2003 AIME I Problems/Problem 5

Problem

Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is $(m + n \pi)/p,$ where $m, n,$ and $p$ are positive integers, and $n$ and $p$ are relatively prime, find $m + n + p.$

Solution

The set can be broken into several parts: the big parallelepiped, the 6 external parallelepipeds, and the 1/8-th spheres (one centered at each vertex).

The volume of the parallelepiped is $60$ cubic units.

The volume of the external parallelepipeds is $2(12)+2(15)+2(20)=94$ .

There are 8 1/8-th spheres, each of radius 1. Together, their volume is $\frac43\pi$ .

The combined volume of these parts is $60+94+\frac43\pi = \frac{464+4\pi}3$ .

$m+n+p = 464+4+3 = 471$

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2003 AIME I Problems/Problem 5

Problem

Solution

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