This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 35

2014 Tuymaada Olympiad, 7
Tags: geometry , geometric transformation , analytic geometry , rectangle , combinatorics unsolved , combinatorics , Tuymaada
Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares. [i](V. Dolnikov)[/i]

2014 Tuymaada Olympiad, 5
Tags: quadratics , function , algebra solved , algebra , Tuymaada
For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist? [i](A. Golovanov)[/i]

2018 Tuymaada Olympiad, 3
Tags: Tuymaada , combinatorics
$n$ rooks and $k$ pawns are arranged on a $100 \times 100$ board. The rooks cannot leap over pawns. For which minimum $k$ is it possible that no rook can capture any other rook? Junior League: $n=2551$ ([i]Proposed by A. Kuznetsov[/i]) Senior League: $n=2550$ ([i]Proposed by N. Vlasova[/i])

2014 Tuymaada Olympiad, 5
Tags: combinatorics unsolved , combinatorics , Tuymaada
There is an even number of cards on a table; a positive integer is written on each card. Let $a_k$ be the number of cards having $k$ written on them. It is known that \[a_n-a_{n-1}+a_{n-2}- \cdots \ge 0 \] for each positive integer $n$. Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by $1$. [i](A. Golovanov)[/i]

2014 Tuymaada Olympiad, 4
Tags: geometry , parallelogram , induction , combinatorics unsolved , combinatorics , Tuymaada
A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set? Seniors) How many ways are there to do that? [asy] size(120); defaultpen(linewidth(0.8)); path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle; for(int i=0;i<=3;i=i+1) { for(int j=0;j<=2;j=j+1) { real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2; draw(shift(shiftx,shifty)*hex); } } [/asy] [i](T. Doslic)[/i]

2014 Tuymaada Olympiad, 6
Tags: trigonometry , geometry unsolved , geometry , Tuymaada
Radius of the circle $\omega_A$ with centre at vertex $A$ of a triangle $\triangle{ABC}$ is equal to the radius of the excircle tangent to $BC$. The circles $\omega_B$ and $\omega_C$ are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other. [i](L. Emelyanov)[/i]

2014 Tuymaada Olympiad, 2
Tags: geometry , geometric transformation , homothety , reflection , geometry unsolved , Tuymaada
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear. [i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]

2014 Tuymaada Olympiad, 8
Tags: combinatorics unsolved , combinatorics , Tuymaada
There are $m$ villages on the left bank of the Lena, $n$ villages on the right bank and one village on an island. It is known that $(m+1,n+1)>1$. Every two villages separated by water are connected by ferry with positive integral number. The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong. [i](K. Kokhas)[/i]

2016 Tuymaada Olympiad, 5
Tags: combinatorics , Tuymaada
Positive numbers are written in the squares of a 10 × 10 table. Frogs sit in five squares and cover the numbers in these squares. Kostya found the sum of all visible numbers and got 10. Then each frog jumped to an adjacent square and Kostya’s sum changed to $10^2$. Then the frogs jumped again, and the sum changed to $10^3$ and so on: every new sum was 10 times greater than the previous one. What maximum sum can Kostya obtain?

2014 Contests, 2
Tags: geometry , geometric transformation , homothety , reflection , geometry unsolved , Tuymaada
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear. [i](A. Akopyan, S. Boev, P. Kozhevnikov)[/i]

2015 Tuymaada Olympiad, 8
Tags: Tuymaada , logic , combinatorics
Four sages stand around a non-transparent baobab. Each of the sages wears red, blue, or green hat. A sage sees only his two neighbors. Each of them at the same time must make a guess about the color of his hat. If at least one sage guesses correctly, the sages win. They could consult before the game started. How should they act to win?

2014 Tuymaada Olympiad, 4
Tags: inequalities , function , Tuymaada
Positive numbers $a,\ b,\ c$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3$. Prove the inequality \[\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}. \] [i](N. Alexandrov)[/i]

2023 Tuymaada Olympiad, 5
Tags: analytic geometry , polynomial , algebra , Tuymaada
A small ship sails on an infinite coordinate sea. At the moment $t$ the ship is at the point with coordinates $(f(t), g(t))$, where $f$ and $g$ are two polynomials of third degree. Yesterday at $14:00$ the ship was at the same point as at $13:00$, and at $20:00$, it was at the same point as at $19:00$. Prove that the ship sails along a straight line.

2014 Tuymaada Olympiad, 2
Tags: geometry , parallelogram , induction , combinatorics unsolved , combinatorics , Tuymaada
A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs. Juniors) How many vertical sides can there be in this set? Seniors) How many ways are there to do that? [asy] size(120); defaultpen(linewidth(0.8)); path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle; for(int i=0;i<=3;i=i+1) { for(int j=0;j<=2;j=j+1) { real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2; draw(shift(shiftx,shifty)*hex); } } [/asy] [i](T. Doslic)[/i]

2023 Tuymaada Olympiad, 3
Tags: geometry , cyclic quadrilateral , reflection , Tuymaada
Point $L$ inside triangle $ABC$ is such that $CL = AB$ and $ \angle BAC + \angle BLC = 180^{\circ}$. Point $K$ on the side $AC$ is such that $KL \parallel BC$. Prove that $AB = BK$

2023 Tuymaada Olympiad, 1
Tags: algebra , Inequality , Tuymaada
Prove that for $a, b, c \in [0;1]$, $$(1-a)(1+ab)(1+ac)(1-abc) \leq (1+a)(1-ab)(1-ac)(1+abc).$$

2019 Tuymaada Olympiad, 6
Tags: number theory , square , Tuymaada
Prove that the expression $$ (1^4+1^2+1)(2^4+2^2+1)\dots(n^4+n^2+1)$$ is not square for all $n \in \mathbb{N}$

2014 Tuymaada Olympiad, 7
Tags: geometry , parallelogram , incenter , analytic geometry , geometry unsolved , Tuymaada
A parallelogram $ABCD$ is given. The excircle of triangle $\triangle{ABC}$ touches the sides $AB$ at $L$ and the extension of $BC$ at $K$. The line $DK$ meets the diagonal $AC$ at point $X$; the line $BX$ meets the median $CC_1$ of trianlge $\triangle{ABC}$ at ${Y}$. Prove that the line $YL$, median $BB_1$ of triangle $\triangle{ABC}$ and its bisector $CC^\prime$ have a common point. [i](A. Golovanov)[/i]

2024 Tuymaada Olympiad, 1
Tags: number theory , NT construction , Tuymaada
Prove that a positive integer of the form $n^4 +1$ can have more than $1000$ divisors of the form $a^4 +1$ with integral $a$.

2013 Tuymaada Olympiad, 4
Tags: inequalities , Tuymaada , three variable inequality
Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then \[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\] [i]A. Golovanov[/i]

2024 Tuymaada Olympiad, 2
Tags: combinatorics , game , Game Theory , Combinatorial games , Tuymaada
Chip and Dale play on a $100 \times 100$ table. In the beginning, a chess king stands in the upper left corner of the table. At each move the king is moved one square right, down or right-down diagonally. A player cannot move in the direction used by his opponent in the previous move. The players move in turn, Chip begins. The player that cannot move loses. Which player has a winning strategy?

2014 Tuymaada Olympiad, 6
Tags: geometry , geometric transformation , analytic geometry , rectangle , combinatorics unsolved , combinatorics , Tuymaada
Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares. [i](V. Dolnikov)[/i]

2014 Tuymaada Olympiad, 1
Tags: modular arithmetic , number theory , greatest common divisor , least common multiple , relatively prime , number theory proposed , Tuymaada
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained? [i](A. Golovanov)[/i]

2014 Tuymaada Olympiad, 8
Tags: calculus , integration , analytic geometry , geometry , modular arithmetic , number theory , Tuymaada
Let positive integers $a,\ b,\ c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$. Prove that \[ g(a, b, c)\ge \sqrt{2abc}\] [i](M. Ivanov)[/i] [hide="Remarks (containing spoilers!)"] 1. It can be proven that $g(a,b,c)\ge \sqrt{3abc}$. 2. The constant $3$ is the best possible, as proved by the equation $g(3,3k+1,3k+2)=9k+5$. [/hide]

2023 Tuymaada Olympiad, 6
Tags: inequalities , geometry , Plane , Tuymaada , Geometry inequality
In the plane $n$ segments with lengths $a_1, a_2, \dots , a_n$ are drawn. Every ray beginning at the point $O$ meets at least one of the segments. Let $h_i$ be the distance from $O$ to the $i$-th segment (not the line!) Prove the inequality \[\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.\]