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: 10

2017 Brazil Team Selection Test, 4
Tags: algebra , functional equation , IMO Shortlist , nice problem , Hi
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2013 Turkey Team Selection Test, 1
Tags: algebra , polynomial , number theory , greatest common divisor , modular arithmetic , number theory proposed , nice problem
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

2017 Estonia Team Selection Test, 6
Tags: algebra , functional equation , IMO Shortlist , nice problem , Hi
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2017 Estonia Team Selection Test, 6
Tags: algebra , functional equation , IMO Shortlist , nice problem , Hi
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2016 IMO Shortlist, A4
Tags: algebra , functional equation , IMO Shortlist , nice problem , Hi
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2013 IFYM, Sozopol, 4
Tags: algebra , polynomial , number theory , greatest common divisor , modular arithmetic , number theory proposed , nice problem
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

2021 Switzerland - Final Round, 6
Tags: function , algebra , nice problem
Let $\mathbb{N}$ be the set of positive integers. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function such that for every positive integer $n \in \mathbb{N}$ $$ f(n) -n<2021 \quad \text{and} \quad f^{f(n)}(n) =n$$ Prove that $f(n)=n$ for infinitely many $n \in \mathbb{N}$

2019 Romania Team Selection Test, 2
Tags: algebra , polynomial , number theory , greatest common divisor , modular arithmetic , number theory proposed , nice problem
Find all pairs of integers $(m,n)$ such that $m^6 = n^{n+1} + n -1$.

Russian TST 2017, P2
Tags: algebra , functional equation , IMO Shortlist , nice problem , Hi
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2006 India Regional Mathematical Olympiad, 5
Tags: geometry , inradius , geometry unsolved , nice problem
Let $ ABCD$ be a quadrilateral in which $ AB$ is parallel to $ CD$ and perpendicular to $ AD; AB \equal{} 3CD;$ and the area of the quadrilateral is $ 4$. if a circle can be drawn touching all the four sides of the quadrilateral, find its radius.