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: 22

2019 Mexico National Olympiad, 1
Tags: number theory , number of divisors , Perfect power , 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]

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

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

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.

2017 Balkan MO Shortlist, A5
Tags: polynomial , algebra , Integer Polynomial , Perfect Powers , 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 $m$ .

2011 Austria Beginners' Competition, 1
Tags: Perfect Powers , 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)

2012 Dutch IMO TST, 1
Tags: number theory , Perfect power , 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 $gcd(n, n @m) = 1$.

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

2016 Latvia Baltic Way TST, 20
Tags: power of prime , Perfect Powers , Perfect power , number theory
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?

OIFMAT III 2013, 4
Tags: Perfect Squares , Perfect Square , Perfect power , 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?

VMEO I 2004, 2
Tags: Fibonacci Numbers , Fibonacci sequence , Fibonacci , number theory , Perfect power , divisibe
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 $p^m$.

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

2012 Dutch IMO TST, 1
Tags: number theory , Perfect power , 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 $gcd(n, n @m) = 1$.

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

2022 Peru MO (ONEM), 4
Tags: number theory , Perfect Powers , Perfect power , Digits
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.

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?

2023 Regional Olympiad of Mexico West, 4
Tags: number theory , Perfect Powers , Perfect power , Perfect Square , perfect cube
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.

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

1999 Greece JBMO TST, 4
Tags: number theory , perfect cube , Perfect Powers , Perfect power
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 Argentina National Olympiad, 1
Tags: number theory , Perfect power , 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 $2$.

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 $$a^2 + b^3 + c^4 = d^5.$$

2022 Saudi Arabia BMO + EGMO TST, 1.1
Tags: number theory , Perfect Powers , Perfect power
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 $1$).