Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 Kazakhstan National Olympiad
The cell of a $(2m +1) \times (2n +1)$ board are painted in two colors - white and black. The unit cell of a row (column) is called dominant on the row (the column) if more than half of the cells that row (column) have the same color as this cell.  Prove that at least $m + n-1$ cells on the board are dominant in both their row and column.
2012 Kazakhstan National Olympiad
For a positive reals $ x_{1},...,x_{n} $ prove inequlity:

$ \frac{1}{x_{1}+1}+...+\frac{1}{x_{n}+1}\le \frac{n}{1+\frac{n}{\frac{1}{x_{1}}+...+\frac{1}{x_{n}}}}$
2012 Kazakhstan National Olympiad
Let $ABCD$ be an inscribed quadrilateral, in  which $\angle BAD<90$. On the rays $AB$ and $AD$ are selected points $K$ and $L$, respectively, such that$ KA = KD, LA = LB$. Let $N$ - the midpoint of $AC$.Prove that if $\angle BNC=\angle DNC $,so $\angle KNL=\angle BCD $
2012 Kazakhstan National Olympiad
There are $n$ balls numbered from $1$ to $n$, and $2n-1$ boxes numbered from $1$ to $2n-1$. For each $i$, ball number $i$ can only be put in the boxes with numbers from $1$ to $2i-1$. Let $k$ be an integer from $1$ to $n$. In how many ways we can choose $k$ balls, $k$ boxes and put these balls in the selected boxes so that each box has exactly one ball?
2012 Kazakhstan National Olympiad
Let $k_{1},k_{2}, k_{3}$ -Excircles triangle $A_{1}A_{2}A_{3}$ with area $S$. $ k_{1}$ touch side $A_{2}A_{3} $ at the point $B_{1}$ Direct $A_{1}B_{1}$ intersect $k_{1}$ at the points $B_{1}$ and $C_{1}$.Let $S_{1}$ - area of ​​the quadrilateral $A_{1}A_{2}C_{1}A_{3}$ Similarly, we define $S_{2}, S_{3}$. Prove that $\frac{1}{S}\le \frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{2}}$
2012 Kazakhstan National Olympiad
Function $ f:\mathbb{R}\rightarrow\mathbb{R} $ such that $f(xf(y))=yf(x)$ for any $x,y$ are real numbers. Prove that $f(-x) = -f(x)$ for all real numbers $x$.
2012 Kazakhstan National Olympiad
The sequence $a_{n}$ defined as follows: $a_{1}=4, a_{2}=17$ and for any $k\geq1$ true equalities

$a_{2k+1}=a_{2}+a_{4}+...+a_{2k}+(k+1)(2^{2k+3}-1)$

$a_{2k+2}=(2^{2k+2}+1)a_{1}+(2^{2k+3}+1)a_{3}+...+(2^{3k+1}+1)a_{2k-1}+k$

Find the smallest $m$ such that $(a_{1}+...a_{m})^{2012^{2012}}-1$ divided $2^{2012^{2012}}$
2012 Kazakhstan National Olympiad
The number $\overline{13\ldots 3}$, with $k>1$ digits $3$, is a prime. Prove that $6\mid k^{2}-2k+3$.
2012 Kazakhstan National Olympiad
We call a $6\times 6$ table consisting of zeros and ones right if the sum of the numbers in each row and each column is equal to $3$. Two right tables are called similar if one can get from one to the other by successive interchanges of rows and columns. Find the largest possible size of a set of pairwise similar right tables.
2012 Kazakhstan National Olympiad
Line $PQ$ is tangent to the incircle  of triangle $ABC$ in such a way that the points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively. On the sides $AB$ and $AC$ are selected points $M$ and $N$, respectively, so that $AM = BP$ and $AN = CQ$. Prove that all lines constructed in this manner $MN$ pass through one point
2012 Kazakhstan National Olympiad
Given the rays $ OP$ and $OQ$.Inside the smaller angle $POQ$ selected points $M$ and $N$, such that $\angle POM=\angle QON $ and $\angle POM<\angle PON $ The circle, which concern the rays $OP$ and $ON$, intersects the second circle, which concern the rays $OM$ and $OQ$ at the points $B$ and $C$. Prove that$\angle POC=\angle QOB $
2012 Kazakhstan National Olympiad
Consider the equation $ax^{2}+by^{2}=1$, where $a,b$ are fixed rational numbers. Prove that either such an equation has no solutions in rational numbers, or it has infinitely many solutions.
2011 Kazakhstan National Olympiad
The quadrilateral $ABCD$ is circumscribed about the circle, touches the sides $AB, BC, CD, DA$ in the points $K, L, M, N,$ respectively. Let $P, Q, R, S$ midpoints of the sides $KL, LM, MN, NK$. Prove that $PR = QS$ if and only if $ABCD$ is inscribed.
2011 Kazakhstan National Olympiad
Determine the smallest possible number $n> 1$ such that there exist positive integers $a_{1}, a_{2}, \ldots, a_{n}$ for which ${a_{1}}^{2}+\cdots +{a_{n}}^{2}\mid (a_{1}+\cdots +a_{n})^{2}-1$.
2011 Kazakhstan National Olympiad
In some cells of a rectangular table $m\times n (m, n> 1)$ is one checker. $Baby$ cut along the lines of the grid this table so that it is split into two equal parts, with the number of pieces on each side were the same. $Carlson$ changed the arrangement of checkers on the board (and on each side of the cage is still worth no more than one pieces). Prove that the $Baby$ may again cut the board into two equal parts containing an equal number of pieces
2011 Kazakhstan National Olympiad
We write in order of increasing number of 1 and all positive integers,which the sum of digits is divisible by $5$. Obtain a sequence of $1, 5, 14, 19. . .$



Prove that the n-th term of the sequence is less than $5n$.
2011 Kazakhstan National Olympiad
Given a non-degenerate triangle $ABC$, let $A_{1}, B_{1}, C_{1}$ be the point of tangency of the incircle with the sides $BC, AC, AB$.  Let $Q$ and $L$ be the intersection of the segment $AA_{1}$ with the incircle and the segment $B_{1}C_{1}$ respectively. Let $M$ be the midpoint of $B_{1}C_{1}$. Let $T$ be the point of intersection of $BC$ and $B_{1}C_{1}$. Let $P$ be the foot of the perpendicular from the point $L$ on the line $AT$. Prove that the points $A_{1}, M, Q, P$ lie on a circle.
2011 Kazakhstan National Olympiad
Given a positive integer $n$. One of the roots of a quadratic equation $x^{2}-ax +2 n = 0$ is equal to

$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}$. Prove that $2\sqrt{2n}\le a\le 3\sqrt{n}$
2011 Kazakhstan National Olympiad
Inscribed in a triangle $ABC$ with the center of the circle $I$ touch the sides $AB$ and $AC$ at points $C_{1}$ and $B_{1}$, respectively. The point $M$ divides the segment $C_{1}B_{1}$ in a 3:1 ratio, measured from $C_{1}$. $N$ - the midpoint of $AC$. Prove that the points $I, M, B_{1}, N$ lie on a circle, if you know that $AC = 3 (BC-AB)$.
2011 Kazakhstan National Olympiad
Given a positive integer $n$. Prove the inequality $\sum\limits_{i=1}^{n}\frac{1}{i(i+1)(i+2)(i+3)(i+4)}<\frac{1}{96}$
2011 Kazakhstan National Olympiad
Prove that there are infinitely many natural numbers, the arithmetic mean and geometric mean of the divisors which are both integers.
2011 Kazakhstan National Olympiad
Determine all pairs of positive real numbers $(a, b)$ for which there exists a function  $ f:\mathbb{R^{+}}\rightarrow\mathbb{R^{+}} $ satisfying for all positive real numbers $x$ the equation

$ f(f(x))=af(x)- bx $
2011 Kazakhstan National Olympiad
Given a real number $a> 0$. How many positive real solutions of the equation is $ a^{x}=x^{a} $
2011 Kazakhstan National Olympiad
Let $w$-circumcircle of triangle $ABC$ with an obtuse angle $C$ and $C '$symmetric point of point $C$ with respect to $AB$. $M$ midpoint of $AB$. $C'M$ intersects $w$ at $N$ ($C '$ between $M$ and $N$). Let $BC'$ second crossing point $w$ in $F$, and $AC'$ again crosses the $w$ at point $E$. $K$-midpoint $EF$. Prove that the lines $AB, CN$ and$ KC'$are concurrent.
2011 Kazakhstan National Olympiad
Given are the odd integers $m> 1$, $k$, and a prime $p$ such that $p> mk +1$. Prove that $p^{2}\mid {\binom{k}{k}}^{m}+{\binom{k+1}{k}}^{m}+\cdots+{\binom{p-1}{k}}^{m}$.
2011 Kazakhstan National Olympiad
On the table lay a pencil, sharpened at one end. The student can rotate the pencil around one of its ends at $45^{\circ}$ clockwise or counterclockwise. Can the student, after a few turns of the pencil, go back to the starting position so that the sharpened end and the not sharpened are reversed?
2011 Kazakhstan National Olympiad
We call a square table of a binary, if at each cell is written a single number 0 or 1. The binary table is called regular if each row and each column exactly two units. Determine the number of regular size tables $n\times n$ ($n> 1$ - given a fixed positive integer). (We can assume that the rows and columns of the tables are numbered: the cases of coincidence in turn, reflect, and so considered different).
2010 Kazakhstan National Olympiad
Triangle $ABC$ is given. Circle $ \omega $ passes through $B$, touch $AC$ in $D$ and intersect sides $AB$ and $BC$ at $P$ and $Q$ respectively. Line $PQ$ intersect $BD$ and $AC$ at $M$ and $N$ respectively. Prove that $ \omega $, circumcircle of $DMN$ and circle, touching $PQ$ in $M$ and passes through B, intersects in one point.
2010 Kazakhstan National Olympiad
Exactly $4n$ numbers in set $A=  \{ 1,2,3,...,6n \} $ of natural numbers painted in red, all other in blue.

Proved that exist $3n$ consecutive natural numbers from $A$,  exactly $2n$ of which numbers is red.
2010 Kazakhstan National Olympiad
Positive real $A$ is given. Find maximum value of $M$ for which inequality



$ \frac{1}{x}+\frac{1}{y}+\frac{A}{x+y} \geq \frac{M}{\sqrt{xy}} $



holds for all $x, y>0$
2010 Kazakhstan National Olympiad
Let $x$- minimal root of equation $x^2-4x+2=0$.

Find two first digits of number $ \{x+x^2+....+x^{20} \}$ after $0$, where $\{a\}$- fractional part of $a$.
2010 Kazakhstan National Olympiad
Arbitrary triangle $ABC$ is given (with $AB<BC$).  Let $M$ - midpoint of $AC$, $N$- midpoint of arc $AC$ of circumcircle $ABC$, which is contains point $B$. Let $I$ - in-center of $ABC$. Proved, that $ \angle IMA = \angle INB$
2010 Kazakhstan National Olympiad
Let numbers $1,2,3,...,2010$ stand in a row at random. Consider row, obtain by next rule:

For any number we sum it and it's number in a row (For example for row $( 2,7,4)$ we consider a row $(2+1;7+2;4+3)=(3;9;7)$ );

Proved, that in resulting row we can found two equals numbers, or two numbers, which is differ by $2010$
2010 Kazakhstan National Olympiad
Triangle $ABC$ is given. Consider ellipse $ \Omega _1$, passes through $C$ with focuses in $A$ and $B$. Similarly define ellipses $ \Omega _2 , \Omega _3$ with focuses $B,C$ and $C,A$ respectively. Prove, that if all ellipses have common point $D$ then $A,B,C,D$ lies on the circle.



Ellipse with focuses $X,Y$, passes through $Z$- locus of point $T$, such that $XT+YT=XZ+YZ$
2010 Kazakhstan National Olympiad
On sides of convex quadrilateral $ABCD$ on external side constructed equilateral triangles $ABK, BCL, CDM, DAN$. Let $P,Q$- midpoints of $BL, AN$ respectively and $X$- circumcenter of $CMD$.

Prove, that $PQ$ perpendicular to $KX$
2010 Kazakhstan National Olympiad
Call $A \in \mathbb{N}^0$ be $number of year$  if all digits of $A$ equals $0$, $1$ or $2$ (in decimal representation).

Prove that exist infinity $N \in \mathbb{N}$, such that $N$ can't presented as $A^2+B$ where $A \in \mathbb{N}^0 ; B$- $number of year$.
2010 Kazakhstan National Olympiad
It is given that for some $n \in \mathbb{N}$ there exists a natural number $a$, such that $a^{n-1} \equiv 1 \pmod{n}$ and that for any prime divisor $p$ of $n-1$ we have $a^{\frac{n-1}{p}} \not \equiv 1 \pmod{n}$.

Prove that $n$ is a prime.
2010 Kazakhstan National Olympiad
Let $n \geq 2$ be an integer. Define  $x_i =1$ or $-1$ for every $i=1,2,3,\cdots, n$.



Call an operation adhesion, if it changes the string $(x_1,x_2,\cdots,x_n)$ to $(x_1x_2, x_2x_3, \cdots ,x_{n-1}x_n, x_nx_1)$ .



Find all integers $n \geq 2$ such that the string  $(x_1,x_2,\cdots, x_n)$ changes to $(1,1,\cdots,1)$ after finitely adhesion operations.
2010 Kazakhstan National Olympiad
Let $ABCD$ be convex quadrilateral, such that exist $M,N$ inside $ABCD$ for which $\angle NAD= \angle MAB; \angle NBC= \angle MBA; \angle MCB=\angle NCD; \angle NDA=\angle MDC$

Prove, that $S_{ABM}+S_{ABN}+S_{CDM}+S_{CDN}=S_{BCM}+S_{BCN}+S_{ADM}+S_{ADN}$, where $S_{XYZ}$-area of triangle $XYZ$
2010 Kazakhstan National Olympiad
For $x;y \geq 0$ prove the inequality:



$\sqrt{x^2-x+1} \sqrt{y^2-y+1}+ \sqrt{x^2+x+1} \sqrt{y^2+y+1} \geq 2(x+y)$
2010 Kazakhstan National Olympiad
Let $O$ be the circumcircle of acute triangle $ABC$, $AD$-altitude of $ABC$ ($ D \in BC$), $ AD \cap CO =E$, $M$-midpoint of $AE$, $F$-feet of perpendicular from $C$ to $AO$.

Proved that point of intersection $OM$ and $BC$ lies on circumcircle of triangle $BOF$
2009 Kazakhstan National Olympiad
Let $S_n$ be number of ordered sets of natural numbers $(a_1;a_2;....;a_n)$ for which $\frac{1}{a_1}+\frac{1}{a_2}+....+\frac{1}{a_n}=1$. Determine

1)$S_{10} mod(2)$.

2)$S_7 mod(2)$.



(1) is first problem in 10 grade, (2)- third in 9 grade.
2009 Kazakhstan National Olympiad
Let in-circle of $ABC$ touch $AB$, $BC$, $AC$ in $C_1$, $A_1$, $B_1$ respectively.

Let $H$- intersection point of altitudes in $A_1B_1C_1$, $I$ and $O$-be in-center and circumcenter of $ABC$ respectively.

Prove, that $I, O, H$ lies on one line.
2009 Kazakhstan National Olympiad
In chess tournament participates $n$ participants ($n >1$). In tournament each of participants plays with each other exactly $1$ game. For each game participant have $1$ point if he wins game, $0,5$ point if game is drow and $0$ points if he lose game.

If after ending of tournament participant have at least $ 75 %

$ of maximum possible points he called $winner$ $of$ $tournament$.

Find maximum possible numbers of $winners$ $of$ $tournament$.
2009 Kazakhstan National Olympiad
Let $a,b,c,d $-reals positive numbers. Prove inequality:

$\frac{a^2+b^2+c^2}{ab+bc+cd}+\frac{b^2+c^2+d^2}{bc+cd+ad}+\frac{a^2+c^2+d^2}{ab+ad+cd}+\frac{a^2+b^2+d^2}{ab+ad+bc} \geq 4$
2009 Kazakhstan National Olympiad
Quadrilateral $ABCD$ inscribed in circle with center $O$. Let lines $AD$ and $BC$ intersects at $M$, lines $AB$ and $CD$- at $N$, lines $AC$ and $BD$ -at $P$, lines $OP$ and $MN$ at $K$.

Proved that $ \angle AKP = \angle PKC$.



As I know, this problem was very short solution by polars, but in olympiad for this solution gives maximum 4 balls (in marking schemes written, that needs to prove all theorems about properties of polars)
2009 Kazakhstan National Olympiad
Let $P(x)$ be polynomial with integer coefficients.

Prove, that if for any natural $k$ holds equality: $ \underbrace{P(P(...P(0)...))}_{n -times}=0$ then $P(0)=0$ or $P(P(0))=0$
2009 Kazakhstan National Olympiad
Prove that for any natural $n \geq 2$, the number $ \underbrace{2^{2^{\cdots^2}}}_{n \textrm{ times}}- \underbrace{2^{2^{\cdots^2}}}_{n-1 \textrm{ times}}$ is divisible by $n$.



I know, that it is a very old problem  :blush: but it is a problem from olympiad.
2009 Kazakhstan National Olympiad
In triangle $ABC$ $AA_1; BB_1; CC_1$-altitudes. Let $I_1$ and $I_2$ be in-centers of triangles $AC_1B_1$ and $CA_1B_1$ respectively. Let in-circle of $ABC$ touch $AC$ in $B_2$.

Prove, that quadrilateral $I_1I_2B_1B_2$ inscribed in a circle.
2009 Kazakhstan National Olympiad
Let $0<a_1 \leq a_2 \leq \cdots\leq a_n $ ($n \geq 3; n \in \mathbb{N}$) be $n$ real numbers. Prove the inequality

$$\frac{a_1^2}{a_2}+\frac{a_2^3}{a_3^2}+\cdots+\frac{a_n^{n+1}}{a_1^n} \geq a_1+a_2+\cdots+a_n$$
2009 Kazakhstan National Olympiad
Is there exist four points on plane, such that distance between any two of them is integer odd number?



May be it is geometry or number theory or combinatoric, I don't know, so it here :blush:
2008 Kazakhstan National Olympiad
Let $ F_n$ be a set of all possible connected figures, that consist of $ n$ unit cells. For each element $ f_n$ of this set, let $ S(f_n)$ be the area of that minimal rectangle that covers $ f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $ max(S(f_n))$,where $ f_n\in F_n$?



Remark: Two cells are called connected if they have a common edge.
2008 Kazakhstan National Olympiad
Suppose that $ B_1$ is the midpoint of the arc $ AC$, containing $ B$, in the circumcircle of $ \triangle ABC$, and let $ I_b$ be the $ B$-excircle's center. Assume that the external angle bisector of $ \angle ABC$ intersects $ AC$ at $ B_2$. Prove that $ B_2I$ is perpendicular to $ B_1I_B$, where $ I$ is the incenter of $ \triangle ABC$.
2008 Kazakhstan National Olympiad
Let $ f(x,y,z)$ be the polynomial with integer coefficients. Suppose that for all reals $ x,y,z$ the following equation holds:

$$ f(x,y,z) = - f(x,z,y) = - f(y,x,z) = - f(z,y,x)

$$

Prove that if $ a,b,c\in\mathbb{Z}$ then $ f(a,b,c)$ takes an even value
2008 Kazakhstan National Olympiad
Find all integer solutions $ (a_1,a_2,\dots,a_{2008})$ of the following equation:



$ (2008-a_1)^2+(a_1-a_2)^2+\dots+(a_{2007}-a_{2008})^2+a_{2008}^2=2008$
2008 Kazakhstan National Olympiad
Let $ \triangle ABC$ be a triangle and let $ K$ be some point on the side $ AB$, so that the tangent line from $ K$ to the incircle of $ \triangle ABC$ intersects the ray $ AC$ at $ L$. Assume that $ \omega$ is tangent to sides $ AB$ and $ AC$,  and to the circumcircle of $ \triangle AKL$. Prove that $ \omega$ is tangent to the circumcircle of $ \triangle ABC$ as well.
2008 Kazakhstan National Olympiad
Find the maximum number of planes in the space, such there are $ 6$ points, that satisfy to the following conditions:

1.Each plane contains at least $ 4$ of them

2.No four points are collinear.
2007 Kazakhstan National Olympiad
Zeros of a fourth-degree polynomial $f (x)$ form an arithmetic progression. Prove that the zeros of $f '(x)$ also form an arithmetic progression.
2007 Kazakhstan National Olympiad
Let $ABC$ be an isosceles triangle with $AC = BC$ and $I$ is the center of the inscribed circle. The point $P$ lies on the circle circumscribed about the triangle $AIB$ and lies inside the triangle $ABC$. Straight lines passing through point $P$ parallel to $CA$ and $CB$ intersect $AB$ at points $D$ and $E$, respectively. The line through $P$ which is parallel to $AB$ intersects $CA$ and $CB$ at points $F$ and $G$, respectively. Prove that the lines $DF$ and $EG$ meet at the circumcircle of triangle $ABC$.
2007 Kazakhstan National Olympiad
Solve in prime numbers the equation

$p(p+1)+q(q+1)=r(r+1)$.
2007 Kazakhstan National Olympiad
Several identical square sheets of paper are laid out on a rectangular table so that their sides are parallel to the edges of the table (sheets may overlap). Prove that you can stick a few pins in such a way that each sheet will be attached to the table exactly by one pin.
2007 Kazakhstan National Olympiad
Convex quadrilateral $ABCD$ with $AB$ not equal to $DC$ is inscribed in a circle. Let $AKDL$ and $CMBN$ be rhombs with same side of $a$. Prove that the points $K, L, M, N$ lie on a circle.
2007 Kazakhstan National Olympiad
Each cell of a $100$ x $100$ board  is painted in one of $100$ different colors so that there are exactly $100$ cells of each color. Prove that there is a row or column in which there are at least $10$ cells of different colors.
2007 Kazakhstan National Olympiad
Let $p$ be a prime such that $2^{p-1}\equiv 1 \pmod{p^2}$. Show that $(p-1)(p!+2^n)$ has at least three distinct prime divisors for each $n\in \mathbb{N}$ .
2007 Kazakhstan National Olympiad
Find all functions $ f :\mathbb{R}\to\mathbb{R} $, satisfying the condition



$f (xf (y) + f (x)) = 2f (x) + xy$



for any real $x$ and $y$.
2005 Kazakhstan National Olympiad
Does there exist a solution in real numbers of the system of equations

$$\left\{

\begin{array}{rcl}

(x - y)(z - t)(z - x)(z - t)^2  = A, \\
(y - z)(t - x)(t - y)(x - z)^2  = B,\\
(x - z)(y - t)(z - t)(y - z)^2  = C,\\
\end{array}

\right.$$

when

a) $A=2, B=8, C=6;$



b) $A=2, B=6, C=8.$?
2005 Kazakhstan National Olympiad
Prove that

$$ab+bc+ca\ge 2(a+b+c)$$

where $a,b,c$ are positive reals such that $a+b+c+2=abc$.
2005 Kazakhstan National Olympiad
Call a set of points in the plane $good$ if any three points of the set are the vertices of an isosceles triangle or if they are collinear. Determine all

$a)$ 6-element $good$ sets

$b)$ 7-element $good$ sets.
2005 Kazakhstan National Olympiad
Find all functions $f :\mathbb{R}\to\mathbb{R}$, satisfying the condition



$f(f(x)+x+y)=2x+f(y)$



for any real $x$ and $y$.
2005 Kazakhstan National Olympiad
Solve equation

$$2^{\tfrac{1}{2}-2|x|} = \left| {\tan x + \frac{1}{2}} \right| + \left| {\tan x - \frac{1}{2}} \right|$$
2005 Kazakhstan National Olympiad
The line parallel to side $AC$ of a right triangle $ABC$ $(\angle C=90^\circ)$ intersects sides $AB$ and $BC$ at $M$ and $N$, respectively, so that the $CN / BN = AC / BC = 2$. Let $O$ be the intersection point of the segments $AN$ and $CM$ and $K$ be a point on the segment $ON$ such that $MO + OK = KN$. The perpendicular line to  $AN$ at point $K$ and the bisector of triangle $ABC$ of $\angle B$ meet at point $T$. Find the angle $\angle MTB$.
2005 Kazakhstan National Olympiad
Exactly one number from the set $\{ -1,0,1 \}$ is written in each unit cell of a $2005 \times 2005$ table, so that the sum of all the entries is $0$. Prove that there exist two rows and two columns of the table, such that the sum of the four numbers written at the intersections of these rows and columns is equal to $0$.
2005 Kazakhstan National Olympiad
Find all polynomials $ P(x)$ with real coefficients such that for every positive integer $ n$ there exists a rational $ r$ with $ P(r)=n$.
2021 Korea - Final Round
An acute triangle $\triangle ABC$ and its incenter $I$, circumcenter $O$ is given. The line that is perpendicular to $AI$ and passes $I$ intersects with $AB$, $AC$ in $D$,$E$. The line that is parallel to $BI$ and passes $D$ and the line that is parallel to $CI$ and passes $E$ intersects in $F$. Denote the circumcircle of $DEF$ as $\omega$, and its center as $K$. $\omega$ and $FI$ intersect in $P$($\neq F$). Prove that $O,K,P$ is collinear.
2021 Korea - Final Round
Positive integer $k(\ge 8)$ is given. Prove that if there exists a pair of positive integers $(x,y)$ that satisfies the conditions below, then there exists infinitely many pairs $(x,y)$.

(1) $ $ $x\mid y^2-3, y\mid x^2-2$

(2) $ $ $gcd\left(3x+\frac{2(y^2-3)}{x},2y+\frac{3(x^2-2)}{y}\right)=k$ $ $
2021 Korea - Final Round
Let $P$ be a set of people. For two people $A$ and $B$, if $A$ knows $B$, $B$ also knows $A$. Each person in $P$ knows $2$ or less people in the set. $S$, a subset of $P$ with $k$ people, is called k-independent set of $P$ if any two people in $S$ don’t know each other. $X_1, X_2, …, X_{4041}$ are 2021-independent sets of $P$ (not necessarily distinct). Show that there exists a 2021-independent set of $P$, $\{v_1, v_2, …, v_{2021}\}$, which satisfies the following condition:





For some integer $1 \le i_1 < i_2 < \cdots < i_{2021} \leq 4041$, $v_1 \in X_{i_1}, v_2 \in X_{i_2}, \ldots, v_{2021} \in X_{i_{2021}}$





Graph WordingThanks to Evan Chen, here's a graph wording of the problem :)



Let $G$ be a finite simple graph with maximum degree at most $2$. Let $X_1, X_2, \ldots, X_{4041}$ be independent sets of size $2021$ (not necessarily distinct). Prove that there exists another independent set $\{v_1, v_2, \ldots, v_{2021}\}$ of size $2021$ and indices $1 \le t_1 < t_2 < \cdots < t_{2021} \le 4041$ such that $v_i \in X_{t_i}$ for all $i$.
2021 Korea - Final Round
There are $n$($\ge 2$) clubs $A_1,A_2,...A_n$ in Korean Mathematical Society. Prove that there exist $n-1$ sets $B_1,B_2,...B_{n-1}$ that satisfy the condition below.

(1) $A_1\cup A_2\cup \cdots A_n=B_1\cup B_2\cup \cdots B_{n-1}$

(2) for any $1\le i<j\le n-1$, $B_i\cap B_j=\emptyset, -1\le\left\vert B_i \right\vert -\left\vert B_j \right\vert\le 1$

(3) for any $1\le i \le n-1$, there exist $A_k,A_j $($1\le k\le j\le n$)such that $B_i\subseteq A_k\cup A_j$
2021 Korea - Final Round
The incenter and $A$-excenter of $\triangle{ABC}$ is $I$ and $O$. The foot from $A,I$ to $BC$ is $D$ and $E$. The intersection of $AD$ and $EO$ is $X$. The circumcenter of $\triangle{BXC}$ is $P$.

Show that the circumcircle of $\triangle{BPC}$ is tangent to the $A$-excircle if $X$ is on the incircle of $\triangle{ABC}$.
2021 Korea - Final Round
Find all functions $f,g: \mathbb{R} \to \mathbb{R}$ such that satisfies

$$f(x^2-g(y))=g(x)^2-y$$for all $x,y \in \mathbb{R}$