KöMaL 2006 A405

The real numbers $a$, $b$, $c$, $x$, $y$, $z$ satisfy $a \ge b \ge c$>0 and $x \ge y \ge z$>0. Prove that $\frac{a^2x^2}{(by+cz)(bz+cy)} + \frac{b^2y^2}{(cz+ax)(cx+az)} + \frac{c^2z^2}{(ax+by)(ay+bx)} \ge \frac34.$