APMO 2002 Problem 4

Let $x, y, z$ be positive numbers such that

$$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$$

Show that

$$\sqrt{x+y z}+\sqrt{y+z x}+\sqrt{z+x y} \geq \sqrt{x y z}+\sqrt{x}+\sqrt{y}+\sqrt{z}$$