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Let  . ,and  such that 
and inequality to prove become
 , similarly:
Now the requirement that x+y+z = 0 guarantees that for (x,y,z) one has
(0,0,0) (0, +, - ) (+,+,-) or (+, - , - ) as positive/negative quantities for the triple!
Thus atleast one of xy,xz, or yz has a product which is positive!
Finally, the triangle inequality ensures the following: a+b>c, a+c>b, and b+c>a
Combining all the above assures us that at least one of the above representations for Q is the sum of two positive quantities.
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Moderation Team
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