Problem

Source: Mediterranean Mathematics Olympiad 2017, Problem 4

Tags: inequalities



Let $x,y,z$ and $a,b,c$ be positive real numbers with $a+b+c=1$. Prove that $$\left(x^2+y^2+z^2\right) \left( \frac{a^3}{x^2+2y^2} + \frac{b^3}{y^2+2z^2} + \frac{c^3}{z^2+2x^2} \right) \ge\frac19.$$