$, respectively. Points $P$ and $Q$ on segments $OL$ and $OM$ respectively are taken such that $OP : PL = MQ : QO$. Prove that $O,K, P,Q$ lie on a circle.
2003 All-Russian Olympiad
Consider a tree (i.e. a connected graph with no cycles) with $n$ vertices.  Its vertices are assigned numbers $x_1,x_2,\cdots x_n$ and each edge is assigned the product of the numbers at its endpoints.  Let $S$ denote the sum of the numbers at the deges.  Prove that $$ 2S \leq \sqrt{n-1}\left(x_1^2+x_2^2+\cdots+x_n^2\right). $$
2003 All-Russian Olympiad
A finite set of points $X$ and an equilateral triangle $T$ are given on a plane. Suppose that every subset $X'$ of $X$ with no more than $9$ elements can be covered by two images of $T$ under translations. Prove that the whole set $X$ can be covered by two images of $T$ under translations.
2003 All-Russian Olympiad
Let $ a_0$ be a natural number. The sequence $ (a_n)$ is deﬁned by $ a_{n+1}=\frac{a_n}{5}$ if $ a_n$ is divisible by $ 5$

and $ a_{n+1}=[a_n \sqrt{5}]$ otherwise . Show that the sequence $ a_n$ is increasing starting from some term.
2003 All-Russian Olympiad
In a triangle $ABC, O$ is the circumcenter and $I$ the incenter. The excircle $\omega_a$ touches rays $AB,AC$ and side $BC$ at $K,M,N$, respectively. Prove that if the midpoint $P$ of $KM$ lies on the circumcircle of $\triangle ABC$, then points $O,N, I$ lie on a line.
2003 All-Russian Olympiad
Find the greatest natural number $N$ such that, for any arrangement of the numbers $1, 2, \ldots, 400$ in a chessboard $20 \times 20$, there exist two numbers in the same row or column, which differ by at least $N.$
2003 All-Russian Olympiad
Let $\alpha , \beta , \gamma , \delta$ be positive numbers such that for all $x$, $\sin{\alpha x}+\sin {\beta x}=\sin {\gamma x}+\sin {\delta x}$. Prove that $\alpha =\gamma$ or $\alpha=\delta$.
2003 All-Russian Olympiad
Let $f(x)$ and $g(x)$ be polynomials with non-negative integer coefficients, and let m be the largest coefficient of $f.$ Suppose that there exist natural numbers $a < b$ such that $f(a) = g(a)$ and $f(b) = g(b)$. Show that if $b > m,$ then $f = g.$
2003 All-Russian Olympiad
Ana and Bora are each given a sufficiently long paper strip, one with letter $A$ written , and the other with letter $B$. Every minute, one of them (not necessarily one after another) writes either on the left or on the right to the word on his/her strip the word written on the other strip. Prove that the day after, one will be able to cut word on Ana's strip into two words and exchange their places, obtaining a palindromic word.
2003 All-Russian Olympiad
The side lengths of a triangle are the roots of a cubic polynomial with rational coefficients. Prove that the altitudes of this triangle are roots of a polynomial of sixth degree with rational coefficients.
2003 All-Russian Olympiad
Is it possible to write a positive integer in every cell of an infinite chessboard, in such a manner that, for all positive integers $m, n$, the sum of numbers in every $m\times n$ rectangle is divisible by $m + n$ ?
2003 All-Russian Olympiad
There are $100$ cities in a country, some of them being joined by roads. Any four cities are connected to each other by at least two roads. Assume that there is no path passing through every city exactly once. Prove that there are two cities such that every other city is connected to at least one of them.
2003 All-Russian Olympiad
The inscribed sphere of a tetrahedron $ABCD$ touches $ABC,ABD,ACD$ and $BCD$ at $D_1,C_1,B_1$ and $A_1$ respectively. Consider the plane equidistant from $A$ and plane $B_1C_1D_1$ (parallel to $B_1C_1D_1$) and the three planes defined analogously for the vertices $B,C,D$. Prove that the circumcenter of the tetrahedron formed by these four planes coincides with the circumcenter of tetrahedron of $ABCD$.
2002 All-Russian Olympiad
Can the cells of a $2002 \times 2002$ table be filled with the numbers from $1$ to $2002^2$ (one per cell) so that for any cell we can find three numbers $a, b, c$ in the same row or column (or the cell itself) with $a = bc$?
2002 All-Russian Olympiad
Point $A$ lies on one ray and points $B,C$ lie on the other ray of an angle with the vertex at $O$ such that $B$ lies between $O$ and $C$.  Let $O_1$ be the incenter of $\triangle OAB$ and $O_2$ be the center of the excircle of $\triangle OAC$ touching side $AC$.  Prove that if $O_1A = O_2A$, then the triangle $ABC$ is isosceles.
2002 All-Russian Olympiad
On a plane are given $6$ red, $6$ blue, and $6$ green points, such that no three of the given points lie on a line.  Prove that the sum of the areas of the triangles whose vertices are of the same color does not exceed quarter the sum of the areas of all triangles with vertices in the given points.
2002 All-Russian Olympiad
A hydra consists of several heads and several necks, where each neck joins two heads.  When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$.  Heracle defeats a hydra by cutting it into two parts which are no joined.  Find the minimum $N$ for which Heracle can defeat any hydra with $100$ necks by no more than $N$ hits.
2002 All-Russian Olympiad
There are eight rooks on a chessboard, no two attacking each other.  Prove that some two of the pairwise distances between the rooks are equal.  (The distance between two rooks is the distance between the centers of their cell.)
2002 All-Russian Olympiad
We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$.  Initially, the pack is situated on the red cell and arranged in an arbitrary order.  In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell.  Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?
2002 All-Russian Olympiad
Let O be the circumcenter of a triangle ABC. Points M and N are choosen on the sides AB and BC respectively so that the angle AOC is two times greater than angle MON. Prove that the perimeter of triangle MBN is not less than the lenght of side AC
2002 All-Russian Olympiad
From the interval $(2^{2n},2^{3n})$ are selected $2^{2n-1}+1$ odd numbers.  Prove that there are two among the selected numbers, none of which divides the square of the other.
2002 All-Russian Olympiad
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$.  Prove that one of the polynomials of degree $3$ has three real roots.
2002 All-Russian Olympiad
A quadrilateral $ABCD$ is inscribed in a circle $\omega$. The tangent to $\omega$ at $A$ intersects the ray $CB$ at $K$, and the tangent to $\omega$ at $B$ intersects the ray $DA$ at $M$. Prove that if $AM=AD$ and $BK=BC$, then $ABCD$ is a trapezoid.
2002 All-Russian Olympiad
Prove that for every integer $n > 10000$ there exists an integer $m$ such that it can be written as the sum of two squares, and $0<m-n<3\sqrt[4]n$.
2002 All-Russian Olympiad
There are 2002 towns in a kingdom. Some of the towns are connected by roads in such a manner that, if all roads from one city closed, one can still travel between any two cities. Every year, the kingdom chooses a non-self-intersecting cycle of roads, founds a new town, connects it by roads with each city from the chosen cycle, and closes all the roads from the original cycle. After several years, no non-self-intersecting cycles remained. Prove that at that moment there are at least 2002 towns, exactly one road going out from each of them.
2002 All-Russian Olympiad
For positive real numbers $a, b, c$ such that $a+b+c=3$, show that:



$$\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab+bc+ca.$$
2002 All-Russian Olympiad
Let $A^\prime$ be the point of tangency of the excircle of a triangle $ABC$ (corrsponding to $A$) with the side $BC$. The line $a$ through $A^\prime$ is parallel to the bisector of $\angle BAC$. Lines $b$ and $c$ are analogously deﬁned. Prove that $a, b, c$ have a common point.
2002 All-Russian Olympiad
On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.
2002 All-Russian Olympiad
Several points are given in the plane. Suppose that for any three of them, there exists an orthogonal coordinate system (determined by the two axes and the unit length) in which these three points have integer coordinates. Prove that there exists an orthogonal coordinate system in which all the given points have integer coordinates.
2002 All-Russian Olympiad
Prove that if $0<x<\frac{\pi}{2}$ and $n>m$, where $n$,$m$ are natural numbers, $$ 2 \left| \sin^n x - \cos^n x \right| \le 3 \left| \sin^m x - \cos^m x \right|.$$
2002 All-Russian Olympiad
There are some markets in a city. Some of them are joined by one-way streets, such that for any market there are exactly two streets to leave it. Prove that the city may be partitioned into $1014$ districts such that streets join only markets from different districts, and by the same one-way for any two districts (either only from first to second, or vice-versa).
2002 All-Russian Olympiad
Determine the smallest natural number which can be represented both as the sum of $2002$ positive integers with the same sum of decimal digits, and as the sum of $2003$ integers with the same sum of decimal digits.
2002 All-Russian Olympiad
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at $O$.  The circumcircles of triangles $AOB$ and $COD$ intersect again at $K$.  Point $L$ is such that the triangles $BLC$ and $AKD$ are similar and equally oriented.  Prove that if the quadrilateral $BLCK$ is convex, then it is tangent [has an incircle].
2002 All-Russian Olympiad
Prove that there exist infinitely many natural numbers $ n$ such that the numerator of $ 1 + \frac {1}{2} + \frac {1}{3} + \frac {1}{4} + ... + \frac {1}{n}$ in the lowest terms is not a power of a prime number.
2001 All-Russian Olympiad
The integers from $1$ to $999999$ are partitioned into two groups: the first group consists of those integers for which the closest perfect square is odd, whereas the second group consists of those for which the closest perfect square is even.  In which group is the sum of the elements greater?
2001 All-Russian Olympiad
The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$.  Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.
2001 All-Russian Olympiad
A point $K$ is taken inside parallelogram $ABCD$ so that the midpoint of $AD$ is equidistant from $K$ and $C$, and the midpoint of $CD$ is equidistant form $K$ and $A$.  Let $N$ be the midpoint of $BK$.  Prove that the angles $NAK$ and $NCK$ are equal.
2001 All-Russian Olympiad
Consider a convex $2000$-gon, no three of whose diagonals have a common point. Each of its diagonals is colored in one of $999$ colors. Prove that there exists a triangle all of whose sides lie on diagonals of the same color. (Vertices of the triangle need not be vertices of the original polygon.)
2001 All-Russian Olympiad
Yura put $2001$ coins of $1$, $2$ or $3$ kopeykas in a row.  It turned out that between any two $1$-kopeyka coins there is at least one coin; between any two $2$-kopeykas coins there are at least two coins; and between any two $3$-kopeykas coins there are at least $3$ coins.  How many $3$-koyepkas coins could Yura put?
2001 All-Russian Olympiad
In a party, there are $2n + 1$  people. It's well known that for every group of $n$ people, there exist a person(out of the group) who knows all them(the $n$ people of the group). Show that there exist a person who knows all the people in the party.
2001 All-Russian Olympiad
Let $N$ be a point on the longest side $AC$ of a triangle $ABC$.  The perpendicular bisectors of $AN$ and $NC$ intersect $AB$ and $BC$ respectively in $K$ and $M$.  Prove that the circumcenter $O$ of $\triangle ABC$ lies on the circumcircle of triangle $KBM$.
2001 All-Russian Olympiad
Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a+b-1$ divides $ n$.
2001 All-Russian Olympiad
Let $A_1, A_2, ... , A_{100}$ be subsets of a line, each a union of $100$ pairwise disjoint closed segments. Prove that the intersection of all hundred sets is a union of at most $9901$ disjoint closed segments.
2001 All-Russian Olympiad
Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.
2001 All-Russian Olympiad
Some towns in a country are connected by two–way roads, so that for any two towns there is a unique path along the roads connecting them. It is known that there is exactly 100 towns which are directly connected to only one town. Prove that we can construct 50 new roads in order to obtain a net in which every two towns will be connected even if one road gets closed.
2001 All-Russian Olympiad
The polynomial $ P(x)=x^3+ax^2+bx+d$ has three distinct real roots. The polynomial $ P(Q(x))$, where $ Q(x)=x^2+x+2001$, has no real roots. Prove that $ P(2001)>\frac{1}{64}$.
2001 All-Russian Olympiad
In a magic square $n \times n$ composed from the numbers $1,2,\cdots,n^2$, the centers of any two squares are joined by a vector going from the smaller number to the bigger one.  Prove that the sum of all these vectors is zero.  (A magic square is a square matrix such that the sums of entries in all its rows and columns are equal.)
2001 All-Russian Olympiad
Points $A_1, B_1, C_1$ inside an acute-angled triangle $ABC$ are selected on the altitudes from $A, B, C$ respectively so that the sum of the areas of triangles $ABC_1, BCA_1$, and $CAB_1$ is equal to the area of triangle $ABC$. Prove that the circumcircle of triangle $A_1B_1C_1$ passes through the orthocenter $H$ of triangle $ABC$.
2001 All-Russian Olympiad
The total mass of $100$ given weights with positive masses equals $2S$.  A natural number $k$ is called middle if some $k$ of the given weights have the total mass $S$.  Find the maximum possible number of middle numbers.
2001 All-Russian Olympiad
There are two families of convex polygons in the plane. Each family has a pair of disjoint polygons. Any polygon from one family intersects any polygon from the other family. Show that there is a line which intersects all the polygons.
2001 All-Russian Olympiad
Participants to an olympiad worked on $n$ problems. Each problem was worth a positive integer number of points, determined by the jury. A contestant gets $0$ points for a wrong answer, and all points for a correct answer to a problem. It turned out after the olympiad that the jury could impose worths of the problems, so as to obtain any (strict) final ranking of the contestants. Find the greatest possible number of contestants.
2001 All-Russian Olympiad
Two monic quadratic trinomials $f(x)$ and $g(x)$ take negative values on disjoint intervals.  Prove that there exist positive numbers $\alpha$ and $\beta$ such that $\alpha f(x) + \beta g(x) > 0$ for all real $x$.
2001 All-Russian Olympiad
Let $ a,b$ be $ 2$ distinct positive interger number such that $ (a^2+ab+b^2)|ab(a+b)$. Prove that: $ |a-b|>\sqrt [3] {ab}$.
2001 All-Russian Olympiad
The $2001$ towns in a country are connected by some roads, at least one road from each town, so that no town is connected by a road to every other city.  We call a set $D$ of towns dominant if every town not in $D$ is connected by a road to a town in $D$.  Suppose that each dominant set consists of at least $k$ towns.  Prove that the country can be partitioned into $2001-k$ republics in such a way that no two towns in the same republic are connected by a road.
2001 All-Russian Olympiad
A sphere with center on the plane of the face $ABC$ of a tetrahedron $SABC$ passes through $A$, $B$ and $C$, and meets the edges $SA$, $SB$, $SC$ again at $A_1$, $B_1$, $C_1$, respectively.  The planes through $A_1$, $B_1$, $C_1$ tangent to the sphere meet at $O$.  Prove that $O$ is the circumcenter of the tetrahedron $SA_1B_1C_1$.
2000 All-Russian Olympiad
Let $a,b,c$ be distinct numbers such that the equations $x^2+ax+1=0$ and $x^2+bx+c=0$ have a common real root, and the equations $x^2+x+a=0$ and $x^2+cx+b$ also have a common real root.  Compute the sum $a+b+c$.
2000 All-Russian Olympiad
Tanya chose a natural number $X\le100$, and Sasha is trying to guess this number.  He can select two natural numbers $M$ and $N$ less than $100$ and ask about $\gcd(X+M,N)$.  Show that Sasha can determine Tanya's number with at most seven questions.
2000 All-Russian Olympiad
Let $O$ be the center of the circumcircle $\omega$ of an acute-angle triangle $ABC$.  A circle $\omega_1$ with center $K$ passes through $A$, $O$, $C$ and intersects $AB$ at $M$ and $BC$ at $N$.  Point $L$ is symmetric to $K$ with respect to line $NM$.  Prove that $BL \perp AC$.
2000 All-Russian Olympiad
Some pairs of cities in a certain country are connected by roads, at least three roads going out of each city.  Prove that there exists a round path consisting of roads whose number is not divisible by $3$.
2000 All-Russian Olympiad
The sequence $a_1 = 1$, $a_2, a_3, \cdots$  is defined as follows: if $a_n - 2$ is a natural number not already occurring on the board, then $a_{n+1} = a_n-2$; otherwise, $a_{n+1} = a_n + 3$.  Prove that every nonzero perfect square occurs in the sequence as the previous term increased by $3$.
2000 All-Russian Olympiad
On some cells of a $2n \times 2n$ board are placed white and black markers (at most one marker on every cell).  We first remove all black markers which are in the same column with a white marker, then remove all white markers which are in the same row with a black one.  Prove that either the number of remaining white markers or that of remaining black markers does not exceed $n^2$.
2000 All-Russian Olympiad
Let $E$ be a point on the median $CD$ of a triangle $ABC$.  The circle $\mathcal S_1$ passing through $E$ and touching $AB$ at $A$ meets the side $AC$ again at $M$. The circle $S_2$ passing through $E$ and touching $AB$ at $B$ meets the side $BC$ at $N$.  Prove that the circumcircle of $\triangle CMN$ is tangent to both $\mathcal S_1$ and $\mathcal S_2$.
2000 All-Russian Olympiad
One hundred natural numbers whose greatest common divisor is $1$ are arranged around a circle.  An allowed operation is to add to a number the greatest common divisor of its two neighhbors.  Prove that we can make all the numbers pairwise copirme in a finite number of moves.
2000 All-Russian Olympiad
Evaluate the sum $$ \left\lfloor \frac{2^0}{3} \right\rfloor  + \left\lfloor \frac{2^1}{3} \right\rfloor + \left\lfloor \frac{2^2}{3} \right\rfloor  + \cdots + \left\lfloor \frac{2^{1000}}{3} \right\rfloor. $$
2000 All-Russian Olympiad
Let $-1 < x_1 < x_2 , \cdots < x_n < 1$ and $x_1^{13} + x_2^{13} + \cdots + x_n^{13} = x_1 + x_2 + \cdots + x_n$.  Prove that if $y_1 < y_2 < \cdots < y_n$, then $$ x_1^{13}y_1 + \cdots + x_n^{13}y_n < x_1y_1 + x_2y_2 + \cdots + x_ny_n. $$
2000 All-Russian Olympiad
In an acute scalene triangle $ABC$ the bisector of the acute angle between the altitudes $AA_1$ and $CC_1$ meets the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The bisector of the angle $B$ intersects the segment joining the orthocenter of $ABC$ and the midpoint of $AC$ at point $R$. Prove that $P$, $B$, $Q$, $R$ lie on a circle.
2000 All-Russian Olympiad
We are given five equal-looking weights of pairwise distinct masses.  For any three weights $A$, $B$, $C$, we can check by a measuring if $m(A) < m(B) < m(C)$, where $m(X)$ denotes the mass of a weight $X$ (the answer is yes or no.)  Can we always arrange the masses of the weights in the increasing order with at most nine measurings?
2000 All-Russian Olympiad
Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$.  Find the greatest possible number of elements of $M$.
2000 All-Russian Olympiad
A perfect number, greater than $6$, is divisible by $3$.  Prove that it is also divisible by $9$.