Let $0<x_{1},x_{2},x_{3},x_{4}\leq\frac{1}{2}$ are real numbers. Prove that $\frac{x_{1}x_{2}x_{3}x_{4}}{(1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})}\leq\frac{x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}}{(1-x_{1})^{4}+(1-x_{2})^{4}+(1-x_{3})^{4}+(1-x_{4})^{4}}$.
Problem
Source: 11-th Taiwanese Mathematical Olympiad 2002
Tags: inequalities, inequalities proposed