Berlov S.L.

100 people came to a party. Then those who had no acquaintances among the attendees left. Then those who had exactly one acquaintance among the remaining people also left. Then, similarly, those who had exactly 2, 3, 4, ..., 99 acquaintances among the remaining people at the time of their departure also left.

What is the maximum number of people that could remain at the end?
It is not difficult to check that if all the attendees, except for two people $A$ and $B$, were acquainted with each other, then at the end, everyone except $A$ and $B$ should have remained, that is, 98 people.

Let's prove that 99 people could not have remained. It is clear that person $A$, who initially had the fewest acquaintances ($k$), would leave at some point. If no one else left, then all the others (except $A$) had more than $k$ acquaintances before $A$ left and fewer than $k+1$ after $A$ left. But then $A$ must have been acquainted with everyone else, meaning $k=99$, which contradicts the strict minimality of $k$.

## Answer

98 people.