Olympiad problems

2009 Kazakhstan National Olympiad, 4

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 \[\frac{a_1^2}{a_2}+\frac{a_2^3}{a_3^2}+\cdots+\frac{a_n^{n+1}}{a_1^n} \geq a_1+a_2+\cdots+a_n\]

2013 Kazakhstan National Olympiad, 1

Tags: number theory unsolved , number theory , Kazakhstan

Find all triples of positive integer $(m,n,k)$ such that $ k^m|m^n-1$ and $ k^n|n^m-1$

2005 Kazakhstan National Olympiad, 4

Tags: function , algebra proposed , algebra , Kazakhstan

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 $y$.

2017 Kazakhstan NMO, Problem 2

Tags: Kazakhstan , 2017 , inequalities

For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$, prove this inequality holds $$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$

2005 Kazakhstan National Olympiad, 3

Tags: combinatorics proposed , combinatorics , Kazakhstan

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 $0$.

2012 Kazakhstan National Olympiad, 1

Tags: geometry , perimeter , geometry unsolved , Kazakhstan

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 $\frac{1}{S}\le \frac{1}{S_{1}}+\frac{1}{S_{2}}+\frac{1}{S_{2}}$

2018 Kazakhstan National Olympiad, 3

Tags: function , algebra , Kazakhstan

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, 2

Tags: Sequence , number theory , algebra , Kazakhstan

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 $b_0 =2$

2015 Kazakhstan National Olympiad, 2

Tags: number theory , Diophantine equation , number theory proposed , Kazakhstan

Solve in positive integers $x^yy^x=(x+y)^z$

2005 Kazakhstan National Olympiad, 4

Tags: algebra , polynomial , algebra proposed , Kazakhstan

Find all polynomials $ P(x)$ with real coefficients such that for every positive integer $ n$ there exists a rational $ r$ with $ P(r)=n$.

2013 Kazakhstan National Olympiad, 1

Tags: inequalities , Kazakhstan , algebra

Find maximum value of $|a^2-bc+1|+|b^2-ac+1|+|c^2-ba+1|$ when $a,b,c$ are reals in $[-2;2]$.

2018 Kazakhstan National Olympiad, 6

Tags: inequalities , geometric inequality , geometry , Kazakhstan

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 $$AM\cdot CM+BM\cdot DM\ge \sqrt{AB\cdot BC\cdot CD\cdot DA}.$$

2014 Contests, 2

Tags: modular arithmetic , number theory unsolved , number theory , Kazakhstan

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, 1

Tags: number theory unsolved , number theory , Kazakhstan

$a_1,a_2,...,a_{2014}$ is a permutation of $1,2,3,...,2014$. What is the greatest number of perfect squares can have a set ${ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?$

2013 Kazakhstan National Olympiad, 2

Tags: modular arithmetic , number theory , relatively prime , number theory unsolved , Kazakhstan

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, 2

Tags: number theory , greatest common divisor , algorithm , function , number theory unsolved , Kazakhstan

Prove that for all natural $n$ there exists $a,b,c$ such that $n=\gcd (a,b)(c^2-ab)+\gcd (b,c)(a^2-bc)+\gcd (c,a)(b^2-ca)$.

2021 Kazakhstan National Olympiad, 1

Tags: Kazakhstan , inequalities proposed , inequalities

Given $a,b,c>0$ such that $$a+b+c+\frac{1}{abc}=\frac{19}{2}$$ What is the greatest value for $a$?