Putnam 1993 B4

The function $K(x,y)$ is positive and continuous for $0 \leq x \leq 1, 0 \leq y \leq 1$, and the functions $f(x)$ and $g(x)$ are positive and continuous for $0 \leq x \leq 1$. Suppose that for all $x$, $0 \leq x \leq 1$, $$\int_0^1 f(y)K(x,y) dy = g(x)$$ and $$\int_0^1 g(y)K(x,y) dy = f(x).$$ Show that $f(x) = g(x)$ for $0 \leq x \leq 1$.