JBMO-SL 2007 A5

The real numbers $x, y, z, m, n$ are positive, such that $m+n \geq 2$. Prove that

$$\begin{gathered} x \sqrt{y z(x+m y)(x+n z)}+y \sqrt{x z(y+m x)(y+n z)}+z \sqrt{x y(z+m x)(x+n y)} \leq \\ \frac{3(m+n)}{8}(x+y)(y+z)(z+x) . \end{gathered}$$