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September 5th, 2013, 11:08 AM   #1
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A rational expression

Let x,y,z be three real numbers which satisfy x+y+z=1; xyz=1; x^2+y^2+z^2=2013

Then find the value of 1/(xy+z) + 1/(xz+y) + 1/(yz+x).

Please help.
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September 5th, 2013, 11:50 AM   #2
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Re: A rational expression

I would first combine terms in the expression for which we are to determine the value:



Using the given values, this becomes:



Next, take:



Square both sides to get:



Hence:





Square again:



Hence:



And so we have:

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September 5th, 2013, 07:02 PM   #3
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Re: A rational expression

I observed that x, y and z are the roots of the equation m - m - 1006m - 1 = 0, but that didn't get me too far.
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