Putnam 1941 B6

Assuming that $f(x)$ is continuous in the interval $(0,1)$ , prove that $$\int_{x=0}^{x=1} \int_{y=x}^{y=1} \int_{z=x}^{z=y} f(x)f(y)f(z) dz dy dx= \frac{1}{6}\left(\int_{0}^{1} f(t) dt\right)^{3}.$$