Problem
2012 AIME I Problem 8
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 \overline{AB} and \overline{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 \tfrac{p}{q}, where p and q are relatively prime positive integers. Find p+q.
![[asy]import cse5; unitsize(10mm); pathpen=black; dotfactor=3; pair A = (0,0), B = (3.8,0), C = (5.876,1.564), D = (2.076,1.564), E = (0,3.8), F = (3.8,3.8), G = (5.876,5.364), H = (2.076,5.364), M = (1.9,0), N = (5.876,3.465); pair[] dotted = {A,B,C,D,E,F,G,H,M,N}; D(A--B--C--G--H--E--A); D(E--F--B); D(F--G); pathpen=dashed; D(A--D--H); D(D--C); dot(dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,W); label("$F$",F,SE); label("$G$",G,NE); label("$H$",H,NW); label("$M$",M,S); label("$N$",N,NE); [/asy]](/static/aops/af154db9d25256bd90239e9f4e37253b234659e0.png)
Leading zeroes must be inputted, so if your answer is 34, then input 034
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