1975 AHSME 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?

\begin{array}{l} \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 \end{array}

\textbf{(A)}\ \text{None are satisfied.} \qquad \textbf{(B)}\ \text{I only} \qquad \textbf{(C)}\ \text{II only} \qquad \textbf{(D)}\ \text{III only} \qquad \textbf{(E)}\ \text{All are satisfied.}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.