Let $x$, $y$ and $z$ be three real numbers. We consider the following implications $\left( P _ { 1 } \right)$ and $\left( P _ { 2 } \right)$:

$$\begin{array} { l l } 
\left( P _ { 1 } \right) & ( x + y + z = 1 ) \Rightarrow \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \geqslant \frac { 1 } { 3 } \right) \\
\left( P _ { 2 } \right) & \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \geqslant \frac { 1 } { 3 } \right) \Rightarrow ( x + y + z = 1 )
\end{array}$$

\section*{Part A}
Is the implication $\left( P _ { 2 } \right)$ true?

\section*{Part B}
In space, we consider the cube $A B C D E F G H$ and we define the orthonormal coordinate system $( A ; \overrightarrow { A B } , \overrightarrow { A D } , \overrightarrow { A E } )$.

\begin{enumerate}
  \item a. Verify that the plane with equation $x + y + z = 1$ is the plane $( B D E )$.\\
b. Show that the line $( A G )$ is orthogonal to the plane $( B D E )$.\\
c. Show that the intersection of the line $( A G )$ with the plane $( B D E )$ is the point $K$ with coordinates $\left( \frac { 1 } { 3 } ; \frac { 1 } { 3 } ; \frac { 1 } { 3 } \right)$.
  \item Is the triangle $B D E$ equilateral?
  \item Let $M$ be a point in space.\\
a. Prove that if $M$ belongs to the plane $( B D E )$, then $A M ^ { 2 } = A K ^ { 2 } + M K ^ { 2 }$.\\
b. Deduce that if $M$ belongs to the plane $( B D E )$, then $A M ^ { 2 } \geqslant A K ^ { 2 }$.\\
c. Let $x$, $y$ and $z$ be arbitrary real numbers. By applying the result of the previous question to the point $M$ with coordinates $( x ; y ; z )$, show that the implication $\left( P _ { 1 } \right)$ is true.
\end{enumerate}