\documentclass[a4paper,11pt]{article}

\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[french]{babel}
\usepackage[a4paper,top=3.2cm,bottom=2cm,left=2cm,right=2cm]{geometry}
\usepackage{graphicx}
\usepackage[export]{adjustbox}
\usepackage{fancyhdr}
\usepackage{qrcode}
\usepackage[table]{xcolor}
\usepackage{amsmath,amssymb}
\usepackage{enumitem}
% Puces nettes (babel-french remplace sinon la puce par un tiret « – » peu pro).
% Vaut pour TOUTES les fiches qui importent ce module. AtBeginDocument pour
% passer après la personnalisation des listes faite par babel-french.
\AtBeginDocument{%
  \setlist[itemize]{leftmargin=1.8em,itemsep=0.5ex,topsep=0.4ex,parsep=0pt}%
  \setlist[itemize,1]{label=\textcolor{red!75!black}{\small\textbullet}}%
  \setlist[itemize,2]{label=\textcolor{red!75!black}{\scriptsize\textbullet}}%
}
% Titres compacts (gain de place vertical, fiches plus denses) — vaut pour tous.
\usepackage{titlesec}
\titlespacing*{\section}{0pt}{1.4ex}{0.6ex}
\titlespacing*{\subsection}{0pt}{1.0ex}{0.3ex}
% multicol : listes denses (calcul mental) sur 2 colonnes — neutre tant qu'il
% n'est pas utilisé, dispo pour tous les générateurs.
\usepackage{multicol}
\setlength{\columnsep}{1.5em}
% needspace : empêche un titre d'exercice de rester orphelin en bas de page
% (réserve quelques lignes ; sinon le titre passe à la page suivante avec sa
% liste). Neutre tant qu'il n'est pas appelé — dispo pour tous les générateurs.
\usepackage{needspace}
\usepackage{tikz}
\usetikzlibrary{decorations.text}

\usepackage{tgheros}
\pagestyle{fancy}
\fancyhf{}
\setlength{\headheight}{75pt}
\setlength{\headsep}{10pt}

\newcommand{\logowouf}{%
\begin{tikzpicture}[baseline={(current bounding box.center)},scale=0.55,
                    line cap=round,line join=round]
    \path[use as bounding box] (-2.20,-1.45) rectangle (2.20,2.20);
    \path[
        postaction={decorate},
        decoration={
            text along path,
            text={|\fontfamily{qhv}\bfseries\fontsize{9}{9}\selectfont|site2wouf.fr},
            text align={center},
            raise=0.03ex
        }
    ] (-1.29,0.48) arc[start angle=160,end angle=20,radius=1.29];
    \begin{scope}[xshift=-0.06cm]
        \fill[black] (-0.66,0.14) ellipse (0.21 and 0.31);
        \fill[black] ( 0.66,0.14) ellipse (0.21 and 0.31);
        \fill[black] (-0.23,0.49) ellipse (0.22 and 0.35);
        \fill[black] ( 0.23,0.49) ellipse (0.22 and 0.35);
        \fill[black]
            (0,-0.10)
            .. controls (-0.36,-0.02) and (-0.63,-0.24) .. (-0.58,-0.58)
            .. controls (-0.52,-0.93) and (-0.16,-1.12) .. (0,-0.90)
            .. controls (0.16,-1.12) and (0.52,-0.93) .. (0.58,-0.58)
            .. controls (0.63,-0.24) and (0.36,-0.02) .. (0,-0.10)
            -- cycle;
    \end{scope}
\end{tikzpicture}%
}
\DeclareUnicodeCharacter{2460}{\textcircled{\scriptsize 1}}
\DeclareUnicodeCharacter{2461}{\textcircled{\scriptsize 2}}
\DeclareUnicodeCharacter{2462}{\textcircled{\scriptsize 3}}
\DeclareUnicodeCharacter{2463}{\textcircled{\scriptsize 4}}
\DeclareUnicodeCharacter{2464}{\textcircled{\scriptsize 5}}
\DeclareUnicodeCharacter{2465}{\textcircled{\scriptsize 6}}
\DeclareUnicodeCharacter{2466}{\textcircled{\scriptsize 7}}
\DeclareUnicodeCharacter{2467}{\textcircled{\scriptsize 8}}
\DeclareUnicodeCharacter{2468}{\textcircled{\scriptsize 9}}
\DeclareUnicodeCharacter{2469}{\textcircled{\scriptsize 10}}
\DeclareUnicodeCharacter{246A}{\textcircled{\scriptsize 11}}
\DeclareUnicodeCharacter{246B}{\textcircled{\scriptsize 12}}
\DeclareUnicodeCharacter{246C}{\textcircled{\scriptsize 13}}
\DeclareUnicodeCharacter{246D}{\textcircled{\scriptsize 14}}
\DeclareUnicodeCharacter{246E}{\textcircled{\scriptsize 15}}
\DeclareUnicodeCharacter{246F}{\textcircled{\scriptsize 16}}
\DeclareUnicodeCharacter{2470}{\textcircled{\scriptsize 17}}
\DeclareUnicodeCharacter{2471}{\textcircled{\scriptsize 18}}
\DeclareUnicodeCharacter{2472}{\textcircled{\scriptsize 19}}
\DeclareUnicodeCharacter{2473}{\textcircled{\scriptsize 20}}
\DeclareUnicodeCharacter{24B6}{\textcircled{\scriptsize A}}
\DeclareUnicodeCharacter{24B7}{\textcircled{\scriptsize B}}
\DeclareUnicodeCharacter{24B8}{\textcircled{\scriptsize C}}
\DeclareUnicodeCharacter{24B9}{\textcircled{\scriptsize D}}
\DeclareUnicodeCharacter{24BA}{\textcircled{\scriptsize E}}
\DeclareUnicodeCharacter{24BB}{\textcircled{\scriptsize F}}
\DeclareUnicodeCharacter{24BC}{\textcircled{\scriptsize G}}
\DeclareUnicodeCharacter{24BD}{\textcircled{\scriptsize H}}
\DeclareUnicodeCharacter{24BE}{\textcircled{\scriptsize I}}
\DeclareUnicodeCharacter{24BF}{\textcircled{\scriptsize J}}
\DeclareUnicodeCharacter{24C0}{\textcircled{\scriptsize K}}
\DeclareUnicodeCharacter{24C1}{\textcircled{\scriptsize L}}
\DeclareUnicodeCharacter{24C2}{\textcircled{\scriptsize M}}
\DeclareUnicodeCharacter{24C3}{\textcircled{\scriptsize N}}
\DeclareUnicodeCharacter{24C4}{\textcircled{\scriptsize O}}
\DeclareUnicodeCharacter{24C5}{\textcircled{\scriptsize P}}
\DeclareUnicodeCharacter{24C6}{\textcircled{\scriptsize Q}}
\DeclareUnicodeCharacter{24C7}{\textcircled{\scriptsize R}}
\DeclareUnicodeCharacter{24C8}{\textcircled{\scriptsize S}}
\DeclareUnicodeCharacter{24C9}{\textcircled{\scriptsize T}}
\DeclareUnicodeCharacter{24CA}{\textcircled{\scriptsize U}}
\DeclareUnicodeCharacter{24CB}{\textcircled{\scriptsize V}}
\DeclareUnicodeCharacter{24CC}{\textcircled{\scriptsize W}}
\DeclareUnicodeCharacter{24CD}{\textcircled{\scriptsize X}}
\DeclareUnicodeCharacter{24CE}{\textcircled{\scriptsize Y}}
\DeclareUnicodeCharacter{24CF}{\textcircled{\scriptsize Z}}
\DeclareUnicodeCharacter{24D0}{\textcircled{\scriptsize a}}
\DeclareUnicodeCharacter{24D1}{\textcircled{\scriptsize b}}
\DeclareUnicodeCharacter{24D2}{\textcircled{\scriptsize c}}
\DeclareUnicodeCharacter{24D3}{\textcircled{\scriptsize d}}
\DeclareUnicodeCharacter{24D4}{\textcircled{\scriptsize e}}
\DeclareUnicodeCharacter{24D5}{\textcircled{\scriptsize f}}
\DeclareUnicodeCharacter{24D6}{\textcircled{\scriptsize g}}
\DeclareUnicodeCharacter{24D7}{\textcircled{\scriptsize h}}
\DeclareUnicodeCharacter{24D8}{\textcircled{\scriptsize i}}
\DeclareUnicodeCharacter{24D9}{\textcircled{\scriptsize j}}

\newcommand{\titre}{Le car de ramassage}
\newcommand{\soustitre}{Probl\`eme --- Fiche n\textdegree 309}

\fancyhead[C]{%
\raisebox{4pt}{%
\begin{minipage}[c]{0.15\textwidth}
    \centering
    \logowouf
\end{minipage}
\begin{minipage}[c]{0.62\textwidth}
    \centering
    {\Large\bfseries \titre\par}
    {\small\color{gray} \soustitre}
\end{minipage}
\begin{minipage}[c]{0.21\textwidth}
    \centering
    \qrcode[height=1.5cm]{https://site2wouf.fr/car_de_ramassage.php?x=309}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0309}
\end{minipage}%
}
}

\begin{document}

\begin{center}
    \small Un car de ramassage doit s'arrêter à chaque carrefour d'un quartier, une fois et une seule. Selon le plan c'est possible ou non — et un simple coloriage permet souvent de le dire d'avance, sans essayer.
\end{center}
\vspace{0.5em}
\sloppy
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6094, line join=round, line cap=round]
\draw[line width=0.5pt] (4.396,8.903) -- (4.628,8.838) -- (4.862,8.778) -- (5.098,8.723) -- (5.334,8.675) -- (5.572,8.631) -- (5.811,8.593) -- (6.051,8.561) -- (6.291,8.535) -- (6.532,8.514) -- (6.774,8.499) -- (7.015,8.489) -- (7.257,8.486) -- (7.498,8.487) -- (7.74,8.495);
\draw[line width=0.5pt] (4.396,8.903) -- (4.428,8.673) -- (4.455,8.441) -- (4.477,8.209) -- (4.493,7.976) -- (4.503,7.743) -- (4.509,7.51) -- (4.508,7.276) -- (4.502,7.043) -- (4.491,6.81) -- (4.474,6.577) -- (4.452,6.345) -- (4.425,6.113) -- (4.391,5.882) -- (4.353,5.653);
\draw[line width=0.5pt] (11.253,8.842) -- (11.207,8.615) -- (11.167,8.386) -- (11.133,8.157) -- (11.104,7.927) -- (11.08,7.696) -- (11.062,7.465) -- (11.049,7.233) -- (11.042,7.001) -- (11.04,6.769) -- (11.043,6.537) -- (11.052,6.305) -- (11.067,6.074) -- (11.087,5.843) -- (11.112,5.612);
\draw[line width=0.5pt] (1.416,8.645) -- (1.624,8.696) -- (1.834,8.742) -- (2.044,8.783) -- (2.256,8.819) -- (2.468,8.85) -- (2.681,8.876) -- (2.895,8.897) -- (3.109,8.913) -- (3.324,8.924) -- (3.538,8.93) -- (3.753,8.931) -- (3.967,8.927) -- (4.182,8.918) -- (4.396,8.903);
\draw[line width=0.5pt] (1.416,8.645) -- (1.39,8.403) -- (1.37,8.16) -- (1.355,7.917) -- (1.346,7.674) -- (1.343,7.431) -- (1.346,7.187) -- (1.354,6.944) -- (1.368,6.7) -- (1.388,6.458) -- (1.413,6.215) -- (1.444,5.974) -- (1.481,5.733) -- (1.524,5.494) -- (1.572,5.255);
\draw[line width=0.5pt] (7.74,8.495) -- (7.985,8.558) -- (8.232,8.615) -- (8.48,8.667) -- (8.729,8.712) -- (8.979,8.752) -- (9.23,8.786) -- (9.482,8.814) -- (9.734,8.836) -- (9.987,8.852) -- (10.24,8.862) -- (10.494,8.866) -- (10.747,8.864) -- (11.0,8.856) -- (11.253,8.842);
\draw[line width=0.5pt] (7.74,8.495) -- (7.699,8.231) -- (7.665,7.965) -- (7.637,7.699) -- (7.615,7.432) -- (7.599,7.164) -- (7.59,6.896) -- (7.587,6.628) -- (7.59,6.36) -- (7.6,6.093) -- (7.616,5.825) -- (7.638,5.558) -- (7.666,5.292) -- (7.701,5.026) -- (7.742,4.762);
\draw[line width=0.5pt] (4.353,5.653) -- (4.603,5.627) -- (4.852,5.595) -- (5.1,5.557) -- (5.348,5.513) -- (5.595,5.463) -- (5.84,5.408) -- (6.084,5.347) -- (6.327,5.28) -- (6.567,5.208) -- (6.806,5.13) -- (7.044,5.046) -- (7.279,4.957) -- (7.511,4.862) -- (7.742,4.762);
\draw[line width=0.5pt] (4.353,5.653) -- (4.339,5.357) -- (4.332,5.062) -- (4.332,4.766) -- (4.338,4.47) -- (4.352,4.174) -- (4.373,3.879) -- (4.401,3.584) -- (4.435,3.29) -- (4.477,2.997) -- (4.525,2.705) -- (4.581,2.415) -- (4.643,2.125) -- (4.712,1.838) -- (4.787,1.552);
\draw[line width=0.5pt] (1.572,5.255) -- (1.765,5.314) -- (1.959,5.368) -- (2.154,5.417) -- (2.351,5.462) -- (2.548,5.502) -- (2.747,5.538) -- (2.946,5.569) -- (3.146,5.595) -- (3.346,5.616) -- (3.547,5.633) -- (3.749,5.645) -- (3.95,5.652) -- (4.151,5.655) -- (4.353,5.653);
\draw[line width=0.5pt] (1.572,5.255) -- (1.512,5.015) -- (1.457,4.773) -- (1.408,4.531) -- (1.365,4.287) -- (1.328,4.042) -- (1.296,3.796) -- (1.27,3.549) -- (1.25,3.302) -- (1.236,3.055) -- (1.228,2.807) -- (1.225,2.56) -- (1.228,2.312) -- (1.238,2.065) -- (1.253,1.818);
\draw[line width=0.5pt] (7.742,4.762) -- (7.971,4.859) -- (8.203,4.951) -- (8.437,5.037) -- (8.673,5.117) -- (8.911,5.193) -- (9.151,5.262) -- (9.392,5.326) -- (9.635,5.384) -- (9.878,5.437) -- (10.123,5.483) -- (10.37,5.524) -- (10.616,5.56) -- (10.864,5.589) -- (11.112,5.612);
\draw[line width=0.5pt] (7.742,4.762) -- (7.725,4.552) -- (7.712,4.341) -- (7.705,4.13) -- (7.703,3.92) -- (7.705,3.709) -- (7.713,3.498) -- (7.725,3.287) -- (7.743,3.077) -- (7.765,2.867) -- (7.793,2.658) -- (7.825,2.45) -- (7.862,2.242) -- (7.904,2.035) -- (7.951,1.83);
\draw[line width=0.5pt] (4.787,1.552) -- (5.008,1.606) -- (5.231,1.655) -- (5.454,1.699) -- (5.679,1.738) -- (5.904,1.771) -- (6.131,1.799) -- (6.358,1.821) -- (6.585,1.839) -- (6.812,1.851) -- (7.04,1.857) -- (7.268,1.859) -- (7.496,1.854) -- (7.724,1.845) -- (7.951,1.83);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.556,8.903) -- (4.534,8.983) -- (4.476,9.042) -- (4.396,9.063) -- (4.316,9.042) -- (4.257,8.983) -- (4.236,8.903) -- (4.257,8.823) -- (4.316,8.765) -- (4.396,8.743) -- (4.476,8.765) -- (4.534,8.823) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.413,8.842) -- (11.391,8.922) -- (11.333,8.981) -- (11.253,9.002) -- (11.173,8.981) -- (11.114,8.922) -- (11.093,8.842) -- (11.114,8.762) -- (11.173,8.704) -- (11.253,8.682) -- (11.333,8.704) -- (11.391,8.762) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.576,8.645) -- (1.554,8.725) -- (1.496,8.783) -- (1.416,8.805) -- (1.336,8.783) -- (1.277,8.725) -- (1.256,8.645) -- (1.277,8.565) -- (1.336,8.506) -- (1.416,8.485) -- (1.496,8.506) -- (1.554,8.565) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.9,8.495) -- (7.878,8.575) -- (7.82,8.634) -- (7.74,8.655) -- (7.66,8.634) -- (7.601,8.575) -- (7.58,8.495) -- (7.601,8.415) -- (7.66,8.357) -- (7.74,8.335) -- (7.82,8.357) -- (7.878,8.415) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.513,5.653) -- (4.491,5.733) -- (4.433,5.791) -- (4.353,5.813) -- (4.273,5.791) -- (4.214,5.733) -- (4.193,5.653) -- (4.214,5.573) -- (4.273,5.514) -- (4.353,5.493) -- (4.433,5.514) -- (4.491,5.573) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.272,5.612) -- (11.251,5.692) -- (11.192,5.751) -- (11.112,5.772) -- (11.032,5.751) -- (10.974,5.692) -- (10.952,5.612) -- (10.974,5.532) -- (11.032,5.474) -- (11.112,5.452) -- (11.192,5.474) -- (11.251,5.532) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.732,5.255) -- (1.711,5.335) -- (1.652,5.394) -- (1.572,5.415) -- (1.492,5.394) -- (1.433,5.335) -- (1.412,5.255) -- (1.433,5.175) -- (1.492,5.117) -- (1.572,5.095) -- (1.652,5.117) -- (1.711,5.175) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.902,4.762) -- (7.88,4.842) -- (7.822,4.9) -- (7.742,4.922) -- (7.662,4.9) -- (7.603,4.842) -- (7.582,4.762) -- (7.603,4.682) -- (7.662,4.623) -- (7.742,4.602) -- (7.822,4.623) -- (7.88,4.682) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.111,1.83) -- (8.089,1.91) -- (8.031,1.968) -- (7.951,1.99) -- (7.871,1.968) -- (7.812,1.91) -- (7.791,1.83) -- (7.812,1.75) -- (7.871,1.691) -- (7.951,1.67) -- (8.031,1.691) -- (8.089,1.75) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.413,1.818) -- (1.391,1.898) -- (1.333,1.956) -- (1.253,1.978) -- (1.173,1.956) -- (1.114,1.898) -- (1.093,1.818) -- (1.114,1.738) -- (1.173,1.679) -- (1.253,1.658) -- (1.333,1.679) -- (1.391,1.738) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.947,1.552) -- (4.926,1.632) -- (4.867,1.691) -- (4.787,1.712) -- (4.707,1.691) -- (4.649,1.632) -- (4.627,1.552) -- (4.649,1.472) -- (4.707,1.413) -- (4.787,1.392) -- (4.867,1.413) -- (4.926,1.472) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.099,8.558) -- (6.22,8.578) -- (6.33,8.634) -- (6.417,8.721) -- (6.473,8.831) -- (6.493,8.952) -- (6.473,9.074) -- (6.417,9.184) -- (6.33,9.271) -- (6.22,9.327) -- (6.099,9.346) -- (6.099,9.346) -- (5.977,9.327) -- (5.867,9.271) -- (5.78,9.184) -- (5.724,9.074) -- (5.705,8.952) -- (5.724,8.831) -- (5.78,8.721) -- (5.867,8.634) -- (5.977,8.578) -- (6.099,8.558) -- cycle;
\node at (6.099,8.952) {\scalebox{1.641}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.902,6.877) -- (5.024,6.896) -- (5.134,6.952) -- (5.221,7.04) -- (5.277,7.149) -- (5.296,7.271) -- (5.277,7.393) -- (5.221,7.503) -- (5.134,7.59) -- (5.024,7.646) -- (4.902,7.665) -- (4.902,7.665) -- (4.78,7.646) -- (4.671,7.59) -- (4.583,7.503) -- (4.528,7.393) -- (4.508,7.271) -- (4.528,7.149) -- (4.583,7.04) -- (4.671,6.952) -- (4.78,6.896) -- (4.902,6.877) -- cycle;
\node at (4.902,7.271) {\scalebox{1.641}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.443,6.822) -- (11.564,6.841) -- (11.674,6.897) -- (11.761,6.984) -- (11.817,7.094) -- (11.837,7.216) -- (11.817,7.338) -- (11.761,7.447) -- (11.674,7.535) -- (11.564,7.591) -- (11.443,7.61) -- (11.443,7.61) -- (11.321,7.591) -- (11.211,7.535) -- (11.124,7.447) -- (11.068,7.338) -- (11.049,7.216) -- (11.068,7.094) -- (11.124,6.984) -- (11.211,6.897) -- (11.321,6.841) -- (11.443,6.822) -- cycle;
\node at (11.443,7.216) {\scalebox{1.641}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.861,8.896) -- (2.983,8.915) -- (3.093,8.971) -- (3.18,9.058) -- (3.236,9.168) -- (3.255,9.29) -- (3.236,9.411) -- (3.18,9.521) -- (3.093,9.608) -- (2.983,9.664) -- (2.861,9.683) -- (2.861,9.683) -- (2.739,9.664) -- (2.629,9.608) -- (2.542,9.521) -- (2.486,9.411) -- (2.467,9.29) -- (2.486,9.168) -- (2.542,9.058) -- (2.629,8.971) -- (2.739,8.915) -- (2.861,8.896) -- cycle;
\node at (2.861,9.29) {\scalebox{1.641}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.748,6.568) -- (1.869,6.587) -- (1.979,6.643) -- (2.066,6.73) -- (2.122,6.84) -- (2.141,6.962) -- (2.122,7.083) -- (2.066,7.193) -- (1.979,7.28) -- (1.869,7.336) -- (1.748,7.356) -- (1.748,7.356) -- (1.626,7.336) -- (1.516,7.28) -- (1.429,7.193) -- (1.373,7.083) -- (1.354,6.962) -- (1.373,6.84) -- (1.429,6.73) -- (1.516,6.643) -- (1.626,6.587) -- (1.748,6.568) -- cycle;
\node at (1.748,6.962) {\scalebox{1.641}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.443,8.812) -- (9.565,8.831) -- (9.675,8.887) -- (9.762,8.974) -- (9.818,9.084) -- (9.837,9.206) -- (9.818,9.327) -- (9.762,9.437) -- (9.675,9.524) -- (9.565,9.58) -- (9.443,9.599) -- (9.443,9.599) -- (9.321,9.58) -- (9.212,9.524) -- (9.125,9.437) -- (9.069,9.327) -- (9.049,9.206) -- (9.069,9.084) -- (9.125,8.974) -- (9.212,8.887) -- (9.321,8.831) -- (9.443,8.812) -- cycle;
\node at (9.443,9.206) {\scalebox{1.641}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.981,6.235) -- (8.102,6.254) -- (8.212,6.31) -- (8.299,6.397) -- (8.355,6.507) -- (8.375,6.629) -- (8.355,6.75) -- (8.299,6.86) -- (8.212,6.947) -- (8.102,7.003) -- (7.981,7.022) -- (7.981,7.022) -- (7.859,7.003) -- (7.749,6.947) -- (7.662,6.86) -- (7.606,6.75) -- (7.587,6.629) -- (7.606,6.507) -- (7.662,6.397) -- (7.749,6.31) -- (7.859,6.254) -- (7.981,6.235) -- cycle;
\node at (7.981,6.629) {\scalebox{1.641}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.184,5.334) -- (6.306,5.353) -- (6.416,5.409) -- (6.503,5.496) -- (6.559,5.606) -- (6.578,5.728) -- (6.559,5.85) -- (6.503,5.959) -- (6.416,6.047) -- (6.306,6.103) -- (6.184,6.122) -- (6.184,6.122) -- (6.062,6.103) -- (5.953,6.047) -- (5.866,5.959) -- (5.81,5.85) -- (5.79,5.728) -- (5.81,5.606) -- (5.866,5.496) -- (5.953,5.409) -- (6.062,5.353) -- (6.184,5.334) -- cycle;
\node at (6.184,5.728) {\scalebox{1.641}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.793,3.232) -- (4.914,3.251) -- (5.024,3.307) -- (5.111,3.394) -- (5.167,3.504) -- (5.186,3.626) -- (5.167,3.748) -- (5.111,3.857) -- (5.024,3.945) -- (4.914,4.0) -- (4.793,4.02) -- (4.793,4.02) -- (4.671,4.0) -- (4.561,3.945) -- (4.474,3.857) -- (4.418,3.748) -- (4.399,3.626) -- (4.418,3.504) -- (4.474,3.394) -- (4.561,3.307) -- (4.671,3.251) -- (4.793,3.232) -- cycle;
\node at (4.793,3.626) {\scalebox{1.641}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.867,5.532) -- (2.989,5.552) -- (3.099,5.608) -- (3.186,5.695) -- (3.242,5.804) -- (3.261,5.926) -- (3.242,6.048) -- (3.186,6.158) -- (3.099,6.245) -- (2.989,6.301) -- (2.867,6.32) -- (2.496,6.32) -- (2.375,6.301) -- (2.265,6.245) -- (2.178,6.158) -- (2.122,6.048) -- (2.103,5.926) -- (2.122,5.804) -- (2.178,5.695) -- (2.265,5.608) -- (2.375,5.552) -- (2.496,5.532) -- cycle;
\node at (2.682,5.926) {\scalebox{1.641}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.848,3.119) -- (1.97,3.138) -- (2.08,3.194) -- (2.167,3.282) -- (2.223,3.391) -- (2.242,3.513) -- (2.223,3.635) -- (2.167,3.745) -- (2.08,3.832) -- (1.97,3.888) -- (1.848,3.907) -- (1.477,3.907) -- (1.356,3.888) -- (1.246,3.832) -- (1.159,3.745) -- (1.103,3.635) -- (1.083,3.513) -- (1.103,3.391) -- (1.159,3.282) -- (1.246,3.194) -- (1.356,3.138) -- (1.477,3.119) -- cycle;
\node at (1.663,3.513) {\scalebox{1.641}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.481,5.314) -- (9.603,5.333) -- (9.712,5.389) -- (9.8,5.476) -- (9.855,5.586) -- (9.875,5.708) -- (9.855,5.83) -- (9.8,5.939) -- (9.712,6.027) -- (9.603,6.083) -- (9.481,6.102) -- (9.11,6.102) -- (8.988,6.083) -- (8.879,6.027) -- (8.791,5.939) -- (8.736,5.83) -- (8.716,5.708) -- (8.736,5.586) -- (8.791,5.476) -- (8.879,5.389) -- (8.988,5.333) -- (9.11,5.314) -- cycle;
\node at (9.295,5.708) {\scalebox{1.641}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.304,2.921) -- (8.425,2.941) -- (8.535,2.997) -- (8.622,3.084) -- (8.678,3.193) -- (8.698,3.315) -- (8.678,3.437) -- (8.622,3.547) -- (8.535,3.634) -- (8.425,3.69) -- (8.304,3.709) -- (7.933,3.709) -- (7.811,3.69) -- (7.701,3.634) -- (7.614,3.547) -- (7.558,3.437) -- (7.539,3.315) -- (7.558,3.193) -- (7.614,3.084) -- (7.701,2.997) -- (7.811,2.941) -- (7.933,2.921) -- cycle;
\node at (8.118,3.315) {\scalebox{1.641}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.508,1.82) -- (6.63,1.839) -- (6.74,1.895) -- (6.827,1.982) -- (6.883,2.092) -- (6.902,2.214) -- (6.883,2.336) -- (6.827,2.445) -- (6.74,2.533) -- (6.63,2.588) -- (6.508,2.608) -- (6.138,2.608) -- (6.016,2.588) -- (5.906,2.533) -- (5.819,2.445) -- (5.763,2.336) -- (5.744,2.214) -- (5.763,2.092) -- (5.819,1.982) -- (5.906,1.895) -- (6.016,1.839) -- (6.138,1.82) -- cycle;
\node at (6.323,2.214) {\scalebox{1.641}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.272,4.955) -- (1.872,4.955) -- (1.872,5.555) -- (1.272,5.555) -- cycle;
\node at (1.572,5.255) {\scalebox{1.641}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (3.685,2.475) -- (4.285,2.475) -- (4.285,3.075) -- (3.685,3.075) -- cycle;
\node at (3.985,2.775) {\scalebox{1.641}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.306,9.839) -- (9.906,9.839) -- (9.906,10.439) -- (9.306,10.439) -- cycle;
\node at (9.606,10.139) {\scalebox{1.641}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.6,7.603) -- (1.2,7.603) -- (1.2,8.203) -- (0.6,8.203) -- cycle;
\node at (0.9,7.903) {\scalebox{1.641}{\small d}};
\node at (4.157,9.476) {\scalebox{1.641}{\small A}};
\node at (11.785,9.16) {\scalebox{1.641}{\small B}};
\node at (0.909,9.001) {\scalebox{1.641}{\small C}};
\node at (8.086,9.01) {\scalebox{1.641}{\small D}};
\node at (3.734,5.685) {\scalebox{1.641}{\small E}};
\node at (11.732,5.616) {\scalebox{1.641}{\small F}};
\node at (0.954,5.208) {\scalebox{1.641}{\small G}};
\node at (8.314,4.523) {\scalebox{1.641}{\small H}};
\node at (8.262,1.293) {\scalebox{1.641}{\small I}};
\node at (0.776,1.421) {\scalebox{1.641}{\small J}};
\node at (4.639,0.95) {\scalebox{1.641}{\small K}};
\draw[line width=0.5pt] (12.553,7.953) -- (12.553,9.053);
\draw[line width=0.5pt] (12.322,8.668) -- (12.553,9.053) -- (12.784,8.668);
\draw[line width=0.5pt] (12.25,8.503) -- (12.855,8.503);
\node at (12.553,9.438) {\scalebox{1.641}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour G, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑧ rue des Jardins & Ⓐ Carrefour Saint-Roch & Ⓘ Placette des Ormes \\ \hline
① rue de la Poste & ⑨ quai aux Fleurs & Ⓑ Place de l'Église & Ⓙ Place du Marché \\ \hline
② rue de la Mairie & ⑩ rue du Moulin & Ⓒ Coin du Lavoir & Ⓚ Place du Vieux-Puits \\ \hline
③ allée des Tilleuls & ⑪ impasse des Roses & Ⓓ Carrefour des Tilleuls & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ chemin des Vignes & ⑫ rue Pasteur & Ⓔ Rond-point de la Poste & \textbf{a} l'école \\ \hline
⑤ rue de l'Église & ⑬ passage du Marché & Ⓕ Place Sainte-Croix & \textbf{b} la gendarmerie \\ \hline
⑥ sentier des Prés & ⑭ rue Neuve & Ⓖ Carrefour du Lavoir & \textbf{c} la mairie \\ \hline
⑦ chemin Creux & \cellcolor{black!32}\bfseries Les carrefours & Ⓗ Carrefour du Moulin & \textbf{d} le cimetière \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\begin{enumerate}
  \item Colorier les carrefours du plan avec \textbf{deux couleurs}, de sorte que deux carrefours \textbf{reliés par une rue} ne soient jamais de la même couleur. Combien y en a-t-il de chaque couleur ?
  \item Le car roule d'un carrefour au suivant. Que peut-on dire de la \textbf{couleur} des arrêts successifs ? Qu'en déduire sur les couleurs des arrêts d'une tournée qui les dessert tous ?
  \item \textbf{Le car peut-il desservir tous les carrefours, une fois chacun, et revenir à l'école ?} Si oui, décrire un trajet ; sinon, démontrer que c'est impossible.
  \item \textbf{Et s'il n'est pas obligé de revenir à l'école ?} Même consigne.
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage — Correction}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6243, line join=round, line cap=round]
\draw[line width=0.5pt] (4.396,8.903) -- (4.628,8.838) -- (4.862,8.778) -- (5.098,8.723) -- (5.334,8.675) -- (5.572,8.631) -- (5.811,8.593) -- (6.051,8.561) -- (6.291,8.535) -- (6.532,8.514) -- (6.774,8.499) -- (7.015,8.489) -- (7.257,8.486) -- (7.498,8.487) -- (7.74,8.495);
\draw[line width=0.5pt] (4.396,8.903) -- (4.428,8.673) -- (4.455,8.441) -- (4.477,8.209) -- (4.493,7.976) -- (4.503,7.743) -- (4.509,7.51) -- (4.508,7.276) -- (4.502,7.043) -- (4.491,6.81) -- (4.474,6.577) -- (4.452,6.345) -- (4.425,6.113) -- (4.391,5.882) -- (4.353,5.653);
\draw[line width=0.5pt] (11.253,8.842) -- (11.207,8.615) -- (11.167,8.386) -- (11.133,8.157) -- (11.104,7.927) -- (11.08,7.696) -- (11.062,7.465) -- (11.049,7.233) -- (11.042,7.001) -- (11.04,6.769) -- (11.043,6.537) -- (11.052,6.305) -- (11.067,6.074) -- (11.087,5.843) -- (11.112,5.612);
\draw[line width=0.5pt] (1.416,8.645) -- (1.624,8.696) -- (1.834,8.742) -- (2.044,8.783) -- (2.256,8.819) -- (2.468,8.85) -- (2.681,8.876) -- (2.895,8.897) -- (3.109,8.913) -- (3.324,8.924) -- (3.538,8.93) -- (3.753,8.931) -- (3.967,8.927) -- (4.182,8.918) -- (4.396,8.903);
\draw[line width=0.5pt] (1.416,8.645) -- (1.39,8.403) -- (1.37,8.16) -- (1.355,7.917) -- (1.346,7.674) -- (1.343,7.431) -- (1.346,7.187) -- (1.354,6.944) -- (1.368,6.7) -- (1.388,6.458) -- (1.413,6.215) -- (1.444,5.974) -- (1.481,5.733) -- (1.524,5.494) -- (1.572,5.255);
\draw[line width=0.5pt] (7.74,8.495) -- (7.985,8.558) -- (8.232,8.615) -- (8.48,8.667) -- (8.729,8.712) -- (8.979,8.752) -- (9.23,8.786) -- (9.482,8.814) -- (9.734,8.836) -- (9.987,8.852) -- (10.24,8.862) -- (10.494,8.866) -- (10.747,8.864) -- (11.0,8.856) -- (11.253,8.842);
\draw[line width=0.5pt] (7.74,8.495) -- (7.699,8.231) -- (7.665,7.965) -- (7.637,7.699) -- (7.615,7.432) -- (7.599,7.164) -- (7.59,6.896) -- (7.587,6.628) -- (7.59,6.36) -- (7.6,6.093) -- (7.616,5.825) -- (7.638,5.558) -- (7.666,5.292) -- (7.701,5.026) -- (7.742,4.762);
\draw[line width=0.5pt] (4.353,5.653) -- (4.603,5.627) -- (4.852,5.595) -- (5.1,5.557) -- (5.348,5.513) -- (5.595,5.463) -- (5.84,5.408) -- (6.084,5.347) -- (6.327,5.28) -- (6.567,5.208) -- (6.806,5.13) -- (7.044,5.046) -- (7.279,4.957) -- (7.511,4.862) -- (7.742,4.762);
\draw[line width=0.5pt] (4.353,5.653) -- (4.339,5.357) -- (4.332,5.062) -- (4.332,4.766) -- (4.338,4.47) -- (4.352,4.174) -- (4.373,3.879) -- (4.401,3.584) -- (4.435,3.29) -- (4.477,2.997) -- (4.525,2.705) -- (4.581,2.415) -- (4.643,2.125) -- (4.712,1.838) -- (4.787,1.552);
\draw[line width=0.5pt] (1.572,5.255) -- (1.765,5.314) -- (1.959,5.368) -- (2.154,5.417) -- (2.351,5.462) -- (2.548,5.502) -- (2.747,5.538) -- (2.946,5.569) -- (3.146,5.595) -- (3.346,5.616) -- (3.547,5.633) -- (3.749,5.645) -- (3.95,5.652) -- (4.151,5.655) -- (4.353,5.653);
\draw[line width=0.5pt] (1.572,5.255) -- (1.512,5.015) -- (1.457,4.773) -- (1.408,4.531) -- (1.365,4.287) -- (1.328,4.042) -- (1.296,3.796) -- (1.27,3.549) -- (1.25,3.302) -- (1.236,3.055) -- (1.228,2.807) -- (1.225,2.56) -- (1.228,2.312) -- (1.238,2.065) -- (1.253,1.818);
\draw[line width=0.5pt] (7.742,4.762) -- (7.971,4.859) -- (8.203,4.951) -- (8.437,5.037) -- (8.673,5.117) -- (8.911,5.193) -- (9.151,5.262) -- (9.392,5.326) -- (9.635,5.384) -- (9.878,5.437) -- (10.123,5.483) -- (10.37,5.524) -- (10.616,5.56) -- (10.864,5.589) -- (11.112,5.612);
\draw[line width=0.5pt] (7.742,4.762) -- (7.725,4.552) -- (7.712,4.341) -- (7.705,4.13) -- (7.703,3.92) -- (7.705,3.709) -- (7.713,3.498) -- (7.725,3.287) -- (7.743,3.077) -- (7.765,2.867) -- (7.793,2.658) -- (7.825,2.45) -- (7.862,2.242) -- (7.904,2.035) -- (7.951,1.83);
\draw[line width=0.5pt] (4.787,1.552) -- (5.008,1.606) -- (5.231,1.655) -- (5.454,1.699) -- (5.679,1.738) -- (5.904,1.771) -- (6.131,1.799) -- (6.358,1.821) -- (6.585,1.839) -- (6.812,1.851) -- (7.04,1.857) -- (7.268,1.859) -- (7.496,1.854) -- (7.724,1.845) -- (7.951,1.83);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.769,8.903) -- (4.74,9.046) -- (4.659,9.167) -- (4.538,9.248) -- (4.396,9.277) -- (4.253,9.248) -- (4.132,9.167) -- (4.051,9.046) -- (4.022,8.903) -- (4.051,8.761) -- (4.132,8.64) -- (4.253,8.559) -- (4.396,8.53) -- (4.538,8.559) -- (4.659,8.64) -- (4.74,8.761) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.626,8.842) -- (11.597,8.985) -- (11.516,9.106) -- (11.395,9.187) -- (11.253,9.215) -- (11.11,9.187) -- (10.989,9.106) -- (10.908,8.985) -- (10.879,8.842) -- (10.908,8.699) -- (10.989,8.578) -- (11.11,8.497) -- (11.253,8.469) -- (11.395,8.497) -- (11.516,8.578) -- (11.597,8.699) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.789,8.645) -- (1.761,8.788) -- (1.68,8.909) -- (1.559,8.99) -- (1.416,9.018) -- (1.273,8.99) -- (1.152,8.909) -- (1.071,8.788) -- (1.043,8.645) -- (1.071,8.502) -- (1.152,8.381) -- (1.273,8.3) -- (1.416,8.272) -- (1.559,8.3) -- (1.68,8.381) -- (1.761,8.502) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.113,8.495) -- (8.085,8.638) -- (8.004,8.759) -- (7.883,8.84) -- (7.74,8.868) -- (7.597,8.84) -- (7.476,8.759) -- (7.395,8.638) -- (7.367,8.495) -- (7.395,8.352) -- (7.476,8.231) -- (7.597,8.15) -- (7.74,8.122) -- (7.883,8.15) -- (8.004,8.231) -- (8.085,8.352) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.726,5.653) -- (4.698,5.795) -- (4.617,5.916) -- (4.496,5.997) -- (4.353,6.026) -- (4.21,5.997) -- (4.089,5.916) -- (4.008,5.795) -- (3.98,5.653) -- (4.008,5.51) -- (4.089,5.389) -- (4.21,5.308) -- (4.353,5.279) -- (4.496,5.308) -- (4.617,5.389) -- (4.698,5.51) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.485,5.612) -- (11.457,5.755) -- (11.376,5.876) -- (11.255,5.957) -- (11.112,5.985) -- (10.969,5.957) -- (10.848,5.876) -- (10.767,5.755) -- (10.739,5.612) -- (10.767,5.469) -- (10.848,5.348) -- (10.969,5.267) -- (11.112,5.239) -- (11.255,5.267) -- (11.376,5.348) -- (11.457,5.469) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.945,5.255) -- (1.917,5.398) -- (1.836,5.519) -- (1.715,5.6) -- (1.572,5.628) -- (1.429,5.6) -- (1.308,5.519) -- (1.227,5.398) -- (1.199,5.255) -- (1.227,5.112) -- (1.308,4.991) -- (1.429,4.91) -- (1.572,4.882) -- (1.715,4.91) -- (1.836,4.991) -- (1.917,5.112) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.115,4.762) -- (8.087,4.905) -- (8.006,5.026) -- (7.885,5.107) -- (7.742,5.135) -- (7.599,5.107) -- (7.478,5.026) -- (7.397,4.905) -- (7.369,4.762) -- (7.397,4.619) -- (7.478,4.498) -- (7.599,4.417) -- (7.742,4.388) -- (7.885,4.417) -- (8.006,4.498) -- (8.087,4.619) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.324,1.83) -- (8.296,1.973) -- (8.215,2.094) -- (8.094,2.175) -- (7.951,2.203) -- (7.808,2.175) -- (7.687,2.094) -- (7.606,1.973) -- (7.578,1.83) -- (7.606,1.687) -- (7.687,1.566) -- (7.808,1.485) -- (7.951,1.457) -- (8.094,1.485) -- (8.215,1.566) -- (8.296,1.687) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.626,1.818) -- (1.597,1.96) -- (1.516,2.081) -- (1.395,2.162) -- (1.253,2.191) -- (1.11,2.162) -- (0.989,2.081) -- (0.908,1.96) -- (0.879,1.818) -- (0.908,1.675) -- (0.989,1.554) -- (1.11,1.473) -- (1.253,1.444) -- (1.395,1.473) -- (1.516,1.554) -- (1.597,1.675) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.16,1.552) -- (5.132,1.695) -- (5.051,1.816) -- (4.93,1.897) -- (4.787,1.925) -- (4.644,1.897) -- (4.523,1.816) -- (4.442,1.695) -- (4.414,1.552) -- (4.442,1.409) -- (4.523,1.288) -- (4.644,1.207) -- (4.787,1.179) -- (4.93,1.207) -- (5.051,1.288) -- (5.132,1.409) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.098,8.558) -- (6.216,8.577) -- (6.324,8.632) -- (6.409,8.717) -- (6.463,8.824) -- (6.482,8.943) -- (6.463,9.062) -- (6.409,9.169) -- (6.324,9.254) -- (6.216,9.309) -- (6.098,9.328) -- (6.098,9.328) -- (5.979,9.309) -- (5.871,9.254) -- (5.786,9.169) -- (5.732,9.062) -- (5.713,8.943) -- (5.732,8.824) -- (5.786,8.717) -- (5.871,8.632) -- (5.979,8.577) -- (6.098,8.558) -- cycle;
\node at (6.098,8.943) {\scalebox{1.602}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.893,6.887) -- (5.012,6.905) -- (5.119,6.96) -- (5.204,7.045) -- (5.258,7.152) -- (5.277,7.271) -- (5.258,7.39) -- (5.204,7.497) -- (5.119,7.582) -- (5.012,7.637) -- (4.893,7.656) -- (4.893,7.656) -- (4.774,7.637) -- (4.667,7.582) -- (4.582,7.497) -- (4.527,7.39) -- (4.508,7.271) -- (4.527,7.152) -- (4.582,7.045) -- (4.667,6.96) -- (4.774,6.905) -- (4.893,6.887) -- cycle;
\node at (4.893,7.271) {\scalebox{1.602}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.433,6.832) -- (11.552,6.851) -- (11.659,6.905) -- (11.744,6.99) -- (11.799,7.098) -- (11.818,7.216) -- (11.799,7.335) -- (11.744,7.442) -- (11.659,7.527) -- (11.552,7.582) -- (11.433,7.601) -- (11.433,7.601) -- (11.314,7.582) -- (11.207,7.527) -- (11.122,7.442) -- (11.068,7.335) -- (11.049,7.216) -- (11.068,7.098) -- (11.122,6.99) -- (11.207,6.905) -- (11.314,6.851) -- (11.433,6.832) -- cycle;
\node at (11.433,7.216) {\scalebox{1.602}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.862,8.896) -- (2.981,8.914) -- (3.088,8.969) -- (3.173,9.054) -- (3.228,9.161) -- (3.246,9.28) -- (3.228,9.399) -- (3.173,9.506) -- (3.088,9.591) -- (2.981,9.646) -- (2.862,9.665) -- (2.862,9.665) -- (2.743,9.646) -- (2.636,9.591) -- (2.551,9.506) -- (2.496,9.399) -- (2.477,9.28) -- (2.496,9.161) -- (2.551,9.054) -- (2.636,8.969) -- (2.743,8.914) -- (2.862,8.896) -- cycle;
\node at (2.862,9.28) {\scalebox{1.602}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.738,6.577) -- (1.857,6.596) -- (1.964,6.65) -- (2.049,6.735) -- (2.104,6.842) -- (2.123,6.961) -- (2.104,7.08) -- (2.049,7.187) -- (1.964,7.272) -- (1.857,7.327) -- (1.738,7.346) -- (1.738,7.346) -- (1.619,7.327) -- (1.512,7.272) -- (1.427,7.187) -- (1.372,7.08) -- (1.354,6.961) -- (1.372,6.842) -- (1.427,6.735) -- (1.512,6.65) -- (1.619,6.596) -- (1.738,6.577) -- cycle;
\node at (1.738,6.961) {\scalebox{1.602}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.444,8.812) -- (9.563,8.831) -- (9.67,8.885) -- (9.755,8.97) -- (9.81,9.077) -- (9.829,9.196) -- (9.81,9.315) -- (9.755,9.422) -- (9.67,9.507) -- (9.563,9.562) -- (9.444,9.581) -- (9.444,9.581) -- (9.325,9.562) -- (9.218,9.507) -- (9.133,9.422) -- (9.078,9.315) -- (9.06,9.196) -- (9.078,9.077) -- (9.133,8.97) -- (9.218,8.885) -- (9.325,8.831) -- (9.444,8.812) -- cycle;
\node at (9.444,9.196) {\scalebox{1.602}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.971,6.244) -- (8.09,6.263) -- (8.197,6.317) -- (8.282,6.403) -- (8.337,6.51) -- (8.356,6.629) -- (8.337,6.747) -- (8.282,6.855) -- (8.197,6.94) -- (8.09,6.994) -- (7.971,7.013) -- (7.971,7.013) -- (7.853,6.994) -- (7.745,6.94) -- (7.66,6.855) -- (7.606,6.747) -- (7.587,6.629) -- (7.606,6.51) -- (7.66,6.403) -- (7.745,6.317) -- (7.853,6.263) -- (7.971,6.244) -- cycle;
\node at (7.971,6.629) {\scalebox{1.602}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.182,5.334) -- (6.301,5.353) -- (6.408,5.408) -- (6.493,5.493) -- (6.548,5.6) -- (6.566,5.719) -- (6.548,5.838) -- (6.493,5.945) -- (6.408,6.03) -- (6.301,6.085) -- (6.182,6.103) -- (6.182,6.103) -- (6.063,6.085) -- (5.956,6.03) -- (5.871,5.945) -- (5.816,5.838) -- (5.797,5.719) -- (5.816,5.6) -- (5.871,5.493) -- (5.956,5.408) -- (6.063,5.353) -- (6.182,5.334) -- cycle;
\node at (6.182,5.719) {\scalebox{1.602}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.783,3.24) -- (4.902,3.259) -- (5.009,3.314) -- (5.094,3.399) -- (5.149,3.506) -- (5.168,3.625) -- (5.149,3.744) -- (5.094,3.851) -- (5.009,3.936) -- (4.902,3.991) -- (4.783,4.009) -- (4.783,4.009) -- (4.664,3.991) -- (4.557,3.936) -- (4.472,3.851) -- (4.418,3.744) -- (4.399,3.625) -- (4.418,3.506) -- (4.472,3.399) -- (4.557,3.314) -- (4.664,3.259) -- (4.783,3.24) -- cycle;
\node at (4.783,3.625) {\scalebox{1.602}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.864,5.532) -- (2.983,5.551) -- (3.09,5.606) -- (3.175,5.691) -- (3.23,5.798) -- (3.249,5.917) -- (3.23,6.036) -- (3.175,6.143) -- (3.09,6.228) -- (2.983,6.283) -- (2.864,6.301) -- (2.502,6.301) -- (2.384,6.283) -- (2.276,6.228) -- (2.191,6.143) -- (2.137,6.036) -- (2.118,5.917) -- (2.137,5.798) -- (2.191,5.691) -- (2.276,5.606) -- (2.384,5.551) -- (2.502,5.532) -- cycle;
\node at (2.683,5.917) {\scalebox{1.602}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.834,3.129) -- (1.953,3.148) -- (2.06,3.203) -- (2.145,3.288) -- (2.2,3.395) -- (2.219,3.514) -- (2.2,3.633) -- (2.145,3.74) -- (2.06,3.825) -- (1.953,3.88) -- (1.834,3.898) -- (1.472,3.898) -- (1.354,3.88) -- (1.246,3.825) -- (1.161,3.74) -- (1.107,3.633) -- (1.088,3.514) -- (1.107,3.395) -- (1.161,3.288) -- (1.246,3.203) -- (1.354,3.148) -- (1.472,3.129) -- cycle;
\node at (1.653,3.514) {\scalebox{1.602}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.479,5.314) -- (9.598,5.333) -- (9.705,5.388) -- (9.79,5.473) -- (9.844,5.58) -- (9.863,5.699) -- (9.844,5.818) -- (9.79,5.925) -- (9.705,6.01) -- (9.598,6.065) -- (9.479,6.083) -- (9.117,6.083) -- (8.998,6.065) -- (8.891,6.01) -- (8.806,5.925) -- (8.751,5.818) -- (8.732,5.699) -- (8.751,5.58) -- (8.806,5.473) -- (8.891,5.388) -- (8.998,5.333) -- (9.117,5.314) -- cycle;
\node at (9.298,5.699) {\scalebox{1.602}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.29,2.93) -- (8.409,2.949) -- (8.516,3.003) -- (8.601,3.089) -- (8.656,3.196) -- (8.674,3.315) -- (8.656,3.433) -- (8.601,3.541) -- (8.516,3.626) -- (8.409,3.68) -- (8.29,3.699) -- (7.928,3.699) -- (7.809,3.68) -- (7.702,3.626) -- (7.617,3.541) -- (7.562,3.433) -- (7.543,3.315) -- (7.562,3.196) -- (7.617,3.089) -- (7.702,3.003) -- (7.809,2.949) -- (7.928,2.93) -- cycle;
\node at (8.109,3.315) {\scalebox{1.602}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.505,1.82) -- (6.624,1.839) -- (6.731,1.893) -- (6.816,1.978) -- (6.871,2.086) -- (6.889,2.205) -- (6.871,2.323) -- (6.816,2.431) -- (6.731,2.516) -- (6.624,2.57) -- (6.505,2.589) -- (6.143,2.589) -- (6.024,2.57) -- (5.917,2.516) -- (5.832,2.431) -- (5.777,2.323) -- (5.758,2.205) -- (5.777,2.086) -- (5.832,1.978) -- (5.917,1.893) -- (6.024,1.839) -- (6.143,1.82) -- cycle;
\node at (6.324,2.205) {\scalebox{1.602}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.272,4.955) -- (1.872,4.955) -- (1.872,5.555) -- (1.272,5.555) -- cycle;
\node at (1.572,5.255) {\scalebox{1.602}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (3.685,2.475) -- (4.285,2.475) -- (4.285,3.075) -- (3.685,3.075) -- cycle;
\node at (3.985,2.775) {\scalebox{1.602}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.306,9.839) -- (9.906,9.839) -- (9.906,10.439) -- (9.306,10.439) -- cycle;
\node at (9.606,10.139) {\scalebox{1.602}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.6,7.603) -- (1.2,7.603) -- (1.2,8.203) -- (0.6,8.203) -- cycle;
\node at (0.9,7.903) {\scalebox{1.602}{\small d}};
\node at (4.157,9.476) {\scalebox{1.602}{\small A}};
\node at (11.785,9.16) {\scalebox{1.602}{\small B}};
\node at (0.909,9.001) {\scalebox{1.602}{\small C}};
\node at (8.086,9.01) {\scalebox{1.602}{\small D}};
\node at (3.734,5.685) {\scalebox{1.602}{\small E}};
\node at (11.732,5.616) {\scalebox{1.602}{\small F}};
\node at (0.954,5.208) {\scalebox{1.602}{\small G}};
\node at (8.314,4.523) {\scalebox{1.602}{\small H}};
\node at (8.262,1.293) {\scalebox{1.602}{\small I}};
\node at (0.776,1.421) {\scalebox{1.602}{\small J}};
\node at (4.639,0.95) {\scalebox{1.602}{\small K}};
\draw[line width=0.5pt] (12.553,7.953) -- (12.553,9.053);
\draw[line width=0.5pt] (12.322,8.668) -- (12.553,9.053) -- (12.784,8.668);
\draw[line width=0.5pt] (12.25,8.503) -- (12.855,8.503);
\node at (12.553,9.438) {\scalebox{1.602}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, B, G, H, K & 5 \\ \hline
clairs & C, D, E, F, I, J & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{6} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 11 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 5 carrefours sombres et 6 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non plus.} Avec un carrefour clair de plus, la tournée doit \textbf{partir et arriver} sur un carrefour clair. Or l'école est en \textbf{G}, qui est sombre.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Une tournée existe pourtant dans ce quartier : il suffirait d'installer l'école sur un carrefour clair. Ce n'est pas le plan qui est en cause, c'est l'endroit d'où l'on part.}\par\addvspace{0.4em}
\end{document}
