IrMO 1990 P1 Q6: Upper Secondary Mathematics Competition Question

We recently had a question where considering the parity of the numbers (odd and evenness) speeded up our proof. Here is a question that is explicitly about the parity of numbers.


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

x₁x₂ + x₂x₃ + x₃x₄ + . . . + x_n-1x_n + x_nx₁ = 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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