Školjka

%V0 Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations $$a^n + pb = b^n + pc = c^n + pa$$ then $a = b = c$. Proposed by Angelo Di Pasquale, Australia

%V0 Let $a_1$, $a_2$, $\ldots$, $a_n$ be distinct positive integers, $n\ge 3$. Prove that there exist distinct indices $i$ and $j$ such that $a_i + a_j$ does not divide any of the numbers $3a_1$, $3a_2$, $\ldots$, $3a_n$. Proposed by Mohsen Jamaali, Iran

%V0 A positive integer $N$ is called balanced, if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P\!\left(x\right) = \left(x+a\right)\left(x+b\right)$. a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P\!\left(1\right)$, $P\!\left(2\right)$, ..., $P\!\left(50\right)$ are balanced. b) Prove that if $P\!\left(n\right)$ is balanced for all positive integers $n$, then $a=b$. Proposed by Jorge Tipe, Peru

%V0 Find all surjective functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ $m \mid n \Leftrightarrow f(m) \mid f(n)$

%V0 Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $$f(f(f(n))) + f(f(n)) + f(n) = 3n$$ for all $n \in \mathbb{N}$.

%V0 Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ $f(f(m)+f(n))=m+n$