CH-SMO 2020

Let $\mathbb N$ be the set of positive integers. Find all functions $f\colon\mathbb N\to \mathbb N$ such that for every $m,n\in \mathbb N$ , $$f(m)+f(n)\mid m+n.$$

We are given $n$ distinct rectangles in the plane. Prove that between the $4n$ interior angles formed by these rectangles at least $4\sqrt n$ are distinct.

Let $\varphi$ denote the Euler phi-function. Prove that for every positive integer $n$ $$2^{n(n+1)} | 32 \cdot \varphi \left( 2^{2^n} - 1 \right).$$