USAMO 2011 Problem 6

Let $A$ be a set with $|A|=225$ , meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$ . Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$ , and give an example for which equality holds.