2011 USAMO 第 6 题

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 $1 \le i < j \le 11$. Prove that $|A_1 \cup A_2 \cup \dots \cup A_{11}| \ge 165$, and give an example for which equality holds.

令 $A$ 为 $|A| 的集合= 225$，表示$A$有225个元素。进一步假设 $A$ 有 11 个子集 $A_1$、$\dots$、$A_{11}$，使得 $|A_i | = 45$ 为 $1 \le i \le 11$ 和 $|A_i \cap A_j| = 9$ 1 $1 \le i < j \le 11$。证明 $|A_1 \cup A_2 \cup \dots \cup A_{11}| \ge 165$，并给出一个等式成立的例子。