Cauchy-Schwarz: Problema 7

Let \(x_{1}, \ldots, x_{n},\) \(y_{1}, \ldots, y_{n} \in \) ℝ such that \( \sum_{i=1}^{n} x_{i}^{2} = \sum_{i=1}^{n} y_{i}^{2}=1.\) Prove that: $$ (x_{1}y_{2} – x_{2}y_{1})^{2} \leq 2 \left| 1-
\sum_{i=1}^{n} x_{i}y_{i} \right|.$$

Korea, 2001