27. At a certain meeting, a total of $12 k$ people attended ($k$ is a positive integer), and each person is acquainted with exactly $3 k+6$ others. Assuming that for any two people, the number of people who are acquainted with both of them is the same, find the number of people who attended the meeting.
(36th IMO Shortlist)
27. Let $12k$ people be $A_{1}, A_{2}, \cdots, A_{12k}$, and for any two people, the number of people who know both of them is $m$. If $A_{t}$ knows both $A_{i}$ and $A_{j} (i \neq j)$, then the triplet $(A_{t}; A_{i}, A_{j})$ is formed. Using two methods to count the number of such triplets, we get $12k C_{3k+6}^{2} = m C_{12k}^{2}$. Thus, $m = \frac{12k C_{3k+6}^{2}}{C_{12k}^{2}} = \frac{(3k+6)(3k+5)}{12k-1} = \frac{9k^2 + 33k + 30}{12k-1}$. Therefore, $4m = \frac{36k^2 + 132k + 120}{12k-1} = 3k + 11 + \frac{3k + 131}{12k-1}$. If $12k-1 > 3k + 131$, i.e., $k \geq 15$, then $\frac{3k + 131}{12k-1}$ is not an integer, leading to a contradiction. Therefore, $1 \leq k \leq 14$. Upon inspection, among $1 \leq k \leq 14$, only $k=3$ makes $\frac{3k + 131}{12k-1} = 4$ an integer. In this case, $12k = 36$, $3k + 6 = 15$, and $m = 6$. The following example shows that $12k = 36$ is feasible. Arrange 36 people in a $6 \times 6$ matrix, and color each person with one of 6 colors: 1, 2, 3, 4, 5, 6, and stipulate that each person knows the other 15 people in the same row, column, or color.
Thus, for any two people $P$ and $Q$: (1) When $P$ and $Q$ are in the same row, the 6 people who know both $P$ and $Q$ are the other 4 people in the same row, 1 person who is the same color as $P$ and in the same column as $Q$, and 1 person who is the same color as $Q$ and in the same column as $P$; (2) When $P$ and $Q$ are in the same column, similarly, there are 6 people who know both $P$ and $Q$; (3) When $P$ and $Q$ are neither in the same row nor in the same column and have different colors, the 6 people who know both $P$ and $Q$ are 2 people who are the same color as $P$ and in the same row or column as $Q$, 2 people who are the same color as $Q$ and in the same row or column as $P$, 1 person who is in the same row as $P$ and in the same column as $Q$, and 1 person who is in the same row as $Q$ and in the same column as $P$; (4) When $P$ and $Q$ are neither in the same row nor in the same column but have the same color, the 6 people who know both $P$ and $Q$ are the other 4 people who are the same color as $P$ and $Q$, 1 person who is in the same row as $P$ and in the same column as $Q$, and 1 person who is in the same row as $Q$ and in the same column as $P$. In summary, the number of people attending the meeting is 36.