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Exactly one negative among six reals

Source: Indian TST Day 2 Problem 2

July 11, 2014
inequalitiestriangle inequalityalgebra unsolvedalgebra

Problem Statement

For j=1,2,3j=1,2,3 let xj,yjx_{j},y_{j} be non-zero real numbers, and let vj=xj+yjv_{j}=x_{j}+y_{j}.Suppose that the following statements hold:
x1x2x3=−y1y2y3x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}
x12+x22+x32=y12+y22+y32x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2
v1,v2,v3v_{1},v_{2},v_{3} satisfy triangle inequality
v12,v22,v32v_{1}^{2},v_{2}^{2},v_{3}^{2} also satisfy triangle inequality.
Prove that exactly one of x1,x2,x3,y1,y2,y3x_{1},x_{2},x_{3},y_{1},y_{2},y_{3} is negative.
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