B4

Part of 2008 Princeton University Math Competition

Problems(3)

2008 PUMaC Geometry B4

Source:

A cube is divided into

27

unit cubes. A sphere is inscribed in each of the corner unit cubes, and another sphere is placed tangent to these

8

spheres. What is the smallest possible value for the radius of the last sphere?

2008 PUMaC Combinatorics B4

Source:

2008 \times 2009

rectangle is divided into unit squares. In how many ways can you remove a pair of squares such that the remainder can be covered with

1 \times 2

2008 PUMaC Algebra B4

Source:

Find the product of the minimum and maximum values of

\frac{3x+1}{9x^2+6x+2}