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Difference between revisions of "2012 AIME I Problems/Problem 8"

(Created page with "==Problem 8== == Solution == == See also == {{AIME box|year=2012|n=I|num-b=7|num-a=9}}")
 
(→‎Problem 8)
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==Problem 8==
 
==Problem 8==
 +
Cube <math>ABCDEFGH,</math> labeled as shown below, has edge length <math>1</math> and is cut by a planing passing through vertex <math>D</math> and the midpoints <math>M</math> and <math>N</math> of <math>\overline{AB}</math> and <math>\overline{CG}</math> respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form <math>\tfrac{p}{q},</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q.</math>
 +
 +
<center><asy>import cse5;
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unitsize(10mm);
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pathpen=black;
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dotfactor=3;
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 +
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);
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D(E--F--B);
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D(F--G);
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pathpen=dashed;
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D(A--D--H);
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D(D--C);
 +
 +
dot(dotted);
 +
label("$A$",A,SW);
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label("$B$",B,S);
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label("$C$",C,SE);
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label("$D$",D,NW);
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label("$E$",E,W);
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label("$F$",F,SE);
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label("$G$",G,NE);
 +
label("$H$",H,NW);
 +
label("$M$",M,S);
 +
label("$N$",N,NE);
 +
 +
</asy></center>
  
 
== Solution ==
 
== Solution ==

Revision as of 17:47, 17 March 2012

Problem 8

Cube $ABCDEFGH,$ labeled as shown below, has edge length $1$ and is cut by a planing 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]

Solution

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
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