### 44th Putnam 1983

**Problem B1**

Let C be a cube side 4, center O. Let S be the sphere center O radius 2. Let A be one of the vertices of the cube. Let R be the set of points in C but not S, which are closer to A than to any other vertex of C. Find the volume of R.

**Solution**

Answer: 8 - 4π/3.

The set of points of C closer to A than to any other vertex is the cube side 2 with A and O as opposite vertices. Evidently the intersection of this cube with S has 1/8 the volume of S. So R has volume 8 - 1/8 4/3 π 2^{3} = 8 - 4π/3 .

44th Putnam 1983

© John Scholes

jscholes@kalva.demon.co.uk

16 Jan 2001