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Difference between revisions of "1995 AIME Problems/Problem 11"

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== Problem ==
 
== Problem ==
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A right rectangular prism <math>\displaystyle P_{}</math> (i.e., a rectangular parallelpiped) has sides of integral length <math>\displaystyle a, b, c,</math> with <math>\displaystyle a\le b\le c.</math>  A plane parallel to one of the faces of <math>\displaystyle P_{}</math> cuts <math>\displaystyle P_{}</math> into two prisms, one of which is similar to <math>\displaystyle P_{},</math> and both of which have nonzero volume.  Given that <math>\displaystyle b=1995,</math> for how many ordered triples <math>\displaystyle (a, b, c)</math> does such a plane exist?
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A right rectangular [[prism]] <math>P_{}</math> (i.e., a rectangular parallelpiped) has sides of integral length <math>a, b, c,</math> with <math>a\le b\le c.</math>  A plane parallel to one of the faces of <math>P_{}</math> cuts <math>P_{}</math> into two prisms, one of which is [[similar]] to <math>P_{},</math> and both of which have nonzero volume.  Given that <math>b=1995,</math> for how many ordered triples <math>(a, b, c)</math> does such a plane exist?
  
 
== Solution ==
 
== Solution ==
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Let <math>P'</math> be the prism similar to <math>P</math>, and let the sides of <math>P'</math> be of length <math>x,y,z</math>, such that <math>x \le y \le z</math>. Then
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<cmath>\frac{x}{a} = \frac{y}{b} = \frac zc < 1.</cmath>
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Note that if the ratio of similarity was equal to <math>1</math>, we would have a prism with zero volume. As one face of <math>P'</math> is a face of <math>P</math>, it follows that <math>P</math> and <math>P'</math> share at least two side lengths in common. Since <math>x < a, y < b, z < c</math>, it follows that the only possibility is <math>y=a,z=b=1995</math>. Then,
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<cmath>\frac{x}{a} = \frac{a}{1995} = \frac{1995}{c} \Longrightarrow ac = 1995^2 = 3^25^27^219^2.</cmath>
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The number of factors of <math>3^25^27^219^2</math> is <math>(2+1)(2+1)(2+1)(2+1) = 81</math>. Only in <math>\left\lfloor \frac {81}2 \right\rfloor = 40</math> of these cases is <math>a < c</math> (for <math>a=c</math>, we end with a prism of zero volume). We can easily verify that these will yield nondegenerate prisms, so the answer is <math>\boxed{040}</math>.
  
 
== See also ==
 
== See also ==
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* [[1995_AIME_Problems/Problem_10|Previous Problem]]
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{{AIME box|year=1995|num-b=10|num-a=12}}
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* [[1995_AIME_Problems/Problem_12|Next Problem]]
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* [[1995 AIME Problems]]
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[[Category:Intermediate Number Theory Problems]]

Revision as of 17:45, 29 July 2008

Problem

A right rectangular prism $P_{}$ (i.e., a rectangular parallelpiped) has sides of integral length $a, b, c,$ with $a\le b\le c.$ A plane parallel to one of the faces of $P_{}$ cuts $P_{}$ into two prisms, one of which is similar to $P_{},$ and both of which have nonzero volume. Given that $b=1995,$ for how many ordered triples $(a, b, c)$ does such a plane exist?

Solution

Let $P'$ be the prism similar to $P$, and let the sides of $P'$ be of length $x,y,z$, such that $x \le y \le z$. Then

\[\frac{x}{a} = \frac{y}{b} = \frac zc < 1.\]

Note that if the ratio of similarity was equal to $1$, we would have a prism with zero volume. As one face of $P'$ is a face of $P$, it follows that $P$ and $P'$ share at least two side lengths in common. Since $x < a, y < b, z < c$, it follows that the only possibility is $y=a,z=b=1995$. Then,

\[\frac{x}{a} = \frac{a}{1995} = \frac{1995}{c} \Longrightarrow ac = 1995^2 = 3^25^27^219^2.\]

The number of factors of $3^25^27^219^2$ is $(2+1)(2+1)(2+1)(2+1) = 81$. Only in $\left\lfloor \frac {81}2 \right\rfloor = 40$ of these cases is $a < c$ (for $a=c$, we end with a prism of zero volume). We can easily verify that these will yield nondegenerate prisms, so the answer is $\boxed{040}$.

See also

1995 AIME (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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