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

2020 OMpD, 2
Tags: combinatorics , perfect power , game strategy , game theory
A pile of $2020$ stones is given. Arnaldo and Bernaldo play the following game: In each move, it is allowed to remove $1, 4, 16, 64, ...$ (any power of $4$) stones from the pile. They make their moves alternately, and the player who can no longer play loses. If Arnaldo is the first to play, who has the winning strategy?

2019 Mexico National Olympiad, 1
Tags: perfect power , number theory , number of divisors , function
An integer number $m\geq 1$ is [i]mexica[/i] if it's of the form $n^{d(n)}$, where $n$ is a positive integer and $d(n)$ is the number of positive integers which divide $n$. Find all mexica numbers less than $2019$. Note. The divisors of $n$ include $1$ and $n$; for example, $d(12)=6$, since $1, 2, 3, 4, 6, 12$ are all the positive divisors of $12$. [i]Proposed by Cuauhtémoc Gómez[/i]

2010 Regional Olympiad of Mexico Center Zone, 2
Tags: perfect power , number theory , prime
Let $p>5$ be a prime number. Show that $p-4$ cannot be the fourth power of a prime number.

1999 Greece JBMO TST, 4
Tags: perfect power , number theory , perfect cube
Examine whether exists $n \in N^*$, such that: (a) $3n$ is perfect cube, $4n$ is perfect fourth power and $5n$ perfect fifth power (b) $3n$ is perfect cube, $4n$ is perfect fourth power, $5n$ perfect fifth power and $6n$ perfect sixth power

2006 China Team Selection Test, 2
Tags: perfect power , modular arithmetic , order
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.

2024 Brazil EGMO TST, 2
Tags: combinatorics , game , game strategy , game theory , perfect power
Let \( m \) and \( n \) be positive integers. Kellem and Carmen play the following game: initially, the number $0$ is on the board. Starting with Kellem and alternating turns, they add powers of \( m \) to the previous number on the board, such that the new value on the board does not exceed \( n \). The player who writes \( n \) wins. Determine, for each pair \( (m, n) \), who has the winning strategy. [b]Note:[/b] A power of \( m \) is a number of the form \( m^k \), where \( k \) is a non-negative integer.

2020 Peru IMO TST, 1
Tags: number theory , perfect power , prime number
Find all pairs $(m,n)$ of positive integers numbers with $m>1$ such that: For any positive integer $b \le m$ that is not coprime with $m$, its posible choose positive integers $a_1, a_2, \cdots, a_n$ all coprimes with $m$ such that: $$m+a_1b+a_2b^2+\cdots+a_nb^n$$ Is a perfect power. Note: A perfect power is a positive integer represented by $a^k$, where $a$ and $k$ are positive integers with $k>1$

2017 Balkan MO Shortlist, A5
Tags: polynomial , algebra , integer polynomial , perfect power
Consider integers $m\ge 2$ and $n\ge 1$. Show that there is a polynomial $P(x)$ of degree equal to $n$ with integer coefficients such that $P(0),P(1),...,P(n)$ are all perfect powers of $m$ .

2022 Saudi Arabia BMO + EGMO TST, 1.1
Tags: perfect power , number theory
Find all positive integers $k$ such that the product of the first $k$ primes increased by $1$ is a power of an integer (with an exponent greater than $1$).

2011 Austria Beginners' Competition, 1
Tags: perfect power , number theory
Let $x$ be the smallest positive integer for which $2x$ is the square of an integer, $3x$ is the third power of an integer, and $5x$ is the fifth power of an integer. Find the prime factorization of $x$. (St. Wagner, Stellenbosch University)

I Soros Olympiad 1994-95 (Rus + Ukr), 11.5
Tags: number theory , perfect power
Prove that for any natural $n>1$ there are infinitely many natural numbers $m$ such that for any nonnegative integers $k_1$,$k_2$, $...$,$k_m$, $$m \ne k_1^n+ k_2^n+... k_n^n,$$

OIFMAT III 2013, 4
Tags: perfect power , perfect square , number theory
Show that there exists a set of infinite positive integers such that the sum of an arbitrary finite subset of these is never a perfect square. What happens if we change the condition from not being a perfect square to not being a perfect power?

2002 Brazil National Olympiad, 1
Tags: perfect square , number theory , perfect cube , perfect power , sum , subset
Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

2019 Belarus Team Selection Test, 1.3
Tags: number theory , diophantine equation , perfect power
Given the equation $$ a^b\cdot b^c=c^a $$ in positive integers $a$, $b$, and $c$. [i](i)[/i] Prove that any prime divisor of $a$ divides $b$ as well. [i](ii)[/i] Solve the equation under the assumption $b\ge a$. [i](iii)[/i] Prove that the equation has infinitely many solutions. [i](I. Voronovich)[/i]

2016 Latvia Baltic Way TST, 20
Tags: power of prime , number theory , perfect power
For what pairs of natural numbers $(a, b)$ is the expression $$(a^6 + 21a^4b^2 + 35a^2b^4 + 7b^6) (b^6 + 21b^4a^2 + 35b^2a^4 + 7a^6)$$ the power of a prime number?

2019 Czech and Slovak Olympiad III A, 3
Tags: number theory , divisibility , perfect power
Let $a,b,c,n$ be positive integers such that the following conditions hold (i) numbers $a,b,c,a+b+c$ are pairwise coprime, (ii) number $(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)$ is a perfect $n$-th power. Prove, that the product $abc$ can be expressed as a difference of two perfect $n$-th powers.

2003 IMO Shortlist, 8
Tags: modular arithmetic , number theory , prime number , perfect power
Let $p$ be a prime number and let $A$ be a set of positive integers that satisfies the following conditions: (i) the set of prime divisors of the elements in $A$ consists of $p-1$ elements; (ii) for any nonempty subset of $A$, the product of its elements is not a perfect $p$-th power. What is the largest possible number of elements in $A$ ?

1985 Bundeswettbewerb Mathematik, 1
Tags: number theory , perfect power
Prove that none of the numbers $11, 111, 1111, ...$ is a square number, cube number or higher power of a natural number.

V Soros Olympiad 1998 - 99 (Russia), 10.1
Tags: number theory , perfect power
Find some natural number $a$ such that $2a$ is a perfect square, $3a$ is a perfect cube, $5a$ is the fifth power of some natural number.

2011 Ukraine Team Selection Test, 3
Tags: perfect power , number theory , 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.

2016 Thailand TSTST, 3
Tags: perfect power , number theory
Determine whether there exists a positive integer $a$ such that $$2015a,2016a,\dots,2558a$$ are all perfect power.

2022 Peru MO (ONEM), 4
Tags: number theory , digit , perfect power
For each positive integer n, the number $R(n) = 11 ... 1$ is defined, which is made up of exactly $n$ digits equal to $1$. For example, $R(5) = 11111$. Let $n > 4$ be an integer for which, by writing all the positive divisors of $R(n)$, it is true that each written digit belongs to the set $\{0, 1\}$. Show that $n$ is a power of an odd prime number. Clarification: A power of an odd prime number is a number of the form $p^a$, where $p$ is an odd prime number and $a$ is a positive integer.

2024 Baltic Way, 16
Tags: perfect power , number theory , divisor
Determine all composite positive integers $n$ such that, for each positive divisor $d$ of $n$, there are integers $k\geq 0$ and $m\geq 2$ such that $d=k^m+1$.

2006 China Team Selection Test, 2
Tags: perfect power , modular arithmetic , order
Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.

1992 IMO Longlists, 60
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).$