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September 5th, 2013, 11:08 AM  #1 
Member Joined: Oct 2011 Posts: 81 Thanks: 0  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. 
September 5th, 2013, 11:50 AM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,204 Thanks: 511 Math Focus: Calculus/ODEs  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: 
September 5th, 2013, 07:02 PM  #3 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,879 Thanks: 1087 Math Focus: Elementary mathematics and beyond  Re: A rational expression
I observed that x, y and z are the roots of the equation m³  m²  1006m  1 = 0, but that didn't get me too far.


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