MathNet 2020 — Math Olympiad Vault — Part 04 Question 74

Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any distinct positive integers $x$, $y$, $z$, $x + y + z$ is a perfect square if and only if $f(x) + f(y) + f(z)$ is a perfect square.