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find n such that x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 <= 0 if sum x_i=0

Source: 1988 Swedish Mathematical Competition p3

March 28, 2021
inequalitiesSumalgebra

Problem Statement

Show that if x1+x2+x3=0x_1+x_2+x_3 = 0 for real numbers x1,x2,x3x_1,x_2,x_3, then x1x2+x2x3+x3x1≤0x_1x_2+x_2x_3+x_3x_1\le 0.
Find all n≥4n \ge 4 for which x1+x2+...+xn=0x_1+x_2+...+x_n = 0 implies x1x2+x2x3+...+xn−1xn+xnx1≤0x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 \le 0.
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