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
See also
2012 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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All AIME Problems and Solutions |