10.4. In an initially empty room, every minute either 2 people enter or 1 person leaves. Can there be exactly 2018 people in the room after 2019 minutes?
Answer: No.

Solution: Every minute, the remainder of the number of people in the room when divided by 3 changes either from 0 to 2, or from 1 to 0, or from 2 to 1. This means that after 3 times, the remainder of the number of people in the room when divided by 3 will not change. 2019 is 673 times 3, so the remainder after 2019 minutes will be the same as at the beginning, i.e., 0. But 2018 is not divisible by 3.