1975 AMC 12 Question 3

Which of the following inequalities are satisfied for all real numbers $a,$ $b,$ $c,$ $x,$ $y,$ $z$ which satisfy $x\lt a,$ $y\lt b,$ and $z\lt c?$

I. $xy+yz+zx\lt ab+bc+ca$

II. $x^2+y^2+z^2\lt a^2+b^2+c^2$

III. $xyz\lt abc$