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Product of x_k is equal to 1 - [IMO LongList 1971]

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January 1, 2011
logarithmsalgebra proposedalgebra

Problem Statement

A system of n numbers x1,x2,…,xnx_1, x_2, \ldots, x_n is given such that x1=log⁡xn−1xn,x2=log⁡xnx1,…,xn=log⁡xn−2xn−1.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 ∏k=1nxk=1.\prod_{k=1}^n x_k =1.
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