National Math Olympiad

The inequality is equivalent to $xy+yz+zx\le x+y+z$. Since $x$ is positive and $y\le 1$ we have $xy\le x$. A similar reasoning shows that $yz\le y$ and $zx\le z$. Hence, the inequality holds.

The equality holds if and only if $xy=x$, $yz=y$ and $zx=z$. If $x=0$, then the third equality implies $z=0$ and the second equality then implies $y=0$. Otherwise, we must have $y=1$ and then the second equation implies $z=1$ and from the third equation we get $x=1$. Hence, the equality holds when $x=y=z=0$ or $x=y=z=1$.