This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 606

2022 Latvia Baltic Way TST, P3
Tags: functional equation , TST
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[ f(f(x))+yf(xy+1) = f(x-f(y)) + xf(y)^2. \]for all real numbers $x$ and $y$.

2018 Peru Cono Sur TST, 3
Tags: geometry , TST
Let $ I $ be the incenter of a triangle $ ABC $ with $ AB \neq AC $, and let $ M $ be the midpoint of the arc $ BAC $ of the circumcircle of the triangle. The perpendicular line to $ AI $ passing through $ I $ intersects line $ BC $ at point $ D $. The line $ MI $ intersects the circumcircle of triangle $ BIC $ at point $ N $. Prove that line $ DN $ is tangent to the circumcircle of triangle $ BIC $.

2019 Macedonia Junior BMO TST, 3
Tags: combinatorics , TST
Define a colouring in tha plane the following way: - we pick a positive integer $m$; - let $K_{1}$, $K_{2}$, ..., $K_{m}$ be different circles with nonzero radii such that $K_{i}\subset K_{j}$ or $K_{j}\subset K_{i}$ if $i \neq j$; - the points in the plane that lie outside an arbitrary circle (one that is amongst the circles we pick) are coloured differently than the points that lie inside the circle. There are $2019$ points in the plane such that any $3$ of them are not collinear. Determine the maximum number of colours which we can use to colour the given points.

2018 Azerbaijan JBMO TST, 3
Tags: number theory , TST , AZE JBMO TST
Find all nonnegative integers $(x,y,z,u)$ with satisfy the following equation: $2^x + 3^y + 5^z = 7^u.$

2023 Austrian MO National Competition, 1
Tags: algebra , functional equation , TST , AZE IzHO TST
Given is a nonzero real number $\alpha$. Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy$$ for all $x, y \in \mathbb{R}$.

2024 Junior Balkan Team Selection Tests - Moldova, 10
Tags: JBMO Shortlist , algebra , AZE JBMO TST , Inequality , TST
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

2024 Moldova Team Selection Test, 7
Tags: inequalities , TST
Prove that $a=2$ is the greatest real number for which the inequality: $$ \frac{x_1}{x_n+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_n}{x_{n-1}+x_1} \ge a $$ holds true for any $n \ge 4$ and any positive real numbers $x_1, x_2,\dots,x_n$.

2021 China Team Selection Test, 3
Tags: inequalities , algebra , TST , China TST
Determine the greatest real number $ C $, such that for every positive integer $ n\ge 2 $, there exists $ x_1, x_2,..., x_n \in [-1,1]$, so that $$\prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}}$$.

2023 Azerbaijan IMO TST, 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 Turkey Team Selection Test, 1
Tags: geometry , TST
In triangle $ABC$, the incenter is $I$ and the circumcenter is $O$. Let $AI$ intersects $(ABC)$ second time at $P$ . The line passes through $I$ and perpendicular to $AI$ intersects $BC$ at $X$. The feet of the perpendicular from $X$ to $IO$ is $Y$. Prove that $A,P,X,Y$ cyclic.

2023 Israel TST, P1
Tags: ratio , TST , number theory
For positive integers $n$, let $f_2(n)$ denote the number of divisors of $n$ which are perfect squares, and $f_3(n)$ denotes the number of positive divisors which are perfect cubes. Prove that for each positive integer $k$ there exists a positive integer $n$ for which $\frac{f_2(n)}{f_3(n)}=k$.

2022 Azerbaijan JBMO TST, A2
Tags: algebra , TST , Inequality , AZE JBMO TST
For positive real numbers $a,b,c$, $\frac{1}{a}+\frac{1}{b} + \frac{1}{c} \ge \frac{3}{abc}$ is true. Prove that: $$ \frac{a^2+b^2}{a^2+b^2+1}+\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1} \ge 2$$

2022 JBMO Shortlist, C5
Tags: combinatorics , JBMO Shortlist , TST , AZE IzHO TST
Let $S$ be a finite set of points in the plane, such that for each $2$ points $A$ and $B$ in $S$, the segment $AB$ is a side of a regular polygon all of whose vertices are contained in $S$. Find all possible values for the number of elements of $S$. Proposed by [i]Viktor Simjanoski, Macedonia[/i]

2022 Greece JBMO TST, 1
Tags: number theory , AZE JBMO TST , JBMO Shortlist , TST
Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

2015 Belarus Team Selection Test, 1
Tags: number theory , Diophantine equation , diophantine , Belarus , TST , 2015
Solve the equation in nonnegative integers $a,b,c$: $3^a+2^b+2015=3c!$ I.Gorodnin

2023 Israel TST, P3
Tags: TST , geometry
In triangle $ABC$ the orthocenter is $H$ and the foot of the altitude from $A$ is $D$. Point $P$ satisfies $AP=HP$, and the line $PA$ is tangent to $(ABC)$. Line $PD$ intersects lines $AB, AC$ at points $X,Y$ respectively. Prove that $\angle YHX = \angle BAC$ or $\angle YHX+\angle BAC= 180^\circ$.

2024 Junior Macedonian Mathematical Olympiad, 1
Tags: algebra , Inequality , TST
Let $a, b$, and $c$ be positive real numbers. Prove that \[\frac{a^4 + 3}{b} + \frac{b^4 + 3}{c} + \frac{c^4 + 3}{a} \ge 12.\] When does equality hold? [i]Proposed by Petar Filipovski[/i]

2024 Germany Team Selection Test, 3
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.

2020 Junior Balkan Team Selection Tests-Serbia, 3#
Tags: Inequality , TST
Given are real numbers $a_1, a_2,...,a_{101}$ from the interval $[-2,10]$ such that their sum is $0$. Prove that the sum of their squares is smaller than $2020$.

2023 Chile TST Ibero., 1
Tags: TST , Chile , number theory
Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers \[ a_n = 4^n c + \frac{4^n - (-1)^n}{5} \] contains at least one perfect square.

2022 Saudi Arabia JBMO TST, 2
Tags: geometry , TST
Let $BB'$, $CC'$ be the altitudes of an acute-angled triangle $ABC$. Two circles passing through $A$ and $C'$ are tangent to $BC$ at points $P$ and $Q$. Prove that $A, B', P, Q$ are concyclic.

1986 China Team Selection Test, 3
Tags: algebra , Number theoretic functions , China , TST , decimal representation , Digits
Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that: [b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$. [b]II.[/b] Find such $A_k$ for $19^{86}.$

2008 Junior Balkan Team Selection Tests - Romania, 3
Tags: TST , number theory
Solve in prime numbers $ 2p^q \minus{} q^p \equal{} 7$.

2011 India IMO Training Camp, 2
Tags: number theory , TST
Prove that for no integer $ n$ is $ n^7 \plus{} 7$ a perfect square.

2025 JBMO TST - Turkey, 5
Tags: number theory , TST
Find all positive integers $n$ such that a positive integer power of $2n^2+4n-1$ equals to a positive integer power of $3n+4$.