\documentclass[a4paper,12pt]{article}
\usepackage[left=1.00cm, right=1.00cm, top=2cm, bottom=1.50cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage[french]{babel}
\usepackage{amsmath, amssymb}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{multicol}
\usepackage{enumitem}
\usepackage{mathrsfs}
\usepackage{tcolorbox,varwidth}
\usepackage{fontspec}

\usepackage{colortbl}
\usepackage{libertinus}

\tcbuselibrary{skins,breakable}
\usepackage{tikz}
\usetikzlibrary{calc}
\usetikzlibrary{shadows}
\usepackage{tabularx, array}
\usepackage{fancyhdr}
\usepackage{setspace}

% Define column type for centered cells
\newcolumntype{Y}{>{\centering\arraybackslash}X}
\newcolumntype{M}[1]{@{}>{\centering\arraybackslash}m{\#1}@{}}

\newcommand{\tb}{\tikz[baseline=-0.6ex]{\fill (0,0) circle (2pt);}~}

\newcommand{\ccc}[1]{
    \begin{tikzpicture}[overlay, remember picture]
        \node[circle, inner sep=3pt, draw=black, outer sep=0pt] at (0.5,0.2) {\#1};
    \end{tikzpicture}
}

\usepackage{bidi}

\newfontfamily\arabicfont[Script=Arabic,Scale=1.1]{Amiri}


% --- Colors ---
\definecolor{maroon}{cmyk}{0,0.87,0.68,0.32}
\definecolor{pBlue}{RGB}{102, 53, 153}

\definecolor{myblue}{HTML}{1D2A6E}
\definecolor{myred}{HTML}{A97E77}
\definecolor{mybrown}{HTML}{B35823}
\definecolor{gold}{HTML}{B48920}
\definecolor{myorange}{HTML}{ED8C2B}
\definecolor{col}{RGB}{45, 136, 119}
\definecolor{MainRed}{rgb}{.8, .1, .1}
\definecolor{winered}{rgb}{0.5,0,0}
\definecolor{JungleGreen}{HTML}{29AB87}

\definecolor{myblue3}{RGB}{36, 57, 126}
\definecolor{mygray}{RGB}{112,121,139}
\definecolor{LemonGlacier}{RGB}{253,255,0}
\colorlet{myblue3}{red!75!black}

\definecolor{lightgray}{gray}{0.6}



% --- Basic Settings ---
\def\professor{R. MOSAID}
\def\classname{2BAC.PC/SVT}
\def\examtitle{Devoir libre 02}
\def\schoolname{\textbf{Lycée :} Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small(Par: Prof. Said AMJAOUCH, Maths en poche )}
\def\province{Direction provinciale de\\ Beni Mellal}
\def\logo{\includegraphics[width=\linewidth]{images/logo-men.png}}
\def\wsite{\color{magenta}\texttt{www.mosaid.xyz}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}
\def\bottommsg{Bonne chance!}
\setstretch{1.3}
\everymath{\displaystyle}

% --- Exercise Theme 11 ---
% --- Theme 11: Chevron arrow banner (as in scanned sheet) ---
\definecolor{arrowblue}{RGB}{0,140,180}
\definecolor{bannerblue}{RGB}{210,240,240}

\newtcolorbox{exothemeeleven}[2][]{%
  enhanced,
  breakable,
  width=\linewidth,
  colback=white,
  colframe=white,
  boxrule=0pt,
  left=0pt, right=0pt, top=5pt, bottom=0pt,
  boxsep=0pt,
  before skip=5pt, after skip=5pt,
  interior style={fill=none, top color=white, bottom color=white},
  title={\#2},
  boxed title style={empty, boxrule=0pt, top=0pt, bottom=0pt},
  attach boxed title to top left={yshift=0pt},
  varwidth boxed title=0.9\linewidth,
  before upper={\parskip=4pt},
  overlay unbroken={
    % Chevron banner background
    \begin{scope}[shift={(frame.north west)}]
      \fill[bannerblue] (0,0) rectangle (\linewidth,0.8);
      \fill[arrowblue] (-0.4,0) -- (0.4,0.4) -- (-0.4,0.8) -- (0,0.4) -- cycle;
      \node[anchor=west, font=\bfseries] at (0.4,0.4)
        {\tcbtitletext};
    \end{scope}
  },
  #1
}


\newcommand{\printexo}[3]{%
  \if\relax\detokenize{\#2}\relax
    \def\fulltitle{Exercice #1}%
  \else
    \def\fulltitle{Exercice #1~#2}%
  \fi
  \begin{exothemeeleven}{\fulltitle}#3\end{exothemeeleven}%
  \vspace{0.2cm}%
}
% --- Header Style 9 ---
\newcommand{\printheadnine}{%

  \arrayrulecolor{lightgray}

  \begin{tabular}{m{0.26\textwidth} m{0.43\textwidth} m{0.26\textwidth}}
      \textbf{\classname} & \centering \textbf{\examtitle}
      & \ddate \\
      \wsite & \centering \textbf{\secondtitle}
      &\hfill
        \begin{tabular}{|c}
          \hline
          \textbf{~~\professor}
        \end{tabular}\\
      \hline
  \end{tabular}

  \arrayrulecolor{black} % restore default if needed
}

\fancyhf{}%
\renewcommand{\headrulewidth}{0pt}%
\renewcommand{\footrulewidth}{0pt}%
\setlength{\headheight}{47pt}%
\setlength{\headsep}{0pt}%
\fancyhead[C]{%
    \printheadnine
}%
\pagestyle{fancy}%
\begin{document}
\vspace*{-0.5cm}
% Exercise 1
\printexo{1}{}{
Soit $(u_n)_n$ la suite numérique définie par :
~$
\begin{cases}
u_0 = \frac{1}{2} \\
u_{n+1} = \frac{u_n}{3 - 2u_n}
\end{cases}
\forall n \in \mathbb{N}
$~
\begin{enumerate}
\item Calculer $u_1$.

\item Montrer par récurrence que $\forall n \in \mathbb{N}$, $0 < u_n \leq \frac{1}{2}$

\item
  \begin{enumerate}[label=(\alph*)]
  \item Montrer que $\forall n \in \mathbb{N}$, $\frac{u_{n+1}}{u_n} \leq \frac{1}{2}$
  \item Déduire la monotonie de $(u_n)_n$.
  \end{enumerate}

\item Montrer que pour tout $n \in \mathbb{N}$, $0 < u_n \leq \left(\frac{1}{2}\right)^{n+1}$, puis calculer la limite de $(u_n)_n$.

\item
  \begin{enumerate}[label=(\alph*)]
  \item Vérifier que pour tout $n$ de $\mathbb{N}$, $\frac{1}{u_{n+1}} - 1 = 3\left(\frac{1}{u_n} - 1\right)$.
  \item En déduire $u_n$ en fonction de $n$ pour tout $n$ de $\mathbb{N}$.
  \end{enumerate}
\end{enumerate}
}

% Exercise 2
\printexo{2}{}{
Soit $h$ la fonction définie sur $\mathbb{R}^+$ par :
~$
h(x) = 4x\sqrt{x} - 3x^2
$~

\begin{enumerate}
\item
  \begin{enumerate}[label=(\alph*)]
  \item Montrer que : $(\forall x > 0)$ : $h'(x) = 6\sqrt{x}(-\sqrt{x} + 1)$.
  \item Dresser le tableau de variations de $h$.
  \item Vérifier que $(\forall x \geq 0)$ : $h(x) - x = -3x(\sqrt{x} - 1)\left(\sqrt{x} - \frac{1}{3}\right)$
  \item Montrer que $\forall x \in \left[\frac{1}{9}, 1\right]$ : $h(x) - x \geq 0$.
  \end{enumerate}

\item Soit la suite $(U_n)_n$ définie par
~$
\begin{cases}
U_{n+1} = h(U_n) & (\forall n \in \mathbb{N}) \\
U_0 = \frac{4}{9}
\end{cases}
$~
  \begin{enumerate}[label=(\alph*)]
  \item Montrer que pour tout $n$ de $\mathbb{N}$ : $\frac{1}{9} \leq U_n \leq 1$.
  \item Montrer que la suite $(U_n)_n$ est croissante.
  \end{enumerate}

\item Déduire que la suite $(U_n)_n$ est convergente et déterminer sa limite.
\end{enumerate}
}

% Exercise 3
\printexo{3}{}{
Soit $f$ la fonction numérique définie sur $I = \mathbb{R}$ par :
~$
f(x) = x - 1 + \frac{2x}{\sqrt{x^2 + 1}}
$~\\
$(C_f)$ sa courbe représentative dans un repère orthonormé.

\begin{enumerate}
\item
  \begin{enumerate}[label=(\alph*)]
  \item Montrer que le point $I(0, -1)$ est un centre de symétrie de $(C_f)$.
  \item Calculer $\lim_{x \to +\infty} f(x)$ et $\lim_{x \to -\infty} f(x)$.
  \item Montrer que la droite $(\Delta)$ : $y = x + 1$ est une asymptote oblique à $(C_f)$ au voisinage de $+\infty$.
  \end{enumerate}

\item
  \begin{enumerate}[label=(\alph*)]
  \item Calculer $f'(x)$ et dresser le tableau de variations de $f$.
  \item Écrire une équation de la tangente $(T)$ à $(C_f)$ au point $I$.
  \item Tracer $(C_f)$, $(T)$ et l'asymptote $(\Delta)$.
  \end{enumerate}
\end{enumerate}
}
\vspace*{-0.7cm}
\begin{center}
\normalsize{ \vskip 2pt \hrule height 2pt \vskip 2pt \RL{\arabicfont ﴿إِنَّ ٱلَّذِينَ ءَامَنُوا۟ وَعَمِلُوا۟ ٱلصَّـٰلِحَـٰتِ أُو۟لَـٰٓئِكَ هُمْ خَيْرُ ٱلْبَرِيَّةِ﴾ (البينة 7) }}
\end{center}
\end{document}