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For any $x_{1},x_{2},\cdots x_{6}$ with $\sum_{i=1}^{6}x_{i}^{2}=1$ and $y_{1},y_{2},\cdots y_{6}$ in $\mathbb{R}$ with $\sum_{i=1}^{6}y_{i}^{2}=1$, do there always exist $z_{1},z_{2},\cdots z_{6}$ in $\mathbb{R}$ with $\sum_{i=1}^{6}z_{i}^{2}=6$ such that $\left|z_{1}z_{2}z_{3}z_{4}\sum_{i=1}^{6}x_{i}z_{i}\sum_{j=1}^{6}y_{j}z_{j}\right|\ge1$?

For special cases such as $x_iy_i\ge0$, one can see it holds. But for the general case, I am stuck. Could anyone help on this question?

Any helpful answer would be greatly appreciated!

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3  
This reminds me of a question I've seen on MO before... –  Yemon Choi Dec 12 '11 at 7:38
2  
Maybe I'm misreading, but it looks like setting $y_1=y_2=y_3=y_4=0$ gives an easy counterexample. –  Brendan McKay Dec 12 '11 at 9:02
    
Oops, I think Brendan you are right. The question that Yemon and I seem to have in mind was slightly different then. –  Suvrit Dec 12 '11 at 9:40
    
@Brendan, oops, the question has been corrected now. Thanks. –  user19892 Dec 12 '11 at 13:20
1  
Yep. I remember seeing what was almost certainly the same unmotivated inequality question. –  Anthony Quas Dec 12 '11 at 19:05
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