Summary: Sponsored by: UGA Math Department and UGA Math Club
Team Round / 45 min / 210 points
October 13, 2007
No calculators are allowed on this test. You do not have to provide proofs;
only the answers matter. Each problem is worth 70 points, for a total of 210 points.
Problem 1. (Spheres) An infinite sequence of spheres is inscribed in a right circular
cone with base 2, with the largest having radius 1, as shown. Find the volume of that
portion of the cone lying outside all the spheres.
Give the answer as a fraction, possibly involving ,
2, etc. In other words, give
the exact answer, not a decimal approximation.