TR-MO 2012 Problem 5

Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$ , where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$ . The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs to $A$ . The set $B \subset P$ is said to be increasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in B$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \geq x_i (1\leq i \leq 2012)$ also belongs to $B$ . Find the maximum possible value of $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$ , where $A$ and $B$ are nonempty decreasing and increasing sets ( $\mid \cdot \mid$ denotes the number of elements of the set).