Problem 7.7. The numbers from 1 to 200 were arranged in a random order on a circle such that the distances between adjacent numbers on the circle are the same.

For any number, the following is true: if you consider 99 numbers standing clockwise from it and 99 numbers standing counterclockwise from it, there will be an equal number of numbers that are smaller than it in both groups. Which number stands opposite the number 113?
Answer: 114.

Solution. Consider the number 2. Less than it is only the number 1. Since it is unique, it cannot be in any of the groups relative to the number 2. Therefore, 1 must be opposite 2.

Consider the number 4. The numbers less than it are three in number - an odd count. This means that one of them should not be in any of the groups relative to the number 4, i.e., it should be opposite. Since 1 and 2 are already standing opposite each other, 3 is opposite 4.

Continuing similar reasoning further, we get that opposite any even number stands an odd number that is one less than it. Thus, opposite the number 114 stands the number 113. And vice versa: opposite 113 stands 114.