2019 IMO Shortlist A5

Let $x_{1}, x_{2}, \ldots, x_{n}$ be different real numbers. Prove that $$\sum_{1 \leq i \leq n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}= \begin{cases}0, & \text { if } n \text { is even } \\ 1, & \text { if } n \text { is odd }\end{cases}$$

令 $x_{1}, x_{2}, \ldots, x_{n}$ 为不同的实数。证明 $$\sum_{1 \leq i \leq n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}= \begin{cases}0, & \text { if } n \text { 为偶数 } \\ 1, & \text { if } n \text { 为奇数 }\end{cases}$$