Difference between revisions of "1995 AIME Problems/Problem 11"
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== Problem == | == Problem == | ||
| − | A right rectangular prism <math> | + | 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 == | ||
| + | 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 | ||
| + | |||
| + | <cmath>\frac{x}{a} = \frac{y}{b} = \frac zc < 1.</cmath> | ||
| + | |||
| + | 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 == | ||
| − | + | {{AIME box|year=1995|num-b=10|num-a=12}} | |
| − | + | ||
| − | + | [[Category:Intermediate Number Theory Problems]] | |
| + | {{MAA Notice}} | ||
Latest revision as of 18:30, 4 July 2013
Problem
A right rectangular prism 
(i.e., a rectangular parallelpiped) has sides of integral length 
with 
A plane parallel to one of the faces of cuts into two prisms, one of which is similar to 
and both of which have nonzero volume. Given that 
for how many ordered triples 
does such a plane exist?
Solution
Let 
be the prism similar to 
, and let the sides of be of length 
, such that 
. Then
![\[\frac{x}{a} = \frac{y}{b} = \frac zc < 1.\]](http://latex.artofproblemsolving.com/8/3/b/83b589d1ce0653b46106e03a94b02113f71ff76c.png)
Note that if the ratio of similarity was equal to 
, we would have a prism with zero volume. As one face of is a face of , it follows that and share at least two side lengths in common. Since 
, it follows that the only possibility is 
. Then,
![\[\frac{x}{a} = \frac{a}{1995} = \frac{1995}{c} \Longrightarrow ac = 1995^2 = 3^25^27^219^2.\]](http://latex.artofproblemsolving.com/3/8/9/389a5f452bc42ca3834beb2a917ba7dc611acb10.png)
The number of factors of 
is 
. Only in 
of these cases is 
(for 
, we end with a prism of zero volume). We can easily verify that these will yield nondegenerate prisms, so the answer is 
.
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
