$A B C D E$ is a pyramid with square base $A B C D$. Point $E$ is directly above $A$ with $A E=1024$ and $A B=640$. The pyramid is cut into two pieces by a horizontal plane parallel to $A B C D$. This horizontal plane is a distance $h$ above the base $A B C D$. The portion of $A B C D E$ that is above the plane is a new pyramid. For how many integers $h$ is the volume of the new pyramid an integer?

![](https://cdn.mathpix.com/cropped/2024_04_17_486fbaaa1172337e46fbg-3.jpg?height=602&width=461&top_left_y=431&top_left_x=889)
We first find an expression for the volume of the top pyramid in terms of $h$.

The volume of any pyramid is equal to $\frac{1}{3} \times$ (area of base) $\times$ (height).

The height of the upper pyramid is $1024-h$.

Suppose the plane intersects $E A$ at $F$ and $E B$ at $G$.

![](https://cdn.mathpix.com/cropped/2024_04_17_698c2a84b151b65f3cc1g-06.jpg?height=591&width=542&top_left_y=1119&top_left_x=843)

Since the plane is parallel to the base, we have that $F G$ is parallel to $A B$.

This means $\triangle E F G$ is similar to $\triangle E A B$, which means $\frac{F G}{A B}=\frac{E F}{E A}$. We know that $A B=640$, $E A=1024$, and $E F=1024-h$, so substitution and rearranging leads to

$$
F G=\frac{(1024-h)(640)}{1024}=\frac{5(1024-h)}{8} .
$$

Next, we show that the base of the new pyramid is a square.

Label by $H$ the point at which the plane intersects $E C$ and $I$ the point at which the plane intersects $E D$.

We have already observed that $F G$ is parallel to $A B$. Similarly, $G H$ is parallel to $B C, H I$ is parallel to $C D$, and $I F$ is parallel to $D A$.

This means $F G$ is parallel to $H I$ and perpendicular to $I F$, which is parallel to $G H$, so $F G H I$ is a rectangle.

To see that $F G H I$ is a square, we will show that $I F=F G$.

We have already observed that $\triangle E F G$ is similar to $\triangle E A B$. For similar reasons, $\triangle E F I$ is similar to $\triangle E A D$.

Since $E$ is directly above $A$, we have that $\angle E A B=\angle E A D=90^{\circ}$.

As well, $A B=A D$, so $\triangle E A B$ is congruent to $\triangle E A D$ by side-angle-side congruence.

It follows that $\triangle E F I$ is similar to $\triangle E F G$, and since they share the side $E F, \triangle E F I$ is congruent to $\triangle E F G$.

Therefore, $F I=F G$, so $F G H I$ is a square.

We can now compute the volume of the square-based pyramid FGHIE.

The area of the base is

$$
F G^{2}=\left(\frac{5(1024-h)}{8}\right)^{2}
$$

and the height is $1024-h$, so the volume is

$$
\frac{1}{3}(1024-h)\left(\frac{5(1024-h)}{8}\right)^{2}=\frac{(1024-h)^{3} 5^{2}}{3 \times 2^{6}}
$$

We now need to count the positive integers $h$ with the property that $\frac{(1024-h)^{3} 5^{2}}{3 \times 2^{6}}$ is an integer.

This is the same as counting the integers $h$ with the property that $3 \times 2^{6}$ is a divisor of $(1024-h)^{3} 5^{2}$.

Since 5 is prime, $5^{2}$ has no factor of 2 and no factor of 3 . Therefore, we need to count integers $h$ with the property that $(1024-h)^{3}$ is a multiple of $3 \times 2^{6}$.

Since 3 is prime, $(1024-h)^{3}$ is a multiple of 3 exactly when $1024-h$ is a multiple of 3 .

Suppose $1024-h$ is a multiple of $2^{2}=4$. Then $(1024-h)^{3}$ is a multiple $2^{6}$.

If $(1024-h)^{3}$ is a multiple of $2^{6}$, then $1024-h$ must be even.

However, if $1024-h$ is a multiple of 2 but not a multiple of 4 , then $(1024-h)^{3}$ cannot have 6 copies of 2 in its prime factorization.

Therefore, $(1024-h)^{6}$ is a multiple of $2^{6}$ exactly when $(1024-h)$ is a multiple of 4.

We now have shown that the volume of the top pyramid is a positive integer exactly when $1024-h$ is a multiple of both 3 and 4 .

This means that the volume of the top pyramid is a positive integer exactly when $1024-h$ is a multiple of 12 .

When $h=4$, we have $1024-h=1024-4=1020=12 \times 85$.

If $h=4+12$, then $1024-h=12 \times 84$.

This continues to give the possibilities $h=4, h=4+12, h=4+2 \times 12, h=4+3 \times 12$, and so on.

The largest value of $h$ is when $1024-h=12$ or $h=1012=4+84 \times 12$.

Thus, the values are $h=4+k \times 12$ where $k$ ranges over the integers 0 through 84 .

There are 85 integers $h$ for which the volume of the upper pyramid is an integer.

ANSWER: 85

## Team Problems