MetaMath '24

Dokažite da za sve $a,b,c > 0$ vrijedi \[\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\geq 2\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a} \right)\geq \dfrac{9}{a+b+c}.\]