\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{tabularx, array, booktabs}
\usepackage{multirow}
\usepackage{array}
\usepackage{setspace}
\usepackage{hyperref}
\hypersetup{colorlinks=true, urlcolor=magenta}



\usepackage{fancyhdr}

% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{TCSF}
\def\examtitle{Devoir 01 (TCSF)}
\def\schoolname{Lycée Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{Notions d'arithmétiques et Calcul vectoriel}}
\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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% Exercise Theme 6: Maroon title bar with triangles
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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{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
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% --- Margin offsets for note placement ---
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% --- Note number placed 1cm from page edge (left margin) ---
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% Document spacing
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% Math display style
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\mathversion{bold}

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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*{-1cm}

\printexo{1}{(12pts)
}{\noindent
\begin{tabular}{>{\hfill}p{0.04\textwidth}|p{0.95\textwidth}}
     & 1. Soient les nombres $x=2420$ et $y=6930$\\
    2& \hspace*{0.5cm}a. Décomposer les deux nombres $x$ et $y$ en produits de facteurs premiers\\
    2& \hspace*{0.5cm}b. Déterminer $pgcd(x,y)$ et $ppcm(x,y)$\\
    2& \hspace*{0.5cm}c. Simplifier $\sqrt{5x}$ et $\sqrt{\frac{35y}{22}}$\\
    2& \hspace*{0.5cm}d. Utiliser l'algorithme d'euclid pour determiner $pgcd(x,y)$\\
    1& 2. Le nombre 371 est il un nombre premier ? justifier.\\
    2& 3. Soit $n\in \mathbb{N}$. Etudier la parité du nombre $a=n^2-n+1$\\
    1& 4. Déterminer l'ensemble des diviseurs de 42\\
\end{tabular}
}

\printexo{2}{(8pts)
}{\noindent
\begin{tabular}{>{\hfill}p{0.04\textwidth}|p{0.95\textwidth}}
    & Soit $ABC$ un triangle. Soient les points $I$, $J$ et $K$ tels que\\
    & \hspace*{1cm}$\overrightarrow{AI}=\frac{1}{2}\overrightarrow{AB}$~~~~;~~~~~
      $\overrightarrow{AJ}=\frac{2}{5}\overrightarrow{AC}$~~~~;~~~~~
      $\overrightarrow{BK}=-2\overrightarrow{BC}$\\
    3& 1. Construire une figure\\
    2& 2. Montrer que
      $\overrightarrow{IJ}=-\frac{1}{2}\overrightarrow{AB}+\frac{2}{5}\overrightarrow{AC}$~~~~et~~~~~
      $\overrightarrow{IK}=\frac{5}{2}\overrightarrow{AB}-2\overrightarrow{AC}$\\
    2& 3. Montrer que $\overrightarrow{IJ}$ et $\overrightarrow{IK}$ sont colinèaires\\
    1& 4. Montrer que les points I, J et K sont alignés.
\end{tabular}
}

% ── Between columns (if two-column)


% ── Repeat section
% Repeat header
\noindent\printheadnine
\printexo{1}{(12pts)
}{\begin{minipage}[t]{0.3\textwidth}

\end{minipage}
\hspace*{0.4cm}
\vline
\begin{minipage}[t]{0.8\textwidth}
\begin{enumerate}
    \item On pose $a=2160,\ b=4860$.
    \begin{enumerate}[label=\alph*),before={\vspace*{-0.25cm}},]
      \item \note{4} Décomposer \( a \) et \( b \) en produit de facteurs premiers puis calculer le pgcd\( (a, b) \) et \(\mathrm{ppcm}(a, b)\).
      \item \note{1} montrer que $\sqrt{a\times b}$ est un entier naturel.
      \item \note{2} Déduire la simplification des nombres \( \dfrac{a}{b}, \sqrt{ab} \).
    \end{enumerate}

    \item \note{1} Soit \( n \in \mathbb{N} \). Montrer que le nombre \( A = 7^{n+2}-7^n \) est multiple de 3.

    \item \note{1} Le nombre $337$ est-il premier?
   \item \begin{enumerate}
       \item \note{1} Montrer que pour $x,y\in\mathbb{N}$, $x+y$ et $x-y$ sont de même parité.
       \item \note{1} Déterminer les diviseurs de $28$.
       \item \note{1} Résoudre $x^2-y^2=28$ en entiers naturels.
    \end{enumerate}

\end{enumerate}
\end{minipage}

}

\printexo{2}{(8pts)
}{Soit $ABC$ un triangle et $G$, $D$ deux points tels que :
~$
3\overrightarrow{AG} = 4\overrightarrow{AB} - 2\overrightarrow{AC}, \quad
\overrightarrow{AD} = \dfrac{4}{5}\overrightarrow{AB}
$~\\
\begin{minipage}[t]{0.3\textwidth}

\end{minipage}
\hspace*{0.4cm}
\vline
\begin{minipage}[t]{0.8\textwidth}
\note{3} \hspace*{1cm} 1. Construire la figure.\\
\note{2} \hspace*{1cm} 2. Montrer que les points $D$, $C$ et $G$ sont alignés.
\end{minipage}

}

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\end{document}