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%\usepackage[metapost]{mfpic}
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\usepackage[pdftex]{graphicx}
\usepackage[arrow,curve,matrix]{xy}
\usepackage{amssymb,amsmath,mathrsfs,amsfonts,amsthm}
\usepackage{verbatim,array,enumerate,epsfig}
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\usepackage{mathrsfs,variations}
\usepackage[bookmarks]{hyperref}
\usepackage{stmaryrd,ulem,wasysym}%,ifsym}
\usepackage[shortlabels]{enumitem}
\usepackage{listings}
\lstset{language=Caml}
\renewcommand\thesection{\Roman{section}}


% Attention~!
\newcommand{\attention}[1]{\begin{minipage}{1.5cm}
\includegraphics[scale=0.2]{Attention.jpg}
\end{minipage}
\begin{minipage}{15.5cm}
#1
\end{minipage}}

% Cochonneries degeu Nico/Denis/Moi
\newcommand{\ver}{\vrule\kern2pt\vrule}
\newcommand{\scp}{\textsc}
\newcommand{\abs}[1]{\left | #1 \right |}
\newcommand{\newdef}{\textbf}%{\underline}


% New
\renewcommand{\leq}{\les}
\renewcommand{\geq}{\ges}
\newcommand{\andegrace}{2020-2021}
\def\DebutAnnee{2020}
\def\FinAnnee{2021}

% Fleches
\newcommand{\too}{\rightarrow}
\newcommand{\fl}{\longrightarrow}
\newcommand{\flto}{\longmapsto}
\renewcommand{\iff}{\Leftrightarrow}
\newcommand{\impl}{\Rightarrow}
\newcommand{\eq}{\Leftrightarrow}

% Ensembles de nombres
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\newcommand{\Q}{\mathbb{Q}}
\newcommand{\U}{\mathbb{U}}
\newcommand{\PP}{\mathcal{P}}
\newcommand{\Sfk}{\mathfrak{S}}
\newcommand{\Pb}{\mathbb{P}}
\newcommand{\ensfini}[1]{[\![ 1,#1]\!]}

% Constantes de base
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%\renewcommand{\i}{{\bf \mathrm{i}}}
\newcommand{\eps}{\varepsilon}
\newcommand{\fy}{\varphi}
\newcommand{\vph}{\varphi}

% Ronds dans l'eau
\renewcommand{\P}{\mathcal{P}}
\renewcommand{\sc}{\mathcal{S}}
\newcommand{\ac}{\mathcal{A}}
\newcommand{\bc}{\mathcal{B}}
\newcommand{\cc}{\mathcal{C}}
\newcommand{\dc}{\mathcal{D}}
\newcommand{\ec}{\mathcal{E}}
\newcommand{\fc}{\mathcal{F}}
\newcommand{\gc}{\mathcal{G}}
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\newcommand{\ic}{\mathcal{I}}
\newcommand{\jc}{\mathcal{J}}
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\newcommand{\lc}{\mathcal{L}}
\newcommand{\mc}{\mathcal{M}}
\newcommand{\nc}{\mathcal{N}}
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\newcommand{\qc}{\mathcal{Q}}
\newcommand{\rc}{\mathcal{R}}
\newcommand{\scal}{\mathcal{S}}
\newcommand{\tc}{\mathcal{T}}
\newcommand{\uc}{\mathcal{U}}
\newcommand{\vc}{\mathcal{V}}
\newcommand{\wc}{\mathcal{W}}
\newcommand{\xc}{\mathcal{X}}
\newcommand{\yc}{\mathcal{Y}}
\newcommand{\zc}{\mathcal{Z}}

% Vecteurs
\newcommand{\bvec}{\overrightarrow}
\newcommand{\eu}{\overrightarrow{e_1}}
\newcommand{\ed}{\overrightarrow{e_2}}
\newcommand{\et}{\overrightarrow{e_3}}
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\newcommand{\veps}{\overrightarrow{\varepsilon}}
\newcommand{\vect}[1]{\left ( \begin{array}{c} #1 \end{array} \right )}
\newcommand{\mvect}[1]{{\scriptsize \left ( \begin{array}{c} #1 \end{array} \right )}}
% Pour Nico :
\newcommand{\pvect}{\overrightarrow{\pc}}
\newcommand{\evect}{\overrightarrow{\ec}}
\newcommand{\fvect}{\overrightarrow{\fc}}
\newcommand{\dvect}{\overrightarrow{\dc}}
\newcommand{\vct}{\overrightarrow}
\newcommand{\coords}[2]{\Big(\!\!\begin{array}{c} #1\\#2\end{array}\!\!\Big)}
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\newcommand{\coooords}[4]{\left(\!\!\begin{array}{c} #1\\#2\\#3\\#4\end{array}\!\!\right)}
%\newcommand{\fl}{\overrightarrow}
\newcommand{\ifl}{\vec{\imath}}
\newcommand{\jfl}{\vec{\jmath}}
\newcommand{\kfl}{\vec{k}}

% Arrays
\newcommand{\dect}[2]{\left | \begin{array}{#1} #2 \end{array} \right |}
\newcommand{\lect}[1]{\left \{ \begin{array}{l} #1 \end{array} \right .}
\newcommand{\defapp}[4]{\left \{ \begin{array}{ccl} #1 & \too & #2\\ #3 & \mapsto & #4 \end{array} \right .}
\newcommand{\matdd}[4]{\left ( \begin{array}{cc} #1 & #2 \\ #3 & #4 \end{array} \right )}
\newcommand{\mattt}[1]{\left ( \begin{array}{ccc} #1 \end{array} \right )}
\newcommand{\matqq}[1]{\left ( \begin{array}{cccc} #1 \end{array} \right )}
%\newcommand{\cooords}[3]{\text{ de coordonn\'ees } \vect{#1\\#2\\#3}}


%Alg\`ebre lin\'eaire
\newcommand{\mat}[2]{\mathcal{M}at_{#1}(#2)}
\newcommand{\mnr}{\mathcal{M}_n(\R)}
\newcommand{\mnk}{\mathcal{M}_n(\K)}
\newcommand{\mpqk}{\mathcal{M}_{p,q}(\K)}
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\newcommand{\mdk}{\mathcal{M}_2(\K)}
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\DeclareMathOperator{\com}{com}

% Geometrie
%\newcommand{\sslash}{//}
%\newcommand{\coords}[2]{\vect{#1\\#2}}
%\renewcommand{\cooords}[3]{\vect{#1\\#2\\#3}}

% Fonctions usuelles
\newcommand{\cotan}{\mathrm{cotan}}
\newcommand{\ch}{\mathrm{ch}}
\newcommand{\sh}{\mathrm{sh}}
\renewcommand{\th}{\mathrm{th}}
\newcommand{\argch}{\mathrm{argch}}
\newcommand{\argsh}{\mathrm{argsh}}
\newcommand{\argth}{\mathrm{argth}}
\newcommand{\sg}{\mathrm{sg}}
% Pour Nico : 
\DeclareMathOperator{\tnh}{th}
\DeclareMathOperator{\Argch}{Argch}
\DeclareMathOperator{\Argsh}{Argsh}
\DeclareMathOperator{\Argth}{Argth}
\DeclareMathOperator{\Arcsin}{Arcsin}
\DeclareMathOperator{\Arccos}{Arccos}
\DeclareMathOperator{\Arctan}{Arctan}

% Probas
\DeclareMathOperator{\Cov}{Cov}


%Analyse vectorielle
\newcommand{\rot}{\vct{\mathrm{rot}}}
\newcommand{\dive}{\mathrm{div}}
\DeclareMathOperator{\grad}{grad}

% Integration
\newcommand{\dx}{~{\rm d}x}
\newcommand{\dy}{~{\rm d}y}
\newcommand{\dt}{~{\rm d}t}
\newcommand{\dr}{~{\rm d}r}
\newcommand{\du}{~{\rm d}u}
\newcommand{\dv}{~{\rm d}v}
\newcommand{\dtheta}{~{\rm d}\theta}
\newcommand{\intc}{\circ\hspace{-0.4cm}\int}

% Intervalles
\newcommand{\seg}[2]{\left[ #1\;;#2\right]}
\newcommand{\ouv}[2]{\left] #1\;;#2\right[}
\newcommand{\fo}[2]{\left[ #1\;;#2\right[}
\newcommand{\of}[2]{\left] #1\;;#2\right]}

%Alg\`ebre 
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\newcommand{\rec}{\Re e}
\DeclareMathOperator{\bary}{bar}
\DeclareMathOperator{\card}{card}
\DeclareMathOperator{\pgcd}{PGCD}
\DeclareMathOperator{\ppcm}{PPCM}
\newcommand{\conj}[1]{\overline{#1}}

% Limites
\newcommand{\tenden}[1]{\displaystyle\mathop{\longrightarrow}_{#1}} 
\newcommand{\tendvers}[2]{\displaystyle\mathop{\longrightarrow}_{#1\rightarrow#2}}
\newcommand{\limiteen}[1]{\displaystyle\mathop{\lim}_{#1}}
\newcommand{\limitevers}[2]{\displaystyle\mathop{\lim}_{#1\rightarrow#2}}

% Op\'erateurs 
\newcommand{\moins}{\smallsetminus}
\newcommand{\ges}{\geqslant}
\newcommand{\les}{\leqslant}

% Op\'erateurs diff\'erentiels
\newcommand{\de}{\partial}
\newcommand{\dbar}{\overline{partial}}
\newcommand{\ddbar}{\partial\overline{\partial}}
\newcommand{\dd}[2]{\dfrac{\partial #1}{\partial #2}}

% Suites
%\newcommand{\sun}{(u_n)_{n\in\N}}
\newcommand{\svn}{(v_n)_{n\in\N}}
\newcommand{\sunp}{(u_n)_{n\in\Np}}


% Enjolivages
\newcommand{\semisubline}{${}_{{}_{{}_{{}_{{}_{}}}}}$}
\newcommand{\subline}{${}_{{}_{{}_{{}_{{}_{{}_{{}_{}}}}}}}$}
\newcommand{\bigsubline}{${}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{}}}}}}}}}$}
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\newcommand{\gigasubline}{${}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{}}}}}}}}}}}}}}}}}$}
\newcommand{\supline}{${}^{{}^{{}^{{}^{{}^{{}^{{}^{}}}}}}}$}
\newcommand{\minisupline}{${}^{{}^{{}^{{}^{{}^{}}}}}$}
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\newcommand{\gigasublined}{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{{}_{}}}}}}}}}}}}}}}}}}
\newcommand{\dsp}{\displaystyle}
%\newcommand{\smiley}{\includegraphics[scale=0.075]{smiley.jpg}}
\newcommand{\cs}{\copyright}
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\newcommand{\fs}{\includegraphics[scale=0.14]{../Entetes/Facile.jpg}}
\newcommand{\ts}{$\bigstar$}
\newcommand{\Chi}{\text{\Large $\chi$}}


% Flemmardise
\newcommand{\ie}{\textit{i.\,e.~}}
\newcommand{\cad}{c'est-\`a-dire~}
\newcommand{\Cad}{C'est-\`a-dire~}
\newcommand{\ssi}{si et seulement si~}
\newcommand{\tbv}{tableau de variations~}
\newcommand{\hdr}{hypoth\`ese de r\'ecurrence}
\newcommand{\cste}{\text{constante}}
\newcommand{\recclc}{La propri\'et\'e est vraie au rang $0$, et elle est h\'er\'editaire, elle est donc vraie pour tout entier $n\in\N$.}
\newcommand{\recdclc}{La propri\'et\'e est vraie aux rangs $0$ et $1$, et elle est doublement h\'er\'editaire, elle est donc vraie pour tout entier $n\in\N$.}
\newcommand{\recinit}[1]{La propri\'et\'e est vraie au rang $#1$, et elle est h\'er\'editaire, elle est donc vraie pour tout entier $n\geq#1$.}
\newcommand{\recinitvar}[2]{La propri\'et\'e est vraie au rang $#1$, et elle est h\'er\'editaire, elle est donc vraie pour tout entier $#2\geq#1$.}
\newcommand{\recdouble}[2]{La propri\'et\'e est vraie aux rangs $#1$ et $#2$, et elle est doublement h\'er\'editaire, elle est donc vraie pour tout entier $n\geq#1$.}
\newcommand{\recfclc}[1]{La propri\'et\'e est fortement h\'er\'editaire, elle est donc vraie pour tout entier $n\geq#1$.}
\newcommand{\CN}{condition n\'ecessaire}
\newcommand{\CS}{condition suffisante}
\newcommand{\CNS}{condition n\'ecessaire et suffisante}
\newcommand{\sev}{sous-espace vectoriel~}
\newcommand{\ev}{espace vectoriel~}
\newcommand{\evs}{espaces vectoriels~}
\newcommand{\sea}{sous-espace affine~}
\newcommand{\sevs}{sous-espaces vectoriels~}
\renewcommand{\Ker}{\mathrm{Ker}}
\renewcommand{\Im}{\mathrm{Im}}



% ====================================================



% Herve's friendship
\addtolength{\columnsep}{1.5cm}
\setlength{\columnseprule}{0.1pt}
\setlength{\parindent}{0cm}
\addtolength{\hoffset}{-3cm}
\addtolength{\textwidth}{6cm}
\addtolength{\voffset}{-2.25cm}
%\addtolength{\voffset}{-3cm}
\addtolength{\textheight}{4.5cm}
%\addtolength{\textheight}{5.5cm}
%\pagestyle{fancy}

% Roch's friendship
\newcommand{\ben}{\begin{enumerate}}
\newcommand{\een}{\end{enumerate}}

% Environnement theoremes
\newcommand{\somefont}{\usefont{T1}{pag}{m}{sl}}
\newcommand{\afont}{\usefont{T1}{phvrrn}{m}{sc}}
\newcommand{\backfont}{\usefont{U}{Starburst}{xl}{n}}%{\usefont{T1}{bch}{m}{n}}
%\definecolor{griser}{gray}{0.9}
\definecolor{thmcolor}{rgb}{0.71,0.14,0.07}
\definecolor{shadethmcolor}{rgb}{0.95,0.95,1}
\setlength{\shadedtextwidth}{16.8cm}
\newtheoremstyle{main}{\topsep}{\topsep}
{\backfont}% Body font, eg \normalfont, \itshape
{\parindent}% Indent amount (empty = no indent, \parindent = para indent)
{\somefont}% Thm head font, eg \scshape, \bfseries
{\textcolor{thmcolor}{.}}% Punctuation after thm head
{\newline}% Space after thm head (\newline = linebreak)
{\textcolor{thmcolor}{\thmname{#1\subline}\thmnote{(\somefont #3)}}}% Thm head spec
\theoremstyle{main}

% Environnement demonstrations
\addto\captionsfrench{\def\proofname{\textsc{D\'emonstration}}}
%\renewcommand{\qedsymbol}{\includegraphics[scale=0.3]{smiley.pdf}}%{$\bullet$}
\definecolor{democolor}{rgb}{0.07,0.14,0.71}
\newenvironment{_demo}{\begin{proof}[\textcolor{democolor}{\proofname}]}{\end{proof}}
\newcommand{\demo}{\begin{_demo}}
\newcommand{\cqfd}{\end{_demo}}

% Theoremes
\newcounter{exemple}
\newcounter{remarque}
\newcounter{application}
\newcounter{notation}
\newshadetheorem{_th}{Th\'eor\`eme}%[section]
\newshadetheorem{_def}[_th]{D\'efinition}
\newshadetheorem{_rk}[_th]{Remarque}
\newshadetheorem{_lm}[_th]{Lemme}
%\newshadetheorem{_ex}[_th]{Exemple}
%\newshadetheorem{_conj}[_th]{Conjecture}
\newshadetheorem{_pd}[_th]{Propri\'et\'e-D\'efinition}
\newshadetheorem{_prop}[_th]{Proposition}
\newshadetheorem{_cor}[_th]{Corollaire}
\newenvironment{rem}{\medskip \refstepcounter{remarque}\textcolor{black}{\bf Remarque \theremarque}\\[0.15cm]}{}
%\newtheorem{_ex}[_th]{Exemple}
%\newshadetheorem{_conj}[_th]{Conjecture}
\newenvironment{exemple}{\medskip \refstepcounter{exemple}\textcolor{black}{\bf Exemple \theexemple}~}{}
\newenvironment{exemples}{\medskip \refstepcounter{exemple}\textcolor{black}{\bf Exemples \theexemple}~}{}
\newenvironment{appl}{\medskip \refstepcounter{application}\textcolor{black}{\bf Application \theapplication}\\}{}
\newenvironment{notation}{\medskip \refstepcounter{notation}\textcolor{black}{\bf Notation \thenotation~~}}{}
\newenvironment{notations}{\medskip \refstepcounter{notation}\textcolor{black}{\bf Notations \thenotation~~}}{}
\newcommand{\thm}{\begin{_th}}
\newcommand{\mht}{\end{_th}}
\newcommand{\pro}{\begin{_prop}}
\newcommand{\orp}{\end{_prop}}
\newcommand{\col}{\begin{_cor}}
\newcommand{\loc}{\end{_cor}}
\newcommand{\dfn}{\begin{_def}}
\newcommand{\nfd}{\end{_def}}
\newcommand{\lem}{\begin{_lm}}
\newcommand{\mel}{\end{_lm}}
%\newcommand{\ex}{\begin{_ex}}
%\newcommand{\xe}{\end{_ex}}
%\newcommand{\conj}{\begin{_conj}}
%\newcommand{\jonc}{\end{_conj}}
\newcommand{\prd}{\begin{_pd}}
\newcommand{\drp}{\end{_pd}}
\newcommand{\rmq}{\begin{rem}}
\newcommand{\qmr}{\end{rem}}
\newcommand{\ex}{\begin{exemple}}
\newcommand{\xe}{\end{exemple}}
\newcommand{\exx}{\begin{exemple}}
\newcommand{\xxe}{\end{exemple}}
\newcommand{\exs}{\begin{exemples}}
\newcommand{\sxe}{\end{exemples}}
\newcommand{\nota}{\begin{notation}}
\newcommand{\aton}{\end{notation}}
\newcommand{\nots}{\begin{notations}}
\newcommand{\ston}{\end{notations}}

% Pour les eleves
\newcommand{\quatrelignes}{.............................................................................................................................................................................${}^{{}^{{}^{{}^{{}^{}}}}}$\\
.............................................................................................................................................................................${}^{{}^{{}^{{}^{{}^{}}}}}$\\
.............................................................................................................................................................................${}^{{}^{{}^{{}^{{}^{}}}}}$\\
.............................................................................................................................................................................${}^{{}^{{}^{{}^{{}^{}}}}}$}
\newcommand{\uneligne}{.............................................................................................................................................................................${}^{{}^{{}^{{}^{{}^{}}}}}$}
\newcommand{\unepetiteligne}{.....................................................................................................................................................${}^{{}^{{}^{{}^{{}^{}}}}}$}
\newcommand{\unezolieligne}{...............................................................................................................................................${}^{{}^{{}^{{}^{{}^{}}}}}$}
\newcommand{\unemicroligne}%{\newline}
{................................................................................................${}^{{}^{{}^{{}^{{}^{}}}}}$}

% Exercices
\newcounter{exo}
\newcounter{partie}
%\newcommand\exo[1][]{\refstepcounter{exo}{\Large \textbf{Exercice~\theexo.~}}}
%\newcommand\pb[1][]{\refstepcounter{exo}{\Large \textbf{Probl\`eme~\theexo.~}}}
%\newcommand\exo[1][]{\refstepcounter{exo}{\textbf{Exercice.~}}}
\newcommand\exo[1][]{\refstepcounter{exo}{\textbf{Exercice~\theexo.~}}}
\newcommand\pb[1][]{\refstepcounter{exo}{\textbf{Probl\`eme~\theexo.~}}}
\newcommand\partie[1][]{\refstepcounter{partie}{\textbf{Partie~\thepartie.~}}}
\newcommand\exoc[1][]{\textbf{Exercice~#1.~}}
\newcommand\cons[1][]{\refstepcounter{exo}{\textbf{Consigne~\theexo.~}}}

% Correction
\definecolor{correct}{rgb}{0.5,0,0.5}
\newcommand{\cor}[1]{\textcolor{correct}{#1}}
\definecolor{anticorrect}{rgb}{0.4,0.7,0.4}
\newcommand{\anticor}[1]{\textcolor{anticorrect}{\it {#1}}}

% Entete
\fancyhead[L]{{\footnotesize  Lyc\'ee Alphonse Daudet  --  MPSI} }\fancyhead[R]{{\footnotesize  Ann\'ee \andegrace{} -- \textsc{Cours -- Option Informatique}}}\renewcommand{\headrulewidth}{0.0pt}
%\fancyfoot[C]{\thepage/\pageref{LastPage}}\renewcommand{\footrulewidth}{0pt}
\fancyfoot[C]{~}

\newcommand{\ocaml}[1]{\mintinline[mathescape]{ocaml}|{#1}|}

% Polyfills
\newcommand{\shadowbox}[1]{\fbox{#1}}

%\pagestyle{empty}
\pagestyle{fancy}
\date{}
\begin{document}

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\vspace{-2cm}
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