Putnam 2004 A6

Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0 \leq x \leq 1, 0 \leq y \leq 1$. Show that $$\int_0^1 \left( \int_0^1 f(x,y) dx \right)^2 dy + \int_0^1 \left( \int_0^1 f(x,y) dy \right)^2 dx \leq \left( \int_0^1 \int_0^1 f(x,y) dx dy \right)^2 + \int_0^1 \int_0^1 [f(x,y)]^2 dx dy.$$