Problem:
Let be the transformation of the coordinate plane that first rotates the plane degrees counterclockwise around the origin and then reflects the plane across the -axis. What is the least positive integer such that performing the sequence of transformations returns the point back to itself?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Denote by the origin of the coordinate plane. Let be the point obtained after the first transformations, where . Observe that, for each integer and for every point , the transformation preserves the distance from to the origin. Thus for every . This means that the point may be uniquely characterized by the counterclockwise angle between the -axis and the ray . (Here is measured in degrees.)
The transformation rotates the plane degrees counterclockwise around the origin and then reflects the plane across the -axis. This means that and
Note that and . It follows that and for every positive integer . Therefore ( 1,0 ) first returns to its starting position when and .
The problems on this page are the property of the MAA's American Mathematics Competitions