Middle European Mathematical Olympiad 2013 Problem T-2

Let $x$, $y$, $z$, $w\in\mathbb{R}\setminus\{0\}$ such that $x+y\neq 0$, $z+w\neq 0$, and $xy+zw\geq 0$. Prove the inequality $$\left(\frac{x+y}{z+w}+\frac{z+w}{x+y}\right)^{-1}+\frac{1}{2}\geq\left(\frac{x}{z}+\frac{z}{x}\right)^{-1}+\left(\frac{y}{w}+\frac{w}{y}\right)^{-1}.$$