ILL 1971 Problem 12

A system of n numbers $x_1, x_2, \ldots, x_n$ is given such that $$x_1 = \log_{x_{n-1}} x_n, x_2 = \log_{x_{n}} x_1, \ldots, x_n = \log_{x_{n-2}} x_{n-1}.$$ Prove that $\prod_{k=1}^n x_k =1.$