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.

2010 APMO, 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.

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 $1$).

V Soros Olympiad 1998 - 99 (Russia), 10.1

Tags: perfect power , number theory

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.

2003 IMO Shortlist, 8

Tags: prime number , perfect power , modular arithmetic , number theory

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 $A$ ?

VMEO I 2004, 2

Tags: fibonacci number , fibonacci sequence , fibonacci , divisible , perfect power , number theory

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 $p^m$.

OIFMAT III 2013, 4

Tags: 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?

2012 ELMO Shortlist, 3

Tags: number theory , additive number theory , extremal combinatorics , perfect power

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]

I Soros Olympiad 1994-95 (Rus + Ukr), 11.5

Tags: perfect power , number theory

Prove that for any natural $n>1$ there are infinitely many natural numbers $m$ such that for any nonnegative integers $k_1$,$k_2$, $...$,$k_m$, $$m \ne k_1^n+ k_2^n+... k_n^n,$$

2020 OMpD, 2

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

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?

1992 IMO Longlists, 60

Tags: perfect power , additive combinatorics , additive number theory , 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).$

2016 Argentina National Olympiad, 1

Tags: number theory , arithmetic sequence , perfect power

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 $2$.

2019 Czech and Slovak Olympiad III A, 3

Tags: divisibility , perfect power , number theory

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.

1992 IMO Shortlist, 15

Tags: perfect power , additive combinatorics , additive number theory , 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).$

2012 ELMO Shortlist, 3

Tags: number theory , additive number theory , extremal combinatorics , perfect power

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]

2012 Dutch IMO TST, 1

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

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 $gcd(n, n @m) = 1$.

2023 Regional Olympiad of Mexico West, 4

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

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.

2006 China Team Selection Test, 2

Tags: order , perfect power , modular arithmetic

Find all positive integer pairs $(a,n)$ such that $\frac{(a+1)^n-a^n}{n}$ is an integer.

II Soros Olympiad 1995 - 96 (Russia), 9.5

Tags: perfect power , number theory

Give an example of four pairwise distinct natural numbers $a$, $b$, $c$ and $d$ such that $$a^2 + b^3 + c^4 = d^5.$$

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: perfect power , power of prime , 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?

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.

2006 China Team Selection Test, 2

Tags: modular arithmetic , order , perfect power

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: digit , perfect power , number theory

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.

2011 Austria Beginners' Competition, 1

Tags: number theory , perfect power

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)