Found problems: 606
2019 Iran Team Selection Test, 1
$S$ is a subset of Natural numbers which has infinite members.
$$S’=\left\{x^y+y^x: \, x,y\in S, \, x\neq y\right\}$$
Prove the set of prime divisors of $S’$ has also infinite members
[i]Proposed by Yahya Motevassel[/i]
2022 Balkan MO Shortlist, A4
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and
\[f(f(x)) + f(f(y)) = f(x + y)f(xy),\]
for all $x, y \in\mathbb{R}$.
2014 Chile TST Ibero, 2
Let $\triangle ABC$ be a triangle and points $P, Q, R$ on the sides $AB, BC,$ and $CA$ respectively, such that:
\[
\frac{AP}{AB} = \frac{BQ}{BC} = \frac{CR}{CA} = \frac{1}{n}
\]
for $n \in \mathbb{N}$. The segments $AQ$ and $CP$ intersect at $D$, the segments $BR$ and $AQ$ intersect at $E$, and the segments $BR$ and $CP$ intersect at $F$. Compute the ratio:
\[
\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)}.
\]
2019 Slovenia Team Selection Test, 2
Determine all non-negative real numbers $a$, for which $f(a)=0$ for all functions $f: \mathbb{R}_{\ge 0}\to \mathbb{R}_{\ge 0} $, who satisfy the equation $f(f(x) + f(y)) = yf(1 + yf(x))$ for all non-negative real numbers $x$ and $y$.
2024 Germany Team Selection Test, 2
Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.
2021 Junior Balkan Team Selection Tests - Romania, P3
The incircle of triangle $ABC$ is tangent to the sides $AB,AC$ and $BC$ at the points $M,N$ and $K$ respectively. The median $AD$ of the triangle $ABC$ intersects $MN$ at the point $L$. Prove that $K,I$ and $L$ are collinear, where $I$ is the incenter of the triangle $ABC$.
2013 USA Team Selection Test, 3
In a table with $n$ rows and $2n$ columns where $n$ is a fixed positive integer, we write either zero or one into each cell so that each row has $n$ zeros and $n$ ones. For $1 \le k \le n$ and $1 \le i \le n$, we define $a_{k,i}$ so that the $i^{\text{th}}$ zero in the $k^{\text{th}}$ row is the $a_{k,i}^{\text{th}}$ column. Let $\mathcal F$ be the set of such tables with $a_{1,i} \ge a_{2,i} \ge \dots \ge a_{n,i}$ for every $i$ with $1 \le i \le n$. We associate another $n \times 2n$ table $f(C)$ from $C \in \mathcal F$ as follows: for the $k^{\text{th}}$ row of $f(C)$, we write $n$ ones in the columns $a_{n,k}-k+1, a_{n-1,k}-k+2, \dots, a_{1,k}-k+n$ (and we write zeros in the other cells in the row).
(a) Show that $f(C) \in \mathcal F$.
(b) Show that $f(f(f(f(f(f(C)))))) = C$ for any $C \in \mathcal F$.
2024 Azerbaijan IMO TST, 2
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.
2017 Greece JBMO TST, 3
Prove that for every positive integer $n$, the number $A_n = 7^{2n} -48n - 1$ is a multiple of $9$.
2023 Serbia Team Selection Test, P1
In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo).
What is the maximal possible number of edges in such a graph?
2011 Saudi Arabia IMO TST, 1
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.
2025 Azerbaijan IZhO TST, 1
An arbitary point $D$ is selected on arc $BC$ not containing $A$ on $(ABC)$. $P$ and $Q$ are the reflections of point $B$ and $C$ with respect to $AD$, respectively. Circumcircles of $ABQ$ and $ACP$ intersect at $E\neq A$. Prove that $A;D;E$ is colinear
2017 China Team Selection Test, 2
In $\varDelta{ABC}$,the excircle of $A$ is tangent to segment $BC$,line $AB$ and $AC$ at $E,D,F$ respectively.$EZ$ is the diameter of the circle.$B_1$ and $C_1$ are on $DF$, and $BB_1\perp{BC}$,$CC_1\perp{BC}$.Line $ZB_1,ZC_1$ intersect $BC$ at $X,Y$ respectively.Line $EZ$ and line $DF$ intersect at $H$,$ZK$ is perpendicular to $FD$ at $K$.If $H$ is the orthocenter of $\varDelta{XYZ}$,prove that:$H,K,X,Y$ are concyclic.
2024 Israel TST, P1
Triangle $ABC$ with $\angle BAC=60^\circ$ is given. The circumcircle of $ABC$ is $\Omega$, and the orthocenter of $ABC$ is $H$. Let $S$ denote the midpoint of the arc $BC$ of $\Omega$ which doesn't contain $A$. Point $P$ was chosen on $\Omega$ so that $\angle HPS=90^\circ$. Prove that there exists a circle that goes through $P$ and $S$ and is tangent to lines $AB$, $AC$.
2015 Turkey Team Selection Test, 8
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$ such that $|AC|>|BC|>|AB|$ and the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. Let the reflection of $A$ with respect to $F$ and $E$ be $F_1$ and $E_1$ respectively. The circle tangent to $BC$ at $D$ and passing through $F_1$ intersects $AB$ a second time at $F_2$ and the circle tangent to $BC$ at $D$ and passing through $E_1$ intersects $AC$ a second time at $E_2$. The midpoints of the segments $|OE|$ and $|IF|$ are $P$ and $Q$ respectively. Prove that \[|AB| + |AC| = 2|BC| \iff PQ\perp E_2F_2 \].
2017 Azerbaijan JBMO TST, 2
Let $ABC$ be isosceles triangle ($AB=BC$) and $K$ and $M$ be the midpoints of $AB$ and $AC,$ respectively.Let the circumcircle of $\triangle BKC$ meets the line $BM$ at $N$ other than $B.$ Let the line passing through $N$ and parallel to $AC$ intersects the circumcircle of $\triangle ABC$ at $A_1$ and $C_1.$ Prove that $\triangle A_1BC_1$ is equilateral.
2023 Romania Team Selection Test, P2
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.
2023 JBMO Shortlist, A2
For positive real numbers $x,y,z$ with $xy+yz+zx=1$, prove that
$$\frac{2}{xyz}+9xyz \geq 7(x+y+z)$$
2013 JBMO TST - Macedonia, 5
Let $ p, r $ be prime numbers, and $ q $ natural. Solve the equation $ (p+q+r)^2=2p^2+2q^2+r^2 $.
2025 Turkey Team Selection Test, 8
A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given.
Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$
\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]
Then for every integer $n\geq s,$ the condition
\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]
is satisfied. Prove that this sequence must be eventually constant.
2023 JBMO TST - Turkey, 1
Let $n,k$ are integers and $p$ is a prime number. Find all $(n,k,p)$ such that
$|6n^2-17n-39|=p^k$
2021 Azerbaijan EGMO TST, 3
Given an integer $k\geq 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\cdots x_k!$ for all positive integers $x_1,x_2,\cdots x_k$.
$Albania$
2021 Israel TST, 2
Let $n>1$ be an integer. Hippo chooses a list of $n$ points in the plane $P_1, \dots, P_n$; some of these points may coincide, but not all of them can be identical. After this, Wombat picks a point from the list $X$ and measures the distances from it to the other $n-1$ points in the list. The average of the resulting $n-1$ numbers will be denoted $m(X)$.
Find all values of $n$ for which Hippo can prepare the list in such a way, that for any point $X$ Wombat may pick, he can point to a point $Y$ from the list such that $XY=m(X)$.
2021 Bolivian Cono Sur TST, 2
Find all posible pairs of positive integers $x,y$ such that $$\text{lcm}(x,y+3001)=\text{lcm}(y,x+3001)$$
2020 JBMO TST of France, 1
Players A and B play a game. They are given a box with $n=>1$ candies. A starts first. On a move, if in the box there are $k$ candies, the player chooses positive integer $l$ so that $l<=k$ and $(l, k) =1$, and eats $l$ candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of $n$.