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: 12

2022 District Olympiad, P1
Tags: Arithmetic Function , functional equation , function , algebra
Let $f:\mathbb{N}^*\rightarrow \mathbb{N}^*$ be a function such that $\frac{x^3+3x^2f(y)}{x+f(y)}+\frac{y^3+3y^2f(x)}{y+f(x)}=\frac{(x+y)^3}{f(x+y)},~(\forall)x,y\in\mathbb{N}^*.$ $a)$ Prove that $f(1)=1.$ $b)$ Find function $f.$

2019 Belarus Team Selection Test, 3.1
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Russian TST 2019, P1
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2019 Hong Kong TST, 1
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2019 Estonia Team Selection Test, 9
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2019 Estonia Team Selection Test, 9
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2019 Philippine TST, 4
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2019 Germany Team Selection Test, 1
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2019 Germany Team Selection Test, 1
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2019 Brazil Team Selection Test, 1
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2019 Thailand TST, 1
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

2018 IMO Shortlist, N1
Tags: number theory , Arithmetic Function , IMO Shortlist , number of divisors , 2018 , divisor
Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.