CGMO 2014 Problem 7

Given a finite nonempty set $X$ with real values, let $f(X) = \frac{1}{|X|} \sum\limits_{a\in X} a$ , where $\left\lvert X \right\rvert$ denotes the cardinality of $X$ . For ordered pairs of sets $(A,B)$ such that $A\cup B = \{1, 2, \dots , 100\}$ and $A\cap B = \emptyset$ where $1\leq |A| \leq 98$ , select some $p\in B$ , and let $A_{p} = A\cup \{p\}$ and $B_{p} = B - \{p\}.$ Over all such $(A,B)$ and $p\in B$ determine the maximum possible value of $(f(A_{p})-f(A))(f(B_{p})-f(B)).$