\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 137}

\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=137}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0137}
\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=0.6075, line join=round, line cap=round]
\draw[line width=0.5pt] (1.595,11.147) -- (1.84,11.045) -- (2.087,10.948) -- (2.336,10.858) -- (2.587,10.773) -- (2.84,10.694) -- (3.095,10.621) -- (3.352,10.554) -- (3.61,10.493) -- (3.87,10.438) -- (4.131,10.389) -- (4.392,10.346) -- (4.655,10.31) -- (4.918,10.28) -- (5.182,10.256);
\draw[line width=0.5pt] (1.595,11.147) -- (1.61,10.911) -- (1.619,10.673) -- (1.623,10.436) -- (1.621,10.199) -- (1.613,9.961) -- (1.6,9.724) -- (1.581,9.487) -- (1.557,9.251) -- (1.527,9.016) -- (1.492,8.781) -- (1.451,8.547) -- (1.404,8.314) -- (1.353,8.083) -- (1.295,7.852);
\draw[line width=0.5pt] (5.182,10.256) -- (5.094,10.086) -- (5.011,9.913) -- (4.931,9.739) -- (4.856,9.563) -- (4.784,9.385) -- (4.717,9.205) -- (4.654,9.024) -- (4.596,8.841) -- (4.542,8.657) -- (4.492,8.472) -- (4.446,8.286) -- (4.405,8.099) -- (4.368,7.911) -- (4.336,7.722);
\draw[line width=0.5pt] (8.206,8.091) -- (8.247,7.863) -- (8.282,7.634) -- (8.312,7.404) -- (8.337,7.173) -- (8.356,6.942) -- (8.369,6.71) -- (8.378,6.479) -- (8.38,6.247) -- (8.378,6.015) -- (8.37,5.783) -- (8.356,5.551) -- (8.337,5.32) -- (8.312,5.09) -- (8.283,4.86);
\draw[line width=0.5pt] (1.295,7.852) -- (1.509,7.81) -- (1.724,7.772) -- (1.94,7.74) -- (2.157,7.713) -- (2.374,7.691) -- (2.592,7.674) -- (2.81,7.662) -- (3.028,7.655) -- (3.247,7.653) -- (3.465,7.657) -- (3.684,7.665) -- (3.901,7.679) -- (4.119,7.698) -- (4.336,7.722);
\draw[line width=0.5pt] (4.336,7.722) -- (4.606,7.79) -- (4.878,7.853) -- (5.151,7.908) -- (5.426,7.957) -- (5.702,8.0) -- (5.978,8.036) -- (6.256,8.066) -- (6.534,8.089) -- (6.812,8.106) -- (7.091,8.116) -- (7.37,8.12) -- (7.649,8.117) -- (7.928,8.107) -- (8.206,8.091);
\draw[line width=0.5pt] (4.336,7.722) -- (4.311,7.531) -- (4.291,7.339) -- (4.275,7.146) -- (4.264,6.953) -- (4.258,6.76) -- (4.256,6.567) -- (4.258,6.374) -- (4.266,6.181) -- (4.277,5.988) -- (4.293,5.796) -- (4.314,5.604) -- (4.339,5.412) -- (4.369,5.222) -- (4.403,5.032);
\draw[line width=0.5pt] (4.403,5.032) -- (4.68,5.062) -- (4.957,5.086) -- (5.235,5.103) -- (5.514,5.114) -- (5.793,5.118) -- (6.071,5.115) -- (6.35,5.106) -- (6.628,5.09) -- (6.906,5.068) -- (7.183,5.04) -- (7.46,5.005) -- (7.735,4.963) -- (8.01,4.915) -- (8.283,4.86);
\draw[line width=0.5pt] (4.403,5.032) -- (4.379,4.815) -- (4.359,4.598) -- (4.345,4.381) -- (4.336,4.163) -- (4.332,3.945) -- (4.333,3.727) -- (4.339,3.509) -- (4.35,3.291) -- (4.367,3.073) -- (4.388,2.856) -- (4.415,2.64) -- (4.447,2.424) -- (4.483,2.209) -- (4.525,1.995);
\draw[line width=0.5pt] (8.283,4.86) -- (8.311,4.617) -- (8.334,4.373) -- (8.351,4.129) -- (8.362,3.884) -- (8.368,3.639) -- (8.367,3.394) -- (8.361,3.149) -- (8.35,2.904) -- (8.332,2.66) -- (8.309,2.416) -- (8.28,2.172) -- (8.245,1.93) -- (8.205,1.688) -- (8.159,1.448);
\draw[line width=0.5pt] (2.135,4.533) -- (2.302,4.543) -- (2.468,4.558) -- (2.633,4.576) -- (2.798,4.598) -- (2.963,4.625) -- (3.127,4.655) -- (3.29,4.689) -- (3.452,4.726) -- (3.614,4.768) -- (3.774,4.813) -- (3.933,4.862) -- (4.091,4.915) -- (4.248,4.972) -- (4.403,5.032);
\draw[line width=0.5pt] (2.135,4.533) -- (2.163,4.339) -- (2.187,4.144) -- (2.206,3.95) -- (2.22,3.754) -- (2.229,3.558) -- (2.234,3.362) -- (2.234,3.166) -- (2.23,2.971) -- (2.221,2.775) -- (2.208,2.579) -- (2.19,2.384) -- (2.167,2.19) -- (2.14,1.996) -- (2.108,1.803);
\draw[line width=0.5pt] (4.525,1.995) -- (4.777,1.917) -- (5.03,1.844) -- (5.285,1.778) -- (5.542,1.717) -- (5.8,1.662) -- (6.059,1.614) -- (6.319,1.572) -- (6.581,1.535) -- (6.843,1.505) -- (7.105,1.481) -- (7.368,1.464) -- (7.632,1.452) -- (7.895,1.447) -- (8.159,1.448);
\draw[line width=0.5pt] (2.108,1.803) -- (2.277,1.843) -- (2.448,1.879) -- (2.618,1.911) -- (2.79,1.939) -- (2.963,1.963) -- (3.135,1.983) -- (3.309,1.999) -- (3.482,2.01) -- (3.656,2.018) -- (3.83,2.022) -- (4.004,2.021) -- (4.178,2.017) -- (4.352,2.008) -- (4.525,1.995);
\draw[line width=0.5pt] (8.159,1.448) -- (8.383,1.425) -- (8.607,1.408) -- (8.832,1.396) -- (9.058,1.389) -- (9.283,1.388) -- (9.508,1.392) -- (9.734,1.401) -- (9.959,1.416) -- (10.183,1.435) -- (10.407,1.46) -- (10.631,1.491) -- (10.853,1.526) -- (11.075,1.567) -- (11.295,1.613);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.755,11.147) -- (1.734,11.227) -- (1.675,11.286) -- (1.595,11.307) -- (1.515,11.286) -- (1.457,11.227) -- (1.435,11.147) -- (1.457,11.067) -- (1.515,11.009) -- (1.595,10.987) -- (1.675,11.009) -- (1.734,11.067) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.342,10.256) -- (5.321,10.336) -- (5.262,10.394) -- (5.182,10.416) -- (5.102,10.394) -- (5.044,10.336) -- (5.022,10.256) -- (5.044,10.176) -- (5.102,10.117) -- (5.182,10.096) -- (5.262,10.117) -- (5.321,10.176) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.366,8.091) -- (8.345,8.171) -- (8.286,8.229) -- (8.206,8.251) -- (8.126,8.229) -- (8.068,8.171) -- (8.046,8.091) -- (8.068,8.011) -- (8.126,7.952) -- (8.206,7.931) -- (8.286,7.952) -- (8.345,8.011) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.455,7.852) -- (1.434,7.932) -- (1.375,7.991) -- (1.295,8.012) -- (1.215,7.991) -- (1.157,7.932) -- (1.135,7.852) -- (1.157,7.772) -- (1.215,7.714) -- (1.295,7.692) -- (1.375,7.714) -- (1.434,7.772) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.496,7.722) -- (4.474,7.802) -- (4.416,7.861) -- (4.336,7.882) -- (4.256,7.861) -- (4.197,7.802) -- (4.176,7.722) -- (4.197,7.642) -- (4.256,7.583) -- (4.336,7.562) -- (4.416,7.583) -- (4.474,7.642) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.563,5.032) -- (4.542,5.112) -- (4.483,5.17) -- (4.403,5.192) -- (4.323,5.17) -- (4.265,5.112) -- (4.243,5.032) -- (4.265,4.952) -- (4.323,4.893) -- (4.403,4.872) -- (4.483,4.893) -- (4.542,4.952) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.443,4.86) -- (8.421,4.94) -- (8.363,4.999) -- (8.283,5.02) -- (8.203,4.999) -- (8.144,4.94) -- (8.123,4.86) -- (8.144,4.78) -- (8.203,4.722) -- (8.283,4.7) -- (8.363,4.722) -- (8.421,4.78) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.295,4.533) -- (2.274,4.613) -- (2.215,4.671) -- (2.135,4.693) -- (2.055,4.671) -- (1.997,4.613) -- (1.975,4.533) -- (1.997,4.453) -- (2.055,4.394) -- (2.135,4.373) -- (2.215,4.394) -- (2.274,4.453) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.685,1.995) -- (4.664,2.075) -- (4.605,2.134) -- (4.525,2.155) -- (4.445,2.134) -- (4.387,2.075) -- (4.365,1.995) -- (4.387,1.915) -- (4.445,1.857) -- (4.525,1.835) -- (4.605,1.857) -- (4.664,1.915) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.268,1.803) -- (2.247,1.883) -- (2.188,1.941) -- (2.108,1.963) -- (2.028,1.941) -- (1.97,1.883) -- (1.948,1.803) -- (1.97,1.723) -- (2.028,1.664) -- (2.108,1.643) -- (2.188,1.664) -- (2.247,1.723) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.455,1.613) -- (11.434,1.693) -- (11.375,1.752) -- (11.295,1.773) -- (11.215,1.752) -- (11.157,1.693) -- (11.135,1.613) -- (11.157,1.533) -- (11.215,1.475) -- (11.295,1.453) -- (11.375,1.475) -- (11.434,1.533) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.319,1.448) -- (8.297,1.528) -- (8.239,1.586) -- (8.159,1.608) -- (8.079,1.586) -- (8.02,1.528) -- (7.999,1.448) -- (8.02,1.368) -- (8.079,1.309) -- (8.159,1.288) -- (8.239,1.309) -- (8.297,1.368) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.447,10.542) -- (3.569,10.561) -- (3.679,10.617) -- (3.766,10.704) -- (3.822,10.814) -- (3.841,10.936) -- (3.822,11.058) -- (3.766,11.167) -- (3.679,11.255) -- (3.569,11.311) -- (3.447,11.33) -- (3.447,11.33) -- (3.325,11.311) -- (3.215,11.255) -- (3.128,11.167) -- (3.072,11.058) -- (3.053,10.936) -- (3.072,10.814) -- (3.128,10.704) -- (3.215,10.617) -- (3.325,10.561) -- (3.447,10.542) -- cycle;
\node at (3.447,10.936) {\scalebox{1.646}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.973,9.058) -- (2.095,9.077) -- (2.205,9.133) -- (2.292,9.22) -- (2.348,9.33) -- (2.367,9.452) -- (2.348,9.573) -- (2.292,9.683) -- (2.205,9.77) -- (2.095,9.826) -- (1.973,9.846) -- (1.973,9.846) -- (1.852,9.826) -- (1.742,9.77) -- (1.655,9.683) -- (1.599,9.573) -- (1.58,9.452) -- (1.599,9.33) -- (1.655,9.22) -- (1.742,9.133) -- (1.852,9.077) -- (1.973,9.058) -- cycle;
\node at (1.973,9.452) {\scalebox{1.646}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.028,8.505) -- (5.15,8.524) -- (5.26,8.58) -- (5.347,8.668) -- (5.403,8.777) -- (5.422,8.899) -- (5.403,9.021) -- (5.347,9.131) -- (5.26,9.218) -- (5.15,9.274) -- (5.028,9.293) -- (5.028,9.293) -- (4.906,9.274) -- (4.797,9.218) -- (4.709,9.131) -- (4.653,9.021) -- (4.634,8.899) -- (4.653,8.777) -- (4.709,8.668) -- (4.797,8.58) -- (4.906,8.524) -- (5.028,8.505) -- cycle;
\node at (5.028,8.899) {\scalebox{1.646}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.771,6.094) -- (8.893,6.113) -- (9.003,6.169) -- (9.09,6.256) -- (9.146,6.366) -- (9.165,6.488) -- (9.146,6.61) -- (9.09,6.719) -- (9.003,6.807) -- (8.893,6.863) -- (8.771,6.882) -- (8.771,6.882) -- (8.65,6.863) -- (8.54,6.807) -- (8.453,6.719) -- (8.397,6.61) -- (8.377,6.488) -- (8.397,6.366) -- (8.453,6.256) -- (8.54,6.169) -- (8.65,6.113) -- (8.771,6.094) -- cycle;
\node at (8.771,6.488) {\scalebox{1.646}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.827,7.661) -- (2.949,7.681) -- (3.059,7.737) -- (3.146,7.824) -- (3.202,7.934) -- (3.221,8.055) -- (3.202,8.177) -- (3.146,8.287) -- (3.059,8.374) -- (2.949,8.43) -- (2.827,8.449) -- (2.827,8.449) -- (2.705,8.43) -- (2.595,8.374) -- (2.508,8.287) -- (2.452,8.177) -- (2.433,8.055) -- (2.452,7.934) -- (2.508,7.824) -- (2.595,7.737) -- (2.705,7.681) -- (2.827,7.661) -- cycle;
\node at (2.827,8.055) {\scalebox{1.646}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.218,8.064) -- (6.34,8.084) -- (6.45,8.139) -- (6.537,8.227) -- (6.593,8.336) -- (6.612,8.458) -- (6.593,8.58) -- (6.537,8.69) -- (6.45,8.777) -- (6.34,8.833) -- (6.218,8.852) -- (6.218,8.852) -- (6.097,8.833) -- (5.987,8.777) -- (5.9,8.69) -- (5.844,8.58) -- (5.824,8.458) -- (5.844,8.336) -- (5.9,8.227) -- (5.987,8.139) -- (6.097,8.084) -- (6.218,8.064) -- cycle;
\node at (6.218,8.458) {\scalebox{1.646}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.652,5.99) -- (4.774,6.009) -- (4.884,6.065) -- (4.971,6.152) -- (5.027,6.262) -- (5.046,6.384) -- (5.027,6.506) -- (4.971,6.615) -- (4.884,6.703) -- (4.774,6.759) -- (4.652,6.778) -- (4.652,6.778) -- (4.531,6.759) -- (4.421,6.703) -- (4.334,6.615) -- (4.278,6.506) -- (4.258,6.384) -- (4.278,6.262) -- (4.334,6.152) -- (4.421,6.065) -- (4.531,6.009) -- (4.652,5.99) -- cycle;
\node at (4.652,6.384) {\scalebox{1.646}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.367,5.106) -- (6.489,5.125) -- (6.599,5.181) -- (6.686,5.268) -- (6.742,5.378) -- (6.761,5.5) -- (6.742,5.621) -- (6.686,5.731) -- (6.599,5.818) -- (6.489,5.874) -- (6.367,5.893) -- (6.367,5.893) -- (6.246,5.874) -- (6.136,5.818) -- (6.049,5.731) -- (5.993,5.621) -- (5.973,5.5) -- (5.993,5.378) -- (6.049,5.268) -- (6.136,5.181) -- (6.246,5.125) -- (6.367,5.106) -- cycle;
\node at (6.367,5.5) {\scalebox{1.646}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.732,3.13) -- (4.854,3.15) -- (4.964,3.206) -- (5.051,3.293) -- (5.107,3.403) -- (5.126,3.524) -- (5.107,3.646) -- (5.051,3.756) -- (4.964,3.843) -- (4.854,3.899) -- (4.732,3.918) -- (4.732,3.918) -- (4.611,3.899) -- (4.501,3.843) -- (4.414,3.756) -- (4.358,3.646) -- (4.339,3.524) -- (4.358,3.403) -- (4.414,3.293) -- (4.501,3.206) -- (4.611,3.15) -- (4.732,3.13) -- cycle;
\node at (4.732,3.524) {\scalebox{1.646}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.94,2.741) -- (9.062,2.76) -- (9.172,2.816) -- (9.259,2.903) -- (9.315,3.013) -- (9.334,3.135) -- (9.315,3.256) -- (9.259,3.366) -- (9.172,3.453) -- (9.062,3.509) -- (8.94,3.528) -- (8.57,3.528) -- (8.448,3.509) -- (8.338,3.453) -- (8.251,3.366) -- (8.195,3.256) -- (8.176,3.135) -- (8.195,3.013) -- (8.251,2.903) -- (8.338,2.816) -- (8.448,2.76) -- (8.57,2.741) -- cycle;
\node at (8.755,3.135) {\scalebox{1.646}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.17,4.989) -- (3.291,5.008) -- (3.401,5.064) -- (3.488,5.151) -- (3.544,5.261) -- (3.564,5.382) -- (3.544,5.504) -- (3.488,5.614) -- (3.401,5.701) -- (3.291,5.757) -- (3.17,5.776) -- (2.799,5.776) -- (2.677,5.757) -- (2.567,5.701) -- (2.48,5.614) -- (2.424,5.504) -- (2.405,5.382) -- (2.424,5.261) -- (2.48,5.151) -- (2.567,5.064) -- (2.677,5.008) -- (2.799,4.989) -- cycle;
\node at (2.984,5.382) {\scalebox{1.646}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.814,2.769) -- (2.935,2.788) -- (3.045,2.844) -- (3.132,2.931) -- (3.188,3.041) -- (3.208,3.163) -- (3.188,3.284) -- (3.132,3.394) -- (3.045,3.481) -- (2.935,3.537) -- (2.814,3.556) -- (2.443,3.556) -- (2.321,3.537) -- (2.211,3.481) -- (2.124,3.394) -- (2.068,3.284) -- (2.049,3.163) -- (2.068,3.041) -- (2.124,2.931) -- (2.211,2.844) -- (2.321,2.788) -- (2.443,2.769) -- cycle;
\node at (2.628,3.163) {\scalebox{1.646}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.563,1.567) -- (6.685,1.587) -- (6.795,1.642) -- (6.882,1.73) -- (6.938,1.839) -- (6.957,1.961) -- (6.938,2.083) -- (6.882,2.193) -- (6.795,2.28) -- (6.685,2.336) -- (6.563,2.355) -- (6.193,2.355) -- (6.071,2.336) -- (5.961,2.28) -- (5.874,2.193) -- (5.818,2.083) -- (5.799,1.961) -- (5.818,1.839) -- (5.874,1.73) -- (5.961,1.642) -- (6.071,1.587) -- (6.193,1.567) -- cycle;
\node at (6.378,1.961) {\scalebox{1.646}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.525,1.212) -- (3.647,1.231) -- (3.757,1.287) -- (3.844,1.374) -- (3.9,1.484) -- (3.919,1.606) -- (3.9,1.728) -- (3.844,1.838) -- (3.757,1.925) -- (3.647,1.981) -- (3.525,2.0) -- (3.155,2.0) -- (3.033,1.981) -- (2.923,1.925) -- (2.836,1.838) -- (2.78,1.728) -- (2.761,1.606) -- (2.78,1.484) -- (2.836,1.374) -- (2.923,1.287) -- (3.033,1.231) -- (3.155,1.212) -- cycle;
\node at (3.34,1.606) {\scalebox{1.646}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.898,1.401) -- (10.02,1.42) -- (10.13,1.476) -- (10.217,1.563) -- (10.273,1.673) -- (10.292,1.794) -- (10.273,1.916) -- (10.217,2.026) -- (10.13,2.113) -- (10.02,2.169) -- (9.898,2.188) -- (9.528,2.188) -- (9.406,2.169) -- (9.296,2.113) -- (9.209,2.026) -- (9.153,1.916) -- (9.134,1.794) -- (9.153,1.673) -- (9.209,1.563) -- (9.296,1.476) -- (9.406,1.42) -- (9.528,1.401) -- cycle;
\node at (9.713,1.794) {\scalebox{1.646}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (0.995,7.552) -- (1.595,7.552) -- (1.595,8.152) -- (0.995,8.152) -- cycle;
\node at (1.295,7.852) {\scalebox{1.646}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.939,1.324) -- (2.539,1.324) -- (2.539,1.924) -- (1.939,1.924) -- cycle;
\node at (2.239,1.624) {\scalebox{1.646}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (4.322,1.961) -- (4.922,1.961) -- (4.922,2.561) -- (4.322,2.561) -- cycle;
\node at (4.622,2.261) {\scalebox{1.646}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.368,6.98) -- (4.968,6.98) -- (4.968,7.58) -- (4.368,7.58) -- cycle;
\node at (4.668,7.28) {\scalebox{1.646}{\small d}};
\node at (1.265,11.672) {\scalebox{1.646}{\small A}};
\node at (5.189,10.876) {\scalebox{1.646}{\small B}};
\node at (8.683,8.487) {\scalebox{1.646}{\small C}};
\node at (0.765,8.174) {\scalebox{1.646}{\small D}};
\node at (4.125,8.305) {\scalebox{1.646}{\small E}};
\node at (3.892,4.681) {\scalebox{1.646}{\small F}};
\node at (8.889,4.732) {\scalebox{1.646}{\small G}};
\node at (1.547,4.337) {\scalebox{1.646}{\small H}};
\node at (4.421,1.384) {\scalebox{1.646}{\small I}};
\node at (1.718,1.321) {\scalebox{1.646}{\small J}};
\node at (11.819,1.281) {\scalebox{1.646}{\small K}};
\node at (8.528,0.95) {\scalebox{1.646}{\small L}};
\draw[line width=0.5pt] (12.595,10.197) -- (12.595,11.297);
\draw[line width=0.5pt] (12.364,10.912) -- (12.595,11.297) -- (12.826,10.912);
\draw[line width=0.5pt] (12.293,10.747) -- (12.898,10.747);
\node at (12.595,11.682) {\scalebox{1.646}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour D, 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 & ⑨ chemin de Ronde & Ⓑ Placette des Ormes & Ⓚ Place du Vieux-Puits \\ \hline
① rue Haute & ⑩ allée des Peupliers & Ⓒ Fontaine Notre-Dame & Ⓛ Rond-point du Stade \\ \hline
② venelle du Chat & ⑪ rue Neuve & Ⓓ Place des Tanneurs & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ allée des Tilleuls & ⑫ rue Pasteur & Ⓔ Carrefour du Moulin & \textbf{a} l'école \\ \hline
④ sentier des Prés & ⑬ chemin du Lavoir & Ⓕ Carrefour de la Mairie & \textbf{b} la boulangerie \\ \hline
⑤ chemin Creux & ⑭ passage du Marché & Ⓖ Rond-point de la Gare & \textbf{c} l'église \\ \hline
⑥ rue de l'Église & ⑮ impasse Sainte-Anne & Ⓗ Place de l'Église & \textbf{d} la bibliothèque \\ \hline
⑦ rue du Moulin & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour Saint-Roch &  \\ \hline
⑧ rue de la Mairie & Ⓐ Coin du Lavoir & Ⓙ Place de la Fontaine &  \\ \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=0.6223, line join=round, line cap=round]
\draw[line width=0.5pt] (1.595,11.147) -- (1.84,11.045) -- (2.087,10.948) -- (2.336,10.858) -- (2.587,10.773) -- (2.84,10.694) -- (3.095,10.621) -- (3.352,10.554) -- (3.61,10.493) -- (3.87,10.438) -- (4.131,10.389) -- (4.392,10.346) -- (4.655,10.31) -- (4.918,10.28) -- (5.182,10.256);
\draw[line width=0.5pt] (1.595,11.147) -- (1.61,10.911) -- (1.619,10.673) -- (1.623,10.436) -- (1.621,10.199) -- (1.613,9.961) -- (1.6,9.724) -- (1.581,9.487) -- (1.557,9.251) -- (1.527,9.016) -- (1.492,8.781) -- (1.451,8.547) -- (1.404,8.314) -- (1.353,8.083) -- (1.295,7.852);
\draw[line width=0.5pt] (5.182,10.256) -- (5.094,10.086) -- (5.011,9.913) -- (4.931,9.739) -- (4.856,9.563) -- (4.784,9.385) -- (4.717,9.205) -- (4.654,9.024) -- (4.596,8.841) -- (4.542,8.657) -- (4.492,8.472) -- (4.446,8.286) -- (4.405,8.099) -- (4.368,7.911) -- (4.336,7.722);
\draw[line width=0.5pt] (8.206,8.091) -- (8.247,7.863) -- (8.282,7.634) -- (8.312,7.404) -- (8.337,7.173) -- (8.356,6.942) -- (8.369,6.71) -- (8.378,6.479) -- (8.38,6.247) -- (8.378,6.015) -- (8.37,5.783) -- (8.356,5.551) -- (8.337,5.32) -- (8.312,5.09) -- (8.283,4.86);
\draw[line width=0.5pt] (1.295,7.852) -- (1.509,7.81) -- (1.724,7.772) -- (1.94,7.74) -- (2.157,7.713) -- (2.374,7.691) -- (2.592,7.674) -- (2.81,7.662) -- (3.028,7.655) -- (3.247,7.653) -- (3.465,7.657) -- (3.684,7.665) -- (3.901,7.679) -- (4.119,7.698) -- (4.336,7.722);
\draw[line width=0.5pt] (4.336,7.722) -- (4.606,7.79) -- (4.878,7.853) -- (5.151,7.908) -- (5.426,7.957) -- (5.702,8.0) -- (5.978,8.036) -- (6.256,8.066) -- (6.534,8.089) -- (6.812,8.106) -- (7.091,8.116) -- (7.37,8.12) -- (7.649,8.117) -- (7.928,8.107) -- (8.206,8.091);
\draw[line width=0.5pt] (4.336,7.722) -- (4.311,7.531) -- (4.291,7.339) -- (4.275,7.146) -- (4.264,6.953) -- (4.258,6.76) -- (4.256,6.567) -- (4.258,6.374) -- (4.266,6.181) -- (4.277,5.988) -- (4.293,5.796) -- (4.314,5.604) -- (4.339,5.412) -- (4.369,5.222) -- (4.403,5.032);
\draw[line width=0.5pt] (4.403,5.032) -- (4.68,5.062) -- (4.957,5.086) -- (5.235,5.103) -- (5.514,5.114) -- (5.793,5.118) -- (6.071,5.115) -- (6.35,5.106) -- (6.628,5.09) -- (6.906,5.068) -- (7.183,5.04) -- (7.46,5.005) -- (7.735,4.963) -- (8.01,4.915) -- (8.283,4.86);
\draw[line width=0.5pt] (4.403,5.032) -- (4.379,4.815) -- (4.359,4.598) -- (4.345,4.381) -- (4.336,4.163) -- (4.332,3.945) -- (4.333,3.727) -- (4.339,3.509) -- (4.35,3.291) -- (4.367,3.073) -- (4.388,2.856) -- (4.415,2.64) -- (4.447,2.424) -- (4.483,2.209) -- (4.525,1.995);
\draw[line width=0.5pt] (8.283,4.86) -- (8.311,4.617) -- (8.334,4.373) -- (8.351,4.129) -- (8.362,3.884) -- (8.368,3.639) -- (8.367,3.394) -- (8.361,3.149) -- (8.35,2.904) -- (8.332,2.66) -- (8.309,2.416) -- (8.28,2.172) -- (8.245,1.93) -- (8.205,1.688) -- (8.159,1.448);
\draw[line width=0.5pt] (2.135,4.533) -- (2.302,4.543) -- (2.468,4.558) -- (2.633,4.576) -- (2.798,4.598) -- (2.963,4.625) -- (3.127,4.655) -- (3.29,4.689) -- (3.452,4.726) -- (3.614,4.768) -- (3.774,4.813) -- (3.933,4.862) -- (4.091,4.915) -- (4.248,4.972) -- (4.403,5.032);
\draw[line width=0.5pt] (2.135,4.533) -- (2.163,4.339) -- (2.187,4.144) -- (2.206,3.95) -- (2.22,3.754) -- (2.229,3.558) -- (2.234,3.362) -- (2.234,3.166) -- (2.23,2.971) -- (2.221,2.775) -- (2.208,2.579) -- (2.19,2.384) -- (2.167,2.19) -- (2.14,1.996) -- (2.108,1.803);
\draw[line width=0.5pt] (4.525,1.995) -- (4.777,1.917) -- (5.03,1.844) -- (5.285,1.778) -- (5.542,1.717) -- (5.8,1.662) -- (6.059,1.614) -- (6.319,1.572) -- (6.581,1.535) -- (6.843,1.505) -- (7.105,1.481) -- (7.368,1.464) -- (7.632,1.452) -- (7.895,1.447) -- (8.159,1.448);
\draw[line width=0.5pt] (2.108,1.803) -- (2.277,1.843) -- (2.448,1.879) -- (2.618,1.911) -- (2.79,1.939) -- (2.963,1.963) -- (3.135,1.983) -- (3.309,1.999) -- (3.482,2.01) -- (3.656,2.018) -- (3.83,2.022) -- (4.004,2.021) -- (4.178,2.017) -- (4.352,2.008) -- (4.525,1.995);
\draw[line width=0.5pt] (8.159,1.448) -- (8.383,1.425) -- (8.607,1.408) -- (8.832,1.396) -- (9.058,1.389) -- (9.283,1.388) -- (9.508,1.392) -- (9.734,1.401) -- (9.959,1.416) -- (10.183,1.435) -- (10.407,1.46) -- (10.631,1.491) -- (10.853,1.526) -- (11.075,1.567) -- (11.295,1.613);
\filldraw[fill=white, draw=black, line width=0.5pt] (1.969,11.147) -- (1.94,11.29) -- (1.859,11.411) -- (1.738,11.492) -- (1.595,11.521) -- (1.453,11.492) -- (1.331,11.411) -- (1.251,11.29) -- (1.222,11.147) -- (1.251,11.004) -- (1.331,10.883) -- (1.453,10.803) -- (1.595,10.774) -- (1.738,10.803) -- (1.859,10.883) -- (1.94,11.004) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.555,10.256) -- (5.527,10.399) -- (5.446,10.52) -- (5.325,10.601) -- (5.182,10.629) -- (5.039,10.601) -- (4.918,10.52) -- (4.837,10.399) -- (4.809,10.256) -- (4.837,10.113) -- (4.918,9.992) -- (5.039,9.911) -- (5.182,9.883) -- (5.325,9.911) -- (5.446,9.992) -- (5.527,10.113) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.579,8.091) -- (8.551,8.234) -- (8.47,8.355) -- (8.349,8.436) -- (8.206,8.464) -- (8.063,8.436) -- (7.942,8.355) -- (7.861,8.234) -- (7.833,8.091) -- (7.861,7.948) -- (7.942,7.827) -- (8.063,7.746) -- (8.206,7.718) -- (8.349,7.746) -- (8.47,7.827) -- (8.551,7.948) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.668,7.852) -- (1.64,7.995) -- (1.559,8.116) -- (1.438,8.197) -- (1.295,8.226) -- (1.152,8.197) -- (1.031,8.116) -- (0.95,7.995) -- (0.922,7.852) -- (0.95,7.71) -- (1.031,7.588) -- (1.152,7.508) -- (1.295,7.479) -- (1.438,7.508) -- (1.559,7.588) -- (1.64,7.71) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.709,7.722) -- (4.681,7.865) -- (4.6,7.986) -- (4.479,8.067) -- (4.336,8.095) -- (4.193,8.067) -- (4.072,7.986) -- (3.991,7.865) -- (3.963,7.722) -- (3.991,7.579) -- (4.072,7.458) -- (4.193,7.377) -- (4.336,7.349) -- (4.479,7.377) -- (4.6,7.458) -- (4.681,7.579) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.776,5.032) -- (4.748,5.175) -- (4.667,5.296) -- (4.546,5.377) -- (4.403,5.405) -- (4.26,5.377) -- (4.139,5.296) -- (4.058,5.175) -- (4.03,5.032) -- (4.058,4.889) -- (4.139,4.768) -- (4.26,4.687) -- (4.403,4.659) -- (4.546,4.687) -- (4.667,4.768) -- (4.748,4.889) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.656,4.86) -- (8.627,5.003) -- (8.546,5.124) -- (8.425,5.205) -- (8.283,5.233) -- (8.14,5.205) -- (8.019,5.124) -- (7.938,5.003) -- (7.909,4.86) -- (7.938,4.717) -- (8.019,4.596) -- (8.14,4.515) -- (8.283,4.487) -- (8.425,4.515) -- (8.546,4.596) -- (8.627,4.717) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.509,4.533) -- (2.48,4.676) -- (2.399,4.797) -- (2.278,4.877) -- (2.135,4.906) -- (1.993,4.877) -- (1.872,4.797) -- (1.791,4.676) -- (1.762,4.533) -- (1.791,4.39) -- (1.872,4.269) -- (1.993,4.188) -- (2.135,4.159) -- (2.278,4.188) -- (2.399,4.269) -- (2.48,4.39) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.898,1.995) -- (4.87,2.138) -- (4.789,2.259) -- (4.668,2.34) -- (4.525,2.369) -- (4.382,2.34) -- (4.261,2.259) -- (4.18,2.138) -- (4.152,1.995) -- (4.18,1.853) -- (4.261,1.731) -- (4.382,1.651) -- (4.525,1.622) -- (4.668,1.651) -- (4.789,1.731) -- (4.87,1.853) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.481,1.803) -- (2.453,1.945) -- (2.372,2.066) -- (2.251,2.147) -- (2.108,2.176) -- (1.965,2.147) -- (1.844,2.066) -- (1.763,1.945) -- (1.735,1.803) -- (1.763,1.66) -- (1.844,1.539) -- (1.965,1.458) -- (2.108,1.429) -- (2.251,1.458) -- (2.372,1.539) -- (2.453,1.66) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.668,1.613) -- (11.64,1.756) -- (11.559,1.877) -- (11.438,1.958) -- (11.295,1.986) -- (11.152,1.958) -- (11.031,1.877) -- (10.95,1.756) -- (10.922,1.613) -- (10.95,1.47) -- (11.031,1.349) -- (11.152,1.268) -- (11.295,1.24) -- (11.438,1.268) -- (11.559,1.349) -- (11.64,1.47) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.532,1.448) -- (8.503,1.591) -- (8.423,1.712) -- (8.301,1.793) -- (8.159,1.821) -- (8.016,1.793) -- (7.895,1.712) -- (7.814,1.591) -- (7.785,1.448) -- (7.814,1.305) -- (7.895,1.184) -- (8.016,1.103) -- (8.159,1.075) -- (8.301,1.103) -- (8.423,1.184) -- (8.503,1.305) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.445,10.542) -- (3.564,10.561) -- (3.671,10.616) -- (3.756,10.701) -- (3.81,10.808) -- (3.829,10.927) -- (3.81,11.046) -- (3.756,11.153) -- (3.671,11.238) -- (3.564,11.293) -- (3.445,11.311) -- (3.445,11.311) -- (3.326,11.293) -- (3.219,11.238) -- (3.134,11.153) -- (3.079,11.046) -- (3.06,10.927) -- (3.079,10.808) -- (3.134,10.701) -- (3.219,10.616) -- (3.326,10.561) -- (3.445,10.542) -- cycle;
\node at (3.445,10.927) {\scalebox{1.607}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.964,9.068) -- (2.083,9.087) -- (2.19,9.141) -- (2.275,9.227) -- (2.33,9.334) -- (2.349,9.453) -- (2.33,9.571) -- (2.275,9.679) -- (2.19,9.764) -- (2.083,9.818) -- (1.964,9.837) -- (1.964,9.837) -- (1.845,9.818) -- (1.738,9.764) -- (1.653,9.679) -- (1.598,9.571) -- (1.58,9.453) -- (1.598,9.334) -- (1.653,9.227) -- (1.738,9.141) -- (1.845,9.087) -- (1.964,9.068) -- cycle;
\node at (1.964,9.453) {\scalebox{1.607}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.019,8.517) -- (5.138,8.536) -- (5.245,8.591) -- (5.33,8.676) -- (5.385,8.783) -- (5.404,8.902) -- (5.385,9.021) -- (5.33,9.128) -- (5.245,9.213) -- (5.138,9.268) -- (5.019,9.287) -- (5.019,9.287) -- (4.9,9.268) -- (4.793,9.213) -- (4.708,9.128) -- (4.653,9.021) -- (4.635,8.902) -- (4.653,8.783) -- (4.708,8.676) -- (4.793,8.591) -- (4.9,8.536) -- (5.019,8.517) -- cycle;
\node at (5.019,8.902) {\scalebox{1.607}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.762,6.103) -- (8.881,6.122) -- (8.988,6.177) -- (9.073,6.262) -- (9.128,6.369) -- (9.147,6.488) -- (9.128,6.607) -- (9.073,6.714) -- (8.988,6.799) -- (8.881,6.853) -- (8.762,6.872) -- (8.762,6.872) -- (8.643,6.853) -- (8.536,6.799) -- (8.451,6.714) -- (8.396,6.607) -- (8.377,6.488) -- (8.396,6.369) -- (8.451,6.262) -- (8.536,6.177) -- (8.643,6.122) -- (8.762,6.103) -- cycle;
\node at (8.762,6.488) {\scalebox{1.607}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.827,7.661) -- (2.945,7.68) -- (3.053,7.735) -- (3.138,7.82) -- (3.192,7.927) -- (3.211,8.046) -- (3.192,8.165) -- (3.138,8.272) -- (3.053,8.357) -- (2.945,8.412) -- (2.827,8.43) -- (2.827,8.43) -- (2.708,8.412) -- (2.601,8.357) -- (2.515,8.272) -- (2.461,8.165) -- (2.442,8.046) -- (2.461,7.927) -- (2.515,7.82) -- (2.601,7.735) -- (2.708,7.68) -- (2.827,7.661) -- cycle;
\node at (2.827,8.046) {\scalebox{1.607}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.219,8.064) -- (6.338,8.083) -- (6.445,8.138) -- (6.53,8.223) -- (6.585,8.33) -- (6.604,8.449) -- (6.585,8.568) -- (6.53,8.675) -- (6.445,8.76) -- (6.338,8.815) -- (6.219,8.833) -- (6.219,8.833) -- (6.1,8.815) -- (5.993,8.76) -- (5.908,8.675) -- (5.854,8.568) -- (5.835,8.449) -- (5.854,8.33) -- (5.908,8.223) -- (5.993,8.138) -- (6.1,8.083) -- (6.219,8.064) -- cycle;
\node at (6.219,8.449) {\scalebox{1.607}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.643,5.999) -- (4.762,6.018) -- (4.869,6.073) -- (4.954,6.158) -- (5.009,6.265) -- (5.027,6.384) -- (5.009,6.503) -- (4.954,6.61) -- (4.869,6.695) -- (4.762,6.749) -- (4.643,6.768) -- (4.643,6.768) -- (4.524,6.749) -- (4.417,6.695) -- (4.332,6.61) -- (4.277,6.503) -- (4.258,6.384) -- (4.277,6.265) -- (4.332,6.158) -- (4.417,6.073) -- (4.524,6.018) -- (4.643,5.999) -- cycle;
\node at (4.643,6.384) {\scalebox{1.607}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.367,5.106) -- (6.486,5.124) -- (6.593,5.179) -- (6.678,5.264) -- (6.733,5.371) -- (6.751,5.49) -- (6.733,5.609) -- (6.678,5.716) -- (6.593,5.801) -- (6.486,5.856) -- (6.367,5.875) -- (6.367,5.875) -- (6.248,5.856) -- (6.141,5.801) -- (6.056,5.716) -- (6.001,5.609) -- (5.982,5.49) -- (6.001,5.371) -- (6.056,5.264) -- (6.141,5.179) -- (6.248,5.124) -- (6.367,5.106) -- cycle;
\node at (6.367,5.49) {\scalebox{1.607}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.723,3.139) -- (4.842,3.158) -- (4.949,3.213) -- (5.034,3.298) -- (5.089,3.405) -- (5.108,3.524) -- (5.089,3.643) -- (5.034,3.75) -- (4.949,3.835) -- (4.842,3.89) -- (4.723,3.909) -- (4.723,3.909) -- (4.604,3.89) -- (4.497,3.835) -- (4.412,3.75) -- (4.357,3.643) -- (4.339,3.524) -- (4.357,3.405) -- (4.412,3.298) -- (4.497,3.213) -- (4.604,3.158) -- (4.723,3.139) -- cycle;
\node at (4.723,3.524) {\scalebox{1.607}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.927,2.75) -- (9.045,2.769) -- (9.153,2.824) -- (9.238,2.909) -- (9.292,3.016) -- (9.311,3.135) -- (9.292,3.254) -- (9.238,3.361) -- (9.153,3.446) -- (9.045,3.501) -- (8.927,3.519) -- (8.565,3.519) -- (8.446,3.501) -- (8.339,3.446) -- (8.254,3.361) -- (8.199,3.254) -- (8.18,3.135) -- (8.199,3.016) -- (8.254,2.909) -- (8.339,2.824) -- (8.446,2.769) -- (8.565,2.75) -- cycle;
\node at (8.746,3.135) {\scalebox{1.607}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.315,5.011) -- (3.434,5.03) -- (3.541,5.084) -- (3.626,5.17) -- (3.681,5.277) -- (3.7,5.396) -- (3.681,5.514) -- (3.626,5.622) -- (3.541,5.707) -- (3.434,5.761) -- (3.315,5.78) -- (2.953,5.78) -- (2.835,5.761) -- (2.727,5.707) -- (2.642,5.622) -- (2.588,5.514) -- (2.569,5.396) -- (2.588,5.277) -- (2.642,5.17) -- (2.727,5.084) -- (2.835,5.03) -- (2.953,5.011) -- cycle;
\node at (3.134,5.396) {\scalebox{1.607}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.8,2.778) -- (2.919,2.797) -- (3.026,2.852) -- (3.111,2.937) -- (3.166,3.044) -- (3.184,3.163) -- (3.166,3.281) -- (3.111,3.389) -- (3.026,3.474) -- (2.919,3.528) -- (2.8,3.547) -- (2.438,3.547) -- (2.319,3.528) -- (2.212,3.474) -- (2.127,3.389) -- (2.072,3.281) -- (2.054,3.163) -- (2.072,3.044) -- (2.127,2.937) -- (2.212,2.852) -- (2.319,2.797) -- (2.438,2.778) -- cycle;
\node at (2.619,3.163) {\scalebox{1.607}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.558,1.567) -- (6.676,1.586) -- (6.784,1.641) -- (6.869,1.726) -- (6.923,1.833) -- (6.942,1.952) -- (6.923,2.071) -- (6.869,2.178) -- (6.784,2.263) -- (6.676,2.318) -- (6.558,2.336) -- (6.196,2.336) -- (6.077,2.318) -- (5.97,2.263) -- (5.885,2.178) -- (5.83,2.071) -- (5.811,1.952) -- (5.83,1.833) -- (5.885,1.726) -- (5.97,1.641) -- (6.077,1.586) -- (6.196,1.567) -- cycle;
\node at (6.377,1.952) {\scalebox{1.607}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.52,1.231) -- (3.639,1.25) -- (3.746,1.304) -- (3.831,1.389) -- (3.886,1.497) -- (3.905,1.615) -- (3.886,1.734) -- (3.831,1.841) -- (3.746,1.926) -- (3.639,1.981) -- (3.52,2.0) -- (3.158,2.0) -- (3.04,1.981) -- (2.932,1.926) -- (2.847,1.841) -- (2.793,1.734) -- (2.774,1.615) -- (2.793,1.497) -- (2.847,1.389) -- (2.932,1.304) -- (3.04,1.25) -- (3.158,1.231) -- cycle;
\node at (3.339,1.615) {\scalebox{1.607}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.894,1.401) -- (10.013,1.419) -- (10.12,1.474) -- (10.206,1.559) -- (10.26,1.666) -- (10.279,1.785) -- (10.26,1.904) -- (10.206,2.011) -- (10.12,2.096) -- (10.013,2.151) -- (9.894,2.17) -- (9.533,2.17) -- (9.414,2.151) -- (9.307,2.096) -- (9.221,2.011) -- (9.167,1.904) -- (9.148,1.785) -- (9.167,1.666) -- (9.221,1.559) -- (9.307,1.474) -- (9.414,1.419) -- (9.533,1.401) -- cycle;
\node at (9.713,1.785) {\scalebox{1.607}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (0.995,7.552) -- (1.595,7.552) -- (1.595,8.152) -- (0.995,8.152) -- cycle;
\node at (1.295,7.852) {\scalebox{1.607}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.939,1.324) -- (2.539,1.324) -- (2.539,1.924) -- (1.939,1.924) -- cycle;
\node at (2.239,1.624) {\scalebox{1.607}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (4.322,1.961) -- (4.922,1.961) -- (4.922,2.561) -- (4.322,2.561) -- cycle;
\node at (4.622,2.261) {\scalebox{1.607}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.368,6.98) -- (4.968,6.98) -- (4.968,7.58) -- (4.368,7.58) -- cycle;
\node at (4.668,7.28) {\scalebox{1.607}{\small d}};
\node at (1.265,11.672) {\scalebox{1.607}{\small A}};
\node at (5.189,10.876) {\scalebox{1.607}{\small B}};
\node at (8.683,8.487) {\scalebox{1.607}{\small C}};
\node at (0.765,8.174) {\scalebox{1.607}{\small D}};
\node at (4.125,8.305) {\scalebox{1.607}{\small E}};
\node at (3.892,4.681) {\scalebox{1.607}{\small F}};
\node at (8.889,4.732) {\scalebox{1.607}{\small G}};
\node at (1.547,4.337) {\scalebox{1.607}{\small H}};
\node at (4.421,1.384) {\scalebox{1.607}{\small I}};
\node at (1.718,1.321) {\scalebox{1.607}{\small J}};
\node at (11.819,1.281) {\scalebox{1.607}{\small K}};
\node at (8.528,0.95) {\scalebox{1.607}{\small L}};
\draw[line width=0.5pt] (12.595,10.197) -- (12.595,11.297);
\draw[line width=0.5pt] (12.364,10.912) -- (12.595,11.297) -- (12.826,10.912);
\draw[line width=0.5pt] (12.293,10.747) -- (12.898,10.747);
\node at (12.595,11.682) {\scalebox{1.607}{\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 & B, C, D, F, J, L & 6 \\ \hline
clairs & A, E, G, H, I, K & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{6} clairs. Il y en a \textbf{autant} de chaque côté.}\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 12 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.} Si l'on retire le carrefour \textbf{E}, le quartier tombe en 2 morceaux : toute route de l'un à l'autre passe par lui. Un circuit qui reviendrait à l'école devrait donc s'y arrêter au moins \textbf{deux fois} — c'est interdit.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{D} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{venelle du Chat ②} vers le nord jusqu'à Coin du Lavoir \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Haute ①}, vers l'est, jusqu'à Placette des Ormes \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Tilleuls ③}, vers le sud, jusqu'à Carrefour du Moulin \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de l'Église ⑥}, vers l'est, jusqu'à Fontaine Notre-Dame \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ④}, vers le sud, jusqu'à Rond-point de la Gare \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Mairie ⑧}, vers l'ouest, jusqu'à Carrefour de la Mairie \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Neuve ⑪}, vers l'ouest, jusqu'à Place de l'Église \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Pasteur ⑫}, vers le sud, jusqu'à Place de la Fontaine \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{passage du Marché ⑭}, vers l'est, jusqu'à Carrefour Saint-Roch \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin du Lavoir ⑬}, vers l'est, jusqu'à Rond-point du Stade \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{impasse Sainte-Anne ⑮}, vers l'est, jusqu'à Place du Vieux-Puits \textbf{(K)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 12 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
