Question 87
Let $N$ be a strictly positive integer. Suppose there exist four subsets $A_{1}$, $A_{2}$, $A_{3}$ and $A_{4}$ of $\{1, \ldots, N\}$, each of cardinality $500$, and suppose that for all $x, y$ in $\{1, \ldots, N\}$ there exists an index $i$ such that $x$ and $y$ are in $A_{i}$. Determine the largest possible value of $N$.