\textbf{Part A: a volume calculation without a coordinate system}\\
We consider an equilateral pyramid SABCD (pyramid with a square base whose lateral faces are all equilateral triangles).\\
The diagonals of the square ABCD measure 24 cm.\\
We denote O the center of the square ABCD.\\
We will admit that $\mathrm { OS } = \mathrm { OA }$.

\begin{enumerate}
  \item Without using a coordinate system, prove that the line (SO) is orthogonal to the plane (ABC).
  \item Deduce the volume, in $\mathrm { cm } ^ { 3 }$, of the pyramid SABCD.
\end{enumerate}

\textbf{Part B: in a coordinate system}\\
We consider the orthonormal coordinate system ( $\mathrm { O } ; \overrightarrow { \mathrm { OA } } , \overrightarrow { \mathrm { OB } } , \overrightarrow { \mathrm { OS } }$ ).

\begin{enumerate}
  \item We denote P and Q the midpoints of the segments [AS] and [BS] respectively.\\
a. Justify that $\vec { n } ( 1 ; 1 ; - 3 )$ is a normal vector to the plane (PQC).\\
b. Deduce a Cartesian equation of the plane (PQC).
  \item Let H be the point of the plane (PQC) such that the line (SH) is orthogonal to the plane (PQC).\\
a. Give a parametric representation of the line (SH).\\
b. Calculate the coordinates of the point H.\\
c. Show then that the length SH, in unit of length, is $\frac { 2 \sqrt { 11 } } { 11 }$.
  \item We will admit that the area of the quadrilateral PQCD, in unit of area, is equal to $\frac { 3 \sqrt { 11 } } { 8 }$. Calculate the volume of the pyramid SPQCD, in unit of volume.
\end{enumerate}

\textbf{Part C: fair sharing}\\
For the birthday of her twin daughters Anne and Fanny, Mrs. Nova has made a beautiful cake in the shape of an equilateral pyramid whose diagonals of the square base measure 24 cm.\\
She is about to share it equally by placing her knife on the apex. That is when Anne stops her and proposes a more original cut:\\
``Place the blade on the midpoint of an edge, parallel to a side of the base, then cut towards the opposite side''.\\
Is this the case? Justify the answer.