$\dfrac{x}{xy+x+1}+\dfrac{y}{yz+y+1}+\dfrac{z}{zx+z+1}$
$=\dfrac{x}{xy+x+xyz}+\dfrac{y}{yz+y+1}+\dfrac{z}{zx+z+xyz}$ (do $xyz=1$)
$=\dfrac{x}{x(yz+y+1)}+\dfrac{y}{yz+y+1}+\dfrac{z}{z(x+1+xy)}$
$=\dfrac{1}{yz+y+1}+\dfrac{y}{yz+y+1}+\dfrac{1}{x+1+xy}$
$=\dfrac{1}{yz+y+1}+\dfrac{y}{yz+y+1}+\dfrac{xyz}{x+xyz+xy}$ Vì $xyz=1$
$=\dfrac{1}{yz+y+1}+\dfrac{y}{yz+y+1}+\dfrac{xyz}{x(1+yz+y)}$
$=\dfrac{1}{yz+y+1}+\dfrac{y}{yz+y+1}+\dfrac{yz}{1+yz+y}$
$=\dfrac{yz+y+1}{yz+y+1}=1$
