2004 IMO Shortlist S22

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold:

*i.)* Each pair of students are in exactly one club.

*ii.)* For each student and each society, the student is in exactly one club of the society.

*iii.)* Each club has an odd number of students. In addition, a club with ${2m+1}$ students ( $m$ is a positive integer) is
in exactly $m$ societies.

Find all possible values of $k$ .

*Proposed by Guihua Gong, Puerto Rico*

一所大学有 $10001$ 学生。有些学生联合起来组成多个俱乐部(一个学生可能属于不同的俱乐部)。有些俱乐部联合起来组成多个社团(一个俱乐部可能属于不同的社团)。总共有 $k$ 个社团。假设以下条件成立：

*i.)* 每对学生都属于同一个俱乐部。

*ii.)* 对于每个学生和每个社团，该学生都属于该社团的一个俱乐部。

*iii.)* 每个俱乐部的学生人数均为奇数。另外，一个拥有${2m+1}$名学生($m$为正整数)的俱乐部是
恰好在 $m$ 社会中。

找出 $k$ 的所有可能值。

*由波多黎各龚桂华提议*