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 Bulgaria JBMO TST, 1
Tags: algebra , TST
Determine all triples $(a,b,c)$ of real numbers such that $$ (2a+1)^2 - 4b = (2b+1)^2 - 4c = (2c+1)^2 - 4a = 5. $$

2025 Junior Balkan Team Selection Tests - Romania, P2
Tags: geometry , TST
Consider the isosceles triangle $ABC$ with $\angle A>90^\circ$ and the circle $\omega$ of radius $AC$ centered at $A.$ Let $M$ be the midpoint of $AC.$ The line $BM$ intersects $\omega$ a second time at $D.$ Let $E$ be a point on $\omega$ such that $BE\perp AC.$ Let $N$ be the intersection of $DE$ and $AC.$ Prove that $AN=2\cdot AB.$

2019 Junior Balkan Team Selection Tests - Romania, 2
Tags: number theory , TST
Determine all positive integers $n$ such that $4k^2 +n$ is a prime number for all non-negative integer $k$ smaller than $n$.

2020 Hong Kong TST, 4
Tags: algebra , TST , 2020 , hong Kong TST , fuctional equation , fe , Reals
Find all real-valued functions $f$ defined on the set of real numbers such that $$f(f(x)+y)+f(x+f(y))=2f(xf(y))$$ for any real numbers $x$ and $y$.

2021 Bolivian Cono Sur TST, 1
Tags: combinatorics , coins , Bolivia , Cono Sur TST , TST
[b]a)[/b] Among $9$ apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making $2$ weighing in a two-course balance? [b]b)[/b] Find the least necessary number of weighing that must be done to cover a false currency between $27$ coins if all the others are true.

2015 Chile TST Ibero, 4
Tags: inequalities , TST , Chile , algebra
Let $x, y \in \mathbb{R}^+$. Prove that: \[ \left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \geq \left( 1 + \frac{2}{x + y} \right)^2. \]

2024 Brazil Team Selection Test, 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 $x, y \in \mathbb R_{>0}$.

2021 Saudi Arabia JBMO TST, 1
Tags: number theory , JBMO Shortlist , TST
Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that $73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.

2018 China Team Selection Test, 3
Tags: geometry , TST
Circle $\omega$ is tangent to sides $AB$,$AC$ of triangle $ABC$ at $D$,$E$ respectively, such that $D\neq B$, $E\neq C$ and $BD+CE<BC$. $F$,$G$ lies on $BC$ such that $BF=BD$, $CG=CE$. Let $DG$ and $EF$ meet at $K$. $L$ lies on minor arc $DE$ of $\omega$, such that the tangent of $L$ to $\omega$ is parallel to $BC$. Prove that the incenter of $\triangle ABC$ lies on $KL$.

2019 Moldova Team Selection Test, 1
Tags: number theory , easy , TST
Let $S$ be the set of all natural numbers with the property: the sum of the biggest three divisors of number $n$, different from $n$, is bigger than $n$. Determine the largest natural number $k$, which divides any number from $S$. (A natural number is a positive integer)

2021 Azerbaijan IZhO TST, 3
Tags: number theory , TST , AZE IzHO TST
For each $n \in N$ let $S(n)$ be the sum of all numbers in the set {1,2,3,…,n} which are relatively prime to $n$. a. Show that $2S(n) $ is not aperfect square for any $n$. b. Given positive integers $m,n$ with odd n, show that the equation $2S(x)=y^n$ has at least one solution $(x,y)$ among positive integers such that $m|x$.

2017 Iran Team Selection Test, 2
Tags: geometry , TST
Let $P$ be a point in the interior of quadrilateral $ABCD$ such that: $$\angle BPC=2\angle BAC \ \ ,\ \ \angle PCA = \angle PAD \ \ ,\ \ \angle PDA=\angle PAC$$ Prove that: $$\angle PBD= \left | \angle BCA - \angle PCA \right |$$ [i]Proposed by Ali Zamani[/i]

2025 Turkey EGMO TST, 5
Tags: geometry , TST
In triangle \( ABC \), the incircle is tangent to side \( BC \) at point \( D \), the excircle opposite vertex \( B \) is tangent to line \( AB \) at point \( X \), and the excircle opposite vertex \( C \) is tangent to line \( AC \) at point \( Y \). If \( T \) is the midpoint of segment \( [AD] \) and \( U \) is the circumcenter of triangle \( AXY \), show that \( UT \perp BC \).

2021 Israel TST, 2
Tags: combinatorial geometry , distances , TST , combinatorics
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 $XY=m(X)$.

1981 Romania Team Selection Tests, 1.
Tags: algebra , polynomial , TST
Consider the polynomial $P(X)=X^{p-1}+X^{p-2}+\ldots+X+1$, where $p>2$ is a prime number. Show that if $n$ is an even number, then the polynomial \[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\] is divisible by $X^2+1$. [i]Mircea Becheanu[/i]

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: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

2004 Germany Team Selection Test, 1
Tags: function , Functional Equations , Closed Formula , Germany , Team Selection Test , TST , 2004
A function $f$ satisfies the equation \[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\] for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.

2018 Polish MO Finals, 4
Tags: number theory , Poland , TST
Let $n$ be a positive integer. Suppose there are exactly $M$ squarefree integers $k$ such that $\left\lfloor\frac nk\right\rfloor$ is odd in the set $\{ 1, 2,\ldots, n\}$. Prove $M$ is odd. An integer is [i]squarefree[/i] if it is not divisible by any square other than $1$.

2016 China Team Selection Test, 2
Tags: combinatorics , number theory , China TST , China , TST
In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?

2024 Romania Team Selection Tests, P4
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 $x, y \in \mathbb R_{>0}$.

2016 JBMO TST - Turkey, 6
Tags: inequalities , algebra , TST
Prove that \[ (x^4+y)(y^4+z)(z^4+x) \geq (x+y^2)(y+z^2)(z+x^2) \] for all positive real numbers $x, y, z$ satisfying $xyz \geq 1$.

2016 Azerbaijan JBMO TST, 3
Tags: combinatorics , TST , AZE JBMO TST
All cells of the $m\times n$ table are colored either white or black such that all corner cells of any rectangle containing the cells of this table with sides greater than one cell are not the same color. For values $m = 2, 3, 4,$ find all $n$ such that the mentioned coloring is possible.

2022 Serbia JBMO TST, 1
Tags: algebra , inequalities , TST
Prove that for all positive real numbers $a$, $b$ the following inequality holds: \begin{align*} \sqrt{\frac{a^2+b^2}{2}}+\frac{2ab}{a+b}\ge \frac{a+b}{2}+ \sqrt{ab} \end{align*} When does equality hold?

2013 IFYM, Sozopol, 5
Tags: number theory , TST
Find all positive integers $n$ satisfying $2n+7 \mid n! -1$.

2020 JBMO TST of France, 4
Tags: algebra , Inequality , TST
$a, b, c$ are real positive numbers for which $a+b+c=3$. Prove that $a^{12}+b^{12}+c^{12}+8(ab+bc+ca) \geq 27$