There are 2017 unoccupied chairs in a row. Every minute, a person arrives and sits in one of them that is empty, and at the same instant, if it is occupied, a person in an adjacent chair stands up and leaves. What is the maximum number of people that can be sitting simultaneously in the row of chairs?
Solution

It is not possible for all chairs to be simultaneously occupied, as the last person to sit down would inevitably sit next to an occupied chair and force, according to the rule in the problem statement, someone to leave. Our goal now is to show a sequence of moves where it is possible for 2016 people to be seated simultaneously. Number the chairs from 1 to 2017. Initially, one person will sit in chair number 1. Next, another person will sit in chair number 3. No one will leave, as chairs 1 and 3 are not adjacent. The next person will sit in chair number 2, and thus the person in chair number 3 will leave. A new person will sit in chair number 4, and then another person will arrive and sit in chair number 3. Now chair number 4 is unoccupied, but the first three chairs are occupied. Suppose all chairs numbered from 1 to $k$, with $k<2016$, are occupied and the remaining chairs are empty. The next person to sit will choose chair $k+2$. Then another person will sit in chair $k+1$ and force the person in chair $k+2$ to leave. Thus, the chairs from 1 to $k+1$ are occupied and the remaining chairs are empty. We can repeat this procedure while $k<2016$, and when it is executed for the last time, we will have the chairs from 1 to 2016 occupied.