Olympiad problems

VMEO III 2006 Shortlist, N13

Prove the following two inequalities: 1) If $n > 49$, then exist positive integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}<1$$ 2) If $n > 4$, then exist integer integers $a, b > 1$ such that $a+b=n$ and $$\frac{\phi (a)}{a}+\frac{\phi (b)}{b}>1$$

2020 Peru Cono Sur TST., P4

Tags: phi function , euler s phi function , perfect square , number theory

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

2020 International Zhautykov Olympiad, 1

Tags: euler s phi function , order , prime number , zhautykov , number theory

Given natural number n such that, for any natural $a,b$ number $2^a3^b+1$ is not divisible by $n$.Prove that $2^c+3^d$ is not divisible by $n$ for any natural $c$ and $d$

2008 Brazil Team Selection Test, 4

Tags: perfect square , phi function , euler s phi function , number theory

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

Dumbest FE I ever created, 1.

Tags: function , number theory , divisibility , euler s phi function , functional equation

Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $m$ and $n$, $$ m^{\phi(n)}+n^{\phi(m)} \mid f(m)^n + f(n)^m$$

2020 Peru Iberoamerican Team Selection Test, P4

Tags: number theory , perfect square , phi function , euler s phi function

Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.

2022 Durer Math Competition Finals, 1

Tags: number theory , euler s phi function , phi function , sequence

Let $c \ge 2$ be a fixed integer. Let $a_1 = c$ and for all $n \ge 2$ let $a_n = c \cdot \phi (a_{n-1})$. What are the numbers $c$ for which sequence $(a_n)$ will be bounded? $\phi$ denotes Euler’s Phi Function, meaning that $\phi (n)$ gives the number of integers within the set $\{1, 2, . . . , n\}$ that are relative primes to $n$. We call a sequence $(x_n)$ bounded if there exist a constant $D$ such that $|x_n| < D$ for all positive integers $n$.

2023 Brazil National Olympiad, 5

Tags: number theory , fermat's little theorem , euler s phi function

An integer $n \geq 3$ is [i]fabulous[/i] when there exists an integer $a$ with $2 \leq a \leq n - 1$ for which $a^n - a$ is divisible by $n$. Find all the [i]fabulous[/i] integers.