Olympiad problems

2018 Tajikistan Team Selection Test, 4

Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that: a/(1-x)+b/(1-y)=1 Prove that: ∛ay+∛bx≤1.

2018 Azerbaijan IZhO TST, 5

Tags: geometry , TST , AZE IzHO TST

Let $\omega$ be the incircle of $\triangle ABC$ and $D,E,F$ be the tangency points on $BC ,CA, AB$. In $\triangle DEF$ let the altitudes from $E,F$ to $FD,DE$ intersect $AB, AC$ at $X ,Y$. Prove that the second intersection of $(AEX)$ and $(AFY)$ lies on $\omega$

2019 Azerbaijan IMO TST, 1

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

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(xy) = yf(x) + x + f(f(y) - f(x)) \] for all $x,y \in \mathbb{R}$.

2023 Estonia Team Selection Test, 6

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.

2025 Azerbaijan IZhO TST, 4

Tags: algebra , functional equation , TST , AZE IzHO TST

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that $$f(f(x)+yg(x))=(x+1)g(y)+f(y)$$ for any $x;y\in\mathbb{Q}$

2014 Belarus Team Selection Test, 3

Tags: algebra , functional equation , TST

Do there exist functions $f$ and $g$, $f : R \to R$, $g : R \to R$ such that $f(x + f(y)) = y^2 + g(x)$ for all real $x$ and $y$ ? (I. Gorodnin)

2024 Brazil Cono Sur TST, 1

Tags: IMO Shortlist , number theory , AZE BMO TST , AZE EGMO TST , TST , BRA Cono Sur TST

For positive integers $n$ and $k \geq 2$, define $E_k(n)$ as the greatest exponent $r$ such that $k^r$ divides $n!$. Prove that there are infinitely many $n$ such that $E_{10}(n) > E_9(n)$ and infinitely many $m$ such that $E_{10}(m) < E_9(m)$.

2024 Chile TST Ibero., 4

Tags: TST , inequalities , algebra , TitusLemma , Chile

Prove that if $ a $, $ b $, and $ c $ are positive real numbers, then the following inequality holds: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

2020 JBMO TST of France, 1

Tags: geometry , TST

Given are four distinct points $A, B, E, P$ so that $P$ is the middle of $AE$ and $B$ is on the segment $AP$. Let $k_1$ and $k_2$ be two circles passing through $A$ and $B$. Let $t_1$ and $t_2$ be the tangents of $k_1$ and $k_2$, respectively, to $A$.Let $C$ be the intersection point of $t_2$ and $k_1$ and $Q$ be the intersection point of $t_2$ and the circumscribed circle of the triangle $ECB$. Let $D$ be the intersection posit of $t_1$ and $k_2$ and $R$ be the intersection point of $t_1$ and the circumscribed circle of triangle $BDE$. Prove that $P, Q, R$ are collinear.

2019 India IMO Training Camp, 3

Tags: indian olympiad , graph , combinatorics , TST

There are $2019$ coins on a table. Some are placed with head up and others tail up. A group of $2019$ persons perform the following operations: the first person chooses any one coin and then turns it over, the second person choses any two coins and turns them over and so on and the $2019$-th person turns over all the coins. Prove that no matter which sides the coins are up initially, the $2019$ persons can come up with a procedure for turning the coins such that all the coins have smae side up at the end of the operations.

2025 Junior Balkan Team Selection Tests - Romania, P4

Tags: number theory , TST

Find all positive integers $n$ such that $2^n-n^2+1$ is a perfect square.

2024 Chile TST Ibero., 5

Tags: TST , Chile , geometry

Let $\triangle ABC$ be an acute-angled triangle. Let $P$ be the midpoint of $BC$, and $K$ the foot of the altitude from $A$ to side $BC$. Let $D$ be a point on segment $AP$ such that $\angle BDC = 90^\circ$. Let $E$ be the second point of intersection of line $BC$ with the circumcircle of $\triangle ADK$. Let $F$ be the second point of intersection of line $AE$ with the circumcircle of $\triangle ABC$. Prove that $\angle AFD = 90^\circ$.

2018 Azerbaijan JBMO TST, 4

Tags: combinatorics , TST , AZE JBMO TST

In the beginning, there are $100$ cards on the table, and each card has a positive integer written on it. An odd number is written on exactly $43$ cards. Every minute, the following operation is performed: for all possible sets of $3$ cards on the table, the product of the numbers on these three cards is calculated, all the obtained results are summed, and this sum is written on a new card and placed on the table. A day later, it turns out that there is a card on the table, the number written on this card is divisible by $2^{2018}.$ Prove that one hour after the start of the process, there was a card on the table that the number written on that card is divisible by $2^{2018}.$

2017 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , Inequality , TST , AZE JBMO TST

a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

2023 Chile TST IMO, 2

Tags: number theory , Chile , TST

Determine the number of pairs of positive integers $ (p, k) $ such that $ p $ is a prime number and $ p^2 + 2^k $ is a perfect square less than 2023. A number is called a perfect square if it is the square of an integer.

2025 Turkey EGMO TST, 2

Tags: inequalities , algebra , TST

Does there exist a sequence of positive real numbers $\{a_i\}_{i=1}^{\infty}$ satisfying: \[ \sum_{i=1}^{n} a_i \geq n^2 \quad \text{and} \quad \sum_{i=1}^{n} a_i^2 \leq n^3 + 2025n \] for all positive integers $n$.

2024 Junior Balkan Team Selection Tests - Moldova, 8

Tags: JBMO Shortlist , combinatorics , AZE JBMO TST , TST

There are $n$ blocks placed on the unit squares of a $n \times n$ chessboard such that there is exactly one block in each row and each column. Find the maximum value $k$, in terms of $n$, such that however the blocks are arranged, we can place $k$ rooks on the board without any two of them threatening each other. (Two rooks are not threatening each other if there is a block lying between them.)

2020 Junior Macedonian National Olympiad, 2

Tags: Inequality , algebra , TST

Let $x, y,$ and $z$ be positive real numbers such that $xy + yz + zx = 27$. Prove that $x + y + z \ge \sqrt{3xyz}$. When does equality hold?

2015 Turkey Team Selection Test, 4

Tags: TST , geometry , geometry proposed , Inversion , tangent

Let $ABC$ be a triangle such that $|AB|=|AC|$ and let $D,E$ be points on the minor arcs $\overarc{AB}$ and $\overarc{AC}$ respectively. The lines $AD$ and $BC$ intersect at $F$ and the line $AE$ intersects the circumcircle of $\triangle FDE$ a second time at $G$. Prove that the line $AC$ is tangent to the circumcircle of $\triangle ECG$.

2017 USA TSTST, 6

Tags: algebra , TSTST 2017 , Tstst , number theory , TST , Fibonacci

A sequence of positive integers $(a_n)_{n \ge 1}$ is of [i]Fibonacci type[/i] if it satisfies the recursive relation $a_{n + 2} = a_{n + 1} + a_n$ for all $n \ge 1$. Is it possible to partition the set of positive integers into an infinite number of Fibonacci type sequences? [i]Proposed by Ivan Borsenco[/i]

2024 Indonesia TST, 2

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 $x, y \in \mathbb R_{>0}$.

2020 Vietnam Team Selection Test, 3

Tags: Vietnam , TST , combinatorics

Suppose $n$ is a positive integer, $4n$ teams participate in a football tournament. In each round of the game, we will divide the $4n$ teams into $2n$ pairs, and each pairs play the game at the same time. After the tournament, it is known that every two teams have played at most one game. Find the smallest positive integer $a$, so that we can arrange a schedule satisfying the above conditions, and if we take one more round, there is always a pair of teams who have played in the game.

Russian TST 2020, P1

Tags: geometry , IMO Shortlist , TST

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)

2017 Greece Team Selection Test, 2

Tags: number theory , Greece , combinatorics , TST , binomial coefficients

Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$, where $n$ is a positive integer.

2024 Brazil Team Selection Test, 4

Tags: geometry , AZE IMO TST , IMO Shortlist , TST

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.