Olympiad problems

2013 JBMO TST - Turkey, 6

Find all positive integers $n$ satisfying $2n+7 \mid n! -1$.

2021 Junior Balkan Team Selection Tests - Romania, P5

Tags: geometry , Romanian TST , TST

Let $I$ be the incenter of triangle $ABC$. The circle of centre $A$ and radius $AI$ intersects the circumcircle of triangle $ABC$ in $M$ and $N$. Prove that the line $MN$ is tangent to the incircle of triangle $ABC$

2019 India IMO Training Camp, P1

Tags: inequalities , india , TST , 2019 , number theory

Let $a_1,a_2,\ldots, a_m$ be a set of $m$ distinct positive even numbers and $b_1,b_2,\ldots,b_n$ be a set of $n$ distinct positive odd numbers such that \[a_1+a_2+\cdots+a_m+b_1+b_2+\cdots+b_n=2019\] Prove that \[5m+12n\le 581.\]

Azerbaijan Al-Khwarizmi IJMO TST 2025, 1

Tags: geometry , TST , AZE Al-Khwarizmi IJMO TST

In isosceles triangle, the condition $AB=AC>BC$ is satisfied. Point $D$ is taken on the circumcircle of $ABC$ such that $\angle CAD=90^{\circ}$.A line parallel to $AC$ which passes from $D$ intersects $AB$ and $BC$ respectively at $E$ and $F$.Show that circumcircle of $ADE$ passes from circumcenter of $DFC$.

2022 Bolivia Cono Sur TST, P4

Tags: geometry , inradius , Pythagorean Triples , number theory , Bolivia , TST

Find all right triangles with integer sides and inradius 6.

2023 Belarus Team Selection Test, 3.1

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.

2024 Azerbaijan BMO TST, 2

Tags: geometry , AZE BMO TST , TST , Miquel point

Let $ABC$ be a triangle with circumcenter $O$. Point $X$ is the intersection of the parallel line from $O$ to $AB$ with the perpendicular line to $AC$ from $C$. Let $Y$ be the point where the external bisector of $\angle BXC$ intersects with $AC$. Let $K$ be the projection of $X$ onto $BY$. Prove that the lines $AK, XO, BC$ have a common point.

2017 Turkey Team Selection Test, 7

Tags: algebra , TST , functional equation

Let $a$ be a real number. Find the number of functions $f:\mathbb{R}\rightarrow \mathbb{R}$ depending on $a$, such that $f(xy+f(y))=f(x)y+a$ holds for every $x, y\in \mathbb{R}$.

2020 Balkan MO Shortlist, N3

Tags: number theory , AZE EGMO TST , TST , BMO Shortlist

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 $x_1,x_2,\cdots x_k$. $Albania$

2024 Chile TST IMO, 4

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

Let $\alpha$ be a real number. Find all the functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $f(f(x+y))=f(x+y) +f(x)f(y)+ \alpha xy$ for all $x,y \in \mathbb{R}$

2019 India IMO Training Camp, P2

Tags: geometry , TST

Let $ABC$ be an acute-angled scalene triangle with circumcircle $\Gamma$ and circumcenter $O$. Suppose $AB < AC$. Let $H$ be the orthocenter and $I$ be the incenter of triangle $ABC$. Let $F$ be the midpoint of the arc $BC$ of the circumcircle of triangle $BHC$, containing $H$. Let $X$ be a point on the arc $AB$ of $\Gamma$ not containing $C$, such that $\angle AXH = \angle AFH$. Let $K$ be the circumcenter of triangle $XIA$. Prove that the lines $AO$ and $KI$ meet on $\Gamma$. [i]Proposed by Anant Mudgal[/i]

2019 Iran Team Selection Test, 2

Tags: Iranian TST , TST , Iran

$a, a_1,a_2,\dots ,a_n$ are natural numbers. We know that for any natural number $k$ which $ak+1$ is square, at least one of $a_1k+1,\dots ,a_n k+1$ is also square. Prove $a$ is one of $a_1,\dots ,a_n$ [i]Proposed by Mohsen Jamali[/i]

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.

2018 China Team Selection Test, 2

Tags: combinatorics , TST

There are $32$ students in the class with $10$ interesting group. Each group contains exactly $16$ students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their $interests-disparity$. Define $S$ as the sum of the $interests-disparity$ of all the couples, $\binom{32}{2}\left ( =\: 496 \right )$ ones in total. Determine the minimal possible value of $S$.

2015 Chile TST Ibero, 1

Tags: algebra , Chile , TST , Iberoamerican 2016 , function

Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$: \[ f(g(n)) = n + 2015, \] \[ g(f(n)) = n^2 + 2015. \]

2018 USA Team Selection Test, 2

Tags: function , algebra , functional equation , TST , probability , random walks

Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$, \[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\] [i]Proposed by Yang Liu and Michael Kural[/i]

2013 USA Team Selection Test, 3

Tags: geometry , circumcircle , TST , USA

Let $ABC$ be a scalene triangle with $\angle BCA = 90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK = BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL = AC$. The circumcircle of triangle $DKL$ intersects segment $AB$ at a second point $T$ (other than $D$). Prove that $\angle ACT = \angle BCT$.

2019 Slovenia Team Selection Test, 2

Tags: TST , inequalities

Prove, that for any positive real numbers $a, b, c$ who satisfy $a^2+b^2+c^2=1$ the following inequality holds. $\sqrt{\frac{1}{a}-a}+\sqrt{\frac{1}{b}-b}+\sqrt{\frac{1}{c}-c} \geq \sqrt{2a}+\sqrt{2b}+\sqrt{2c}$

2016 Dutch IMO TST, 4

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

Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.

2024 Junior Balkan Team Selection Tests - Moldova, 6

Tags: geometry , TST

In the isosceles triangle $ABC$, with $AB=BC$, points $X$ and $Y$ are the midpoints of the sides $AB$ and $AC$, respectively. Point $Z$ is the foot of the perpendicular from $B$ to $CX$. Prove that the circumcenter of the triangle $XYZ$ is of the line $AC$.

2025 JBMO TST - Turkey, 1

Tags: geometry , TST

Let $ABCD$ be a cyclic quadrilateral and let the intersection point of lines $AB$ and $CD$ be $E$. Let the points $K$ and $L$ be arbitrary points on sides $CD$ and $AB$ respectively, which satisfy the conditions $$\angle KAD = \angle KBC \quad \text{and} \quad \angle LDA = \angle LCB.$$ Prove that $EK = EL$.

2024 Indonesia TST, G

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.

2021 Romania Team Selection Test, 2

Tags: number theory , Romanian TST , TST , BritishMathematicalOlympiad

For any positive integer $n>1$, let $p(n)$ be the greatest prime factor of $n$. Find all the triplets of distinct positive integers $(x,y,z)$ which satisfy the following properties: $x,y$ and $z$ form an arithmetic progression, and $p(xyz)\leq 3.$

2016 Junior Balkan Team Selection Test, 1

Tags: geometry , TST

Let rightangled $\triangle ABC$ be given with right angle at vertex $C$. Let $D$ be foot of altitude from $C$ and let $k$ be circle that touches $BD$ at $E$, $CD$ at $F$ and circumcircle of $\triangle ABC$ at $G$. $a.)$ Prove that points $A$, $F$ and $G$ are collinear. $b.)$ Express radius of circle $k$ in terms of sides of $\triangle ABC$.

2013 Chile TST Ibero, 1

Tags: number theory , TST , Chile , Density

Prove that the equation \[ x^z + y^z = z^z \] has no solutions in postive integers.