Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 IOM
Let $a_1, . . . , a_n$ be positive integers satisfying the inequality

$\sum_{i=1}^{n}\frac{1}{a_n}\le \frac{1}{2}$.

Every year, the government of Optimistica publishes its Annual Report with n economic indicators. For each $i = 1, . . . , n$,the possible values of the $i-th$ indicator are $1, 2, . . . , a_i$. The Annual Report is said to be optimistic if at least $n - 1$ indicators have higher values than in the previous report. Prove that the government can publish optimistic Annual Reports in an infinitely long sequence.
2016 IOM
Let $A_1A_2 . . . A_n$ be a cyclic convex polygon whose circumcenter is strictly in its interior. Let $B_1, B_2, ..., B_n$ be arbitrary points on the sides $A_1A_2, A_2A_3, ..., A_nA_1$, respectively, other than the vertices. Prove that

$\frac{B_1B_2}{A_1A_3}+ \frac{B_2B_3}{A_2A_4}+...+\frac{B_nB_1}{A_nA_2}>1$.
2016 IOM
A convex quadrilateral $ABCD$ has right angles at $A$ and $C$. A point $E$ lies on the extension of the side $AD$ beyond $D$ so that$\angle ABE =\angle ADC$. The point $K$ is symmetric to the point $C$ with respect to point $A$. Prove that$\angle ADB =\angle AKE$ .
2016 IOM
Let $r(x)$ be a polynomial of odd degree with real coefficients. Prove that there exist only finitely many (or none at all) pairs of polynomials $p(x) $ and $q(x)$ with real coefficients satisfying the equation $(p(x))^3 + q(x^2) = r(x)$.
2016 IOM
In a country with $n$ cities, some pairs of cities are connected by one-way flights operated by one of two companies $A$ and $B$. Two cities can be connected by more than one flight in either direction. An $AB$-word $w$ is called implementable if there is a sequence of connected flights whose companies’ names form the word $w$. Given that every $AB$-word of length $ 2^n $ is implementable, prove that every finite $AB$-word is implementable. (An $AB$-word of length $k$ is an arbitrary sequence of $k$ letters $A $ or $B$; e.g. $ AABA $ is a word of length $4$.)
2021 International Zhautykov Olympiad
Prove that there exists a positive integer $n$, such that the remainder of $3^n$ when divided by $2^n$ is greater than $10^{2021} $.
2021 International Zhautykov Olympiad
In a convex cyclic hexagon $ABCDEF$, $BC=EF$ and $CD=AF$. Diagonals $AC$ and $BF$ intersect at point $Q,$ and diagonals $EC$ and $DF$ intersect at point $P.$ Points $R$ and $S$ are marked on the segments $DF$ and $BF$ respectively so that $FR=PD$ and $BQ=FS.$ The segments $RQ$ and $PS$ intersect at point $T.$ Prove that the line $TC$ bisects the diagonal $DB$.
2021 International Zhautykov Olympiad
Let $n\ge 2$ be an integer. Elwyn is given an $n\times n$ table filled with real numbers (each cell of the table contains exactly one number). We define a rook set as a set of $n$ cells of the table situated in $n$ distinct rows as well as in n distinct columns. Assume that, for every rook set, the sum of $n$ numbers in the cells forming the set is nonnegative.



By a move, Elwyn chooses a row, a column, and a real number $a,$ and then he adds $a$ to each number in the chosen row, and subtracts $a$ from each number in the chosen column (thus, the number at the intersection of the chosen row and column does not change). Prove that Elwyn can perform a sequence of moves so that all numbers in the table become nonnegative.
2021 International Zhautykov Olympiad
Let there be an incircle of triangle $ABC$, and 3 circles each inscribed between incircle and angles of $ABC$.

Let $r, r_1, r_2, r_3$ be radii of these circles ($r_1, r_2, r_3 < r$). Prove that $$r_1+r_2+r_3 \geq r$$
2021 International Zhautykov Olympiad
On a party with $99$ guests, hosts Ann and Bob play a game (the hosts are not regarded as guests). There are $99$ chairs arranged in a circle; initially, all guests hang around those chairs. The hosts take turns alternately. By a turn, a host orders any standing guest to sit on an unoccupied chair $c$. If some chair adjacent to $c$ is already occupied, the same host orders one guest on such chair to stand up (if both chairs adjacent to $c$ are occupied, the host chooses exactly one of them). All orders are carried out immediately. Ann makes the first move; her goal is to fulfill, after some move of hers, that at least $k$ chairs are occupied. Determine the largest $k$ for which Ann can reach the goal, regardless of Bob's play.
2021 International Zhautykov Olympiad
Let $P(x)$ be a nonconstant polynomial of degree $n$ with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials $Q(x)$ of degree less than $n$ with rational coefficients such that $P(x)$ divides $P(Q(x))$

a) is finite

b) does not exceed $n$.
2020 International Zhautykov Olympiad
Given natural number n such that, for any natural $a,b$ number $2^a3^b+1$ is not divisible by $n$.Prove that $2^c+3^d$ is not divisible by $n$ for any natural $c$ and $d$
2020 International Zhautykov Olympiad
Each of $2k+1$ distinct 7-element subsets of the 20 element set intersects with exactly $k$ of them. Find the maximum possible value of $k$.
2020 International Zhautykov Olympiad
Given convex hexagon $ABCDEF$, inscribed in the circle. Prove that

$AC*BD*DE*CE*EA*FB \geq 27 AB * BC * CD * DE * EF * FA$
2020 International Zhautykov Olympiad
In a scalene triangle $ABC$ $I$ is the incentr and $CN$ is the bisector of angle $C$. The line $CN$ meets the circumcircle of $ABC$ again at $M$. The line $l$ is parallel to $AB$ and touches the incircle of $ABC$. The point $R$ on $l$ is such. That $CI \bot IR$. The circumcircle of $MNR$ meets the line $IR$ again at S. Prpve that $AS=BS$.
2020 International Zhautykov Olympiad
Let $Z$ be the set of all integers. Find all the function $f: Z->Z$ such that

$f(4x+3y)=f(3x+y)+f(x+2y)$

For all integers $x,y$
2020 International Zhautykov Olympiad
Some squares of a $n \times n$ tabel ($n>2$) are black, the rest are withe. In every white square we write the number of all the black squares having at least one common vertex with it. Find the maximum possible sum of all these numbers.
2019 International Zhautykov OIympiad
Prove that there exist at least $100!$ ways to write $100!$ as sum of elements of set {$1!,2!,3!...99!$}

(each number in sum can be two or more times)
2019 International Zhautykov OIympiad
Find the biggest real number $C$, such that for every different positive real numbers $a_1,a_2...a_{2019}$ that satisfy inequality :

$\frac{a_1}{|a_2-a_3|} + \frac{a_2}{|a_3-a_4|} + ... + \frac{a_{2019}}{|a_1-a_2|} > C$
2019 International Zhautykov OIympiad
Triangle $ABC$ is given. The median $CM$ intersects the circumference  of $ABC$ in $N$. $P$ and $Q$ are chosen on the rays $CA$ and $CB$ respectively, such that $PM$ is parallel to $BN$ and $QM$ is parallel to $AN$. Points $X$ and $Y$ are chosen on the segments $PM$ and $QM$ respectively, such that both $PY$ and $QX$ touch the circumference of $ABC$. Let $Z$ be intersection of $PY$ and $QX$. Prove that, the quadrilateral $MXZY$ is circumscribed.
2019 International Zhautykov OIympiad
Triangle $ABC$ with $AC=BC$ given and point $D$ is chosen on the side $AC$. $S1$ is a circle that touches $AD$ and extensions of $AB$ and $BD$ with radius $R$ and center $O_1$. $S2$ is a circle that touches $CD$ and extensions of $BC$ and $BD$ with radius $2R$ and center $O_2$. Let $F$ be intersection of the extension of $AB$ and tangent at $O_2$ to circumference of $BO_1O_2$. Prove that $FO_1=O_1O_2$.
2019 International Zhautykov OIympiad
Natural number $n>1$ is given. Let $I$ be a set of integers that are relatively prime to $n$. Define the function $f:I=>N$. We call a function $k-periodic$ if for any $a,b$ , $f(a)=f(b)$ whenever $ k|a-b $. We know that $f$ is $n-periodic$. Prove that minimal period of $f$ divides all other periods.

Example: if $n=6$ and $f(1)=f(5)$ then minimal period is 1, if $f(1)$ is not equal to $f(5)$ then minimal period is 3.
2019 International Zhautykov OIympiad
We define two types of operation on polynomial of third degree:

a) switch places of the coefficients of polynomial(including zero coefficients), ex:

$ x^3+x^2+3x-2 $ => $ -2x^3+3x^2+x+1$

b) replace the polynomial $P(x)$ with $P(x+1)$

If limitless amount of operations is allowed,

is it possible from $x^3-2$ to get $x^3-3x^2+3x-3$ ?
2018 International Zhautykov Olympiad
Let $\alpha,\beta,\gamma$ measures of angles of opposite to the sides of triangle with measures $a,b,c$ respectively.Prove that

$$2(cos^2\alpha+cos^2\beta+cos^2\gamma)\geq \frac{a^2}{b^2+c^2}+\frac{b^2}{a^2+c^2}+\frac{c^2}{a^2+b^2}$$
2018 International Zhautykov Olympiad
Let $N,K,L$ be points on $AB,BC,CA$ such that $CN$ bisector of angle $\angle ACB$ and $AL=BK$.Let $BL\cap AK=P$.If $I,J$ be incenters of triangles $\triangle BPK$ and $\triangle ALP$ and $IJ\cap CN=Q$ prove that $IQ=JP$
2018 International Zhautykov Olympiad
Prove that there exist infinitely pairs $(m,n)$ such that $m+n$ divides $(m!)^n+(n!)^m+1$
2018 International Zhautykov Olympiad
Crocodile chooses $1$ x $4$ tile from $2018$ x $2018$ square.The bear has tilometer that checks $3$x$3$ square of $2018$ x $2018$ is there any of choosen cells by crocodile.Tilometer says "YES" if there is at least one choosen cell among checked $3$ x $3$ square.For what is the smallest number of such questions the Bear can certainly get an affirmative answer?
2018 International Zhautykov Olympiad
Find all real numbers $a$ such that there exist $f:\mathbb{R} \to \mathbb{R}$ with $$f(x-f(y))=f(x)+a[y]$$for all $x,y\in \mathbb{R}$
2018 International Zhautykov Olympiad
In a circle with a radius $R$ a convex hexagon is inscribed. The diagonals $AD$ and $BE$,$BE$ and $CF$,$CF$ and $AD$ of the hexagon intersect at the points $M$,$N$ and$K$, respectively. Let $r_1,r_2,r_3,r_4,r_5,r_6$ be the radii of circles inscribed in triangles $ ABM,BCN,CDK,DEM,EFN,AFK$ respectively. Prove that.$$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$.
2017 International Zhautykov Olympiad
Let $ABC$ be a non-isosceles triangle with circumcircle $\omega$ and let $H,  M$ be orthocenter and midpoint of $AB$ respectively. Let $P,Q$ be points on the arc $AB$ of $\omega$ not containing $C$ such that $\angle ACP=\angle BCQ < \angle ACQ$.Let $R,S$ be the foot of altitudes from $H$ to $CQ,CP$ respectively. Prove that thé points $P,Q,R,S$ are concyclic and $M$ is the center of this circle.
2017 International Zhautykov Olympiad
Find all functions  $f:R \rightarrow R$ such that $$(x+y^2)f(yf(x))=xyf(y^2+f(x))$$, where $x,y \in \mathbb{R}$
2017 International Zhautykov Olympiad
Rectangle  on a checked paper with length of a unit square side being $1$ Is divided into domino figures( two unit square sharing a common edge). Prove that you colour all corners of squares on the edge of rectangle and inside rectangle with $3$ colours such that for any two corners with distance $1$ the following conditions hold: they are coloured in different colour if the line connecting the two corners is on the border of two domino figures and coloured in same colour if the line connecting the two corners is inside a domino figure.
2017 International Zhautykov Olympiad
Let $(a_n)$ be sequnce of positive integers such that first $k$ members $a_1,a_2,...,a_k$ are distinct positive integers, and for each $n>k$, number $a_n$ is the smallest positive integer that can't be represented as a sum of several (possibly one) of the numbers $a_1,a_2,...,a_{n-1}$. Prove that $a_n=2a_{n-1}$ for all sufficently large $n$.
2017 International Zhautykov Olympiad
For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.
2017 International Zhautykov Olympiad
Let $ABCD$ be the regular tetrahedron, and $M, N$ points in space. Prove that: $AM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN$
2016 International Zhautykov Olympiad
A quadrilateral $ABCD$  is inscribed in a circle with center $O$. It's diagonals meet at $M$.The circumcircle of $ABM$ intersects the sides  $AD$ and $BC$ at $N$ and $K$ respectively. Prove that areas of $NOMD$ and $KOMC$ are equal.
2016 International Zhautykov Olympiad
$a_1,a_2,...,a_{100}$ are permutation of $1,2,...,100$. $S_1=a_1, S_2=a_1+a_2,...,S_{100}=a_1+a_2+...+a_{100}$Find the maximum number of perfect squares from $S_i$
2016 International Zhautykov Olympiad
There are $60$ towns in $Graphland$ every two countries of which are connected by only a directed way. Prove that we can color four towns to red and four towns to green such that every way between green and red towns are directed from red to green
2016 International Zhautykov Olympiad
Find all $k>0$ for which a strictly decreasing function $g:(0;+\infty)\to(0;+\infty)$ exists such that $g(x)\geq kg(x+g(x))$ for all positive $x$.
2016 International Zhautykov Olympiad
A convex hexagon $ABCDEF$ is given such that $AB||DE$, $BC||EF$, $CD||FA$. The point $M, N, K$ are common points of the lines $BD$ and $AE$, $AC$ and $DF$, $CE$ and $BF$ respectively. Prove that perpendiculars drawn from $M, N, K$ to lines $AB, CD, EF$ respectively concurrent.
2016 International Zhautykov Olympiad
We call a positive integer $q$ a $convenient \quad denominator$ for a real number $\alpha$ if

$\displaystyle |\alpha - \dfrac{p}{q}|<\dfrac{1}{10q}$ for some integer $p$. Prove that if two irrational numbers $\alpha$ and

$\beta$ have the same set of convenient denominators then either $\alpha+\beta$ or $\alpha- \beta$ is an integer.
2015 International Zhautykov Olympiad
Each point with integral coordinates in the plane is coloured white or blue. Prove that one can choose a colour so that for every positive integer $ n $ there exists a triangle of area $ n $ having its vertices of the chosen colour.
2015 International Zhautykov Olympiad
Inside the triangle $ ABC $ a point $ M $ is given. The line $ BM $ meets the side $ AC $ at $ N $. The point $ K $ is symmetrical to $ M $ with respect to $ AC $. The line $ BK $  meets $ AC $ at $ P $. If $ \angle AMP = \angle CMN $, prove that $ \angle ABP=\angle CBN $.
2015 International Zhautykov Olympiad
Find all functions $ f\colon \mathbb{R} \to \mathbb{R} $ such that $ f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy) $, for all $ x,y \in \mathbb{R} $.
2015 International Zhautykov Olympiad
Determine the maximum integer $ n $ such that for each positive integer $ k \le \frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $.
2015 International Zhautykov Olympiad
Let $ A_n $ be the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that every two neighbouring terms of each subsequence have different parity,and $ B_n $ the set of partitions of the sequence $ 1,2,..., n $ into several subsequences such that all the terms of each subsequence have the same parity ( for example,the partition $ {(1,4,5,8),(2,3),(6,9),(7)} $ is an element of $ A_9  $,and the partition $ {(1,3,5),(2,4),(6)} $ is an element of $ B_6 $ ).

Prove that for every positive integer $ n $ the sets $ A_n $ and $ B_{n+1} $ contain the same number of elements.
2015 International Zhautykov Olympiad
The area of a convex pentagon $ABCDE$ is $S$, and the circumradii of the triangles $ABC$, $BCD$, $CDE$, $DEA$, $EAB$ are $R_1$, $R_2$, $R_3$, $R_4$, $R_5$. Prove the inequality

$$ R_1^4+R_2^4+R_3^4+R_4^4+R_5^4\geq {4\over 5\sin^2 108^\circ}S^2. $$
2014 International Zhautykov Olympiad
Points $M$, $N$, $K$ lie on the sides $BC$, $CA$, $AB$ of a triangle $ABC$, respectively, and are different from its vertices. The triangle $MNK$ is called beautiful if $\angle BAC=\angle KMN$ and $\angle ABC=\angle KNM$. If in the triangle $ABC$ there are two beautiful triangles with a common vertex, prove that the triangle $ABC$ is right-angled.



Proposed by Nairi M. Sedrakyan, Armenia
2014 International Zhautykov Olympiad
Does there exist a function $f: \mathbb R \to \mathbb R $  satisfying the following conditions:

(i) for each real $y$  there is a real  $x$ such that $f(x)=y$ , and

(ii)  $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?



Proposed by Igor I. Voronovich, Belarus
2014 International Zhautykov Olympiad
Given are 100 different positive integers. We call a pair of numbers good if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.)



Proposed by Alexander S. Golovanov, Russia
2014 International Zhautykov Olympiad
Does there exist a polynomial $P(x)$ with integral coefficients such that   $P(1+\sqrt 3) = 2+\sqrt 3$ and $P(3+\sqrt 5) = 3+\sqrt 5 $?



Proposed by Alexander S. Golovanov, Russia
2014 International Zhautykov Olympiad
Let $U=\{1, 2,\ldots, 2014\}$. For positive integers $a$, $b$, $c$ we denote by $f(a, b, c)$ the number of ordered 6-tuples of sets $(X_1,X_2,X_3,Y_1,Y_2,Y_3)$ satisfying the following conditions:

(i) $Y_1 \subseteq X_1 \subseteq U$ and $|X_1|=a$;

(ii) $Y_2 \subseteq X_2 \subseteq U\setminus Y_1$ and $|X_2|=b$;

(iii) $Y_3 \subseteq X_3 \subseteq U\setminus (Y_1\cup Y_2)$ and $|X_3|=c$.

Prove that $f(a,b,c)$ does not change when $a$, $b$, $c$ are rearranged.



Proposed by Damir A. Yeliussizov, Kazakhstan
2014 International Zhautykov Olympiad
Four segments divide a convex quadrilateral into nine quadrilaterals. The points of intersections of these segments lie on the diagonals of the quadrilateral (see figure). It is known that the quadrilaterals 1, 2, 3, 4 admit inscribed circles. Prove that the quadrilateral 5 also has an inscribed circle.

[asy]

pair A,B,C,D,E,F,G,H,I,J,K,L;

A=(-4.0,4.0);B=(-1.06,4.34);C=(-0.02,4.46);D=(4.14,4.93);E=(3.81,0.85);F=(3.7,-0.42);

G=(3.49,-3.05);H=(1.37,-2.88);I=(-1.46,-2.65);J=(-2.91,-2.52);K=(-3.14,-1.03);L=(-3.61,1.64);

draw(A--D);draw(D--G);draw(G--J);draw(J--A);

draw(A--G);draw(D--J);

draw(B--I);draw(C--H);draw(E--L);draw(F--K);



pair R,S,T,U,V;

R=(-2.52,2.56);S=(1.91,2.58);T=(-0.63,-0.11);U=(-2.37,-1.94);V=(2.38,-2.06);

label("1",R,N);label("2",S,N);label("3",T,N);label("4",U,N);label("5",V,N);

[/asy]



Proposed by Nairi M. Sedrakyan, Armenia
2013 International Zhautykov Olympiad
Given a trapezoid   $ABCD$ ($AD \parallel BC$) with $\angle ABC > 90^\circ$  . Point $M$ is chosen on the lateral side $AB$. Let  $O_1$ and  $O_2$ be the circumcenters of the triangles  $MAD$ and $MBC$, respectively. The circumcircles of the  triangles  $MO_1D$ and $MO_2C$ meet again at the point $N$. Prove that the line $O_1O_2$ passes through the point $N$.
2013 International Zhautykov Olympiad
Find all odd positive integers $n>1$ such that there is a permutation $a_1, a_2, a_3, \ldots, a_n$ of the numbers $1, 2,3, \ldots, n$ where $n$ divides one of the numbers $a_k^2 - a_{k+1} - 1$ and $a_k^2 - a_{k+1} + 1$ for each $k$, $1 \leq k \leq n$ (we assume $a_{n+1}=a_1$).
2013 International Zhautykov Olympiad
Let $a, b, c$, and $d$ be positive real numbers such that $abcd = 1$. Prove that

$$\frac{(a-1)(c+1)}{1+bc+c} + \frac{(b-1)(d+1)}{1+cd+d} + \frac{(c-1)(a+1)}{1+da+a} + \frac{(d-1)(b+1)}{1+ab+b} \geq 0.$$

Proposed by Orif Ibrogimov, Uzbekistan.
2013 International Zhautykov Olympiad
A quadratic trinomial  $p(x)$ with real coefficients is given. Prove that there is a positive integer $n$ such that the equation $p(x) = \frac{1}{n}$ has no rational roots.
2013 International Zhautykov Olympiad
Given convex hexagon $ABCDEF$ with $AB \parallel DE$, $BC \parallel EF$, and $CD \parallel FA$ . The distance between the lines $AB$ and $DE$ is equal to the distance between the lines $BC$ and $EF$ and to the distance between the lines $CD$ and $FA$. Prove that the sum $AD+BE+CF$ does not exceed the perimeter of hexagon $ABCDEF$.
2013 International Zhautykov Olympiad
A $10 \times 10$ table consists of $100$ unit cells. A block is a $2 \times 2$ square consisting of $4$ unit cells of the table. A set $C$ of $n$ blocks covers the table (i.e. each cell of the table is covered by some block of $C$ ) but no $n -1$ blocks of $C$ cover the table. Find the largest possible value of $n$.
2012 International Zhautykov Olympiad
An acute triangle $ABC$ is given. Let $D$ be an arbitrary inner point of the side $AB$. Let $M$ and $N$ be the feet of the perpendiculars from $D$ to $BC$ and $AC$, respectively. Let $H_1$ and $H_2$ be the orthocentres of triangles $MNC$ and $MND$, respectively. Prove that the area of the quadrilateral $AH_1BH_2$ does not depend on the position of $D$ on $AB$.
2012 International Zhautykov Olympiad
A set of (unit) squares of a $n\times n$ table is called convenient if each row and each column of the table contains at least two squares belonging to the set. For each $n\geq 5$ determine the maximum $m$ for which there exists a convenient  set made of $m$ squares, which becomes inconvenient  when any of its squares is removed.
2012 International Zhautykov Olympiad
Let $P, Q,R$ be three polynomials with real coefficients such that $$P(Q(x)) + P(R(x))=\text{constant}$$ for all $x$. Prove that $P(x)=\text{constant}$ or $Q(x)+R(x)=\text{constant}$ for all $x$.
2012 International Zhautykov Olympiad
Do there exist integers $m, n$ and a function $f\colon \mathbb R \to \mathbb R$ satisfying simultaneously the following two conditions?



$\bullet$ i) $f(f(x))=2f(x)-x-2$ for any $x \in \mathbb R$;

$\bullet$ ii) $m \leq n$ and $f(m)=n$.
2012 International Zhautykov Olympiad
Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\angle BAD + \angle CDA$ if $B'C' = AB+CD$.
2012 International Zhautykov Olympiad
Find all integer solutions of the equation the equation $2x^2-y^{14}=1$.
2011 International Zhautykov Olympiad
Given is trapezoid $ABCD$, $M$ and $N$ being the midpoints of the bases of $AD$ and $BC$, respectively.

a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment $MN$.

b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line $MN$?
2011 International Zhautykov Olympiad
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfy the equality,

$$f(x+f(y))=f(x-f(y))+4xf(y)$$for any $x,y\in\mathbb{R}$.
2011 International Zhautykov Olympiad
Let $\mathbb{N}$ denote the set of all positive integers. An ordered pair $(a;b)$ of numbers $a,b\in\mathbb{N}$ is called interesting, if for any $n\in\mathbb{N}$ there exists $k\in\mathbb{N}$ such that the number $a^k+b$ is divisible by $2^n$. Find all interesting ordered pairs of numbers.
2011 International Zhautykov Olympiad
Find the maximum number of sets which simultaneously satisfy the following conditions:



i) any of the sets consists of $4$ elements,



ii) any two different sets have exactly $2$ common elements,



iii) no two elements are common to all the sets.
2011 International Zhautykov Olympiad
Let $n$ be integer, $n>1.$ An element of the set $M=\{ 1,2,3,\ldots,n^2-1\}$ is called good if there exists some element $b$ of $M$ such that $ab-b$ is divisible by $n^2.$ Furthermore, an element $a$ is called very good if $a^2-a$ is divisible by $n^2.$ Let $g$ denote the number of good elements in $M$ and $v$ denote the number of very good elements in $M.$ Prove that

$$v^2+v \leq g \leq n^2-n.$$
2011 International Zhautykov Olympiad
Diagonals of a cyclic quadrilateral $ABCD$  intersect at point $K.$ The midpoints of diagonals $AC$ and $BD$ are $M$ and $N,$ respectively. The circumscribed circles $ADM$ and $BCM$ intersect at points $M$ and $L.$ Prove that the points $K ,L ,M,$ and $ N$ lie on a circle. (all points are supposed to be different.)
2010 International Zhautykov Olympiad
Find all primes $p,q$ such that $p^3-q^7=p-q$.
2010 International Zhautykov Olympiad
In a cyclic quadrilateral  $ABCD$ with $AB=AD$  points  $M$,$N$  lie on the sides $BC$  and $CD$  respectively so that $MN=BM+DN$ . Lines $AM$  and $AN$  meet the circumcircle of  $ABCD$ again at points $P$  and $Q$  respectively. Prove that the orthocenter of the triangle $APQ$  lies on the segment $MN$ .