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Geometry Difficulty 4.8 AIME Find the answer Canada

Cube $ABCDEFGH$ has edge length 100. Point $P$ is on $AB$ , point $Q$ is on $AD$ , and point $R$ is on $AF$ , as shown, so that $AP = x$ , $AQ = x+1$ and $AR = \dfrac{x+1}{2x}$ for some integer $x$ .For how many integers $x$ is the volume of triangular-based pyramid $APQR$ between 0.04% and 0.08% of the volume of cube $ABCDEFGH$ ? (The
volume of a pyramid is equal to one-third of the area of its base times
its height.)

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Triangular-based pyramid $APQR$ can be thought of as having triangular base $\triangle APQ$ and height $AR$ . Since this pyramid is built at a vertex of the cube, then $\triangle APQ$ is right-angled at $A$ and $AR$ is perpendicular to the base. The area of $\triangle APQ$ is $\$\dfrac{1}{2} \times AP \times$ AQ =
$\dfrac{1}{2}x(x+1)$ . The height of the pyramid is

Figure for this problem $\dfrac{x+1}{2x}$ . Thus, the volume of the pyramid is

Figure for this problem $\dfrac{1}{3} \times \dfrac{1}{2}x(x+1) \times \dfrac{x+1}{2x}$ $which equals$ $\dfrac{(x+1)^2}{12}$ . Since the cube has edge length 100, its volume is

Figure for this problem 100^3 $or$ 1\,000\,000 $. Now, 1% of 1 000 000 is$ $\dfrac{1}{100}$ $of 1 000 000 or 10 000. Thus, 0.01% of 1 000 000 is$ $\dfrac{1}{100}$ of 10 000 or 100. This tells us that 0.04% of 1 000 000 is 400, and 0.08% of 1 000 000 is 800. We want to determine the number of integers

Figure for this problem x $for which$ $\dfrac{(x+1)^2}{12}$ $is between 400 and 800. This is equivalent to determining the number of integers$ x $for which$ (x+1)^2 $is between$ 12 $\times$ 400 = 4800 $and$ 12 $\times$ 800 = 9600 $. Since$ $\sqrt{4800} \approx$ 69.28 $and$ $\sqrt{9600} \approx$ 97.98, then the perfect squares between 4800 and 9600 are

Figure for this problem 70^2, 71^2, 72^2, $\ldots,$ 96^2,
97^2. These are the possible values for

Figure for this problem (x+1)^2 and so the possible values for

Figure for this problem x $are$ 69, 70, 71, $\ldots,$ 95, 96 $. There are$ 96 - 69 + 1 = 28 $values for$ x$.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.