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% --- Basic Settings ---
% Language variables for French exams
\def\professor{R. MOSAID}  % Should be R. MOSAID not {{R. MOSAID}}
\def\classname{1BAC.SM}
\def\examtitle{Devoir 02 - S01, Par Pr: M. ENSSAH}
\def\schoolname{\textbf{Lycée :} Lycée : 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 definitions ---

% Exam-specific settings and custom definitions
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% Edit this file to customize your exam settings

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

% Page layout adjustments
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% Custom colors for this exam
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% ----------------------------------------------------------------------
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% ----------------------------------------------------------------------

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% --- Header Style 9 (fr) ---
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\begin{document}



% Exercise 1 (originally 1)
\printexo{1}{}{
Soit \( ABC \) un triangle dans le plan \( \mathcal{P} \).\\
Soient \( I \) et \( J \) deux points du plan \( \mathcal{P} \) tels que :
~$
\overrightarrow{BI} = \alpha \overrightarrow{BC} \quad \text{et} \quad \overrightarrow{AJ} = \alpha \overrightarrow{AC} \quad \text{avec} \quad \alpha \in ]0; 1[.
$~
\begin{enumerate}[tight]
    \item Montrer que :
    ~$
    \overrightarrow{IC} = (1 - \alpha)\overrightarrow{BC}.
    $~

    \item En déduire que \( I \) est le barycentre du système pondéré :
    ~$
    \left\{\left(C; 1\right) ; \left(B; \frac{1}{\alpha} - 1\right)\right\}.
    $~

    \item Montrer que :
    ~$
    J = \text{bar}\left\{\left(C; 1\right) ; \left(A; \frac{1}{\alpha} - 1\right)\right\}.
    $~

    \item On considère \( G \) le barycentre du système :
    ~$
    \left\{\left(C; 1\right) ; \left(B; \frac{1}{\alpha} - 1\right) ; \left(A; \frac{1}{\alpha} - 1\right)\right\}.\\
    $~
    Vérifier que :
    ~$
    \overrightarrow{AG} = \frac{1}{2 - \alpha} \overrightarrow{AI}.
    $~

    \item Montrer que les points \( J, G \) et \( B \) sont alignés.

    \item Soit \( K \) le milieu du segment \([AB]\). Montrer que les droites \((AI), (BJ)\) et \((CK)\) sont concourantes en un point à déterminer.

    \item Déterminer l'ensemble :
    $
    \Omega = \left\{ M \in \mathcal{P} \Bigm| 2\left\|(1 - \alpha)\overrightarrow{MA} + (1 - \alpha)\overrightarrow{MB} + \alpha\overrightarrow{MC}\right\| = (2 - \alpha)\left\|\overrightarrow{MA} + \overrightarrow{MB}\right\|\right\}.
    $
\end{enumerate}
}

% Exercise 2 (originally 2)
\printexo{2}{}{
Soit \( f \) et \( g \) les fonctions numériques définies par :
~$
f(x) = -x^2 + 2x + 1 \quad \text{et} \quad g(x) = \sqrt{x - 1}.
$~\\
\( \mathcal{C}_f \) et \( \mathcal{C}_g \) les courbes représentatives de \( f \) et \( g \) dans un repère orthonormé \((O; \vec{i}, \vec{j})\).
\begin{enumerate}[tight]
    \item
    \begin{enumerate}[tight]
        \item Vérifier que \( f(2) = g(2) \) puis interpréter graphiquement le résultat.
        \item Tracer les courbes \( \mathcal{C}_f \) et \( \mathcal{C}_g \).
    \end{enumerate}

    \item Résoudre graphiquement l'inéquation :
    ~$
    x^2 - 2x - 1 + \sqrt{x - 1} < 0.
    $~

    \item Déterminer graphiquement les images des intervalles \([0; 1]\) et \([1; 2]\) par la fonction \( f \).

    \item Soit \( h \) la fonction numérique définie par :
    ~$
    h(x) = \sqrt{-x^2 + 2x}.
    $~
    \begin{enumerate}[tight]
        \item Déterminer \( D_h \) ensemble de définition de \( h \).
        \item Vérifier que :
        ~$
        (\forall x \in D_h) \quad h(x) = g \circ f(x).
        $~
        \item Étudier les variations de la fonction \( h \) sur les intervalles \([0; 1]\) et \([1; 2]\).
    \end{enumerate}
\end{enumerate}
}

% Exercise 3 (originally 3)
\printexo{3}{}{
On considère la fonction \( \psi \) définie par :
~$
\psi(x) = \left( x - 2E\left(\frac{x}{2}\right) \right) \left( 2E\left(\frac{x}{2}\right) - x + 2 \right),
$~
où \( E \) désigne la fonction partie entière.
\begin{enumerate}[tight]
    \item Montrer que \( \psi \) est périodique de période 2.
    \item Simplifier l'expression de \( \psi \) sur l'intervalle \([0; 2[\).
    \item Construire la courbe \((C_\psi)\) sur l'intervalle \([-4; 4]\) dans un repère \((O; \vec{i}, \vec{j})\).
\end{enumerate}
}

% Exercise 4 (originally 4)
\printexo{4}{}{
Soit \( m \) un réel strictement positif et la fonction \( f \) définie par :
~$
f_m(x) = \frac{x^2}{m} - 2x.
$~
\begin{enumerate}[tight]
    \item Dresser le tableau de variations de la fonction \( f_m \) puis déduire que :
    ~$
    (\forall x \in \mathbb{R}) : \frac{x^2}{m} + m \geq 2x.
    $~

    \item Soient \( a, b \) et \( c \) des nombres réels de \( \mathbb{R}^+ \). Déduire que :
    ~$
    \frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq a + b + c.
    $~
\end{enumerate}
}








\end{document}