Problem

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2015

Tags: inequalities, algebra



Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \leq \frac{a^2+b^2+c^2}{2}$$