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: 40

2010 APMO, 2
Tags: number theory , perfect power , additive number theory
For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.

1992 IMO Shortlist, 15
Tags: number theory , perfect power , additive combinatorics , additive number theory
Does there exist a set $ M$ with the following properties? [i](i)[/i] The set $ M$ consists of 1992 natural numbers. [i](ii)[/i] Every element in $ M$ and the sum of any number of elements have the form $ m^k$ $ (m, k \in \mathbb{N}, k \geq 2).$

2013 Kyiv Mathematical Festival, 5
Tags: perfect power , number theory , perfect square
Do there exist positive integers $a \ne b$ such that $ a+b$ is a perfect square and $a^3 +b^3$ is a fourth power of an integer?

2012 Dutch IMO TST, 1
Tags: perfect power , number theory , prime , gcd , greatest common divisor
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ . Show that for every integer $n > 1$, the following holds: $n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.

II Soros Olympiad 1995 - 96 (Russia), 9.5
Tags: number theory , perfect power
Give an example of four pairwise distinct natural numbers $a$, $b$, $c$ and $d$ such that $$a^2 + b^3 + c^4 = d^5.$$

2016 Argentina National Olympiad, 1
Tags: perfect power , number theory , arithmetic sequence
Find an arithmetic progression of $2016$ natural numbers such that neither is a perfect power but its multiplication is a perfect power. Clarification: A perfect power is a number of the form $n^k$ where $n$ and $k$ are both natural numbers greater than or equal to $2$.

2012 Dutch IMO TST, 1
Tags: perfect power , number theory , prime , gcd , greatest common divisor
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ . Show that for every integer $n > 1$, the following holds: $n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.

2016 Latvia National Olympiad, 1
Tags: number theory , perfect power
Given that $x$, $y$ and $z$ are positive integers such that $x^3y^5z^6$ is a perfect 7th power of a positive integer, show that also $x^5y^6z^3$ is a perfect 7th power.

2022 New Zealand MO, 2
Tags: perfect power , number theory , power of number
Is it possible to pair up the numbers $0, 1, 2, 3,... , 61$ in such a way that when we sum each pair, the product of the $31$ numbers we get is a perfect f ifth power?

2012 ELMO Shortlist, 3
Tags: number theory , additive number theory , extremal combinatorics , perfect power
Let $s(k)$ be the number of ways to express $k$ as the sum of distinct $2012^{th}$ powers, where order does not matter. Show that for every real number $c$ there exists an integer $n$ such that $s(n)>cn$. [i]Alex Zhu.[/i]

2012 ELMO Shortlist, 3
Tags: number theory , additive number theory , extremal combinatorics , perfect power
Let $s(k)$ be the number of ways to express $k$ as the sum of distinct $2012^{th}$ powers, where order does not matter. Show that for every real number $c$ there exists an integer $n$ such that $s(n)>cn$. [i]Alex Zhu.[/i]

2010 Contests, 2
Tags: number theory , perfect power , additive number theory
For a positive integer $k,$ call an integer a $pure$ $k-th$ $power$ if it can be represented as $m^k$ for some integer $m.$ Show that for every positive integer $n,$ there exists $n$ distinct positive integers such that their sum is a pure $2009-$th power and their product is a pure $2010-$th power.

2023 Regional Olympiad of Mexico West, 4
Tags: number theory , perfect square , perfect cube , perfect power
Prove that you can pick $15$ distinct positive integers between $1$ and $2023$, such that each one of them and the sum between some of them is never a perfect square, nor a perfect cube or any other greater perfect power.

VMEO I 2004, 2
Tags: fibonacci sequence , fibonacci , number theory , perfect power , fibonacci number , divisible
The Fibonacci numbers $(F_n)_{n=1}^{\infty}$ are defined as follows: $$F_1 = F_2 = 1, F_n = F_{n-2} + F_{n-1}, n = 3, 4, ...$$ Assume $p$ is a prime greater than $3$. With $m$ being a natural number greater than $3$, find all $n$ numbers such that $F_n$ is divisible by $p^m$.

2016 Latvia National Olympiad, 1
Tags: number theory , perfect power
Given that $x$ and $y$ are positive integers such that $xy^{433}$ is a perfect 2016-power of a positive integer, prove that $x^{433}y$ is also a perfect 2016-power.