USAMO 2015 Problem 3

Let $S = \left\{ 1,2,\dots,n \right\}$ , where $n \ge 1$ . Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$ , we then write $f(T)$ for the number of subsets of $T$ that are blue.

Determine the number of colorings that satisfy the following condition: for any subsets $T_1$ and $T_2$ of $S$ , $$f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2).$$