Prove that $a=2$ is the greatest real number for which the inequality: $$ \frac{x_1}{x_n+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_n}{x_{n-1}+x_1} \ge a $$holds true for any $n \ge 4$ and any positive real numbers $x_1, x_2,\dots,x_n$.
Source: Moldova TST 2024 P7
Tags: inequalities, TST
Prove that $a=2$ is the greatest real number for which the inequality: $$ \frac{x_1}{x_n+x_2}+\frac{x_2}{x_1+x_3}+\dots+\frac{x_n}{x_{n-1}+x_1} \ge a $$holds true for any $n \ge 4$ and any positive real numbers $x_1, x_2,\dots,x_n$.