Olympiad problems

2022 Germany Team Selection Test, 1

Given a triangle $ABC$ and three circles $x$, $y$ and $z$ such that $A \in y \cap z$, $B \in z \cap x$ and $C \in x \cap y$. The circle $x$ intersects the line $AC$ at the points $X_b$ and $C$, and intersects the line $AB$ at the points $X_c$ and $B$. The circle $y$ intersects the line $BA$ at the points $Y_c$ and $A$, and intersects the line $BC$ at the points $Y_a$ and $C$. The circle $z$ intersects the line $CB$ at the points $Z_a$ and $B$, and intersects the line $CA$ at the points $Z_b$ and $A$. (Yes, these definitions have the symmetries you would expect.) Prove that the perpendicular bisectors of the segments $Y_a Z_a$, $Z_b X_b$ and $X_c Y_c$ concur.

2023 Serbia Team Selection Test, P2

Tags: geometry , Serbia , TST , circumcircle

A circle centered at $A$ intersects sides $AC$ and $AB$ of $\triangle ABC$ at $E$ and $F$, and the circumcircle of $\triangle ABC$ at $X$ and $Y$. Let $D$ be the point on $BC$ such that $AD$, $BE$, $CF$ concur. Let $P=XE\cap YF$ and $Q=XF\cap YE$. Prove that the foot of the perpendicular from $D$ to $EF$ lies on $PQ$.

2025 Azerbaijan IZhO TST, 2

Tags: combinatorics , TST , AZE IzHO TST

You are given a word consisting of letters $a;b;c$ You can apply 3 operations on this word: [b]1)[/b] You can write any $3$ letter long $\text{subword}$ in reverse. ($\text{xyz}\rightarrow \text{zyx}$) [b]2)[/b] You can add same $2$ letters between any $2$ consecutive letters. ($\text{xyxy}\rightarrow \text{xy}$[b]zz[/b]$\text{xy}$) [b]3)[/b] You can remove any $\text{subword}$ in the given form $\text{xyyx}$ ($\text{x}$[b]yzzy[/b]$\text{xy}\rightarrow\text{xxy}$) Given these $3$ operations, can you go from $\text{abccab}$ to $\text{baccba}$? (Note: A $\text{subword}$ is a string of consecutive letters from the given word)

2023 Indonesia 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.

2017 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , TST , AZE JBMO TST

Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number

2023 Indonesia 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.

2022 Bolivia Cono Sur TST, P6

Tags: geometry , Bolivia , TST , Spiral Similarity , Angle Chasing

On $\triangle ABC$ let points $D,E$ on sides $AB,BC$ respectivily such that $AD=DE=EC$ and $AE \ne DC$. Let $P$ the intersection of lines $AE, DC$, show that $\angle ABC=60$ if $AP=CP$.

2018 Greece JBMO TST, 4

Tags: number theory , TST

Find all positive integers $x,y,z$ with $z$ odd, which satisfy the equation: $$2018^x=100^y + 1918^z$$

2024 Azerbaijan BMO TST, 6

Tags: geometry , AZE BMO TST , TST , BMO Shortlist

Let $ABC$ be an acute triangle ($AB < BC < AC$) with circumcircle $\Gamma$. Assume there exists $X \in AC$ satisfying $AB=BX$ and $AX=BC$. Points $D, E \in \Gamma$ are taken such that $\angle ADB<90^{\circ}$, $DA=DB$ and $BC=CE$. Let $P$ be the intersection point of $AE$ with the tangent line to $\Gamma$ at $B$, and let $Q$ be the intersection point of $AB$ with tangent line to $\Gamma$ at $C$. Show that the projection of $D$ onto $PQ$ lies on the circumcircle of $\triangle PAB$.

2018 Azerbaijan JBMO TST, 1

Tags: geometry , TST , AZE JBMO TST

Let $\triangle ABC$ be an acute triangle. Let us denote the foot of the altitudes from the vertices $A, B$ and $C$ to the opposite sides by $D, E$ and $F,$ respectively, and the intersection point of the altitudes of the triangle $ABC$ by $H.$ Let $P$ be the intersection of the line $BE$ and the segment $DF.$ A straight line passing through $P$ and perpendicular to $BC$ intersects $AB$ at $Q.$ Let $N$ be the intersection of the segment $EQ$ with the perpendicular drawn from $A.$ Prove that $N$ is the midpoint of segment $AH.$

2022 Israel TST, 1

Tags: combinatorics , TST , Israel

Bilbo, Gandalf, and Nitzan play the following game. First, Nitzan picks a whole number between $1$ and $2^{2022}$ inclusive and reveals it to Bilbo. Bilbo now compiles a string of length $4044$ built from the three letters $a,b,c$. Nitzan looks at the string, chooses one of the three letters $a,b,c$, and removes from the string all instances of the chosen letter. Only then is the string revealed to Gandalf. He must now guess the number Nitzan chose. Can Bilbo and Gandalf work together and come up with a strategy beforehand that will always allow Gandalf to guess Nitzan's number correctly, no matter how he acts?

2004 Germany Team Selection Test, 4

Tags: Germany , TST , Team Selection Test , equation , integer equation

Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation \[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\] Show that at least two of these integers are equal to each other.

2018 Polish MO Finals, 6

Tags: number theory , Poland , TST , combinatorics

A prime $p>3$ is given. Let $K$ be the number of such permutations $(a_1, a_2, \ldots, a_p)$ of $\{ 1, 2, \ldots, p\}$ such that $$a_1a_2+a_2a_3+\ldots + a_{p-1}a_p+a_pa_1$$ is divisible by $p$. Prove $K+p$ is divisible by $p^2$.

2016 JBMO TST - Turkey, 1

Tags: algebra , system of equations , TST

Find all pairs $(x, y)$ of real numbers satisfying the equations \begin{align*} x^2+y&=xy^2 \\ 2x^2y+y^2&=x+y+3xy. \end{align*}

2024 Azerbaijan IMO TST, 3

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

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , TST

Determine positive integers $a$ and $b$ co-prime such that $a^2+b = (a-b)^3$ .

2015 Turkey Team Selection Test, 6

Tags: Turkey , TST , 2015 , number theory

Prove that there are infinitely many positive integers $n$ such that $(n!)^{n+2015}$ divides $(n^{2})!$.

2023 Israel TST, P3

Tags: TST , number theory , Polynomials , composition , algebra

Given a polynomial $P$ and a positive integer $k$, we denote the $k$-fold composition of $P$ by $P^{\circ k}$. A polynomial $P$ with real coefficients is called [b]perfect[/b] if for each integer $n$ there is a positive integer $k$ so that $P^{\circ k}(n)$ is an integer. Is it true that for each perfect polynomial $P$, there exists a positive $m$ so that for each integer $n$ there is $0<k\leq m$ for which $P^{\circ k}(n)$ is an integer?

2023 Indonesia TST, 1

Tags: number theory , IMO Shortlist , AZE IMO TST , TST

A number is called [i]Norwegian[/i] if it has three distinct positive divisors whose sum is equal to $2022$. Determine the smallest Norwegian number. (Note: The total number of positive divisors of a Norwegian number is allowed to be larger than $3$.)

2023 Dutch IMO TST, 1

Tags: number theory , TST

Find all prime numbers $p$ such that the number $$3^p+4^p+5^p+9^p-98$$ has at most $6$ positive divisors.

2019 Junior Balkan Team Selection Tests - Moldova, 2

Tags: TST

The numeric sequence $(a_n)_{n\geq1}$ verifies the relation $a_{n+1} = \frac{n+2}{n} \cdot (a_n-1)$ for any $n\in N^*$.Show that $a_n \in Z$ for any $n\in N^*$ ,if $a_1\in Z$.

Azerbaijan Al-Khwarizmi IJMO TST 2025, 2

Tags: Inequality , algebra , TST , AZE Al-Khwarizmi IJMO TST , inequalities proposed

For $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2 \geq 3$,show that: $\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9} \geq \frac{4}{3}$.

2023 Balkan MO Shortlist, C1

Tags: combinatorics , AZE BMO TST , TST

Joe and Penny play a game. Initially there are $5000$ stones in a pile, and the two players remove stones from the pile by making a sequence of moves. On the $k$-th move, any number of stones between $1$ and $k$ inclusive may be removed. Joe makes the odd-numbered moves and Penny makes the even-numbered moves. The player who removes the very last stone is the winner. Who wins if both players play perfectly?

2021 Bolivia Ibero TST, 4

Tags: geometry , Bolivia , TST , isosceles

On a isosceles triangle $\triangle ABC$ with $AB=BC$ let $K,M$ be the midpoints of $AB,AC$ respectivily. Let $(CKB)$ intersect $BM$ at $N \ne M$, the line through $N$ parallel to $AC$ intersects $(ABC)$ at $A_1,C_1$. Show that $\triangle A_1BC_1$ is equilateral.

2016 IMO Shortlist, C2

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

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]