AHSME 1975 Problem 3

Which of the following inequalities are satisfied for all real numbers $a, b, c, x, y, z$ which satisfy the conditions $x < a, y < b$, and $z < c$? $\text{I}. \ xy + yz + zx < ab + bc + ca \\ \text{II}. \ x^2 + y^2 + z^2 < a^2 + b^2 + c^2 \\ \text{III}. \ xyz < abc$ $\textbf{(A)}\ \text{None are satisfied.} \textbf{(B)}\ \text{I only} \textbf{(C)}\ \text{II only} \textbf{(D)}\ \text{III only} \textbf{(E)}\ \text{All are satisfied.}$