CEMC Pascal, 2023

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 $\dfrac{x+1}{2x}$.

Thus, the volume of the pyramid is $\dfrac{1}{3} \times \dfrac{1}{2}x(x+1) \times \dfrac{x+1}{2x}$ which equals $\dfrac{(x+1)^2}{12}$.

Since the cube has edge length 100, its volume is $100^3$ 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 $x$ 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 $70^2, 71^2, 72^2, \ldots, 96^2, 97^2$.

These are the possible values for $(x+1)^2$ and so the possible values for
$x$ are $69, 70, 71, \ldots, 95, 96$.

There are $96 - 69 + 1 = 28$ values
for $x$.