Problem

Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2016

Tags: algebra, inequalities



Let $x$, $y$ and $z$ be positive real numbers such that $\sqrt{xy} + \sqrt{yz} + \sqrt{zx} = 3$. Prove that $\sqrt{x^3+x} + \sqrt{y^3+y} + \sqrt{z^3+z} \geq \sqrt{6(x+y+z)}$