Question 70
Let $x_{1}, x_{2}, \ldots, x_{n}$ be different real numbers. Prove that
$$
\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}= \begin{cases}0, & \text{ if } n \text{ is even } \\ 1, & \text{ if } n \text{ is odd }\end{cases}
$$
$$
\sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}= \begin{cases}0, & \text{ if } n \text{ is even } \\ 1, & \text{ if } n \text{ is odd }\end{cases}
$$