Prove that

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May 20th 2010, 11:46 PM
dhiab

Prove that

If a , b and c are three positf real numbers and :
$ x=\frac{a}{a+b} $
$ y=\frac{b}{b+c} $
$ z=\frac{c}{c+a} $
Prove that : $x+y+z> 1$
May 21st 2010, 12:21 AM
pickslides

$ \frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}=\dots $ ??
May 21st 2010, 12:47 AM
aman_cc

Without the loss of generality we can assume
a >= b >= c > 0

Under this what can you say about x,y,z ?

Quote:

Originally Posted by dhiab

If a , b and c are three positf real numbers and :
$ x=\frac{a}{a+b} $
$ y=\frac{b}{b+c} $
$ z=\frac{c}{c+a} $
Prove that : $x+y+z> 1$
May 21st 2010, 05:48 AM
roninpro

It may also help to scale the numbers so that $a+b+c=1$ and sharpen the lower bound to $x+y+z\geq \frac{3}{2}$ .

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