Problem:
Let P1​ be a regular r-gon and P2​ be a regular s-gon (r≥s≥3) such that each interior angle of P1​ is 5859​ as large as each interior angle of P2​. What is the largest possible value of s?
Solution:
In a regular n-gon, each interior angle has radian measure (n−2)π/n. The information in the problem says
5859​=(rr−2​π)/(ss−2​π)=rs−2rrs−2s​(*)
Solving for r gives
r=118−s116s​
Since r must be positive, we must have s≤117. Indeed, if s=117​ then we find r=116⋅117 and equation (*) will be satisfied.
The problems on this page are the property of the MAA's American Mathematics Competitions