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China Western Mathematical Olympiad 2015 ,Problem 5

Source: China Yinchuan Aug 17, 2015

August 17, 2015
inequalitiesgeometry

Problem Statement

Let a,b,c,da,b,c,d are lengths of the sides of a convex quadrangle with the area equal to SS, set S={x1,x2,x3,x4}S =\{x_1, x_2,x_3,x_4\} consists of permutations xix_i of (a,b,c,d)(a, b, c, d). Prove that S≤12(x1x2+x3x4).S \leq \frac{1}{2}(x_1x_2+x_3x_4).
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