RU-ALLRUSSIAN 1990 Problem 10

Let $\mathbf{x}_1$ , $\mathbf{x}_2$ , ..., $\mathbf{x}_n$ be positive reals with sum 1. Show that $\mathbf{x}_1^{2} / (\mathbf{x}_1 + \mathbf{x}_2) + \mathbf{x}_2^{2} / (\mathbf{x}_2 + \mathbf{x}_3) + \ldots + \mathbf{x}_n$ . $\frac{1}{2} /(\mathbf{x}_{\mathrm{n - 1}} + \mathbf{x}_{\mathrm{n}}) + \mathbf{x}_{\mathrm{n}}^{2} / (\mathbf{x}_{\mathrm{n}} + \mathbf{x}_{\mathrm{1}}) \geq 1 / 2$ .