\section*{Problem 3 - 131033}

Consider a four-sided pyramid with a square base. The vertices of this base are the points \(A, B, C\), and \(D\). The apex of the pyramid is \(S\). All eight edges have the length \(a\).

\(E\) and \(F\) are the midpoints of the edges \(S B\) and \(S C\), respectively. A plane through the points \(A, E, F\), and \(D\) divides the pyramid into two sub-bodies.

Calculate the ratio of the volumes of these two sub-bodies!
}

Calculation of the lower partial body:

The lower partial body can be understood, suitably divided, as a triangular prism \(T_{1}\) and two slidable pyramids \(T_{2}\). The base area of the prism is \(G=\frac{a}{2} \cdot \frac{a}{2}\) and the prism height is \(h=\frac{a}{4} \sqrt{2}\). Therefore,

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-1333.jpg?height=431&width=696&top_left_y=236&top_left_x=246)

For the upper partial body, the remaining volume is

\[
V_{\text {above }}=V_{\text {pyramid }}-V_{\text {below }}=\frac{1}{16} a^{3} \sqrt{2}
\]

and a ratio of \(V_{\text {above }}: V_{\text {below }}=3: 5\).