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May 11, 2020

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

Check the Answer


Answer: is 40.

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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