CGMO 2013 Problem 5

For any given positive numbers $a_1,a_2,\ldots,a_n$ , prove that there exist positive numbers $x_1,x_2,\ldots,x_n$ satisfying $\sum_{i=1}^n x_i=1$ , such that for any positive numbers $y_1,y_2,\ldots,y_n$ with $\sum_{i=1}^n y_i=1$ , the inequality $\sum_{i=1}^n \frac{a_ix_i}{x_i+y_i}\ge \frac{1}{2}\sum_{i=1}^n a_i$ holds.