11.5. At a diplomatic reception, there are 99 persons, each of whom has heard of no fewer than $n$ other attendees. If A has heard of B, it does not automatically mean that B has heard of A. For what minimum $n$ is it guaranteed that there will be two attendees who have heard of each other?
Answer. When $n=50$.

Solution. Let's call a situation where one of the guests has heard about another a half-acquaintance. If each guest has heard about at least 50 other participants at the reception, then there are at least $99 \cdot 50$ half-acquaintances, which is more than the total number of pairs of guests at the reception, equal to $\frac{\mathbf{9} \boldsymbol{9}}{\mathbf{2}} \underline{\underline{\mathbf{8}}} \boldsymbol{9 4}$: Therefore, in some pair, there are at least two half-acquaintances, and the guests in this pair have heard about each other.

Let's provide an example where each has heard about exactly 49 other guests, but there are no two guests who have heard about each other. Arranging the guests in a circle, we assume that each has heard only about the next 49 guests in the clockwise direction. In this case, each has been heard about only by the 49 guests located in front of them in the clockwise direction. These two sets do not intersect, so for each guest, there is no other guest who has heard about them and whom they have heard about.

Criteria. Proof that $n$ is not greater than $50: 4$ points. Example that $n$ is not less than 50: 3 points.