1. A row of 50 people, all of different heights, stands. Exactly 15 of them are taller than their left neighbor. How many people can be taller than their right neighbor? (Provide all possible answers and prove that there are no others)
Answer: 34.

Solution. Let's call 15 people who are taller than their left neighbor - tall people.

If someone has a tall person to their right, then they are shorter than them, otherwise, they are taller.

There are a total of 49 pairs of adjacent people.

Tall people are on the right in exactly 15 pairs. In the remaining $49-15=34$ pairs, the person on the right is not a tall person. It is in these pairs that the person is taller than their neighbor on the right.