\textbf{Exercise 4 — Candidates who have followed the specialization course}

For each of the following statements, say whether it is true or false by justifying the answer. One point is awarded for each correct justified answer. An unjustified answer will not be taken into account and the absence of an answer is not penalized.

\begin{itemize}
  \item Consider the system $\left\{ \begin{array} { l l l l } n & \equiv & 1 & { [ 5 ] } \\ n & \equiv & 3 & { [ 4 ] } \end{array} \right.$ with unknown $n$ a relative integer.
\end{itemize}

Statement 1: If $n$ is a solution of this system then $n - 11$ is divisible by 4 and by 5.\\
Statement 2: For all relative integer $k$, the integer $11 + 20 k$ is a solution of the system.\\
Statement 3: If a relative integer $n$ is a solution of the system then there exists a relative integer $k$ such that $n = 11 + 20 k$.

\begin{itemize}
  \item An automaton can be in one of two states A or B. At each second it can either remain in the state it is in or change it, with probabilities given by the probabilistic graph below.\\
For all natural number $n$, we denote $a _ { n }$ the probability that the automaton is in state A after $n$ seconds and $b _ { n }$ the probability that the automaton is in state B after $n$ seconds. Initially, the automaton is in state B.
\end{itemize}

Consider the following algorithm:

\begin{center}
\begin{tabular}{|l|l|}
\hline
\begin{tabular}{l}
Variables: Initialization: \\
Processing: \\
Output: \\
\end{tabular} & \begin{tabular}{l}
$a$ and $b$ are real numbers \\
$a$ takes the value 0 \\
$b$ takes the value 1 \\
For $k$ going from 1 to 10 \\
$a$ takes the value $0.8 a + 0.3 b$ \\
$b$ takes the value $1 - a$ \\
End For \\
Display $a$ \\
Display $b$ \\
\end{tabular} \\
\hline
\end{tabular}
\end{center}

Statement 4: On output, this algorithm displays the values of $a _ { 10 }$ and $b _ { 10 }$.\\
Statement 5: After 4 seconds, the automaton has an equal chance of being in state $A$ or being in state $B$.