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Equation with 1s and -1s

Source: 1990 IrMO Paper 1 Problem 6

September 30, 2017
algebra

Problem Statement

Let nn be a natural number, and suppose that the equation x1x2+x2x3+x3x4+x4x5+⋯+xn−1xn+xnx1=0x_1x_2+x_2x_3+x_3x_4+x_4x_5+\dots +x_{n-1}x_n+x_nx_1=0 has a solution with all the xix_is equal to ±1\pm 1. Prove that nn is divisible by 44.
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