2000 CMO 第 3 题

A table tennis club hosts a series of doubles matches following several rules:

(i) each player belongs to two pairs at most;

(ii) every two distinct pairs play one game against each other at most;

(iii) players in the same pair do not play against each other when they pair with others respectively.

Every player plays a certain number of games in this series. All these distinct numbers make up a set called the “*set of games*”. Consider a set $A=\{a_1,a_2,\ldots ,a_k\}$ of positive integers such that every element in $A$ is divisible by $6$ . Determine the minimum number of players needed to participate in this series so that a schedule for which the corresponding *set of games* is equal to set $A$ exists.

乒乓球俱乐部举办一系列双打比赛，遵循以下几条规则：

(i) 每位选手最多属于两对；

(ii) 每两对不同的配对最多进行一场比赛；

(iii) 同一对的玩家在分别与其他人配对时不会互相对抗。

每个玩家都会在这个系列中玩一定数量的游戏。所有这些不同的数字组成了一个称为“*游戏集*”的集合。考虑一个正整数集合 $A=\{a_1,a_2,\ldots ,a_k\}$ ，使得 $A$ 中的每个元素都可以被 $6$ 整除。确定参加该系列赛所需的最少玩家数量，以便存在相应的*游戏集*等于集合$A$的时间表。