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

1989 IMO Longlists, 93
Tags: modular arithmetic , prime number , sequence , power of number , number theory
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

2013 IFYM, Sozopol, 3
Tags: number theory , power of number , divisibility , prime number
Let $a$ and $b$ be two distinct natural numbers. It is known that $a^2+b|b^2+a$ and that $b^2+a$ is a power of a prime number. Determine the possible values of $a$ and $b$.

2006 Junior Tuymaada Olympiad, 2
Tags: number theory , prime , power , power of number
Ten different odd primes are given. Is it possible that for any two of them, the difference of their sixteenth powers to be divisible by all the remaining ones ?

2011 Ukraine Team Selection Test, 3
Tags: number theory , perfect power , power of number
Given a positive integer $ n> 2 $. Prove that there exists a natural $ K $ such that for all integers $ k \ge K $ on the open interval $ ({{k} ^{n}}, \ {{(k + 1)} ^{n}}) $ there are $n$ different integers, the product of which is the $n$-th power of an integer.

1989 IMO Shortlist, 30
Tags: prime number , sequence , power of number , modular arithmetic , number theory
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

1989 IMO, 5
Tags: modular arithmetic , prime number , sequence , power of number , number theory
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

2012 Polish MO Finals, 1
Tags: number theory , rational number , power of number
Decide, whether exists positive rational number $w$, which isn't integer, such that $w^w$ is a rational number.

1989 Spain Mathematical Olympiad, 4
Tags: number theory , power of number , perfect square
Show that the number $1989$ as well as each of its powers $1989^n$ ($n \in N$), can be expressed as a sum of two positive squares in at least two ways.

2022 New Zealand MO, 2
Tags: number theory , perfect power , 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?

2022 Cono Sur, 3
Tags: number theory , digit , power of number
Prove that for every positive integer $n$ there exists a positive integer $k$, such that each of the numbers $k, k^2, \dots, k^n$ have at least one block of $2022$ in their decimal representation. For example, the numbers 4[b]2022[/b]13 and 544[b]2022[/b]1[b]2022[/b] have at least one block of $2022$ in their decimal representation.