Olympiad problems

1999 Greece JBMO TST, 4

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

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.

2010 Contests, 2

Tags: perfect power , additive number theory , 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.

2024 Brazil EGMO TST, 2

Tags: game , game strategy , game theory , perfect power , combinatorics

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.

2016 Thailand TSTST, 3

Tags: number theory , perfect power

Determine whether there exists a positive integer $a$ such that $$2015a,2016a,\dots,2558a$$ are all perfect power.

2016 Latvia National Olympiad, 1

Tags: perfect power , number theory

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.

2013 Kyiv Mathematical Festival, 5

Tags: number theory , perfect power , 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: number theory , perfect power , prime , gcd , greatest common divisor

For all positive integers $a$ and $b$, we dene $a @ b = \frac{a - b}{gcd(a, b)}$ . 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 $gcd(n, n @m) = 1$.

2002 Brazil National Olympiad, 1

Tags: number theory , subset , perfect cube , perfect power , sum , perfect square

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.

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?

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.

2019 Mexico National Olympiad, 1

Tags: function , number theory , perfect power , number of divisors

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]

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 $k>1$

2024 Baltic Way, 16

Tags: perfect power , divisor , number theory

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 $d=k^m+1$.

2017 Balkan MO Shortlist, A5

Tags: algebra , perfect power , polynomial , integer polynomial

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 $m$ .