Let $x_1, x_2, …, x_n$ and $y_1, y_2, …,y_n$ be positive reals such that $$x_1 + x_2 +.. + x_n \ge y_i \ge x^2_i$$for all $i = 1, 2, …, n$. Prove that $$\frac{x_1}{x_1y_1 + x_2}+ + \frac{x_2}{x_2y_2 + x_3} + ...+ \frac{x_n}{x_ny_n + x_1}> \frac{1}{2n}.$$