Školjka

%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 Prove that there is a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $\forall n \in \mathbb{N}$ $$f(f(n))=n^2 \text{.}$$

%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 Let $n$ be a positive integer. Show that the numbers $$\binom{2^n - 1}{0},\; \binom{2^n - 1}{1},\; \binom{2^n - 1}{2},\; \ldots,\; \binom{2^n - 1}{2^{n - 1} - 1}$$ are congruent modulo $2^n$ to $1$, $3$, $5$, $\ldots$, $2^n - 1$ in some order. Proposed by Duskan Dukic, Serbia

%V0 For every integer $k \geq 2,$ prove that $2^{3k}$ divides the number $$\binom{2^{k + 1}}{2^{k}} - \binom{2^{k}}{2^{k - 1}}$$ but $2^{3k + 1}$ does not. Author: unknown author, Poland

%V0 Find all positive integers $n$ such that there exists a unique integer $a$ such that $0\leq a < n!$ with the following property: $$n!\mid a^n + 1$$