KöMaL 2007 A425

Let $n \ge$2 and let $a_1,a_2,\ldots,a_n, x_1,x_2,\ldots,x_n$ be positive real numbers such that $a_1$+...+$a_n$=$x_1$+...+$x_n$=1. Prove that $2\sum_{1\le i<j\le n}x_ix_j \le \frac{n-2}{n-1} + \sum_{i=1}^n \frac{a_ix_i^2}{1-a_i}\,.$