1/x+y+1/y+z+1/z+x-1/x+y+z
=1/x+y+z*(z/x+y+x/y+z+y/z+x)+2/x+y+z
>=1/x+y+z*(x+y+z)^2/2(xy+yz+zx)+2/x+y+z
=x+y+z/2+2/x+y+z>=2 Q.E.D.