USAMO 1974 Problem 2

Show that for any positive reals $\mathrm{x}, \mathrm{y}, \mathrm{z}$ we have $\mathrm{x}^{\mathrm{x}} \mathrm{y}^{\mathrm{y}} \mathrm{z}^{\mathrm{z}} \geq(\mathrm{xyz})^{\mathrm{a}}$, where $\mathrm{a}$ is the arithmetic mean of $\mathrm{x}, \mathrm{y}, \mathrm{z}$.