Problem:
Cube ABCDEFGH, labeled as shown below, has edge length 1 and is cut by a plane passing through vertex D and the midpoints M and N of AB and CG, respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form qp​, where p and q are relatively prime positive integers. Find p+q.
Solution:
Let the plane DMN intersect BF at point P and the extension of BC at point K. Because M is the midpoint of AB and MB is parallel to CD, it follows that B is the midpoint of CK and therefore CK=2. Hence the volume of the pyramid DCNK is equal to 31​[CDN]⋅CK=31​⋅21​⋅21​⋅2=61​. Because DCNK is similar to MBPK with a constant of proportionality of 2, the volume of MBPK is 231​⋅61​=481​, and the volume of DCNMBP is 61​−481​=487​. Thus the volume of the larger solid is 1−487​=4841​, and p+q=89​.
The problems on this page are the property of the MAA's American Mathematics Competitions
