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Tricky Shapiro-like inequality

Source: Moldova TST 2024 P7

June 9, 2024
inequalitiesTST

Problem Statement

Prove that a=2a=2 is the greatest real number for which the inequality: x1xn+x2+x2x1+x3+⋯+xnxn−1+x1≥a \frac{x_1}{x_n+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_n}{x_{n-1}+x_1} \ge a holds true for any n≥4n \ge 4 and any positive real numbers x1,x2,…,xnx_1, x_2,\dots,x_n.
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