JBMO-SL 2011 A2

Let $x, y, z$ be positive real numbers. Prove that:

$$\frac{x+2 y}{z+2 x+3 y}+\frac{y+2 z}{x+2 y+3 z}+\frac{z+2 x}{y+2 z+3 x} \leq \frac{3}{2}$$