Problem:
Let the set consist of the twelve vertices of a regular -gon. A subset of is called communal if there is a circle such that all points of are inside the circle, and all points of not in are outside of the circle. How many communal subsets are there? (Note that the empty set is a communal subset.)
Solution:
It is clear that and the empty set are both communal. Let be the circle that passes through the points of . A proper subset of is communal if and only if there is an arc of such that is the set of points of that lie on that arc. To see this let be a communal subset of , and let be a circle so that the points of are inside of , and the points of not in are outside of . Because the portion of that is inside of is an arc, the points of must be the points of that lie on this arc.
Now let be the set of points in that lie on an of . If necessary, extend slightly so that no points of are endpoints of . Let and be the endpoints of , and let be the midpoint of . Let be the circle with center passing through and . Then is the arc of inside of , all points of are inside of , and all points of not in are outside of . This shows that is communal.
Let , and let be a communal subset with points. Then each rotation of through an angle of results in another communal subset of elements. With successive rotations this process generates communal subsets of elements. Thus the number of communal subsets is .
The problems on this page are the property of the MAA's American Mathematics Competitions