Roller coaster - Colegiul National C. Negruzzi (Iași - Roumanie)

A disc with diameter d rolls on a disc with diameter D (0<d<D) without slipping until it reaches its original position.
a) How many times will the small disc rotate until it makes a complete revolution around the disc with the diameter D? What is the distance travelled by the center of the small disc in this case?
b) Assume that, in the initial position, the point A is the common tangent point. What is the minimum number of rotations that the small disc will make around its center until A becomes a common tangent point again? What is the distance travelled by the center of the small disc in this case?
c) Respond to points (a) and (b) if the disc with the diameter d rolls: on an equilateral triangle with sides of length L (L=D) or on a square with sides of length L or on a regular hexagon with sides of length L.
d) Reset the requirements of the previous paragraphs to: the disc with the diameter d rolls without slipping inside a circle with the diameter D or inside an equilateral triangle with sides of length L or inside a square with sides of length L or inside a hexagon with sides of length L.