Let $x_1, x_2, ..., x_n$ be positive reals. Prove that $$\sum_{k=1}^n \frac{x_k(2x_k - x_{k+1} - x_{k+2})}{x_{k+1} + x_{k+2}} \ge 0$$In the sum, cyclic indices have been taken, that is, $x_{n+1} = x_1$ and $x_{n+2} = x_2$.
Source: 2009 Cuba 2.7
Tags: algebra, inequalities
Let $x_1, x_2, ..., x_n$ be positive reals. Prove that $$\sum_{k=1}^n \frac{x_k(2x_k - x_{k+1} - x_{k+2})}{x_{k+1} + x_{k+2}} \ge 0$$In the sum, cyclic indices have been taken, that is, $x_{n+1} = x_1$ and $x_{n+2} = x_2$.