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x_1y_1 + x_2y_2 + x_3y_3 <1 if x_1 + y_2 = x_2 + y_3 = x_3 + y_1 =1

Source: Mathematics Regional Olympiad of Mexico Center Zone 2014 P2

November 11, 2021
algebrainequalities

Problem Statement

Let x1x_1, x2x_2,x3x_3, y1y_1, y2y_2, and y3y_3 be positive real numbers, such that x1+y2=x2+y3=x3+y1=1x_1 + y_2 = x_2 + y_3 = x_3 + y_1 =1. Prove that x1y1+x2y2+x3y3<1 x_1y_1 + x_2y_2 + x_3y_3 <1
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