Middle European Mathematical Olympiad 2015 Problem T-1

Prove that for all positive real numbers $a$, $b$, $c$ such that $abc = 1$ the following inequality holds: $$\frac{a}{2b + c^2} + \frac{b}{2c + a^2} + \frac{c}{2a + b^2} \leq \frac{a^2 + b^2 + c^2}{3}.$$