Question 51
Let $x_1$, $x_2$, ..., $x_n$ be positive reals with sum $1$. Show that
$$
\frac{x_1^2}{x_1 + x_2} + \frac{x_2^2}{x_2 + x_3} + \cdots + \frac{x_n^2}{x_n + x_1} \geq \frac{1}{2}.
$$
$$
\frac{x_1^2}{x_1 + x_2} + \frac{x_2^2}{x_2 + x_3} + \cdots + \frac{x_n^2}{x_n + x_1} \geq \frac{1}{2}.
$$