Problem 5.2. Several people were seated around a round table such that the distances between adjacent people are the same. One of them was given a sign with the number 1, and then everyone was given signs with numbers 2, 3, and so on, in a clockwise direction.

The person with the sign numbered 31 noticed that the distance from them to the person with the sign numbered 7 is the same as the distance to the person with the sign numbered 14. How many people in total sat at the table?
Answer: 41.

Solution. For such a situation to be possible, people from the 31st to the 14th need to be counted in a circle in the direction of decreasing numbers, and from the 31st to the 7th - in a circle in the direction of increasing numbers.

There are 16 people sitting between the 31st and the 14th. Therefore, there are also 16 people between the 31st and the 7th. Among them, 6 have numbers from 1 to 6, so the remaining $16-6=10$ people are those with numbers from the 32nd. The largest number is 41.