JBMO-SL 2016 A5

Let $x, y, z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$.

a) Prove the inequality

$$x+y+z \geq \sqrt{\frac{x y+1}{2}}+\sqrt{\frac{y z+1}{2}}+\sqrt{\frac{z x+1}{2}}$$

b) (Added by the problem selecting committee) When does the equality hold?