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\prod^n_{i=0} x^{n-1}_i/ \prod_{j \ne i}(x_i - x_j) is fixed

Source: IFYM - XI International Festival of Young Mathematicians Sozopol 2022, Theme for 11-12 grade, 4th round p4

November 12, 2022
algebra

Problem Statement

Let nn be a natural number. To prove that the value of the expression ∏i=0nxin−1∏j≠i(xi−xj)\prod^n_{i=0}\frac{x^{n-1}_i}{\prod_{j \ne i}(x_i - x_j)} does not depend on the choice of the different real numbers x0,x1,...,xnx_0, x_1, ... , x_n.
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