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Inequality with reals a,b,c,d and x_i, y_i

Source: Chinese Mathematical Olympiad 2003 Problem 6

February 18, 2012

inequalitiesinequalities proposed

Problem Statement

Suppose

a,b,c,d

are positive reals such that

ab+cd=1

and

x_i,y_i

are real numbers such that

x_i^2+y_i^2=1

for

i=1,2,3,4

. Prove that

(ax_1+bx_2+cx_3+dx_4)^2+(ay_4+by_3+cy_2+dy_1)^2\le 2\left(\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}\right).

Li Shenghong