Problem

Source: FKMO 2022 Problem 6

Tags: combinatorics, Sets, interval



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 $n(A, B)$.