Question 135
Let $x_1,\ldots,x_n$ be distinct real numbers, where $n\ge2$. Prove that
$$\sum_{i=1}^n\prod_{j\ne i}\frac{1-x_ix_j}{x_i-x_j}=\begin{cases}0,&n\text{ even},\\1,&n\text{ odd}.\end{cases}$$
$$\sum_{i=1}^n\prod_{j\ne i}\frac{1-x_ix_j}{x_i-x_j}=\begin{cases}0,&n\text{ even},\\1,&n\text{ odd}.\end{cases}$$