Question 164
Let $\mathbb{N}^* = \{1,2,3,...\}$ be the set of positive integers. Find all functions $f: \mathbb{N}^* \rightarrow \mathbb{N}^*$ such that
$$
f(f(m)^2 + 2f(n)^2) = m^2 + 2n^2, \text{ for all } m, n \in \mathbb{N}^*.
$$
$$
f(f(m)^2 + 2f(n)^2) = m^2 + 2n^2, \text{ for all } m, n \in \mathbb{N}^*.
$$