MathDB

One right pyramid has a base that is a regular hexagon with side length

1

, and the height of the pyramid is

8

. Two other right pyramids have bases that are regular hexagons with side length

4

, and the heights of those pyramids are both

7

. The three pyramids sit on a plane so that their bases are adjacent to each other and meet at a single common vertex. A sphere with radius

4

rests above the plane supported by these three pyramids. The distance that the center of the sphere is from the plane can be written as

\frac{p\sqrt{q}}{r}

, where

p, q

, and

r

are relatively prime positive integers, and

q

is not divisible by the square of any prime. Find

p+q+r

Problem Statement