Problem:
Point is in the exterior of the regular -sided polygon , and is an equilateral triangle. What is the largest value of for which , and are consecutive vertices of a regular polygon?
Solution:
Let be the number of sides of the polygon determined by , and . The degree measures of the interior angles of the three polygons are , and . If , the polygons fit together at their common vertex , thus
This can be rewritten in the form
It is clear that , so that is a decreasing function of . The largest value of is , obtained when .
The problems on this page are the property of the MAA's American Mathematics Competitions