Problem

Source: Indonesian MO (INAMO) 2020, Day 1, Problem 3, or APMO 2019 Problem 1

Tags: algebra, functional equation, APMO, 2020, Divisibility, Indonesia MO, Indonesian MO



The wording is just ever so slightly different, however the problem is identical. Problem 3. Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $n^2 + f(n)f(m)$ is a multiple of $f(n) + m$ for all natural numbers $m, n$.