1995 AIME Problems/Problem 11

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 $\boxed{040}$ .

1995 AIME (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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1995 AIME Problems/Problem 11

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