Problem:
A right circular cone has base radius and height . The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making complete rotations. The value of can be written in the form , where and are positive integers and is not divisible by the square of any prime. Find .
Solution:
The slant height of the cone is . This is the radius of the circle described by the cone as it rolls on the table. The circumference of this circle is . The circumference of the base of the cone is . The cone makes complete rotations in rolling back to its original position, and so
Thus
and .
The problems on this page are the property of the MAA's American Mathematics Competitions