2002 CMO 第 3 题

In a competition there are $18$ teams and in each round $18$ teams are divided into $9$ pairs where the $9$ matches are played coincidentally. There are $17$ rounds, so that each pair of teams play each other exactly once. After $n$ rounds, there always exists $4$ teams such that there was exactly one match played between these teams in those $n$ rounds. Find the maximum value of $n$ .

在一场比赛中，有 $18$ 球队，每轮 $18$ 球队被分为 $9$ 对，其中 $9$ 比赛同时进行。每轮比赛金额为 17 美元，因此每对球队只对战一次。在 $n$ 轮之后，总是存在 $4$ 球队，因此这些球队在这 $n$ 轮中恰好进行了一场比赛。求 $n$ 的最大值。