n hoops on a circle
Source: Flanders Math Olympiad 2020 p4
December 24, 2022
combinatorics
Problem Statement
There are hoops on a circle.
Rik numbers all hoops with a natural number so that all numbers from to occur exactly once. Then he makes one walk from hoop to hoop. He starts in hoop and then follows the following rule: if he gets to hoop , then he walks to the hoop that places clockwise without getting into the intermediate hoops. The walk ends when Rik has to walk to a hoop he has already been to. The length of the walk is the number of hoops he passed on the way.
For example, for Rik can take a walk of length as the hoops are numbered as shown in the figure.
https://cdn.artofproblemsolving.com/attachments/2/a/3d4b7edbba4d145c7e00368f9b794f39572dc5.png
(a) Determine for every even how Rik can number the hoops so that he has one walk of length .(b) Determine for every odd how Rik can number the hoops so that he has one walk of length .(c) Show that for an odd there is no such numbering of the hoops that Rik can make a walk of length .