MathNet 2011 Question 89

Let $x$, $y$, $z$ be positive real numbers. Prove that:
$$
\frac{x+2y}{z+2x+3y} + \frac{y+2z}{x+2y+3z} + \frac{z+2x}{y+2z+3x} \leq \frac{3}{2}
$$