Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 Japan MO Finals
Prove that the polynomial $f(x)=(x^2+1)(x^2+2^2)\cdots (x^2+n^2)+1$ cannot be expressed as a product of two polynomials with integer coefficients with degree greater than $1$.
1999 Japan MO Finals
All sides of a convex hexagon $ABCDEF$ are $1$. Let $M,m$ be the maximum and minimum possible values of three diagonals $AD,BE,CF$. Find the range of $M$, $m$.
1998 Japan MO Finals
Let $ p \geq 3$ be a prime, and let $ p$ points $ A_{0}, \ldots, A_{p-1}$ lie on a circle in that order. Above the point $ A_{1+\cdots+k-1}$ we write the number $ k$ for $ k=1, \ldots, p$ (so $ 1$ is written above $ A_{0}$). How many points have at least one number written above them?
1998 Japan MO Finals
A country has $ 1998$ airports connected by some direct flights. For any three airports, some two are not connected by a direct flight. What is the maximum number of direct flights that can be offered?
1998 Japan MO Finals
Let $ P_{1}, \ldots P_{n}$ be the sequence of vertices of a closed polygons whose sides may properly intersect each other at points other than the vertices. The external angle at $ P_{i}$ is defined as $ 180^\circ$ minus the angle of rotation about $ P_{i}$ required to bring the ray $ P_{i}P_{i-1}$ onto the ray $ P_{i}P_{i+1}$, taken in the range ($ 0^\circ, 360^\circ$). (Here $ P_{0}=P_{n}$ and $ P_{1}=P_{n+1}$). Prove that if the sum of the external angles is a multiple of $ 720^\circ$, then the number of self-intersections is odd.
1998 Japan MO Finals
Let $ c_{n, m}$ be the number of permutations of $ \{1, \ldots, n\}$ which can be written as the product of $ m$ transpositions of the form $ (i, i+1)$ for some $ i=1, \ldots, n-1$ but not of $ m-1$ suct transpositions. Prove that for all $ n \in \mathbb{N}$,

$$ \sum_{m=0}^\infty c_{n, m}t^{m}= \prod_{i=1}^{n}(1+t+\cdots+t^{i-1}). $$
1998 Japan MO Finals
On each of $ 12$ points around a circle we place a disk with one white side and one black side. We may perform the following move: select a black disk, and reverse its two neighbors. Find all initial configurations from which some sequence of such moves leads to the position where all disks but one are white.
1997 Japan MO Finals
Take 10 points inside the circle with diameter 5. Prove that for any these 10 points there exist two points whose distance is less than 2.
1997 Japan MO Finals
Prove that



$ \frac{\left(b+c-a\right)^{2}}{\left(b+c\right)^{2}+a^{2}}+\frac{\left(c+a-b\right)^{2}}{\left(c+a\right)^{2}+b^{2}}+\frac{\left(a+b-c\right)^{2}}{\left(a+b\right)^{2}+c^{2}}\geq\frac35$



for any positive real numbers $ a$, $ b$, $ c$.
1997 Japan MO Finals
Call the Graph the set which composed of several vertices $P_1,\ \cdots P_2$ and several edges $($segments$)$ connecting two points among these vertices. Now let $G$ be a graph with 9 vertices and satisfies the following condition.



Condition: Even if we select any five points from the vertices in $G,$ there exist at least two edges whose endpoints are included in the set of 5 points.



What is the minimum possible numbers of edges satisfying the condition?
1997 Japan MO Finals
Let $A,B,C,D$ be points in space which are not on a same plane and no any 3 points are not colinear.

Suppose that the sum of the segments $AX+BX+CX+DX$ is minimized at $X=X_0$ which is different from $A,B,C,D.$ Prove that $\angle{AX_0B}=\angle{CX_0D}.$
1997 Japan MO Finals
The letters $A$ or $B$ are assigned on the points divided equally into $2^{n}\ (n=1,\ 2,\cdots)$ parts of a circumference.If you choose $n$ letters from any succesively arranging points directed clockwise, prove that there exists the way of assignning for which the line of letters are mutually distinct.
1996 Japan MO Finals
A plane is partitioned into triangles. Let $\mathcal{T}_0$ denote the set of vertices of triangles in the partition. Let $ABC$ be a triangle with $A,B,C\in\mathcal{T}_0$ and $\theta$ be its smallest angle. Assume that no point of $\mathcal{T}_0$ lies inside the circumcircle of $\triangle ABC$. Prove that there exists a triangle $\sigma$ in the partition such that its intersection with $\triangle ABC$ is nonempty and whose every angle is greater than $\theta$.
1996 Japan MO Finals
Let $m,n$ be positive integers with $(m,n)=1$. Find $(5^m+7^m,5^n+7^n)$.
1996 Japan MO Finals
Let $x>1$ be a real number which is not an integer. For each $n\in\mathbb{N}$, let $a_n=\lfloor x^{n+1}\rfloor - x\lfloor x^n\rfloor$. Prove that the sequence $(a_n)$ is not periodic.
1996 Japan MO Finals
Let $\theta $ be the maximum of the six angles between six edges of a regular tetrahedron in space and a fixed plane. When the tetrahedron is rotated in space, find the maximum of $\theta$.
1996 Japan MO Finals
Let $q$ be a real number with $\frac{1+\sqrt{5}}{2}<q<2$. If a positive integer $n$ is represented in binary system as $2^k+a_{k-1}2^{k-1}+\cdots +2a_1+a_0$, where $a_i\in\{0,1\}$, define

$$p_n=q^k+a_{k-1}q^{k-1}+\cdots +qa_1+a_0. $$

Prove that there exist infinitely many positive integers $k$ with the property that there is no $l\in\mathbb{N}$ such that $p_{2k}<p_l< p_{2k+1}$ .
1995 Japan MO Finals
Let $n\geq 2$ be integers  and $r$ be positive integers such that $r$ is not the multiple of $n$, and let $g$ be the greatest common measure of $n$ and $r$. Prove that $$ \sum_{i=1}^{n-1} \left\{\frac{ri}{n} \right\}\ =\frac{1}{2}(n-g).  $$ where $\left\{x\right\}$ is the fractional part, that is to say, which means the value by subtracting $x$ from the maximum integer value which is equal or less than $x$.
1995 Japan MO Finals
Find all the non-constant rational function of $f(x)$ and real numbers $a$ satisfying $\{f(x)\}^2-a=f(x^2).$

Here a rational function of $x$ is the equation expressed by the ratio of two polynominals of $x.$
1995 Japan MO Finals
Given a convex pentagon $ABCDE.$ Let $S,R$ be the intersection points of $AC$ or $AD$ and $BE$ respectively, and let the intersection points $T,P$ of $CA$ or $CE$ and $BD$ respectively.Let $Q$ be the intersection point of $CE$ and $AD.$ If all of $\triangle{ASR},\ \triangle{BTS},\ \triangle{CPT},\ \triangle{DQP},$ and $\triangle{ERQ}$ have the area of $1,$ then find the area of the following pentagons.



(1) The pentagon $PQRST.$

(2) The pentagon $ABCDE.$
1995 Japan MO Finals
The sequence $\{a_1,\ a_2,\ \cdots\}$ is defined by $a_{2n}=a_n,\ a_{2n+1}=(-1)^n.$ A point $P$ moves on the coordinate plane as follows.



(1) Let $P_0$ be the origin, $P$ moves in a distance of $1$ from $P_0$ toward in the positive direction of  $x$-axis, Denote this point by $P_i.$



(2) After $P$ has moved to $P_i,$ it turns $90^\circ$ to the left and moves in a distance of $1$ when $a_i=1,$ and turns $90^\circ$ to the right and moves in a distance of $1$ when $a_i=-1.$ Denote this point by $P_{i+1},$where $i=1,2,\cdots.$ Prove that $P$ can't pass on the same segment more than two times.
1995 Japan MO Finals
Let $k,n$ be integers such that $1\leq k\leq n,$and let $a_1,\ a_2,\ \cdots,\ a_k$ be numbers satisfying the following equations.



$$ \begin{cases} a_1+\cdots\cdots +a_k=n \\ a_1^2+\cdots\cdots +a_k^2=n\\ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \\ a_1^k+\cdots\cdots +a_k^k=n \end{cases} $$

Prove that $$ (x+a_1)(x+a_2)\cdots(x+a_k)=x^k+\ _n C_1\ x^{k-1}+\ _nC_2\ x^{k-2}+\cdots+\ _nC_k.  $$

where $_iC_j$ is a binomial coefficient which means $\frac{i\cdot (i-1)\cdots(i-j+1)}{j\cdot (j-1)\cdots 2\cdot 1}.$
1994 Japan MO Finals
For any positive integer $n$, let $a_n$ denote the closest integer to $\sqrt{n}$, and let $b_n=n+a_n$. Determine the increasing sequence $(c_n)$ of positive integers which do not occur in the sequence $(b_n)$.
1994 Japan MO Finals
Five points, no three collinear, are given on the plane. Let $l_1,l_2,\ldots ,l_{10}$ be the lengths of the ten segments joining any two of the given points. Prove that if $l_1^2,\ldots ,l_9^2$ are rational numbers, then $l_{10}^2$ is also a rational number.
1994 Japan MO Finals
Let $P_0$ be a point in the plane of triangle $A_0A_1A_2$. Define $P_i\ (i=1,\ldots ,6)$ inductively as the point symmetric to $P_{i-1}$ with respect to $A_k$, where $k$ is the remainder when $i$ is divided by $3$.



a) Prove that $P_6\equiv P_1$.

b) Find the locus of points $P_0$ for which $P_iP_{i+1}$ does not meet the interior of $\triangle A_0A_1A_2$ for $0\le i\le 5$.
1994 Japan MO Finals
In a triangle $ABC$, $M$ is the midpoint of $BC$. Given that $\angle MAC=15^{\circ}$, find the maximum possible value of $\angle ABC$.
1994 Japan MO Finals
In a deck of $N$ cards, the cards are denoted by $1$ to $N$. These cards are dealt to $N$ people twice. A person $X$ wins a prize if there is no person $Y$ who got a card with a smaller number than $X$ both times. Determine the expected number of prize winners.
1993 Japan MO Finals
Call a word  forming by alphabetical small letters $a,b,c,\ \cdots x,y,z$ and a periodic word arranging by a certain word more than two times repeatedly.For example kyonkyon is eight-letter periodic word. Given two words $W_1,\ W_2$ which have the same number of letters and have a different first letter,

if you will remove the letter, $W_1$ and $W_2$ will be same word.Prove that either $W_1$ or $W_2$ is not periodic.
1993 Japan MO Finals
Denote by $d(n)$ the largest odd divisor of positive integers $n$ and define $D(n),\ T(n)$ as follows.

$$ D(n)=d(1)+d(2)+\cdots +d(n),\ \ T(n)=1+2+\cdots +n.  $$

Prove that there existed infinitely positive integers $n$ such that $3D(n)=2T(n).$
1993 Japan MO Finals
$ x$ students took an exam with $ y$ problems. Each student solved a half of the problem and the number of person those who solved the problem is equal for each problem.For any two students exactly three problems could be solved by the two persons. Find all pairs of $ (x,\ y)$  satisfying this condition, then show an example to prove that it is possiible the case in terms of $ a,\ b$ as follows:

Note that every student solve $ (a)$ or don't solve $ (b)$.
1993 Japan MO Finals
Given five radii $l_1,\ \cdots l_5$ of a sphere $S,$ no three of these radii are on a same plane.

Choose a pair to endpoint from each radius $l_1,\ \cdots l_5.$ Find the number of choices such that five points are in a hemisphere among 32 choices of an endpoint.
1993 Japan MO Finals
Prove that  there existed a positive number $C,$ irrelevant to $n$ and $a_1,\ a_2,\ \cdots a_n,$ satisfying the following condition.



Condition: For arbiterary positive numbers $n$ and arbiterary real numbers $a_1,\ \cdots a_n,$ the following inequality holds.



$$ \max_{0\leq x\leq 2}\prod_{j=1}^n |x-a_j|\leq C^n \max_{0\leq x\leq 1}\prod_{j=1}^n |x-a_j|. $$
1992 Japan MO Finals
Let $ x,\ y$ be relatively prime numbers such that $ xy\neq 1$. For positive even integer $ n$, prove that $ x + y$ isn't a divisor of $ x^n + y^n$.
1992 Japan MO Finals
Suppose that $ D,\ E$ are points on $ AB,\ AC$ of $ \triangle{ABC}$ with area $ 1$ respectively and $ P$ is $ BE\cap CD$. When $ D,\ E$ move on $ AB,\ AC$ with satisfying the condition $ [BCED] = 2\triangle{PBC}$, find the maximum area of $ \triangle{PDE}$.
1992 Japan MO Finals
Prove the inequality $ \sum_{k=1}^{n-1} \frac{n}{n-k}\cdot \frac{1}{2^{k-1}}<4\ (n\geq 2)$.
1992 Japan MO Finals
Let $ A$ be a $ m\times n\ (m\neq n)$ matrix with the entries $ 0$ and $ 1$. Suppose that if $ f$ is injective such that $ f: \{1,\ 2,\ \cdots ,\ m\}\longrightarrow \{1,\ 2,\ \cdots ,\ n\}$, then there exists $ 1\leq i\leq m$ such that $ (i,\ f(i))$ element is $ 0$.



Prove that there exist $ S\subseteq \{1,\ 2,\ \cdots ,\ m\}$ and $ T\subseteq \{1,\ 2,\ \cdots ,\ n\}$ satisfying the condition:



$ i)$ if $ i\in{S},\ j\in{T}$, then $ (i,\ j)$ entry is $ 0$.



$ ii)$ $ card\ S + card\ T > n$.
1992 Japan MO Finals
Suppose that $ n\geq 2$ be integer and $ a_1,\ a_2,\ a_3,\ a_4$ satisfy the following condition:



i) $ \ n$ and $ a_i\ (i = 1,\ 2,\ 3,\ 4)$ are relatively prime.



ii) $ \ (ka_1)_n + (ka_2)_n + (ka_3)_n + (ka_4)_n = 2n$ for $ k = 1,\ 2,\ \cdots ,\ n - 1$.



Note that $ (a)_n$ expresses the divisor when $ a$ is divided by $ n$.



Prove that $ (a_1)_n,\ (a_2)_n,\ (a_3)_n,\ (a_4)_n$ can be divided into two pair with sum $ n$.
1991 Japan MO Finals
Let $ P,\ Q,\ R$ be the points such that $ BP: PC=CQ: QA=AR: RB=t: 1-t\ (0<t<1)$ for a triangle $ ABC$.

Denote $ K$ by the area of the triangle with segments $ AP,\ BQ,\ CR$ as side lengths and $ L$ by triangle $ ABC$, find $ \frac{K}{L}$ in terms of $ t$.
1991 Japan MO Finals
Let $ N$ be the set of the whole of positive integers. The mapping from $ N$ to $ N$ is defined as follows: $ p(1)=2,\ p(2)=3,\ p(3)=4,\ p(4)=1,\ \ \ p(n)=n\ (n\geq 5)$,

$ q(1)=3,\ q(2)=4,\ q(3)=2,\ q(4)=1,\ \ \ p(n)=n\ (n\geq 5)$. Answer the following questions.



(1) If you make a mapping $ f: N\rightarrow N$ sucessfully, we have $ f$ such that $ f(f(n))=p(n)+2$. Give an example.



(2) Prove that it is impossible that  $ f(f(n))=q(n)+2$ holds in regardless of any definition for $ f: N\rightarrow N$.
1991 Japan MO Finals
Let $ A$ be a positive 16 digit integer.If you take out some consecutive digits integers among $ A$, prove that we can make the product of the numbers be square number. For example if some digit of $ A$ is 4, you may take out only the digit.
1991 Japan MO Finals
A rectangular of a $ 10*14$ is divided into small 140 unit squares and painted in red and white like chess board as below.

We put $ 0$ or $ 1$ in the square such that each row and column has an odd numbers of $ 1$.

Prove that the number of $ 1$ contained in red-painted square is even.



The pattern arranged by a red and a white square alternatively.



RWRWRWRW.....

WRWRWRWR.....

RWRWRWRW.....

WRWRWRWR.....
1991 Japan MO Finals
Let $ A$ be a set of $ n\geq 2$ points on a plane. Prove that there exists a circle which contains at least $ \left[\frac{n}{3}\right]$ points of $ A$ among circles (involving perimeter) with some end points taken from $ A$ as the diameter, where $ [x]$ is the greatest integer which is less than or equal to $ x$.
2019 Kazakhstan National Olympiad
Prove for any positives $a,b,c$ the inequality $$

\sqrt[3]{\dfrac{a}{b}}+\sqrt[5]{\dfrac{b}{c}}+\sqrt[7]{\dfrac{c}{a}}>\dfrac{5}{2}$$
2019 Kazakhstan National Olympiad
The set Φ consists of a finite number of points on the plane. The distance between any two points from Φ is at least $\sqrt{2}$. It is known that a regular triangle with side lenght $3$ cut out of paper can cover all points of Φ. What is the greatest number of points that Φ can consist of?
2019 Kazakhstan National Olympiad
Let $p$ be a prime number of the form $4k+1$ and $\frac{m}{n}$ is an irreducible fraction such that

$$\sum_{a=2}^{p-2} \frac{1}{a^{(p-1)/2}+a^{(p+1)/2}}=\frac{m}{n}.$$Prove that $p|m+n$.





(Fixed, thanks Pavel)
2019 Kazakhstan National Olympiad
Find all positive integers $n,k,a_1,a_2,...,a_k$ so that $n^{k+1}+1$ is divisible by $(na_1+1)(na_2+1)...(na_k+1)$
2019 Kazakhstan National Olympiad
Given a checkered rectangle of size n × m. Is it always possible to mark $3$ or $4$ nodes of a rectangle so that at least one of the marked nodes lay on each straight line containing the side of the rectangle, and the non-self-intersecting polygon with vertices at these nodes has an area equal to



$$\dfrac{1}{2}\min \left ( \text{gcd}(n, m), \dfrac{n+m}{\text{gcd}(n, m)} \right)$$?
2019 Kazakhstan National Olympiad
The tangent line $l$ to the circumcircle of an acute triangle $ABC$ intersects the lines $AB, BC$, and $CA$ at points $C', A'$ and $B'$, respectively. Let $H$ be the orthocenter of a triangle $ABC$. On the straight lines A'H, B′H and C'H, respectively, points $A_1, B_1$ and $C_1$ (other than $H$) are marked such that $AH = AA_1, BH = BB_1$ and $CH = CC_1$. Prove that the circumcircles of triangles $ABC$ and $A_1B_1C_1$ are tangent.
2018 Kazakhstan National Olympiad
In an equilateral trapezoid, the point $O$ is the midpoint of the base $AD$. A circle with a center at a point $O$ and a radius $BO$ is tangent to a straight line $AB$. Let the segment $AC$ intersect this circle at point $K(K \ne C)$, and let $M$ is a point such that $ABCM$ is a parallelogram. The circumscribed circle of a triangle $CMD$ intersects the segment $AC$ at a point $L(L\ne C)$. Prove that $AK=CL$.
2018 Kazakhstan National Olympiad
The natural number $m\geq 2$ is given.Sequence of natural numbers $(b_0,b_1,\ldots,b_m)$ is called concave if $b_k+b_{k-2}\le2b_{k-1}$ for all $2\le k\le m.$ Prove that there exist not greater than $2^m$ concave sequences starting with $b_0 =1$ or $b_0 =2$
2018 Kazakhstan National Olympiad
Is there exist a function $f:\mathbb {N}\to \mathbb {N}$ with for $\forall m,n \in \mathbb {N}$ $$f\left(mf\left(n\right)\right)=f\left(m\right)f\left(m+n\right)+n ?$$
2018 Kazakhstan National Olympiad
Prove that for all reas $a,b,c,d\in(0,1)$ we have $$\left(ab-cd\right)\left(ac+bd\right)\left(ad-bc\right)+\min{\left(a,b,c,d\right)} < 1.$$
2018 Kazakhstan National Olympiad
Given set $S = \{ xy\left( {x + y} \right)\; |\; x,y \in \mathbb{N}\}$.Let $a$ and $n$ natural numbers such that $a+2^k\in S$ for all $k=1,2,3,...,n$.Find the greatest value of $n$.
2018 Kazakhstan National Olympiad
Inside of convex quadrilateral $ABCD$ found a point $M$ such that $\angle AMB=\angle ADM+\angle BCM$ and $\angle AMD=\angle ABM+\angle DCM$.Prove that $$AM\cdot CM+BM\cdot DM\ge \sqrt{AB\cdot BC\cdot CD\cdot DA}.$$
2015 Kazakhstan National Olympiad
Prove that $$\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{(n+1)^2} < n \cdot \left(1-\frac{1}{\sqrt[n]{2}}\right).$$
2015 Kazakhstan National Olympiad
Solve in positive integers



$x^yy^x=(x+y)^z$
2015 Kazakhstan National Olympiad
A rectangle is said to be $ inscribed$ in a triangle if all its vertices lie on the sides of the triangle. Prove that the locus of the centers (the meeting points of the diagonals) of all inscribed in an acute-angled triangle rectangles are three concurrent unclosed segments.
2015 Kazakhstan National Olympiad
$P_k(n) $ is the product of all positive divisors of $n$ that are divisible by $k$ (the empty product is equal to $1$). Show that $P_1(n)P_2(n)\cdots P_n(n)$ is a perfect square, for any positive integer $n$.
2015 Kazakhstan National Olympiad
Find all possible $\{ x_1,x_2,...x_n \}$ permutations of $ \{1,2,...,n \}$ so that  when $1\le i \le n-2 $ then we have  $x_i < x_{i+2}$ and when $1 \le i \le n-3$ then we have  $x_i < x_{i+3}$ . Here $n \ge 4$.
2015 Kazakhstan National Olympiad
The quadrilateral $ABCD$  has an incircle of diameter $d$ which touches $BC$ at $K$ and touches $DA$ at $L$. Is it always true that the harmonic mean of $AB$ and $CD$ is equal to $KL$ if and only if the geometric mean of $AB$ and $CD$ is equal to $d$?
2014 Kazakhstan National Olympiad
$a_1,a_2,...,a_{2014}$ is a permutation of $1,2,3,...,2014$. What is the greatest number of perfect squares can have a set ${ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?$
2014 Kazakhstan National Olympiad
Do there exist positive integers $a$ and $b$ such that $a^n+n^b$ and $b^n+n^a$ are relatively prime for all natural $n$?
2014 Kazakhstan National Olympiad
The triangle $ABC$ is inscribed in a circle $w_1$. Inscribed in a triangle circle touchs the sides $BC$ in a point $N$. $w_2$ — the circle inscribed in a segment $BAC$ circle of $w_1$, and passing through a point $N$. Let points $O$ and $J$ — the centers of circles $w_2$ and an extra inscribed circle (touching side $BC$) respectively. Prove, that lines $AO$ and $JN$ are parallel.
2014 Kazakhstan National Olympiad
Given a scalene triangle $ABC$. Incircle of $\triangle{ABC{}}$ touches the sides $AB$ and $BC$ at points $C_1$ and $A_1$ respectively, and excircle of $\triangle{ABC}$ (on side $AC$) touches $AB$ and $BC$  at points $ C_2$ and $A_2$ respectively. $BN$ is bisector of $\angle{ABC}$ ($N$ lies on $BC$). Lines $A_1C_1$ and $A_2C_2$ intersects the line $AC$ at points $K_1$ and $K_2$ respectively. Let circumcircles of $\triangle{BK_1N}$ and $\triangle{BK_2N}$ intersect circumcircle of a $\triangle{ABC}$ at points $P_1$ and $P_2$ respectively. Prove that $AP_1$=$CP_2$
2014 Kazakhstan National Olympiad
$\mathbb{Q}$ is set of all rational numbers. Find all functions $f:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$ such that for all $x$, $y$, $z$ $\in\mathbb{Q}$  satisfy

$f(x,y)+f(y,z)+f(z,x)=f(0,x+y+z)$
2014 Kazakhstan National Olympiad
Prove that, for all $n\in\mathbb{N}$, on $ [n-4\sqrt{n}, n+4\sqrt{n}]$ exists natural number $k=x^3+y^3$ where $x$, $y$ are nonnegative integers.
2013 Kazakhstan National Olympiad
On the board written numbers from 1 to 25 . Bob can pick any three of them say $a,b,c$  and  replace by  $a^3+b^3+c^3$ . Prove that last number on the board can not be $2013^3$.
2013 Kazakhstan National Olympiad
Prove that for all natural $n$ there exists $a,b,c$ such that $n=\gcd (a,b)(c^2-ab)+\gcd (b,c)(a^2-bc)+\gcd (c,a)(b^2-ca)$.
2013 Kazakhstan National Olympiad
Given triangle ABC with incenter I. Let P,Q be  point on circumcircle such that $\angle API=\angle CPI$ and $\angle BQI=\angle CQI$.Prove that $BP,AQ$ and $OI$ are concurrent.
2013 Kazakhstan National Olympiad
a)Does there exist for any rational number $\frac{a}{b}$ some rational numbers $x_1,x_2,....x_n$ such that

$x_1*x_2*....*x_n=1$ and $x_1+x_2+....+x_n=\frac{a}{b}$

a)Does there exist for any rational number  $\frac{a}{b}$  some rational numbers $x_1,x_2,....x_n$ such that

$x_1*x_2*....*x_n=\frac{a}{b}$ and $x_1+x_2+....+x_n=1$
2013 Kazakhstan National Olympiad
Let for  natural numbers $a,b,c$  and any natural $n$ we have that

$(abc)^n$ divides $ ((a^n-1)(b^n-1)(c^n-1)+1)^3$.   Prove that then $a=b=c$.
2013 Kazakhstan National Olympiad
Let $ABCD$ be cyclic quadrilateral. Let $AC$ and $BD$ intersect at $R$, and let $AB$ and $CD$ intersect at $K$.  Let $M$ and $N$ are points on $AB$ and $CD$ such that $\frac{AM}{MB}=\frac{CN}{ND}$. Let $P$ and $Q$ be the intersections of $MN$ with the diagonals of $ABCD$.  Prove that circumcircles of triangles $KMN$ and $PQR$ are tangent at a fixed point.
2013 Kazakhstan National Olympiad
Find maximum value of

$|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.
2013 Kazakhstan National Olympiad
Let in triangle $ABC$ incircle touches sides $AB,BC,CA$ at $C_1,A_1,B_1$ respectively. Let $\frac {2}{CA_1}=\frac {1}{BC_1}+\frac {1}{AC_1}$ .Prove that if $X$ is intersection of incircle and $CC_1$ then $3CX=CC_1$
2013 Kazakhstan National Olympiad
How many non-intersecting pairs  of  paths we have from (0,0) to (n,n) so that path can move two ways:top or right?
2013 Kazakhstan National Olympiad
Find all triples of positive integer $(m,n,k)$ such that $ k^m|m^n-1$ and $ k^n|n^m-1$
2013 Kazakhstan National Olympiad
Consider the following sequence : $a_1=1 ; a_n=\frac{a_[{\frac{n}{2}]}}{2}+\frac{a_[{\frac{n}{3}]}}{3}+\ldots+\frac{a_[{\frac{n}{n}]}}{n}$. Prove that $ a_{2n}< 2*a_{n } (\forall n\in\mathbb{N})$
2012 Kazakhstan National Olympiad
Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.
2012 Kazakhstan National Olympiad
Given two circles $k_{1}$ and $k_{2}$ with centers $O_{1}$ and $O_{2}$ that intersect at the points $A$ and $B$.Passes through A two lines that intersect the circle $k_{1}$ at the points $N_{1}$and $M_{1}$, and the circle $k_{2}$ at the points $N_{2}$ and $M_{2}$ (points $A, N_{1},M_{1}$ in colinear). Denote the midpoints of the segments $N_{1}N_{2}$ and $M_{1}M_{2]}$  , through $N$ and $M$.Prove that:

$a)$ Points $M,N,A$ and $B$ lie on a circle

$b)$The center of the circle passing through $M,N,A$ and $B$ lies in the middle of the segment $O_{1}O_{2}$
2012 Kazakhstan National Olympiad
Let $ a,b,c,d>0$ for which the following conditions::

$a)$ $(a-c)(b-d)=-4$

$b)$ $\frac{a+c}{2}\geq\frac{a^{2}+b^{2}+c^{2}+d^{2}}{a+b+c+d}$

Find the minimum of expression $a+c$
2012 Kazakhstan National Olympiad
Do there exist a infinite sequence of positive integers $(a_{n})$ ,such that for any $n\ge 1$ the relation $ a_{n+2}=\sqrt{a_{n+1}}+a_{n} $?
2012 Kazakhstan National Olympiad
Given an inscribed quadrilateral $ABCD$, which marked the midpoints of the points $M, N, P, Q$ in this order. Let diagonals $AC$ and $BD$ intersect at point $O$. Prove that the triangle $OMN, ONP, OPQ, OQM$ have the same radius of the circles