MEMO 2013 T2

Let $x, y, z, w$ be nonzero real numbers such that $x+y \ne 0$ , $z+w \ne 0$ , and $xy+zw \ge 0$ . Prove that $$\left( \frac{x+y}{z+w} + \frac{z+w}{x+y} \right) ^{-1} + \frac{1}{2} \ge \left( \frac{x}{z} + \frac{z}{x} \right) ^{-1} + \left( \frac{y}{w} + \frac{w}{y} \right) ^{-1}$$