\textbf{For candidates who have followed the specialization course}

The two parts are independent.

\section*{Part A}
A laboratory studies the evolution of a population of parasitic insects on plants.\\
This evolution has two stages: a larval stage and an adult stage which is the only one during which insects can reproduce.\\
Observation of the evolution of this population leads to proposing the following model.\\
Each week:
\begin{itemize}
  \item Each adult gives birth to 2 larvae then $75\%$ of adults die.
  \item $25\%$ of larvae die and $50\%$ of larvae become adults.
\end{itemize}
For all natural integer $n$, we denote $\ell _ { n }$ the number of larvae and $a _ { n }$ the number of adults after $n$ weeks.\\
For all natural integer $n$, we denote $X _ { n }$ the column matrix defined by: $X _ { n } = \binom { \ell _ { n } } { a _ { n } }$

\begin{enumerate}
  \item Show that, for all natural integer $n$, $X _ { n + 1 } = A X _ { n }$ where $A$ is the matrix:
$$A = \left( \begin{array} { c c } 
0.25 & 2 \\
0.5 & 0.25
\end{array} \right)$$
  \item We denote $U$ and $V$ the column matrices: $U = \binom { 2 } { 1 }$ and $V = \binom { a } { 1 }$, where $a$ is a real number.\\
a. Show that $A U = 1.25 U$.\\
b. Determine the real number $a$ such that $A V = - 0.75 V$.
\end{enumerate}

In questions 3 and 4, the real number $a$ is fixed so that it is the solution of $A V = - 0.75 V$.

\begin{enumerate}
  \setcounter{enumi}{2}
  \item It is admitted that there exist two real numbers $\alpha$ and $\beta$ such that: $X _ { 0 } = \alpha U + \beta V$ and $\alpha > 0$.\\
a. Show that, for all natural integer $n$, $X _ { n } = \alpha ( 1.25 ) ^ { n } U + \beta ( - 0.75 ) ^ { n } V$.\\
b. Deduce that for all natural integer $n$:
$$\left\{ \begin{array} { l } \ell _ { n } = 2 ( 1.25 ) ^ { n } \left( \alpha - \beta ( - 0.6 ) ^ { n } \right) \\ a _ { n } = ( 1.25 ) ^ { n } \left( \alpha + \beta ( - 0.6 ) ^ { n } \right) . \end{array} \right.$$
  \item Show that $\lim _ { n \rightarrow + \infty } \frac { \ell _ { n } } { a _ { n } } = 2$. Interpret this result in the context of the exercise.
\end{enumerate}

\section*{Part B}
\begin{enumerate}
  \item We consider the equation $( E ) : 19 x - 6 y = 1$. Determine the number of couples of integers ( $x ; y$ ) solutions of the equation $( E )$ and satisfying $2000 \leqslant x \leqslant 2100$.
  \item Let $n$ be a natural integer. Show that the integers ( $2 n + 3$ ) and ( $n + 3$ ) are coprime if and only if $n$ is not a multiple of 3.
\end{enumerate}