Olympiad problems

2021 Kazakhstan National Olympiad, 5

Let $a$ be a positive integer. Prove that for any pair $(x,y)$ of integer solutions of equation $$x(y^2-2x^2)+x+y+a=0$$ we have: $$|x| \leqslant a+\sqrt{2a^2+2}$$

2015 Kazakhstan National Olympiad, 1

Tags: inequalities , inequalities proposed , Kazakhstan

Prove that $$\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{(n+1)^2} < n \cdot \left(1-\frac{1}{\sqrt[n]{2}}\right).$$

2022 Kazakhstan National Olympiad, 1

Tags: geometry , ratio , Kazakhstan

$CH$ is an altitude in a right triangle $ABC$ $(\angle C = 90^{\circ})$. Points $P$ and $Q$ lie on $AC$ and $BC$ respectively such that $HP \perp AC$ and $HQ \perp BC$. Let $M$ be an arbitrary point on $PQ$. A line passing through $M$ and perpendicular to $MH$ intersects lines $AC$ and $BC$ at points $R$ and $S$ respectively. Let $M_1$ be another point on $PQ$ distinct from $M$. Points $R_1$ and $S_1$ are determined similarly for $M_1$. Prove that the ratio $\frac{RR_1}{SS_1}$ is constant.

2010 Kazakhstan National Olympiad, 3

Tags: inequalities , inequalities proposed , Kazakhstan

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 $x, y>0$

2013 Kazakhstan National Olympiad, 2

Tags: algorithm , algebra unsolved , algebra , Kazakhstan

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 $x_1+x_2+....+x_n=1$

2017 Kazakhstan NMO, Problem 1

Tags: Kazakhstan , geometry , 2017 , projective geometry

The non-isosceles triangle $ABC$ is inscribed in the circle ω. The tangent to this circle at the point $C$ intersects the line $AB$ at the point $D$. Let the bisector of the angle $CDB$ intersect the segments $AC$ and $BC$ at the points $K$ and $L$, respectively. On the side $AB$, the point $M$ is taken such that $AK / BL = AM / BM$. Let the perpendiculars from the point $M$ to the lines $KL$ and $DC$ intersect the lines $AC$ and $DC$ at the points $P$ and $Q$, respectively. Prove that the angle $CQP$ is half of the angle $ACB$.

2014 Kazakhstan National Olympiad, 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 $n$?

2020 Kazakhstan National Olympiad, 1

Tags: Kazakhstan , combinatorics , floor function

There are $n$ lamps and $k$ switches in a room. Initially, each lamp is either turned on or turned off. Each lamp is connected by a wire with $2020$ switches. Switching a switch changes the state of a lamp, that is connected to it, to the opposite state. It is known that one can switch the switches so that all lamps will be turned on. Prove, that it is possible to achieve the same result by switching the switches no more than $ \left \lfloor \dfrac{k}{2} \right \rfloor$ times. [i]Proposed by T. Zimanov[/i]

2022 Kazakhstan National Olympiad, 6

Tags: combinatorics , numbers in a table , table , Kazakhstan , construction

Numbers from $1$ to $49$ are randomly placed in a $35 \times 35$ table such that number $i$ is used exactly $i$ times. Some random cells of the table are removed so that table falls apart into several connected (by sides) polygons. Among them, the one with the largest area is chosen (if there are several of the same largest areas, a random one of them is chosen). What is the largest number of cells that can be removed that guarantees that in the chosen polygon there is a number which occurs at least $15$ times?

2010 Contests, 2

Tags: combinatorics proposed , combinatorics , Kazakhstan

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.

2023 Kazakhstan National Olympiad, 4

Tags: inequalities , algebra , Kazakhstan

Given $x,y>0$ such that $x^2y^2+2x^3y=1$. Find the minimum value of sum $x+y$

2020 Kazakhstan National Olympiad, 2

Tags: algebra , Kazakhstan , inequalities

Let $x_1, x_2, ... , x_n$ be a real numbers such that\\ 1) $1 \le x_1, x_2, ... , x_n \le 160$ 2) $x^{2}_{i} + x^{2}_{j} + x^{2}_{k} \ge 2(x_ix_j + x_jx_k + x_kx_i)$ for all $1\le i < j < k \le n$ Find the largest possible $n$.

2011 Kazakhstan National Olympiad, 3

Tags: number theory unsolved , number theory , Kazakhstan

Given are the odd integers $m> 1$, $k$, and a prime $p$ such that $p> mk +1$. Prove that $p^{2}\mid {\binom{k}{k}}^{m}+{\binom{k+1}{k}}^{m}+\cdots+{\binom{p-1}{k}}^{m}$.

2013 Kazakhstan National Olympiad, 1

Tags: geometry , incenter , circumcircle , geometric transformation , geometry unsolved , Kazakhstan

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.

2009 Kazakhstan National Olympiad, 6

Tags: algebra , polynomial , abstract algebra , algebra proposed , Kazakhstan

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 $P(P(0))=0$

2014 Kazakhstan National Olympiad, 3

Tags: inequalities , number theory proposed , number theory , Kazakhstan

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.

2012 Kazakhstan National Olympiad, 2

Tags: function , algebra unsolved , algebra , Kazakhstan

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 $x$.

2018 Kazakhstan National Olympiad, 5

Tags: number theory , Kazakhstan

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 $n$.

2022 Kazakhstan National Olympiad, 4

Tags: geometry , concurrency , Kazakhstan

$P$ and $Q$ are points on angle bisectors of two adjacent angles. Let $PA$, $PB$, $QC$ and $QD$ be altitudes on the sides of these adjacent angles. Prove that lines $AB$, $CD$ and $PQ$ are concurrent.

2008 Kazakhstan National Olympiad, 1

Tags: inequalities , number theory proposed , number theory , Kazakhstan

Find all integer solutions $ (a_1,a_2,\dots,a_{2008})$ of the following equation: $ (2008\minus{}a_1)^2\plus{}(a_1\minus{}a_2)^2\plus{}\dots\plus{}(a_{2007}\minus{}a_{2008})^2\plus{}a_{2008}^2\equal{}2008$

2010 Kazakhstan National Olympiad, 4

Tags: inequalities , geometry , inequalities proposed , Kazakhstan

For $x;y \geq 0$ prove the inequality: $\sqrt{x^2-x+1} \sqrt{y^2-y+1}+ \sqrt{x^2+x+1} \sqrt{y^2+y+1} \geq 2(x+y)$

2017 Kazakhstan NMO, Problem 6

Tags: number theory , Kazakhstan

Show that there exist infinitely many composite positive integers $n$ such that $n$ divides $2^{\frac{n-1}{2}}+1$

2017 Kazakhstan National Olympiad, 6

Tags: number theory , Kazakhstan

Show that there exist infinitely many composite positive integers $n$ such that $n$ divides $2^{\frac{n-1}{2}}+1$

2010 Kazakhstan National Olympiad, 4

Tags: algebra proposed , algebra , Kazakhstan

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 $a$.

2012 Kazakhstan National Olympiad, 2

Tags: geometry , circumcircle , geometry unsolved , Kazakhstan

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 $