Question 29
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 $1 \le i \le 2012$. 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 $y_i \le x_i$ ($1 \le i \le 2012$) 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 $y_i \ge x_i$ ($1 \le i \le 2012$) also belongs to $B$. Find the maximal possible value of $f(A, B) = \frac{|A \cap B|}{|A| \cdot |B|}$, where $A$ and $B$ are nonempty decreasing and increasing sets ($| \cdot |$ denotes the number of elements of the set).