Question 111
A *partition* of a set $S$ is a set of pairwise disjoint subsets of $S$ whose union is $S$. Let $P$ be a partition of the set $\{1, 2, ..., 2024\}$ into 2-element sets, satisfying the following condition: For every set $\{a, b\}$ in $P$, either $|a - b| = 1$ or $|a - b| = 506$. Assume that $\{1518, 1519\}$ belongs to $P$. Determine the number that pairs off with $505$ to form a set in $P$.
The Problem Selection Committee
The Problem Selection Committee