MathDB

Sum of products is n mod 2

Source: IMO 2019 SL A5

September 22, 2020

algebraProductIMO ShortlistIMO Shortlist 2019determinant

Problem Statement

Let

x_1, x_2, \dots, 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}}=\left\{\begin{array}{ll} 0, & \text { if } n \text { is even; } \\ 1, & \text { if } n \text { is odd. } \end{array}\right.