Olympiad problems

2010 Regional Olympiad of Mexico Center Zone, 2

Let $p>5$ be a prime number. Show that $p-4$ cannot be the fourth power of a prime number.

2003 IMO Shortlist, 8

Tags: modular arithmetic , number theory , prime numbers , Perfect Powers , IMO Shortlist

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 $A$ ?

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.

2012 ELMO Shortlist, 3

Tags: number theory , Additive Number Theory , Extremal combinatorics , Perfect Powers

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]

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 $m$ .

1992 IMO Longlists, 60

Tags: number theory , Perfect Powers , Additive combinatorics , Additive Number Theory , IMO Shortlist , IMO Longlist

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).$

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)

2019 Belarus Team Selection Test, 1.3

Tags: number theory , Diophantine equation , Perfect Powers

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]

2019 Czech and Slovak Olympiad III A, 3

Tags: number theory , Divisibility , Perfect Powers , national olympiad

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.

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.

2020 Peru IMO TST, 1

Tags: Peru , TST , number theory , Perfect Powers , prime numbers

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 $k>1$

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?

2010 APMO, 2

Tags: number theory , APMO , Perfect Powers , 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.

2012 ELMO Shortlist, 3

Tags: number theory , Additive Number Theory , Extremal combinatorics , Perfect Powers

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]

2020 OMpD, 2

Tags: combinatorics , Perfect Powers , 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?

2016 Latvia National Olympiad, 1

Tags: number theory , Perfect Powers

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.

2010 Contests, 2

Tags: number theory , APMO , Perfect Powers , 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.

2016 Latvia National Olympiad, 1

Tags: number theory , Perfect Powers

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.

1992 IMO Shortlist, 15

Tags: number theory , Perfect Powers , Additive combinatorics , Additive Number Theory , IMO Shortlist , IMO Longlist

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).$

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.

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.

2002 Brazil National Olympiad, 1

Tags: number theory , Perfect Squares , perfect cube , Perfect Powers , Subset , Sum

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.

2024 Brazil EGMO TST, 2

Tags: combinatorics , game , game strategy , Game Theory , Perfect Powers

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.

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

2024 Baltic Way, 16

Tags: number theory , number theory proposed , Perfect Powers , Divisors

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 $d=k^m+1$.