Olympiad problems

VMEO I 2004, 2

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

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 $p^m$.

2010 Regional Olympiad of Mexico Center Zone, 2

Tags: perfect power , prime , number theory

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

2012 ELMO Shortlist, 3

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

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]

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.

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.

2010 Contests, 2

Tags: number theory , perfect power , 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.

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.

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 $2$.

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.

2022 Saudi Arabia BMO + EGMO TST, 1.1

Tags: number theory , 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 $1$).

2019 Mexico National Olympiad, 1

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

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]