If $a,b,c$ are positive, prove that $\frac{2}{a+b}+\frac{2}{b+c}+\frac{2}{c+a} \geq \frac{9}{a+b+c}$ If $a,b,c$ are positive real numbers, prove that 
$$\frac{2}{a+b}+\frac{2}{b+c}+ \frac{2}{c+a}≥ \frac{9}{a+b+c}$$

Let $a+b+c = s$. Then we have to prove
$$\dfrac{1}{s-a} +  \dfrac{1}{s-b} + \dfrac{1}{s-c} \geq \dfrac{9}{2s},$$
or, equivalently, 
$$\dfrac{3}{\dfrac{1}{s-a} +  \dfrac{1}{s-b} + \dfrac{1}{s-c}} \leq \dfrac{2s}{3}.$$
Note that the LHS is the harmonic mean of $s-a,s-b,s-c$ and the RHS is the arithmetic mean of the same numbers. This inequality is true by the AM-HM inequality.
