Baltic Way 1992 · Problem 13

Solution: Since $\left(x_{i}+y_{i}\right)^{2} \geq 4 x_{i} y_{i}$, it is sufficient to prove that

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

This can easily be done by induction using the fact that $a+\frac{1}{a} \geq 2$ for any $a>0$. It also follows directly from the Cauchy-Schwarz inequality.