Prove that

May 2009
596
31
ALGERIA
If a , b and c are three positf real numbers and :
\(\displaystyle
x=\frac{a}{a+b}
\)
\(\displaystyle
y=\frac{b}{b+c}
\)
\(\displaystyle
z=\frac{c}{c+a}
\)
Prove that : \(\displaystyle x+y+z> 1\)
 

pickslides

MHF Helper
Sep 2008
5,237
1,625
Melbourne
\(\displaystyle

\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}=\dots
\) ??
 
Apr 2009
678
140
Without the loss of generality we can assume
a >= b >= c > 0

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

If a , b and c are three positf real numbers and :
\(\displaystyle
x=\frac{a}{a+b}
\)
\(\displaystyle
y=\frac{b}{b+c}
\)
\(\displaystyle
z=\frac{c}{c+a}
\)
Prove that : \(\displaystyle x+y+z> 1\)
 
Nov 2009
485
184
It may also help to scale the numbers so that \(\displaystyle a+b+c=1\) and sharpen the lower bound to \(\displaystyle x+y+z\geq \frac{3}{2}\).