Question

Given that $\mathcal{O}$ is a regular octahedron, that $\mathcal{C}$ is the cube whose vertices are the centers of the faces of $\mathcal{O},$ and that the ratio of the volume of $\mathcal{O}$ to that of $\mathcal{C}$ is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, find $m + n.$

Answer

Solution:

Place the octahedron's vertices at $(\pm 1, 0, 0),$ $(0, \pm 1, 0),$ $(0, 0, \pm 1).$ It is two square pyramids glued along the square with vertices $(\pm 1, 0, 0)$ and $(0, \pm 1, 0),$ which has area $2,$ and each pyramid has height $1,$ so $V_{\mathcal{O}} = 2 \cdot \frac{1}{3} \cdot 2 \cdot 1 = \frac{4}{3}.$

Each face centroid is the average of that face's three vertices, e.g. $\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\right),$ so the cube has vertices $\left(\pm\frac{1}{3}, \pm\frac{1}{3}, \pm\frac{1}{3}\right).$ Its edge is $\frac{2}{3}$ and its volume is $\frac{8}{27}.$

The ratio is $\frac{4/3}{8/27} = \frac{9}{2},$ so $m + n = 9 + 2 = 11.$

2005 AIME II Problem 10

Solution:

Problem 10 in Other Years