MathDB

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Part of 2002 Taiwan National Olympiad

Problems(1)

$0<x_1,x_2,x_3,x_4\leq\frac{1}{2}$

Source: 11-th Taiwanese Mathematical Olympiad 2002

1/22/2007
Let 0<x1,x2,x3,x4≤120<x_{1},x_{2},x_{3},x_{4}\leq\frac{1}{2} are real numbers. Prove that x1x2x3x4(1−x1)(1−x2)(1−x3)(1−x4)≤x14+x24+x34+x44(1−x1)4+(1−x2)4+(1−x3)4+(1−x4)4\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}}.
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