MathNet 2024-part-05 — 168 problems (EN–VI) Question 62

Let $\mathbb{N}$ be the set of all positive integers. Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that $mf(m) + (f(f(m)) + n)^2$ divides $4m^4 + n^2 f(f(n))^2$ for all positive integers $m, n$.