11.1 A quirky person didn't mind the effort and recorded 2023 numbers in a circle such that each number is equal to the product of its two neighbors. What is the maximum number of different numbers that could have been used?
Solution: If there is a zero among the numbers, then its neighbors are also zeros, so all the recorded numbers will be equal to zero. We will assume that there are no zeros among the numbers. Let $a$ and $b$ be two adjacent numbers. Then on the other side of $a$ stands $a / b$, and on the other side of $b$ stands $b / a$. In other words, two numbers at a distance of 3 are reciprocals of each other, and then numbers at a distance of 6 are equal. Since 2023 is coprime with 6, it follows that all numbers are equal in any case.

## Criteria:

- Only the answer with an example - 0 points.
- Points are not deducted for the absence of an example where all numbers are equal.