BALTIC-WAY 1992 Problem 13

Prove that for any positive $x_{1}, x_{2}, \ldots, x_{n}$ and $y_{1}, y_{2}, \ldots, y_{n}$ the inequality

$$\sum_{i=1}^{n} \frac{1}{x_{i} y_{i}} \geq \frac{4 n^{2}}{\sum_{i=1}^{n}\left(x_{i}+y_{i}\right)^{2}}$$

holds.