MathNet 2020 — Math Olympiad Vault — Part 02 Question 74

Let $C$ be a positive integer. Find all functions $f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*}$ such that for all integers $a$ and $b$ with sum $a+b \geqslant C$, the integer $a+f(b)$ divides $a^{2}+b f(a)$.