Question 19
Let $A$ be a set with $|A| = 225$, meaning that $A$ has $225$ elements. Suppose further that there are eleven subsets $A_1, \dots, A_{11}$ of $A$ such that $|A_i| = 45$ for $1 \le i \le 11$ and $|A_i \cap A_j| = 9$ for $1 \le i \lt j \le 11$. Prove that $|A_1 \cup A_2 \cup \dots \cup A_{11}| \ge 165$, and give an example for which equality holds.