\documentclass[a4paper,12pt]{article}

% ─────────────────────────────────────────────────────────────────
% default_preamble.tex — packages + command definitions
% Everything here is placed BEFORE \begin{document}.
% ─────────────────────────────────────────────────────────────────

\usepackage[left=1.5cm,right=1.5cm,top=2cm,bottom=1.5cm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{fontspec}
\usepackage{amsmath, amsfonts, amssymb, amsthm}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{multicol}
\usepackage{enumitem}
\usepackage{mathrsfs}
\usepackage{colortbl}
\usepackage{varwidth}
\usepackage[most]{tcolorbox}
\tcbuselibrary{skins,breakable}
\usepackage{tikz,tkz-euclide}
\usetikzlibrary{calc}
\usetikzlibrary{shadows}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}
\usepackage{tabularx, array, booktabs}
\usepackage{multirow}
\usepackage{array}
\usepackage{setspace}
\usepackage{hyperref}
\hypersetup{colorlinks=true, urlcolor=magenta}

% Arabic support for bottom message
\usepackage{bidi}
\newfontfamily\arabicfont[Script=Arabic,Scale=1.1]{Amiri}


\usepackage{fancyhdr}

% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{1BAC.SM}
\def\examtitle{Devoir 02 (1BAC.SM)}
\def\schoolname{Lycée Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{}}
\def\province{}
\def\logo{\includegraphics[width=\linewidth]{images/logo-men.png}}
\def\wsite{\href{https://www.mosaid.xyz}{\textcolor{magenta}{\texttt{www.mosaid.xyz}}}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}

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      M{0.25\textwidth}
      }
      \textbf{\classname} & \textbf{\examtitle} & \ddate \\
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      \hline
  \end{tabularx}
  \arrayrulecolor{black}
}


% Exercise Theme 6: Maroon title bar with triangles
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  after skip=0pt,
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  title=#2,
  #1
}

\newcommand{\printexo}[3]{%
  \if\relax\detokenize{#2}\relax
    \def\fulltitle{Exercice #1}%
  \else
    \def\fulltitle{Exercice #1~#2}%
  \fi
  \begin{exothemesix}[]{\fulltitle}#3\end{exothemesix}%
  \vspace{0.2cm}%
}



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\definecolor{lightgray}{gray}{0.6}
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\definecolor{bannerblue}{RGB}{210,240,240}

% Exam-specific settings and custom definitions
% This file is generated in the current working directory
% Edit this file to customize your exam settings



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

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}


\newcommand{\circled}[1]{%
  \tikz[baseline=(char.base)]{
    \node[shape=circle,draw,inner sep=1pt, font=\bf] (char) {#1};}%
}


% --- Margin offsets for note placement ---
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% --- Note number placed 1cm from page edge (right margin) ---
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% --- Note number placed 1cm from page edge (left margin) ---
\newcommand{\note}[1]{%
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% Document spacing
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% Math display style
\everymath{\displaystyle}

\mathversion{bold}

%\AtBeginDocument{\fontsize{15}{17}\selectfont}

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% Custom theorem environments (uncomment if needed)
% \newtheorem{theorem}{Theorem}
% \newtheorem{lemma}{Lemma}
% \newtheorem{corollary}{Corollary}

% Custom exercise environments (uncomment if needed)
% \newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
% \newenvironment{remark}{\begin{proof}[Remark]}{\end{proof}}

% Custom commands for this exam (add your own below)
% \newcommand{\answerline}[1]{\underline{\hspace{#1}}}
% \newcommand{\points}[1]{\hfill(\mbox{#1 points})}

% Additional packages (uncomment if needed)
% \usepackage{siunitx} % For SI units
% \usepackage{chemformula} % For chemical formulas
% \usepackage{circuitikz} % For circuit diagrams

% Page layout adjustments
% \setlength{\parindent}{0pt}
% \setlength{\parskip}{1em}

% Custom colors for this exam
% \definecolor{myblue}{RGB}{0,100,200}
% \definecolor{mygreen}{RGB}{0,150,0}

\def\nex{01}

% ----------------------------------------------------------------------
% ADD YOUR CUSTOM DEFINITIONS BELOW THIS LINE
% ----------------------------------------------------------------------

% Example:
\newcommand{\dif}{\mathrm{d}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}


% ─────────────────────────────────────────────────────────────────
% default_wrapper.tex — the document body.
% Placed AFTER preamble / colors / definitions in _assemble().
% ─────────────────────────────────────────────────────────────────

\begin{document}



% ── Top of document (user block) ────────────────────────────────


% ── Header command (exam class only; article class uses fancyhdr)


% ── Two-column start


% ── Main exercises
\vspace*{-0.75cm}

\printexo{1}{}{\noindent Soit \(f\) la fonction numérique définie sur l'intervalle \(]0; 1[\) par : \(f(x) = \left(1 + \frac{1}{x}\right)\left(1 + \frac{1}{1-x}\right)\)
\begin{enumerate}[label=\textbf{\arabic*)}]
    \item Montrer que : \((\forall x \in ]0; 1[) \quad f(x) = 1 - \frac{2}{x^2 - x}\)

    \item Soit \(x\) et \(y\) deux éléments distincts de l'intervalle \(]0; 1[\).
    \begin{enumerate}[label=\textbf{\alph*)}]
        \item Montrer que : \(\frac{f(x) - f(y)}{x - y} = \frac{2(x + y - 1)}{(x^2 - x)(y^2 - y)}\)
        \item Etudier les variations de la fonction \(f\) sur chacun des intervalles : \(\left]0; \frac{1}{2}\right]\) et \(\left[\frac{1}{2}; 1\right[\)
    \end{enumerate}

    \item Soit \(a\) et \(b\) deux réels strictement positifs tels que : \(a + b = 1\) \\
    En utilisant ce qui précède montrer que : \(\left(1 + \frac{1}{a}\right)\left(1 + \frac{1}{b}\right) \geq 9\)
\end{enumerate}

}

\printexo{2}{}{\noindent On considère l'application \(f\) définie par :
\(f : \begin{cases}
\mathbb{R} \to \mathbb{R} \\
x \mapsto \sqrt{4x^2 - 4x + 3}
\end{cases}\)

\begin{enumerate}[label=\textbf{\arabic*)}]
    \item Montrer que : \((\forall x \in \mathbb{R}) \; f(1 - x) = f(x)\). \(f\) est-elle injective ?

    \item Montrer que : \(f(\mathbb{R}) \subset [\sqrt{2}; +\infty[\). \(f\) est-elle surjective ?

    \item Soit \(g\) l'application définie par :
    \(g : \begin{cases}
    \left[\frac{1}{2}; +\infty\right[ \to [\sqrt{2}; +\infty[ \\
    x \mapsto f(x)
    \end{cases}\) \\
    Montrer que \(g\) réalise une bijection de \(\left[\frac{1}{2}; +\infty\right[\) sur \([\sqrt{2}; +\infty[\) puis définir \(g^{-1}\).

    \item En déduire que l'équation \(g(x) = 2026\) admet une solution unique dans \([1; +\infty[\)
\end{enumerate}

}

\printexo{3}{}{\vspace{0.2cm}
\noindent Soit \(n \in \mathbb{N}^*\). On considère la fonction \(f\) définie sur \(\mathbb{R}\) par : \(f(x) = E(nx) - \sum_{k=0}^{n-1} E\left(x + \frac{k}{n}\right)\)

\begin{enumerate}[label=\textbf{\arabic*)}]
    \item Montrer que \(f\) est périodique de période \(T = \frac{1}{n}\).

    \item Calculer \(f(x)\) pour tout \(x \in \left[0, \frac{1}{n}\right[\)

    \item En Déduire que : \((\forall x \in \mathbb{R})(\forall n \in \mathbb{N}^*) \; E(nx) = \sum_{k=0}^{n-1} E\left(x + \frac{k}{n}\right)\).
\end{enumerate}
}

% ── Between columns (if two-column)


% ── Repeat section



% ── Bottom message — inside column 2 (last column)


% ── Two-column end


% ── Bottom of document (user block) ─────────────────────────────


% Bottom message
% Bottom message template
% ────────────────────────────────────────────────────────────
%  • Paste your Arabic text between the ↓↑ markers below
%  • For non-Arabic content, remove the \RL{\arabicfont ... }
%    wrapper — the surrounding \begin{center} block stays.
%  • The document's own font is unaffected: \arabicfont is
%    scoped only to this block.
% ────────────────────────────────────────────────────────────

\begin{center}
  \normalsize{%
    \vskip 3pt \hrule height 2pt \vskip 2pt
    \RL{\arabicfont
      % ↓ Paste your Arabic text here ↓
﴿وَٱللَّهُ يَدْعُوٓا۟ إِلَىٰ دَارِ ٱلسَّلَـٰمِ وَيَهْدِى مَن يَشَآءُ إِلَىٰ صِرَٰطٍ مُّسْتَقِيمٍ﴾ (يونس 25)
      % ↑ Paste your Arabic text here ↑
    }%
  }%
\end{center}

\end{document}