We consider a cube $ABCDEFGH$.\\
The point M is the midpoint of $[\mathrm{BF}]$, I is the midpoint of [BC], the point N is defined by the relation $\overrightarrow{\mathrm{CN}} = \frac{1}{2}\overrightarrow{\mathrm{GC}}$ and the point P is the center of the face ADHE.

\section*{Part A:}
\begin{enumerate}
  \item Justify that the line (MN) intersects the segment [BC] at its midpoint I.
  \item Construct, on the figure provided in the appendix, the cross-section of the cube by the plane (MNP).
\end{enumerate}

\section*{Part B:}
We equip space with the orthonormal coordinate system ($A; \overrightarrow{AB}, \overrightarrow{AD}, \overrightarrow{AE}$).

\begin{enumerate}
  \item Justify that the vector $\vec{n}\left(\begin{array}{l}1\\2\\2\end{array}\right)$ is a normal vector to the plane (MNP).

Deduce a Cartesian equation of the plane (MNP).\\
  \item Determine a system of parametric equations of the line (d) passing through G and orthogonal to the plane (MNP).\\
  \item Show that the line (d) intersects the plane (MNP) at the point K with coordinates $\left(\frac{2}{3}; \frac{1}{3}; \frac{1}{3}\right)$. Deduce the distance GK.\\
  \item We admit that the four points M, E, D and I are coplanar and that the area of the quadrilateral MEDI is $\frac{9}{8}$ square units.\\
Calculate the volume of the pyramid GMEDI.
\end{enumerate}