Problem 1106

Let's assume that participant A has $m$ acquaintances $a_{1}, a_{2}, \cdots, a_{m}$. Since they are all familiar with A, they do not know each other (why?). Since they do not know each other, every pair $\left(a_{i}, a_{j}\right)$, apart from A, must have another common acquaintance, and these acquaintances, of course, do not know A (why?). Furthermore, for different $\left(a_{i}, a_{j}\right)$, the other acquaintances are also different. Therefore, the number of participants who do not know A is no less than $C_{m}^{2}$. On the other hand, every person who does not know A has two common acquaintances with A, and since these people all know A, they must be among $a_{1}, a_{2}, \cdots, a_{m}$, and for different people who do not know A, the two common acquaintances $\left(a_{i}, a_{j}\right)$ will not be entirely the same (why?). Therefore, it can be concluded that the number of people who do not know A is no more than $C_{m}^{2}$. Combining the above, we know that the number of participants who do not know A is exactly $C_{m}^{2}$. Thus, the total number of participants $n$ should satisfy the equation
$$n=1+m+C_{m}^{2}=1+m+\frac{1}{2} m(m+1)$$

where 1 is A himself, $m$ is the number of A's acquaintances, and $\frac{1}{2} m(m-1)$ is the number of people who do not know A.

Examining this positive integer quadratic equation in $m$, we find that it has only one positive root. This indicates that for each participant, the number of acquaintances $m$ is a constant.