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\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}

% French language support
\usepackage[french]{babel}



% Two-page layout (conditionally included)



% --- Colors ---
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\definecolor{pBlue}{RGB}{102, 53, 153}

\definecolor{myblue}{HTML}{1D2A6E}
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\definecolor{bannerblue}{RGB}{210,240,240}

% --- Basic Settings ---
% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{1BAC.SF}
\def\examtitle{Série Logique (1BAC.SF)}
\def\schoolname{Taghzirt}
\def\academicyear{2025/2026}
\def\subject{Mathématiques}
\def\duration{2h}
\def\secondtitle{\small{visit ~~\wsite~~ for more!}}
\def\province{Direction provinciale\newline de Beni Mellal}
\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~~}


% 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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\newcolumntype{M}[1]{@{}>{\centering\arraybackslash}m{#1}@{}}

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\newcommand{\circled}[1]{%
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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}}


% Exercise Theme 1: TikZ shadow title
\newcommand{\exothemeone}[1]{%
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\begin{tikzpicture}[baseline=(text.base)]
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    \node[] at (text) {\textbf{#1}};
\end{tikzpicture}\\[0.2cm]
}

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

% --- Header Style 9 (fr)  ---
\newcommand{\printheadnine}{%
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      M{0.48\textwidth}
      M{0.25\textwidth}
      }
      \textbf{\classname} & \textbf{\examtitle} & \ddate \\
      \wsite & \secondtitle & \hfill \begin{tabular}{|c|}\hline\professor\end{tabular}  \\
      \hline
  \end{tabularx}
  \arrayrulecolor{black}
}

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\fancyhead[C]{%
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}%
\pagestyle{fancy}




\begin{document}



% Two-column layout
\setlength{\columnsep}{0.5cm}
\setlength{\columnseprule}{2pt}
\renewcommand{\columnseprulecolor}{\color{black}}
\begin{multicols}{2}[\raggedcolumns]
\vspace*{-\topskip}
\vspace*{-0.5cm}
% Exercise 1 (originally 1)
\printexo{1}{(Déduction)}{
\vspace*{-0.75cm}
\begin{enumerate}
    \item Soit $x,y\in\mathbb{R}$. Montrer que :\\
    ~\(
    \begin{cases}
    x>2 \\
    y>2
    \end{cases}
    \implies
    \dfrac{1}{x}+\dfrac{1}{y}<\dfrac{1}{2}.
    \)~

    \item Soit $x\in\mathbb{R}^{+}$. Montrer que :\\
    ~\(
    \dfrac{\sqrt{x}}{1+\sqrt{x}}=1-\dfrac{1}{1+\sqrt{x}}\implies x=1.
    \)~

    \item Soit $a,b\in\mathbb{R}$. Montrer que :\\
    ~\(
    \begin{cases}
    a>1 \\
    b>1
    \end{cases}
    \implies
    \dfrac{1}{a}+\dfrac{1}{b}<1+\dfrac{1}{ab}.
    \)~

    \item Soit $x\in\mathbb{R}$. Montrer que :\\
    ~\(
    x>3\implies\dfrac{x+1}{x-1}>2.
    \)~

    \item Soit $x,y\in\mathbb{R}^{+*}$. Montrer que :\\
    ~\(
    \begin{cases}
    x>y \\
    xy>1
    \end{cases}
    \implies
    \dfrac{1}{x}+\dfrac{1}{y}<x+y.
    \)~

    \item Soit $a\in\mathbb{R}$. Montrer que :\\
    ~\(
    a>2\implies a^{2}-3a+2>0.
    \)~
\end{enumerate}
\vspace*{-0.25cm}
}

% Exercise 2 (originally 2)
\printexo{2}{(Contraposée)}{
\vspace*{-0.75cm}
\begin{enumerate}
    \item Montrer que :\\
    ~\(
    x\neq 3\quad\text{et}\quad y\neq 3\implies 3x+3y-xy\neq 6.
    \)~

    \item Soit $n\in\mathbb{N}$. Montrer que si $n^{2}$ est divisible par $3$, alors $n$ est divisible par $3$.

    \item Soit $n\in\mathbb{N}$. Montrer que si $n^{2}$ est pair, alors $n$ est pair.

    \item Montrer que :\\
    ~\(
    x\neq 2\quad\text{et}\quad y\neq 2\implies xy-2x-2y\neq -4.
    \)~

    \item Soit $n\in\mathbb{N}$. Montrer que \\si $n$ n'est pas divisible par $5$, alors $n^{2}$ n'est pas divisible par $5$.

\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{(Équivalences)}{
\vspace*{-0.5cm}
\begin{enumerate}
    \item Soit $x\in\mathbb{R}$. Montrer que :\\
    ~\(
    |x-3|\leqslant 2\quad\Leftrightarrow\quad 1\leqslant x\leqslant 5.
    \)~

    \item Soit $x\in\mathbb{R}\setminus\{-1\}$. Montrer que :\\
    ~\(
    \dfrac{x-1}{x+1}\geqslant 0\quad\Leftrightarrow\quad x\in\,]-\infty,-1[\,\cup\,[1,+\infty[.
    \)~

    \item Soit $x\in\mathbb{R}$. Montrer que :\\
    ~\(
    |x+2|<3\quad\Leftrightarrow\quad -5<x<1.
    \)~

    \item Soit $x\in\mathbb{R}\setminus\{2\}$. Montrer que :\\
    ~\(
    \dfrac{x+1}{x-2}\leqslant 0\quad\Leftrightarrow\quad x\in\,]-1,2[.
    \)~

    \item Soit $x\in\mathbb{R}$. Montrer que :\\
    ~\(
    x^{2}-5x+6\leqslant 0\quad\Leftrightarrow\quad 2\leqslant x\leqslant 3.
    \)~

    \item Soit $x\in\mathbb{R}^{+*}$. Montrer que :\\
    ~\(
    \sqrt{x}\geqslant 2\quad\Leftrightarrow\quad x\geqslant 4.
    \)~
\end{enumerate}
}

% Exercise 4 (originally 4)
\printexo{4}{(Disjonction des cas)}{
\vspace*{-0.5cm}
\begin{enumerate}[tight]
    \item Résoudre dans $\mathbb{R}$ l'inéquation :
    \(
    |x-2|+3x-5\geqslant 0.
    \)

    \item Montrer que $\forall x\in\mathbb{R}$ :
    ~\(
    |x-2|\leqslant x^{2}-x+2.
    \)~

    \item Résoudre dans $\mathbb{R}$ l'inéquation :
    ~\(
    |x+1|-2x\geqslant 3.
    \)~

    \item Montrer que $\forall x\in\mathbb{R}$ :
    ~\(
    |x|\leqslant x^{2}+1.
    \)~

    \item Résoudre dans $\mathbb{R}$ l'équation :
    ~\(
    |2x-1|=|x+3|.
    \)~

    \item Montrer que $\forall x\in\mathbb{R}$ :
    ~\(
    |x-1|+|x+1|\geqslant 2.
    \)~
\end{enumerate}
}

% Exercise 5 (originally 5)
\printexo{5}{(Absurde)}{
\vspace*{-0.5cm}
\begin{enumerate}
    \item Soit $n\in\mathbb{N}^{*}$. Montrer que $\sqrt{n^{2}+3}$ n'est pas un entier.

    \item Montrer que $\forall n\in\mathbb{N},\ \dfrac{n+2}{n+3}\notin\mathbb{N}$.

    \item Montrer que $\sqrt{2}$ est irrationnel.

    \item Soit $n\in\mathbb{N}^{*}$. Montrer que $\sqrt{n^{2}+n+1}$ n'est pas un entier.

    \item Montrer que $\forall n\in\mathbb{N}^{*},\ \dfrac{n}{n+1}\notin\mathbb{N}$.

    \item Montrer qu'il n'existe aucun entier $n\geqslant 2$ tel que $n^{2}-2$ soit un carré parfait.
\end{enumerate}
}

% Exercise 6 (originally 6)
\printexo{6}{(Récurrence)}{
\vspace*{-0.5cm}
\begin{enumerate}
    \item Montrer par récurrence que \\ $\forall n\in\mathbb{N}$ :
    ~\(
    \sum_{k=1}^{n}k(k+1)=\dfrac{n(n+1)(n+2)}{3}.
    \)~

    \item Montrer que \\ $\forall n\in\mathbb{N},\ 4^{n}-1$ est divisible par $3$.

    \item Soit $a>0$. Montrer par récurrence que \\ $\forall n\in\mathbb{N}$ :
    ~\(
    (1+a)^{n}\geqslant 1+na.
    \)~

    \item Montrer par récurrence que \\ $\forall n\in\mathbb{N}$ :
    ~\(
    \sum_{k=1}^{n}k^{2}=\dfrac{n(n+1)(2n+1)}{6}.
    \)~

    \item Montrer que \\ $\forall n\in\mathbb{N},\ 3^{n}-1$ est divisible par $2$.

    \item Soit $a>1$. Montrer par récurrence que \\ $\forall n\in\mathbb{N}$ :
    ~\(
    a^{n}\geqslant 1+n(a-1).
    \)~
\end{enumerate}
}




\end{multicols}


\end{document}