Prove that 1/(a+b)+1/(b+c)+1/(a+c) is bigger or equal to 4.5 where a, b and c are positive and their sum 1.
If x,y,z are positive numbers then by AM-GM we have
$x+y+z\geq 3 (x y z)^{1/3}$
and
$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\geq 3(x y z)^{-1/3}$
Multiply these two equations to get
$(x+y+z) \left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \geq 9$
Now replace $x=a+b$, $y=b+c$, $z=c+a$