Problem

Source: Moldova TST Problem 6

Tags: inequalities



Let $a,b,c$ be positive real numbers such that $abc=1$. Prove the following inequality: $a^3+b^3+c^3+\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2} \geq \frac{9}{2}$.