A rectangular prism has side lengths of 26, 20, and 8 units. Construct straight gable roofs over the $26 \times 8$ faces, where the "slant" edges are 13 units long, and the "horizontal" edge is 20 units. What is the volume of the part of the rectangular prism that remains after removing the mirror images of the gable roofs on the faces of the rectangular prism?
Note. The phrase "straight ridge roof" (as many have pointed out) was not appropriate from either a mathematical or an architectural perspective. The overwhelming majority of the submissions thought of what the editorial board did.

Let the vertices of the $26 \times 8$ rectangle be denoted by $A, B, C$, and $D$, and the endpoints of the 20-unit edge of the ridge roof by $E$ and $F$, as shown in the figure. The projection of $E F$ onto the plane of the rectangle is $E^{\prime} F^{\prime}$.
The
ridge roof
symmetry
implies that $E^{\prime} F^{\prime}$ falls on the midline of the rectangle, and its extension intersects the $B C$ side at $G$!

In the right triangle $E^{\prime} G B$,

$$
\begin{gathered}
E^{\prime} G=\frac{26-20}{2}=3, \quad B G=4, \quad E B=\sqrt{4^{2}+3^{2}}=5 \\
E E^{\prime}=\sqrt{E B^{2}-E^{\prime} B^{2}}=\sqrt{13^{2}-5^{2}}=12
\end{gathered}
$$

this is the height of the ridge roof. Let's draw planes through $E$ and $F$ parallel to the $20 \times 8$ side of the rectangular prism.

The planes divide the ridge roof into a triangular prism with a 20-unit height and two congruent rectangular-based pyramids. Their volumes can be easily written down.

The volume of the prism: $\frac{8 \cdot 12}{2} \cdot 20=960$ cubic units. The combined volume of the two pyramids: $2 \cdot \frac{3 \cdot 8 \cdot 12}{3}=192$. The total volume: 1152.

If we reflect the ridge roofs onto the rectangles - since their combined height is $24>20$ greater than the edge of the rectangular prism -, the two reflections will have a common part.

Determine the volume of the common part!

This will be a ridge roof whose height is 2 units, $\frac{1}{6}$ of the original height. The base area of the triangular prism becomes $\frac{1}{36}$ of the original, while its height remains 20 units, so its volume decreases to $\frac{1}{36}$ of the original volume.

The base area of the rectangular-based pyramid becomes $\frac{1}{36}$ of the original, and its height $\frac{1}{6}$ of the original, so its volume becomes $\frac{1}{216}$ of the original prism volume.

The volume of the rectangular prism: $26 \cdot 20 \cdot 8=4160$ cubic units.

The volume of the overlapping parts when reflected: $2 \cdot \frac{96}{216}+\frac{960}{36} \approx 27,55$.

The sought volume: $4160-2 \cdot(1152-27,55) \approx 1911,1$ cubic units.

Márton Hablicsek (Fazekas M. Fóv. Gyak. Gimn., 8th grade)

![](https://cdn.mathpix.com/cropped/2024_05_02_fcaa6a05ee076fe5c8afg-1.jpg?height=615&width=569&top_left_y=1557&top_left_x=797)