Prove that

• 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 $x+y+z\geq \frac{3}{2}$.