CWMO 2003 Problem 7

The non-negative numbers $x_1, x_2, \ldots, x_5$ satisfy $\sum_{i = 1}^5 \frac {1}{1 + x_i} = 1$ . Prove that $\sum_{i = 1}^5 \frac {x_i}{4 + x_i^2} \leq 1$ .