Let n be a natural number, and suppose that the equation

x

_{1}x

_{2}+ x

_{2}x

_{3}+ x

_{3}x

_{4}+ . . . + x

_{n-1}x

_{n}+ x

_{n}x

_{1}= 0

has a solution with each of the x

_{i}'s equal to either +1 or -1. Prove that n is divisible by 4.

[IrMO 1990 Paper 1 Question 6][edited slightly]

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