2024 USAMO 第 2 题

Let $S_1, S_2, \ldots, S_{100}$ be finite sets of integers whose intersection is not empty. For each non-empty $T \subseteq\left\{S_1, S_2, \ldots, S_{100}\right\}$, the size of the intersection of the sets in $T$ is a multiple of the number of sets in $T$. What is the least possible number of elements that are in at least 50 sets?

令 $S_1, S_2, \ldots, S_{100}$ 为交集不为空的有限整数集。对于每个非空 $T \subseteq\left\{S_1, S_2, \ldots, S_{100}\right\}$，$T$ 中集合的交集大小是 $T$ 中集合数量的倍数。至少 50 个集合中的元素的最少数量是多少？