Problem:
Define a regular -pointed star to be the union of line segments such that
the points are coplanar and no three of them are collinear,
each of the line segments intersects at least one of the other line segments at a point other than an endpoint,
all of the angles at are congruent,
all of the line segments are congruent, and
the path turns counterclockwise at an angle of less than at each vertex.
There are no regular -pointed, -pointed, or -pointed stars. All regular -pointed stars are similar, but there are two non-similar regular -pointed stars. How many non-similar regular -pointed stars are there?
Solution:
Let be the circle determined by , and . Because the path turns counterclockwise at an angle of less than at and and must be on the same side of line . Note that , and so . Thus is on . Similarly, because , and are on must be too, and, in general, are all on . The fact that the minor arcs , and are congruent implies that , and are equally spaced on .
Thus any regular -pointed star can be constructed by choosing equally spaced points on a circle, and numbering them consecutively from to . For positive integers , the path consisting of line segments whose vertices are numbered modulo will be a regular -pointed star if and only if and is relatively prime to . This is because if or , the resulting path will be a polygon; and if is not relatively prime to , not every vertex will be included in the path. Also, for any choice of that yields a regular -pointed star, any two such stars will be similar because a dilation of one of the stars about the center of its circle will yield the other.
Because , numbers that are relatively prime to are those that are multiples of neither nor . There are multiples of that are less than or equal to ; there are multiples of that are less than or equal to ; and there are numbers less than or equal to that are multiples of both and . Hence there are numbers that are less than and relatively prime to , and of them are between and , inclusive. Because yields the same path as (and also because one of these two paths turns clockwise at each vertex), there are non-similar regular -pointed stars.
The problems on this page are the property of the MAA's American Mathematics Competitions