Question 49
If $\mathbb{N}$ is the set of positive integers, find all functions $f: \mathbb{N} \to \mathbb{N}$ such that
(i) For any positive integer $a \gt 2021^{2021}$ it holds that $f(a) \le a$.
(ii) $\frac{a^2 f(b) + b^2 f(a)}{f(a) + f(b)}$ is a positive integer for all $a, b \in \mathbb{N}$.
(i) For any positive integer $a \gt 2021^{2021}$ it holds that $f(a) \le a$.
(ii) $\frac{a^2 f(b) + b^2 f(a)}{f(a) + f(b)}$ is a positive integer for all $a, b \in \mathbb{N}$.