Question 41
Let $x, y, z, w \in \mathbb{R} \setminus \{0\}$ such that $x+y \neq 0$, $z+w \neq 0$, and $x y+z w \geq 0$. Prove the inequality
$$
\left(\frac{x+y}{z+w}+\frac{z+w}{x+y}\right)^{-1}+\frac{1}{2} \geq \left(\frac{x}{z}+\frac{z}{x}\right)^{-1}+\left(\frac{y}{w}+\frac{w}{y}\right)^{-1}
$$
$$
\left(\frac{x+y}{z+w}+\frac{z+w}{x+y}\right)^{-1}+\frac{1}{2} \geq \left(\frac{x}{z}+\frac{z}{x}\right)^{-1}+\left(\frac{y}{w}+\frac{w}{y}\right)^{-1}
$$