Olympiad problems

2024 Romania Team Selection Tests, P1

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

2022 IMO Shortlist, C3

Tags: combinatorics , IMO Shortlist , AZE IMO TST , TST

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

2024 Chile TST Ibero., 1

Tags: TST , Chile , algebra

Determine all integers $ x $ for which the expression $ x^2 + 10x + 160 $ is a perfect square.

2025 Israel TST, P1

Tags: TST

A sequence starts at some rational number $x_1>1$, and is subsequently defined using the recurrence relation \[x_{n+1}=\frac{x_n\cdot n}{\lfloor x_n\cdot n\rfloor }\] Show that $k>0$ exists with $x_k=1$.

2023 Azerbaijan IZhO TST, 2

Tags: algebra , polynomial , TST , AZE IzHO TST

P(x) is polynomial such that, polynomial P(P(x)) is strictly monotone in all real number line. Prove that polynomial P(x) is also strictly monotone in all real number line.

2016 Azerbaijan JBMO TST, 4

Tags: TST

Find all positive integers n such that $ (n ^{ 2} + 11n - 4) n! + 33.13 ^ n + 4 $ is the perfect square

2019 Iran Team Selection Test, 2

Tags: TST , Iranian TST , geometry , Iran

In a triangle $ABC$, $\angle A$ is $60^\circ$. On sides $AB$ and $AC$ we make two equilateral triangles (outside the triangle $ABC$) $ABK$ and $ACL$. $CK$ and $AB$ intersect at $S$ , $AC$ and $BL$ intersect at $R$ , $BL$ and $CK$ intersect at $T$. Prove the radical centre of circumcircle of triangles $BSK, CLR$ and $BTC$ is on the median of vertex $A$ in triangle $ABC$. [i]Proposed by Ali Zamani[/i]

2025 Junior Balkan Team Selection Tests - Romania, P5

Tags: combinatorics , TST

Let $n\geqslant 3$ be a positive integer and $\mathcal F$ be a family of at most $n$ distinct subsets of the set $\{1,2,\ldots,n\}$ with the following property: we can consider $n$ distinct points in the plane, labelled $1,2,\ldots,n$ and draw segments connecting these points such that points $i$ and $j$ are connected if and only if $i{}$ belongs to $j$ subsets in $\mathcal F$ for any $i\neq j.$ Determine the maximal value that the sum of the cardinalities of the subsets in $\mathcal{F}$ can take.

2022 JBMO Shortlist, G1

Tags: geometry , JBMO Shortlist , TST

Let $ABCDE$ be a cyclic pentagon such that $BC = DE$ and $AB$ is parallel to $DE$. Let $X, Y,$ and $Z$ be the midpoints of $BD, CE,$ and $AE$ respectively. Show that $AE$ is tangent to the circumcircle of the triangle $XYZ$. Proposed by [i]Nikola Velov, Macedonia[/i]

2022 China Team Selection Test, 6

Tags: combinatorics , colorings , graph theory , Hall s marriage theorem , TST , China TST

Let $m$ be a positive integer, and $A_1, A_2, \ldots, A_m$ (not necessarily different) be $m$ subsets of a finite set $A$. It is known that for any nonempty subset $I$ of $\{1, 2 \ldots, m \}$, \[ \Big| \bigcup_{i \in I} A_i \Big| \ge |I|+1. \] Show that the elements of $A$ can be colored black and white, so that each of $A_1,A_2,\ldots,A_m$ contains both black and white elements.

2017 Balkan MO Shortlist, A6

Tags: functional equation , algebra , BMO Shortlist , TST , AZE IMO TST

Find all functions $f : \mathbb R\to\mathbb R $ such that \[f(x+yf(x^2))=f(x)+xf(xy)\] for all real numbers $x$ and $y$.

2020 USA EGMO Team Selection Test, 6

Tags: algebra , polynomial , Integer Polynomial , Periodic sequence , modular arithmetic , TST

Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.

2023 Israel TST, P1

Tags: combinatorics , TST , regular polygon

A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he draws. How many fish can Bob guarantee getting, no matter Alice's goodwill?

2016 Serbia Additional Team Selection Test, 2

Tags: combinatorics , geometry , TST

Let $ABCD$ be a square with side $4$. Find, with proof, the biggest $k$ such that no matter how we place $k$ points into $ABCD$, such that they are on the interior but not on the sides, we always have a square with sidr length $1$, which is inside the square $ABCD$, such that it contains no points in its interior(they can be on the sides).

Azerbaijan Al-Khwarizmi IJMO TST 2025, 3

Tags: number theory , TST

Let $a$ and $b$ be integers such that $a - b = a^2c - b^2d$ for some consecutive integers $c$ and $d$. Prove that $|a - b|$ is a perfect square.

2021 Junior Balkan Team Selection Tests - Romania, P2

Tags: number theory , prime numbers , Romanian TST , TST

Find all the pairs of positive integers $(x,y)$ such that $x\leq y$ and \[\frac{(x+y)(xy-1)}{xy+1}=p,\]where $p$ is a prime number.

2020 Azerbaijan IZHO TST, 4

Tags: number theory , TST , AZE IzHO TST

Consider an odd prime number $p$ and $p$ consecutive positive integers $m_1,m_2,…,m_p$. Choose a permutation $\sigma$ of $1,2,…,p$ . Show that there exist two different numbers $k,l\in{(1,2,…,p)}$ such that $p\mid{m_k.m_{\sigma(k)}-m_l.m_{\sigma(l)}}$

2022 IMO Shortlist, A2

Tags: algebra , IMO Shortlist , AZE IMO TST , TST

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

2016 Chile TST IMO, 1

Tags: TST , Chile , Iberoamerican

An equilateral triangle with side length 20 is subdivided using parallels to its sides into $ 20^2 = 400 $ smaller equilateral triangles of side length 1. Some segments of length 1, which are edges of these small triangles, must be colored red in such a way that no small triangle has all three of its edges colored red. Determine the maximum number of segments of length 1 that can be colored red.

2017 Azerbaijan JBMO TST, 1

Tags: inequalities , algebra , TST , AZE JBMO TST

Let $x,y,z,t$ be positive numbers.Prove that $\frac{xyzt}{(x+y)(z+t)}\leq\frac{(x+z)^2(y+t)^2}{4(x+y+z+t)^2}.$

2008 Greece JBMO TST, 2

Tags: algebra , Inequality , TST

If $a,b,c$ are positive real numbers, prove that $\frac{a^2b^2}{a+b}+\frac{b^2c^2}{b+c}+\frac{c^2a^2}{c+a}\le \frac{a^3+b^3+c^3}{2}$

2023 Azerbaijan IMO TST, 2

Tags: combinatorics , IMO Shortlist , AZE IMO TST , TST

In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn: [list] [*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller. [*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter. [/list] We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.

2018 Iranian Geometry Olympiad, 2

Tags: geometry , IGO , TST , AZE IzHO TST

In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$. Proposed by Fatemeh Sajadi

2023 Hong Kong Team Selection Test, Problem 4

Tags: algbera , TST

Let $x$, $y$, $z$ be real numbers such that $x+y+z \ne 0$. Find the minimum value of $\frac{|x|+|x+4y|+|y+7z|+2|z|}{|x+y+z|}$

2004 Germany Team Selection Test, 2

Tags: function , algebra proposed , algebra , Functional Equations , Germany , TST , Team Selection Test

Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties: (a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$. (b) We have $f\left(2\right) = 0$. (c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$. [b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.