Problem

Source: ISL 2019 N4

Tags: number theory, IMO Shortlist, IMO Shortlist 2019, functional equation, Hi



Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.