PAMO 2005 Problem 1

For any positive real numbers $a,b$ and $c$ , prove: $$\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \geq \dfrac{2}{a+b} + \dfrac{2}{b+c} + \dfrac{2}{c+a} \geq \dfrac{9}{a+b+c}$$