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Very cute algebra

Source: Kyiv City MO 2022 Round 2, Problem 8.3, 9.3

January 30, 2022
algebra

Problem Statement

Nonzero real numbers x1,x2,…,xnx_1, x_2, \ldots, x_n satisfy the following condition:
x1−1x2=x2−1x3=…=xn−1−1xn=xn−1x1x_1 - \frac{1}{x_2} = x_2 - \frac{1}{x_3} = \ldots = x_{n-1} - \frac{1}{x_n} = x_n - \frac{1}{x_1}
Determine all nn for which x1,x2,…,xnx_1, x_2, \ldots, x_n have to be equal.
(Proposed by Oleksii Masalitin, Anton Trygub)
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