# Right Rectangular Prism | AIME I, 1995 | Question 11

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1995 based on Right Rectangular Prism.

## Right Rectangular Prism - AIME I, 1995

A right rectangular prism P (that is rectangular parallelopiped) has sides of integral length a,b,c with $a\leq b \leq c$, 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 has non-zero volume, given that b=1995, find number of ordered tuples (a,b,c) does such a plane exist.

• is 107
• is 40
• is 840
• cannot be determined from the given information

### Key Concepts

Integers

Divisibility

Algebra

AIME I, 1995, Question 11

Geometry Vol I to IV by Hall and Stevens

## Try with Hints

First hint

Let Q be similar to P

Let sides of Q be x,y,z for $x \leq y \leq z$

then $\frac{x}{a}=\frac{y}{b}=\frac{z}{c} < 1$

Second Hint

As one face of Q is face of P

or, P and Q has at least two side lengths in common

or, x <a, y<b, z<c

or, y=a, z=b=1995

or, $\frac{x}{a}=\frac{a}{1995}=\frac{1995}{c}$

or, $ac=1995^{2}=(3)^{2}(5)^{2}(7)^{2}(19)^{2}$

Final Step

or, number of factors of $(3)^{2}(5)^{2}(7)^{2}(19)^{2}$=(2+1)(2+1)(2+1)(2+1)=81

or, $[\frac{81}{2}]=40$ for a <c.

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