2012 AIME I Problems/Problem 8
Problem 8
Cube 
labeled as shown below, has edge length 
and is cut by a planing passing through vertex 
and the midpoints 
and 
of 
and 
respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form 
where 
and 
are relatively prime positive integers. Find 
![[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]](http://latex.artofproblemsolving.com/a/f/1/af154db9d25256bd90239e9f4e37253b234659e0.png)
Solution
Define a coordinate system with at the origin and 
and 
on the 
, 
, and 
axes respectively. The 
and 
It follows that the plane going through 
and has equation 
Let 
be the intersection of this plane and edge 
and let 
Now since 
is on the plane. Also, lies on the extensions of segments 
and 
so the solid 
is a right triangular pyramid. Note also that pyramid 
is similar to 
with scale factor 
and thus the volume of solid 
which is one of the solids bounded by the cube and the plane, is ![$[DPCN] - [MPBQ] = [DPCN] - \left(\frac{1}{2}\right)^3[DPCN] = \frac{7}{8}[DPCN].$](http://latex.artofproblemsolving.com/f/3/8/f38a345d7d2ca1ed8772089bed1fdda122eb1b04.png)
But the volume of is simply the volume of a pyramid with base and height which is 
So ![$[DMBCQN] = \frac{7}{8} \cdot \frac{1}{6} = \frac{7}{48}.$](http://latex.artofproblemsolving.com/6/6/3/6636ae83385b2bf1620ca7a5bba37147b2ac9c15.png)
Note, however, that this volume is less than and thus this solid is the smaller of the two solids. The desired volume is then ![$[ABCDEFGH] - [DMBCQN] = 1 - \frac{7}{48} = \frac{41}{48} \rightarrow p+q = \boxed{089.}$](http://latex.artofproblemsolving.com/a/e/4/ae4820579b6396a4110984789dcffcfcc2084e7c.png)
See also
| 2012 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
