Problem:
Let P1β be a regular r-gon and P2β be a regular s-gon (rβ₯sβ₯3) such that each interior angle of P1β is 5859β as large as each interior angle of P2β. What is the largest possible value of s?
Solution:
In a regular n-gon, each interior angle has radian measure (nβ2)Ο/n. The information in the problem says
5859β=(rrβ2βΟ)/(ssβ2βΟ)=rsβ2rrsβ2sβ(*)
Solving for r gives
r=118βs116sβ
Since r must be positive, we must have sβ€117. Indeed, if s=117β then we find r=116β
117 and equation (*) will be satisfied.
The problems on this page are the property of the MAA's American Mathematics Competitions