SILK-ROAD 2014 Problem 4

Find all $f:N\rightarrow N$ , such that $\forall m,n\in N$ $2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n)$

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.