AIME I #5 2008

[stupidityismygam]

A right circular cone has base radius [unparseable or potentially dangerous latex formula] and height [unparseable or potentially dangerous latex formula]. The cone lies on its side on a flat table. As the cone rolls on he 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 [unparseable or potentially dangerous latex formula] complete rotations. The value of [unparseable or potentially dangerous latex formula] can be written in the form [unparseable or potentially dangerous latex formula], where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are positive integers and [unparseable or potentially dangerous latex formula] is not divisible by the square of any prime. Find [unparseable or potentially dangerous latex formula].

[Celsion]

The slant height of a cone is [unparseable or potentially dangerous latex formula]. Radius of the circle is equal to the slant height.
Length of path:
[unparseable or potentially dangerous latex formula]
Since the cone made [unparseable or potentially dangerous latex formula] complete rotations, the length of the path is [unparseable or potentially dangerous latex formula] times the circumference of the base of the cone, [unparseable or potentially dangerous latex formula].
Set the two equal:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
Square root of both sides:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]