Ren Posted December 4, 2011 Share Posted December 4, 2011 Let a,b, and c be positive real numbers. Prove that [math]\sqrt{\frac {a}{a+b}}+ \sqrt{\frac {b}{b+c}}+ \sqrt{\frac {c}{c+a}} \leq 2\sqrt{1+\frac {abc}{(a+b)(b+c)(c+a)}}[/math] I've tried squaring both sides, and that eventually lead me to nothing. Then I tried taking the common denominator of the left side, but I got nothing either... There has to be some trick involved that I'm not seeing... Anyone has any suggestions or input, that will be great! Link to comment Share on other sites More sharing options...
the tree Posted December 4, 2011 Share Posted December 4, 2011 (edited) Try making these substitutions: [math]\left( \frac{a}{a+b} \right) =x^2 , \, \left( \frac{b}{b+c} \right)=y^2 , \, \left( \frac{c}{c+a}\right)=z^2[/math] It should be fairly easy from there. Edited December 4, 2011 by the tree Link to comment Share on other sites More sharing options...
uncool Posted December 4, 2011 Share Posted December 4, 2011 the tree: I don't think it's quite so simple just from there. There is a constraint on x^2, y^2, and z^2 that is necessary to prove the relationship. Namely, ((1/x^2) - 1)((1/y^2) - 1)((1/z^2) - 1) = 1. =Uncool- Link to comment Share on other sites More sharing options...
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