It is known that x_1,x_2,\ldots,x_n\ge 0 and \displaystyle x_1+x_2+\ldots +x_n=\frac{1}{2}. Prove that

\displaystyle\frac{1-x_1}{1+x_1}\cdot\frac{1-x_2}{1+x_2}\cdot\ldots\cdot\frac{1-x_n}{1+x_n}\ge\frac{1}{3}.

I placed a proof on a separate page.