CGMO 2021 Problem 6

Given a finite set $S$ , $P(S)$ denotes the set of all the subsets of $S$ . For any $f:P(S)\rightarrow \mathbb{R}$ ,prove the following inequality: $$\sum_{A\in P(S)}\sum_{B\in P(S)}f(A)f(B)2^{\left| A\cap B \right|}\geq 0.$$