VN-TST 2006

Prove that for all real numbers $x,y,z \in [1,2]$ the following inequality always holds: $$(x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 6(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}).$$ When does the equality occur?