KZ-MO 2001 Problem 3

For positive numbers $x_1, x_2, \ldots, x_n$ $(n \geq 1)$ the following equality holds $$\frac {1} {{1 + x_1}} + \frac {1} {{1 + x_2}} + \ldots + \frac {1} {{1 + x_n}} = 1.$$ Prove that $x_1 \cdot x_2 \cdot \ldots \cdot x_n \geq (n-1) ^ n.$