Question

In a regular tetrahedron, the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

Answer

Solution:

Use position vectors, and let $G = \frac{A + B + C + D}{4}$ be the centroid of the tetrahedron. The center of the face opposite $A$ is $\frac{B + C + D}{3} = \frac{4G - A}{3} = G - \frac{1}{3}(A - G),$ so each face center is the image of the opposite vertex under the homothety centered at $G$ with ratio $-\frac{1}{3}.$

Hence the smaller tetrahedron is similar to the larger with ratio $\frac{1}{3},$ and its volume is $\left(\frac{1}{3}\right)^3 = \frac{1}{27}$ of the larger. Thus $m + n = 1 + 27 = 28.$

2003 AIME II Problem 4

Solution:

Problem 4 in Other Years