1. The numbers 1, 2, 3, 4, 5, 6, 7 were written in a circle in some order. We will call a written number good if it is equal to the sum of the two numbers written next to it. What is the maximum possible number of good numbers among the written ones? (E. Bakayev)
Answer: 3. Solution: If the numbers are written, for example, in the order 2, 7, 5, 6, 1, 4, 3, then the numbers 7, 6, and 4 will be good. It remains to show that there cannot be more than three good numbers.

Note that a good number is greater than both of its neighbors, so two good numbers cannot stand next to each other. Therefore, the number following a good number clockwise should not be good, and the numbers following different good numbers should be different and not good. Consequently, among all the written numbers, the good ones do not exceed half, and thus, there are no more than three.