FKMO 2022 Problem 6

Set $X$ is called fancy if it satisfies all of the following conditions:

• The number of elements of $X$ is $2022$ .
• Each element of $X$ is a closed interval contained in $[0, 1]$ .
• For any real number $r \in [0, 1]$ , the number of elements of $X$ containing $r$ is less than or equal to $1011$ .

For fancy sets $A, B$ , and intervals $I \in A, J \in B$ , denote by $n(A, B)$ the number of pairs $(I, J)$ such that $I \cap J \neq \emptyset$ . Determine the maximum value of $n(A, B)$ .