\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 173}

\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=173}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0173}
\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=0.6058, line join=round, line cap=round]
\draw[line width=0.5pt] (4.737,11.543) -- (4.968,11.476) -- (5.202,11.414) -- (5.436,11.358) -- (5.672,11.308) -- (5.91,11.263) -- (6.148,11.223) -- (6.387,11.189) -- (6.627,11.161) -- (6.867,11.138) -- (7.108,11.121) -- (7.35,11.11) -- (7.591,11.105) -- (7.832,11.105) -- (8.073,11.111);
\draw[line width=0.5pt] (4.737,11.543) -- (4.792,11.307) -- (4.842,11.068) -- (4.887,10.829) -- (4.925,10.589) -- (4.958,10.348) -- (4.986,10.106) -- (5.008,9.863) -- (5.024,9.62) -- (5.034,9.377) -- (5.038,9.134) -- (5.037,8.89) -- (5.03,8.647) -- (5.018,8.404) -- (5.0,8.162);
\draw[line width=0.5pt] (11.334,11.249) -- (11.287,11.038) -- (11.244,10.825) -- (11.207,10.612) -- (11.175,10.397) -- (11.147,10.182) -- (11.125,9.966) -- (11.108,9.75) -- (11.096,9.534) -- (11.089,9.317) -- (11.087,9.1) -- (11.09,8.883) -- (11.098,8.667) -- (11.112,8.451) -- (11.13,8.235);
\draw[line width=0.5pt] (8.073,11.111) -- (8.303,11.156) -- (8.534,11.196) -- (8.765,11.231) -- (8.998,11.26) -- (9.231,11.283) -- (9.464,11.302) -- (9.698,11.314) -- (9.932,11.321) -- (10.167,11.323) -- (10.401,11.319) -- (10.635,11.31) -- (10.869,11.295) -- (11.102,11.275) -- (11.334,11.249);
\draw[line width=0.5pt] (8.073,11.111) -- (8.032,10.889) -- (7.997,10.666) -- (7.966,10.443) -- (7.941,10.218) -- (7.921,9.994) -- (7.907,9.768) -- (7.897,9.543) -- (7.893,9.317) -- (7.895,9.091) -- (7.901,8.866) -- (7.913,8.641) -- (7.93,8.416) -- (7.953,8.191) -- (7.981,7.967);
\draw[line width=0.5pt] (5.0,8.162) -- (5.213,8.181) -- (5.427,8.194) -- (5.641,8.203) -- (5.855,8.207) -- (6.07,8.205) -- (6.284,8.199) -- (6.498,8.188) -- (6.712,8.171) -- (6.925,8.15) -- (7.138,8.123) -- (7.35,8.092) -- (7.561,8.055) -- (7.772,8.014) -- (7.981,7.967);
\draw[line width=0.5pt] (5.0,8.162) -- (4.993,7.895) -- (4.994,7.627) -- (5.0,7.36) -- (5.013,7.093) -- (5.032,6.826) -- (5.057,6.559) -- (5.089,6.294) -- (5.127,6.029) -- (5.171,5.765) -- (5.221,5.502) -- (5.277,5.24) -- (5.34,4.98) -- (5.409,4.722) -- (5.483,4.465);
\draw[line width=0.5pt] (1.374,8.031) -- (1.63,8.08) -- (1.886,8.123) -- (2.144,8.16) -- (2.402,8.19) -- (2.662,8.215) -- (2.921,8.233) -- (3.181,8.246) -- (3.442,8.252) -- (3.702,8.252) -- (3.962,8.247) -- (4.222,8.235) -- (4.482,8.216) -- (4.741,8.192) -- (5.0,8.162);
\draw[line width=0.5pt] (1.374,8.031) -- (1.377,7.777) -- (1.385,7.524) -- (1.399,7.27) -- (1.42,7.017) -- (1.446,6.764) -- (1.479,6.512) -- (1.517,6.261) -- (1.561,6.011) -- (1.611,5.762) -- (1.667,5.514) -- (1.729,5.268) -- (1.796,5.023) -- (1.869,4.78) -- (1.948,4.539);
\draw[line width=0.5pt] (7.981,7.967) -- (8.201,8.021) -- (8.423,8.069) -- (8.645,8.112) -- (8.869,8.15) -- (9.093,8.182) -- (9.319,8.209) -- (9.544,8.231) -- (9.771,8.248) -- (9.997,8.259) -- (10.224,8.265) -- (10.451,8.265) -- (10.678,8.26) -- (10.904,8.25) -- (11.13,8.235);
\draw[line width=0.5pt] (7.981,7.967) -- (7.994,7.723) -- (8.013,7.479) -- (8.038,7.235) -- (8.069,6.991) -- (8.105,6.749) -- (8.147,6.507) -- (8.195,6.267) -- (8.248,6.027) -- (8.307,5.789) -- (8.371,5.553) -- (8.441,5.318) -- (8.517,5.085) -- (8.598,4.853) -- (8.684,4.624);
\draw[line width=0.5pt] (8.684,4.624) -- (8.723,4.407) -- (8.757,4.188) -- (8.786,3.969) -- (8.81,3.749) -- (8.828,3.529) -- (8.842,3.308) -- (8.85,3.087) -- (8.853,2.866) -- (8.851,2.644) -- (8.843,2.423) -- (8.83,2.203) -- (8.813,1.982) -- (8.79,1.762) -- (8.761,1.543);
\draw[line width=0.5pt] (1.948,4.539) -- (2.198,4.495) -- (2.448,4.457) -- (2.7,4.425) -- (2.953,4.399) -- (3.206,4.379) -- (3.459,4.365) -- (3.713,4.356) -- (3.966,4.354) -- (4.22,4.358) -- (4.474,4.367) -- (4.727,4.383) -- (4.98,4.404) -- (5.232,4.432) -- (5.483,4.465);
\draw[line width=0.5pt] (5.483,4.465) -- (5.708,4.512) -- (5.935,4.552) -- (6.162,4.588) -- (6.39,4.618) -- (6.618,4.643) -- (6.848,4.663) -- (7.077,4.677) -- (7.307,4.685) -- (7.537,4.689) -- (7.767,4.687) -- (7.997,4.679) -- (8.227,4.666) -- (8.456,4.648) -- (8.684,4.624);
\draw[line width=0.5pt] (8.761,1.543) -- (8.922,1.534) -- (9.084,1.528) -- (9.245,1.527) -- (9.406,1.529) -- (9.568,1.535) -- (9.729,1.544) -- (9.89,1.558) -- (10.051,1.575) -- (10.211,1.596) -- (10.37,1.621) -- (10.529,1.649) -- (10.687,1.681) -- (10.845,1.717) -- (11.001,1.757);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.897,11.543) -- (4.875,11.623) -- (4.817,11.682) -- (4.737,11.703) -- (4.657,11.682) -- (4.598,11.623) -- (4.577,11.543) -- (4.598,11.463) -- (4.657,11.405) -- (4.737,11.383) -- (4.817,11.405) -- (4.875,11.463) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.494,11.249) -- (11.473,11.329) -- (11.414,11.388) -- (11.334,11.409) -- (11.254,11.388) -- (11.196,11.329) -- (11.174,11.249) -- (11.196,11.169) -- (11.254,11.11) -- (11.334,11.089) -- (11.414,11.11) -- (11.473,11.169) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.233,11.111) -- (8.212,11.191) -- (8.153,11.249) -- (8.073,11.271) -- (7.993,11.249) -- (7.935,11.191) -- (7.913,11.111) -- (7.935,11.031) -- (7.993,10.972) -- (8.073,10.951) -- (8.153,10.972) -- (8.212,11.031) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.29,8.235) -- (11.269,8.315) -- (11.21,8.373) -- (11.13,8.395) -- (11.05,8.373) -- (10.991,8.315) -- (10.97,8.235) -- (10.991,8.155) -- (11.05,8.096) -- (11.13,8.075) -- (11.21,8.096) -- (11.269,8.155) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.16,8.162) -- (5.138,8.242) -- (5.08,8.3) -- (5.0,8.322) -- (4.92,8.3) -- (4.861,8.242) -- (4.84,8.162) -- (4.861,8.082) -- (4.92,8.023) -- (5.0,8.002) -- (5.08,8.023) -- (5.138,8.082) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.534,8.031) -- (1.513,8.111) -- (1.454,8.17) -- (1.374,8.191) -- (1.294,8.17) -- (1.235,8.111) -- (1.214,8.031) -- (1.235,7.951) -- (1.294,7.892) -- (1.374,7.871) -- (1.454,7.892) -- (1.513,7.951) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.141,7.967) -- (8.119,8.047) -- (8.061,8.106) -- (7.981,8.127) -- (7.901,8.106) -- (7.842,8.047) -- (7.821,7.967) -- (7.842,7.887) -- (7.901,7.829) -- (7.981,7.807) -- (8.061,7.829) -- (8.119,7.887) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.844,4.624) -- (8.823,4.704) -- (8.764,4.763) -- (8.684,4.784) -- (8.604,4.763) -- (8.546,4.704) -- (8.524,4.624) -- (8.546,4.544) -- (8.604,4.486) -- (8.684,4.464) -- (8.764,4.486) -- (8.823,4.544) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.108,4.539) -- (2.087,4.619) -- (2.028,4.678) -- (1.948,4.699) -- (1.868,4.678) -- (1.809,4.619) -- (1.788,4.539) -- (1.809,4.459) -- (1.868,4.401) -- (1.948,4.379) -- (2.028,4.401) -- (2.087,4.459) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.643,4.465) -- (5.622,4.545) -- (5.563,4.604) -- (5.483,4.625) -- (5.403,4.604) -- (5.345,4.545) -- (5.323,4.465) -- (5.345,4.385) -- (5.403,4.327) -- (5.483,4.305) -- (5.563,4.327) -- (5.622,4.385) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.161,1.757) -- (11.14,1.837) -- (11.081,1.895) -- (11.001,1.917) -- (10.921,1.895) -- (10.862,1.837) -- (10.841,1.757) -- (10.862,1.677) -- (10.921,1.618) -- (11.001,1.597) -- (11.081,1.618) -- (11.14,1.677) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.921,1.543) -- (8.9,1.623) -- (8.841,1.682) -- (8.761,1.703) -- (8.681,1.682) -- (8.623,1.623) -- (8.601,1.543) -- (8.623,1.463) -- (8.681,1.405) -- (8.761,1.383) -- (8.841,1.405) -- (8.9,1.463) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.438,11.186) -- (6.559,11.205) -- (6.669,11.261) -- (6.756,11.348) -- (6.812,11.458) -- (6.832,11.58) -- (6.812,11.702) -- (6.756,11.811) -- (6.669,11.899) -- (6.559,11.955) -- (6.438,11.974) -- (6.438,11.974) -- (6.316,11.955) -- (6.206,11.899) -- (6.119,11.811) -- (6.063,11.702) -- (6.044,11.58) -- (6.063,11.458) -- (6.119,11.348) -- (6.206,11.261) -- (6.316,11.205) -- (6.438,11.186) -- cycle;
\node at (6.438,11.58) {\scalebox{1.651}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.4,9.5) -- (5.522,9.519) -- (5.632,9.575) -- (5.719,9.662) -- (5.775,9.772) -- (5.794,9.894) -- (5.775,10.016) -- (5.719,10.125) -- (5.632,10.213) -- (5.522,10.269) -- (5.4,10.288) -- (5.4,10.288) -- (5.279,10.269) -- (5.169,10.213) -- (5.082,10.125) -- (5.026,10.016) -- (5.006,9.894) -- (5.026,9.772) -- (5.082,9.662) -- (5.169,9.575) -- (5.279,9.519) -- (5.4,9.5) -- cycle;
\node at (5.4,9.894) {\scalebox{1.651}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.501,9.33) -- (11.623,9.349) -- (11.732,9.405) -- (11.82,9.492) -- (11.875,9.602) -- (11.895,9.724) -- (11.875,9.845) -- (11.82,9.955) -- (11.732,10.042) -- (11.623,10.098) -- (11.501,10.118) -- (11.501,10.118) -- (11.379,10.098) -- (11.269,10.042) -- (11.182,9.955) -- (11.126,9.845) -- (11.107,9.724) -- (11.126,9.602) -- (11.182,9.492) -- (11.269,9.405) -- (11.379,9.349) -- (11.501,9.33) -- cycle;
\node at (11.501,9.724) {\scalebox{1.651}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.681,11.314) -- (9.803,11.333) -- (9.913,11.389) -- (10.0,11.476) -- (10.056,11.586) -- (10.075,11.708) -- (10.056,11.83) -- (10.0,11.939) -- (9.913,12.026) -- (9.803,12.082) -- (9.681,12.102) -- (9.681,12.102) -- (9.56,12.082) -- (9.45,12.026) -- (9.363,11.939) -- (9.307,11.83) -- (9.288,11.708) -- (9.307,11.586) -- (9.363,11.476) -- (9.45,11.389) -- (9.56,11.333) -- (9.681,11.314) -- cycle;
\node at (9.681,11.708) {\scalebox{1.651}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.291,9.137) -- (8.413,9.157) -- (8.523,9.213) -- (8.61,9.3) -- (8.666,9.409) -- (8.685,9.531) -- (8.666,9.653) -- (8.61,9.763) -- (8.523,9.85) -- (8.413,9.906) -- (8.291,9.925) -- (8.291,9.925) -- (8.169,9.906) -- (8.06,9.85) -- (7.973,9.763) -- (7.917,9.653) -- (7.897,9.531) -- (7.917,9.409) -- (7.973,9.3) -- (8.06,9.213) -- (8.169,9.157) -- (8.291,9.137) -- cycle;
\node at (8.291,9.531) {\scalebox{1.651}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.524,8.187) -- (6.646,8.206) -- (6.755,8.262) -- (6.842,8.349) -- (6.898,8.459) -- (6.918,8.581) -- (6.898,8.702) -- (6.842,8.812) -- (6.755,8.899) -- (6.646,8.955) -- (6.524,8.975) -- (6.524,8.975) -- (6.402,8.955) -- (6.292,8.899) -- (6.205,8.812) -- (6.149,8.702) -- (6.13,8.581) -- (6.149,8.459) -- (6.205,8.349) -- (6.292,8.262) -- (6.402,8.206) -- (6.524,8.187) -- cycle;
\node at (6.524,8.581) {\scalebox{1.651}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.48,5.951) -- (5.601,5.97) -- (5.711,6.026) -- (5.798,6.113) -- (5.854,6.223) -- (5.873,6.345) -- (5.854,6.466) -- (5.798,6.576) -- (5.711,6.663) -- (5.601,6.719) -- (5.48,6.739) -- (5.48,6.739) -- (5.358,6.719) -- (5.248,6.663) -- (5.161,6.576) -- (5.105,6.466) -- (5.086,6.345) -- (5.105,6.223) -- (5.161,6.113) -- (5.248,6.026) -- (5.358,5.97) -- (5.48,5.951) -- cycle;
\node at (5.48,6.345) {\scalebox{1.651}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.167,8.246) -- (3.289,8.265) -- (3.399,8.321) -- (3.486,8.408) -- (3.542,8.518) -- (3.561,8.64) -- (3.542,8.761) -- (3.486,8.871) -- (3.399,8.958) -- (3.289,9.014) -- (3.167,9.033) -- (3.167,9.033) -- (3.045,9.014) -- (2.936,8.958) -- (2.849,8.871) -- (2.793,8.761) -- (2.773,8.64) -- (2.793,8.518) -- (2.849,8.408) -- (2.936,8.321) -- (3.045,8.265) -- (3.167,8.246) -- cycle;
\node at (3.167,8.64) {\scalebox{1.651}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.906,5.931) -- (2.027,5.951) -- (2.137,6.007) -- (2.224,6.094) -- (2.28,6.204) -- (2.3,6.325) -- (2.28,6.447) -- (2.224,6.557) -- (2.137,6.644) -- (2.027,6.7) -- (1.906,6.719) -- (1.906,6.719) -- (1.784,6.7) -- (1.674,6.644) -- (1.587,6.557) -- (1.531,6.447) -- (1.512,6.325) -- (1.531,6.204) -- (1.587,6.094) -- (1.674,6.007) -- (1.784,5.951) -- (1.906,5.931) -- cycle;
\node at (1.906,6.325) {\scalebox{1.651}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.696,8.23) -- (9.818,8.249) -- (9.928,8.305) -- (10.015,8.392) -- (10.071,8.502) -- (10.09,8.623) -- (10.071,8.745) -- (10.015,8.855) -- (9.928,8.942) -- (9.818,8.998) -- (9.696,9.017) -- (9.326,9.017) -- (9.204,8.998) -- (9.094,8.942) -- (9.007,8.855) -- (8.951,8.745) -- (8.932,8.623) -- (8.951,8.502) -- (9.007,8.392) -- (9.094,8.305) -- (9.204,8.249) -- (9.326,8.23) -- cycle;
\node at (9.511,8.623) {\scalebox{1.651}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.765,5.954) -- (8.887,5.973) -- (8.997,6.029) -- (9.084,6.116) -- (9.14,6.226) -- (9.159,6.348) -- (9.14,6.47) -- (9.084,6.579) -- (8.997,6.667) -- (8.887,6.722) -- (8.765,6.742) -- (8.395,6.742) -- (8.273,6.722) -- (8.163,6.667) -- (8.076,6.579) -- (8.02,6.47) -- (8.001,6.348) -- (8.02,6.226) -- (8.076,6.116) -- (8.163,6.029) -- (8.273,5.973) -- (8.395,5.954) -- cycle;
\node at (8.58,6.348) {\scalebox{1.651}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.641,2.683) -- (8.763,2.702) -- (8.873,2.758) -- (8.96,2.846) -- (9.016,2.955) -- (9.035,3.077) -- (9.016,3.199) -- (8.96,3.309) -- (8.873,3.396) -- (8.763,3.452) -- (8.641,3.471) -- (8.271,3.471) -- (8.149,3.452) -- (8.039,3.396) -- (7.952,3.309) -- (7.896,3.199) -- (7.877,3.077) -- (7.896,2.955) -- (7.952,2.846) -- (8.039,2.758) -- (8.149,2.702) -- (8.271,2.683) -- cycle;
\node at (8.456,3.077) {\scalebox{1.651}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.906,4.356) -- (4.028,4.376) -- (4.138,4.431) -- (4.225,4.519) -- (4.281,4.628) -- (4.3,4.75) -- (4.281,4.872) -- (4.225,4.982) -- (4.138,5.069) -- (4.028,5.125) -- (3.906,5.144) -- (3.535,5.144) -- (3.414,5.125) -- (3.304,5.069) -- (3.217,4.982) -- (3.161,4.872) -- (3.142,4.75) -- (3.161,4.628) -- (3.217,4.519) -- (3.304,4.431) -- (3.414,4.376) -- (3.535,4.356) -- cycle;
\node at (3.721,4.75) {\scalebox{1.651}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.243,4.676) -- (7.365,4.696) -- (7.475,4.751) -- (7.562,4.839) -- (7.618,4.948) -- (7.637,5.07) -- (7.618,5.192) -- (7.562,5.302) -- (7.475,5.389) -- (7.365,5.445) -- (7.243,5.464) -- (6.872,5.464) -- (6.751,5.445) -- (6.641,5.389) -- (6.554,5.302) -- (6.498,5.192) -- (6.478,5.07) -- (6.498,4.948) -- (6.554,4.839) -- (6.641,4.751) -- (6.751,4.696) -- (6.872,4.676) -- cycle;
\node at (7.058,5.07) {\scalebox{1.651}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.038,1.556) -- (10.16,1.575) -- (10.269,1.631) -- (10.357,1.718) -- (10.413,1.828) -- (10.432,1.95) -- (10.413,2.072) -- (10.357,2.181) -- (10.269,2.269) -- (10.16,2.324) -- (10.038,2.344) -- (9.667,2.344) -- (9.545,2.324) -- (9.436,2.269) -- (9.349,2.181) -- (9.293,2.072) -- (9.273,1.95) -- (9.293,1.828) -- (9.349,1.718) -- (9.436,1.631) -- (9.545,1.575) -- (9.667,1.556) -- cycle;
\node at (9.853,1.95) {\scalebox{1.651}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (5.183,4.165) -- (5.783,4.165) -- (5.783,4.765) -- (5.183,4.765) -- cycle;
\node at (5.483,4.465) {\scalebox{1.651}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.181,6.526) -- (4.781,6.526) -- (4.781,7.126) -- (4.181,7.126) -- cycle;
\node at (4.481,6.826) {\scalebox{1.651}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.972,10.615) -- (5.572,10.615) -- (5.572,11.215) -- (4.972,11.215) -- cycle;
\node at (5.272,10.915) {\scalebox{1.651}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.583,12.04) -- (6.183,12.04) -- (6.183,12.64) -- (5.583,12.64) -- cycle;
\node at (5.883,12.34) {\scalebox{1.651}{\small d}};
\node at (4.451,12.094) {\scalebox{1.651}{\small A}};
\node at (11.767,11.693) {\scalebox{1.651}{\small B}};
\node at (8.211,11.715) {\scalebox{1.651}{\small C}};
\node at (11.72,8.426) {\scalebox{1.651}{\small D}};
\node at (4.462,8.471) {\scalebox{1.651}{\small E}};
\node at (0.765,8.147) {\scalebox{1.651}{\small F}};
\node at (8.376,8.445) {\scalebox{1.651}{\small G}};
\node at (9.031,4.11) {\scalebox{1.651}{\small H}};
\node at (1.385,4.279) {\scalebox{1.651}{\small I}};
\node at (5.14,3.949) {\scalebox{1.651}{\small J}};
\node at (11.372,1.26) {\scalebox{1.651}{\small K}};
\node at (8.941,0.95) {\scalebox{1.651}{\small L}};
\draw[line width=0.5pt] (12.634,10.593) -- (12.634,11.693);
\draw[line width=0.5pt] (12.403,11.308) -- (12.634,11.693) -- (12.865,11.308);
\draw[line width=0.5pt] (12.332,11.143) -- (12.937,11.143);
\node at (12.634,12.078) {\scalebox{1.651}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour J, 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 rue Haute & Ⓑ Place Sainte-Croix & Ⓚ Coin du Lavoir \\ \hline
① rue de la Mairie & ⑩ rue de l'Église & Ⓒ Carrefour de la Mairie & Ⓛ Place aux Herbes \\ \hline
② rue du Puits & ⑪ boulevard du Parc & Ⓓ Place de l'Église & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ rue du Moulin & ⑫ rue du Stade & Ⓔ Carrefour du Lavoir & \textbf{a} l'école \\ \hline
④ venelle du Chat & ⑬ rue du Four & Ⓕ Place du Vieux-Puits & \textbf{b} la salle des fêtes \\ \hline
⑤ rue Pasteur & ⑭ rue des Écoles & Ⓖ Rond-point des Vaches & \textbf{c} le lavoir \\ \hline
⑥ rue Verte & ⑮ impasse des Roses & Ⓗ Carrefour Saint-Roch & \textbf{d} le gymnase \\ \hline
⑦ rue du Pont & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Place des Tanneurs &  \\ \hline
⑧ allée des Tilleuls & Ⓐ Placette des Ormes & Ⓙ Carrefour des Tilleuls &  \\ \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=0.6205, line join=round, line cap=round]
\draw[line width=0.5pt] (4.737,11.543) -- (4.968,11.476) -- (5.202,11.414) -- (5.436,11.358) -- (5.672,11.308) -- (5.91,11.263) -- (6.148,11.223) -- (6.387,11.189) -- (6.627,11.161) -- (6.867,11.138) -- (7.108,11.121) -- (7.35,11.11) -- (7.591,11.105) -- (7.832,11.105) -- (8.073,11.111);
\draw[line width=0.5pt] (4.737,11.543) -- (4.792,11.307) -- (4.842,11.068) -- (4.887,10.829) -- (4.925,10.589) -- (4.958,10.348) -- (4.986,10.106) -- (5.008,9.863) -- (5.024,9.62) -- (5.034,9.377) -- (5.038,9.134) -- (5.037,8.89) -- (5.03,8.647) -- (5.018,8.404) -- (5.0,8.162);
\draw[line width=0.5pt] (11.334,11.249) -- (11.287,11.038) -- (11.244,10.825) -- (11.207,10.612) -- (11.175,10.397) -- (11.147,10.182) -- (11.125,9.966) -- (11.108,9.75) -- (11.096,9.534) -- (11.089,9.317) -- (11.087,9.1) -- (11.09,8.883) -- (11.098,8.667) -- (11.112,8.451) -- (11.13,8.235);
\draw[line width=0.5pt] (8.073,11.111) -- (8.303,11.156) -- (8.534,11.196) -- (8.765,11.231) -- (8.998,11.26) -- (9.231,11.283) -- (9.464,11.302) -- (9.698,11.314) -- (9.932,11.321) -- (10.167,11.323) -- (10.401,11.319) -- (10.635,11.31) -- (10.869,11.295) -- (11.102,11.275) -- (11.334,11.249);
\draw[line width=0.5pt] (8.073,11.111) -- (8.032,10.889) -- (7.997,10.666) -- (7.966,10.443) -- (7.941,10.218) -- (7.921,9.994) -- (7.907,9.768) -- (7.897,9.543) -- (7.893,9.317) -- (7.895,9.091) -- (7.901,8.866) -- (7.913,8.641) -- (7.93,8.416) -- (7.953,8.191) -- (7.981,7.967);
\draw[line width=0.5pt] (5.0,8.162) -- (5.213,8.181) -- (5.427,8.194) -- (5.641,8.203) -- (5.855,8.207) -- (6.07,8.205) -- (6.284,8.199) -- (6.498,8.188) -- (6.712,8.171) -- (6.925,8.15) -- (7.138,8.123) -- (7.35,8.092) -- (7.561,8.055) -- (7.772,8.014) -- (7.981,7.967);
\draw[line width=0.5pt] (5.0,8.162) -- (4.993,7.895) -- (4.994,7.627) -- (5.0,7.36) -- (5.013,7.093) -- (5.032,6.826) -- (5.057,6.559) -- (5.089,6.294) -- (5.127,6.029) -- (5.171,5.765) -- (5.221,5.502) -- (5.277,5.24) -- (5.34,4.98) -- (5.409,4.722) -- (5.483,4.465);
\draw[line width=0.5pt] (1.374,8.031) -- (1.63,8.08) -- (1.886,8.123) -- (2.144,8.16) -- (2.402,8.19) -- (2.662,8.215) -- (2.921,8.233) -- (3.181,8.246) -- (3.442,8.252) -- (3.702,8.252) -- (3.962,8.247) -- (4.222,8.235) -- (4.482,8.216) -- (4.741,8.192) -- (5.0,8.162);
\draw[line width=0.5pt] (1.374,8.031) -- (1.377,7.777) -- (1.385,7.524) -- (1.399,7.27) -- (1.42,7.017) -- (1.446,6.764) -- (1.479,6.512) -- (1.517,6.261) -- (1.561,6.011) -- (1.611,5.762) -- (1.667,5.514) -- (1.729,5.268) -- (1.796,5.023) -- (1.869,4.78) -- (1.948,4.539);
\draw[line width=0.5pt] (7.981,7.967) -- (8.201,8.021) -- (8.423,8.069) -- (8.645,8.112) -- (8.869,8.15) -- (9.093,8.182) -- (9.319,8.209) -- (9.544,8.231) -- (9.771,8.248) -- (9.997,8.259) -- (10.224,8.265) -- (10.451,8.265) -- (10.678,8.26) -- (10.904,8.25) -- (11.13,8.235);
\draw[line width=0.5pt] (7.981,7.967) -- (7.994,7.723) -- (8.013,7.479) -- (8.038,7.235) -- (8.069,6.991) -- (8.105,6.749) -- (8.147,6.507) -- (8.195,6.267) -- (8.248,6.027) -- (8.307,5.789) -- (8.371,5.553) -- (8.441,5.318) -- (8.517,5.085) -- (8.598,4.853) -- (8.684,4.624);
\draw[line width=0.5pt] (8.684,4.624) -- (8.723,4.407) -- (8.757,4.188) -- (8.786,3.969) -- (8.81,3.749) -- (8.828,3.529) -- (8.842,3.308) -- (8.85,3.087) -- (8.853,2.866) -- (8.851,2.644) -- (8.843,2.423) -- (8.83,2.203) -- (8.813,1.982) -- (8.79,1.762) -- (8.761,1.543);
\draw[line width=0.5pt] (1.948,4.539) -- (2.198,4.495) -- (2.448,4.457) -- (2.7,4.425) -- (2.953,4.399) -- (3.206,4.379) -- (3.459,4.365) -- (3.713,4.356) -- (3.966,4.354) -- (4.22,4.358) -- (4.474,4.367) -- (4.727,4.383) -- (4.98,4.404) -- (5.232,4.432) -- (5.483,4.465);
\draw[line width=0.5pt] (5.483,4.465) -- (5.708,4.512) -- (5.935,4.552) -- (6.162,4.588) -- (6.39,4.618) -- (6.618,4.643) -- (6.848,4.663) -- (7.077,4.677) -- (7.307,4.685) -- (7.537,4.689) -- (7.767,4.687) -- (7.997,4.679) -- (8.227,4.666) -- (8.456,4.648) -- (8.684,4.624);
\draw[line width=0.5pt] (8.761,1.543) -- (8.922,1.534) -- (9.084,1.528) -- (9.245,1.527) -- (9.406,1.529) -- (9.568,1.535) -- (9.729,1.544) -- (9.89,1.558) -- (10.051,1.575) -- (10.211,1.596) -- (10.37,1.621) -- (10.529,1.649) -- (10.687,1.681) -- (10.845,1.717) -- (11.001,1.757);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.11,11.543) -- (5.081,11.686) -- (5.0,11.807) -- (4.879,11.888) -- (4.737,11.917) -- (4.594,11.888) -- (4.473,11.807) -- (4.392,11.686) -- (4.363,11.543) -- (4.392,11.4) -- (4.473,11.279) -- (4.594,11.198) -- (4.737,11.17) -- (4.879,11.198) -- (5.0,11.279) -- (5.081,11.4) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.708,11.249) -- (11.679,11.392) -- (11.598,11.513) -- (11.477,11.594) -- (11.334,11.622) -- (11.192,11.594) -- (11.07,11.513) -- (10.99,11.392) -- (10.961,11.249) -- (10.99,11.106) -- (11.07,10.985) -- (11.192,10.904) -- (11.334,10.876) -- (11.477,10.904) -- (11.598,10.985) -- (11.679,11.106) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.447,11.111) -- (8.418,11.253) -- (8.337,11.374) -- (8.216,11.455) -- (8.073,11.484) -- (7.931,11.455) -- (7.809,11.374) -- (7.729,11.253) -- (7.7,11.111) -- (7.729,10.968) -- (7.809,10.847) -- (7.931,10.766) -- (8.073,10.737) -- (8.216,10.766) -- (8.337,10.847) -- (8.418,10.968) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.503,8.235) -- (11.475,8.378) -- (11.394,8.499) -- (11.273,8.58) -- (11.13,8.608) -- (10.987,8.58) -- (10.866,8.499) -- (10.785,8.378) -- (10.757,8.235) -- (10.785,8.092) -- (10.866,7.971) -- (10.987,7.89) -- (11.13,7.862) -- (11.273,7.89) -- (11.394,7.971) -- (11.475,8.092) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.373,8.162) -- (5.344,8.305) -- (5.263,8.426) -- (5.142,8.507) -- (5.0,8.535) -- (4.857,8.507) -- (4.736,8.426) -- (4.655,8.305) -- (4.626,8.162) -- (4.655,8.019) -- (4.736,7.898) -- (4.857,7.817) -- (5.0,7.789) -- (5.142,7.817) -- (5.263,7.898) -- (5.344,8.019) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.747,8.031) -- (1.719,8.174) -- (1.638,8.295) -- (1.517,8.376) -- (1.374,8.404) -- (1.231,8.376) -- (1.11,8.295) -- (1.029,8.174) -- (1.001,8.031) -- (1.029,7.888) -- (1.11,7.767) -- (1.231,7.686) -- (1.374,7.658) -- (1.517,7.686) -- (1.638,7.767) -- (1.719,7.888) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.354,7.967) -- (8.326,8.11) -- (8.245,8.231) -- (8.124,8.312) -- (7.981,8.341) -- (7.838,8.312) -- (7.717,8.231) -- (7.636,8.11) -- (7.608,7.967) -- (7.636,7.825) -- (7.717,7.704) -- (7.838,7.623) -- (7.981,7.594) -- (8.124,7.623) -- (8.245,7.704) -- (8.326,7.825) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.058,4.624) -- (9.029,4.767) -- (8.948,4.888) -- (8.827,4.969) -- (8.684,4.997) -- (8.541,4.969) -- (8.42,4.888) -- (8.34,4.767) -- (8.311,4.624) -- (8.34,4.481) -- (8.42,4.36) -- (8.541,4.279) -- (8.684,4.251) -- (8.827,4.279) -- (8.948,4.36) -- (9.029,4.481) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.321,4.539) -- (2.293,4.682) -- (2.212,4.803) -- (2.091,4.884) -- (1.948,4.912) -- (1.805,4.884) -- (1.684,4.803) -- (1.603,4.682) -- (1.575,4.539) -- (1.603,4.396) -- (1.684,4.275) -- (1.805,4.194) -- (1.948,4.166) -- (2.091,4.194) -- (2.212,4.275) -- (2.293,4.396) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.857,4.465) -- (5.828,4.608) -- (5.747,4.729) -- (5.626,4.81) -- (5.483,4.838) -- (5.34,4.81) -- (5.219,4.729) -- (5.139,4.608) -- (5.11,4.465) -- (5.139,4.322) -- (5.219,4.201) -- (5.34,4.12) -- (5.483,4.092) -- (5.626,4.12) -- (5.747,4.201) -- (5.828,4.322) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.374,1.757) -- (11.346,1.9) -- (11.265,2.021) -- (11.144,2.102) -- (11.001,2.13) -- (10.858,2.102) -- (10.737,2.021) -- (10.656,1.9) -- (10.628,1.757) -- (10.656,1.614) -- (10.737,1.493) -- (10.858,1.412) -- (11.001,1.384) -- (11.144,1.412) -- (11.265,1.493) -- (11.346,1.614) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.134,1.543) -- (9.106,1.686) -- (9.025,1.807) -- (8.904,1.888) -- (8.761,1.917) -- (8.618,1.888) -- (8.497,1.807) -- (8.416,1.686) -- (8.388,1.543) -- (8.416,1.4) -- (8.497,1.279) -- (8.618,1.198) -- (8.761,1.17) -- (8.904,1.198) -- (9.025,1.279) -- (9.106,1.4) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.437,11.186) -- (6.555,11.205) -- (6.663,11.26) -- (6.748,11.345) -- (6.802,11.452) -- (6.821,11.571) -- (6.802,11.689) -- (6.748,11.797) -- (6.663,11.882) -- (6.555,11.936) -- (6.437,11.955) -- (6.437,11.955) -- (6.318,11.936) -- (6.211,11.882) -- (6.125,11.797) -- (6.071,11.689) -- (6.052,11.571) -- (6.071,11.452) -- (6.125,11.345) -- (6.211,11.26) -- (6.318,11.205) -- (6.437,11.186) -- cycle;
\node at (6.437,11.571) {\scalebox{1.612}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.391,9.509) -- (5.51,9.527) -- (5.617,9.582) -- (5.702,9.667) -- (5.757,9.774) -- (5.775,9.893) -- (5.757,10.012) -- (5.702,10.119) -- (5.617,10.204) -- (5.51,10.259) -- (5.391,10.278) -- (5.391,10.278) -- (5.272,10.259) -- (5.165,10.204) -- (5.08,10.119) -- (5.025,10.012) -- (5.006,9.893) -- (5.025,9.774) -- (5.08,9.667) -- (5.165,9.582) -- (5.272,9.527) -- (5.391,9.509) -- cycle;
\node at (5.391,9.893) {\scalebox{1.612}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.492,9.34) -- (11.61,9.359) -- (11.718,9.413) -- (11.803,9.498) -- (11.857,9.605) -- (11.876,9.724) -- (11.857,9.843) -- (11.803,9.95) -- (11.718,10.035) -- (11.61,10.09) -- (11.492,10.109) -- (11.492,10.109) -- (11.373,10.09) -- (11.265,10.035) -- (11.18,9.95) -- (11.126,9.843) -- (11.107,9.724) -- (11.126,9.605) -- (11.18,9.498) -- (11.265,9.413) -- (11.373,9.359) -- (11.492,9.34) -- cycle;
\node at (11.492,9.724) {\scalebox{1.612}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.682,11.314) -- (9.801,11.333) -- (9.908,11.387) -- (9.993,11.472) -- (10.048,11.58) -- (10.066,11.698) -- (10.048,11.817) -- (9.993,11.924) -- (9.908,12.01) -- (9.801,12.064) -- (9.682,12.083) -- (9.682,12.083) -- (9.563,12.064) -- (9.456,12.01) -- (9.371,11.924) -- (9.316,11.817) -- (9.297,11.698) -- (9.316,11.58) -- (9.371,11.472) -- (9.456,11.387) -- (9.563,11.333) -- (9.682,11.314) -- cycle;
\node at (9.682,11.698) {\scalebox{1.612}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.282,9.147) -- (8.401,9.166) -- (8.508,9.22) -- (8.593,9.305) -- (8.648,9.413) -- (8.666,9.531) -- (8.648,9.65) -- (8.593,9.758) -- (8.508,9.843) -- (8.401,9.897) -- (8.282,9.916) -- (8.282,9.916) -- (8.163,9.897) -- (8.056,9.843) -- (7.971,9.758) -- (7.916,9.65) -- (7.897,9.531) -- (7.916,9.413) -- (7.971,9.305) -- (8.056,9.22) -- (8.163,9.166) -- (8.282,9.147) -- cycle;
\node at (8.282,9.531) {\scalebox{1.612}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.523,8.187) -- (6.642,8.206) -- (6.749,8.26) -- (6.834,8.345) -- (6.889,8.452) -- (6.908,8.571) -- (6.889,8.69) -- (6.834,8.797) -- (6.749,8.882) -- (6.642,8.937) -- (6.523,8.956) -- (6.523,8.956) -- (6.404,8.937) -- (6.297,8.882) -- (6.212,8.797) -- (6.157,8.69) -- (6.139,8.571) -- (6.157,8.452) -- (6.212,8.345) -- (6.297,8.26) -- (6.404,8.206) -- (6.523,8.187) -- cycle;
\node at (6.523,8.571) {\scalebox{1.612}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.47,5.959) -- (5.589,5.978) -- (5.696,6.032) -- (5.781,6.117) -- (5.836,6.225) -- (5.855,6.343) -- (5.836,6.462) -- (5.781,6.569) -- (5.696,6.655) -- (5.589,6.709) -- (5.47,6.728) -- (5.47,6.728) -- (5.351,6.709) -- (5.244,6.655) -- (5.159,6.569) -- (5.104,6.462) -- (5.086,6.343) -- (5.104,6.225) -- (5.159,6.117) -- (5.244,6.032) -- (5.351,5.978) -- (5.47,5.959) -- cycle;
\node at (5.47,6.343) {\scalebox{1.612}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.168,8.246) -- (3.286,8.264) -- (3.394,8.319) -- (3.479,8.404) -- (3.533,8.511) -- (3.552,8.63) -- (3.533,8.749) -- (3.479,8.856) -- (3.394,8.941) -- (3.286,8.996) -- (3.168,9.015) -- (3.168,9.015) -- (3.049,8.996) -- (2.942,8.941) -- (2.856,8.856) -- (2.802,8.749) -- (2.783,8.63) -- (2.802,8.511) -- (2.856,8.404) -- (2.942,8.319) -- (3.049,8.264) -- (3.168,8.246) -- cycle;
\node at (3.168,8.63) {\scalebox{1.612}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.896,5.939) -- (2.015,5.958) -- (2.122,6.013) -- (2.208,6.098) -- (2.262,6.205) -- (2.281,6.324) -- (2.262,6.443) -- (2.208,6.55) -- (2.122,6.635) -- (2.015,6.689) -- (1.896,6.708) -- (1.896,6.708) -- (1.778,6.689) -- (1.67,6.635) -- (1.585,6.55) -- (1.531,6.443) -- (1.512,6.324) -- (1.531,6.205) -- (1.585,6.098) -- (1.67,6.013) -- (1.778,5.958) -- (1.896,5.939) -- cycle;
\node at (1.896,6.324) {\scalebox{1.612}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.693,8.23) -- (9.812,8.248) -- (9.919,8.303) -- (10.004,8.388) -- (10.059,8.495) -- (10.077,8.614) -- (10.059,8.733) -- (10.004,8.84) -- (9.919,8.925) -- (9.812,8.98) -- (9.693,8.999) -- (9.331,8.999) -- (9.212,8.98) -- (9.105,8.925) -- (9.02,8.84) -- (8.965,8.733) -- (8.946,8.614) -- (8.965,8.495) -- (9.02,8.388) -- (9.105,8.303) -- (9.212,8.248) -- (9.331,8.23) -- cycle;
\node at (9.512,8.614) {\scalebox{1.612}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.752,5.961) -- (8.871,5.98) -- (8.978,6.035) -- (9.063,6.12) -- (9.118,6.227) -- (9.136,6.346) -- (9.118,6.465) -- (9.063,6.572) -- (8.978,6.657) -- (8.871,6.712) -- (8.752,6.73) -- (8.39,6.73) -- (8.271,6.712) -- (8.164,6.657) -- (8.079,6.572) -- (8.024,6.465) -- (8.005,6.346) -- (8.024,6.227) -- (8.079,6.12) -- (8.164,6.035) -- (8.271,5.98) -- (8.39,5.961) -- cycle;
\node at (8.571,6.346) {\scalebox{1.612}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.646,2.693) -- (8.765,2.712) -- (8.872,2.766) -- (8.958,2.851) -- (9.012,2.958) -- (9.031,3.077) -- (9.012,3.196) -- (8.958,3.303) -- (8.872,3.388) -- (8.765,3.443) -- (8.646,3.462) -- (8.285,3.462) -- (8.166,3.443) -- (8.058,3.388) -- (7.973,3.303) -- (7.919,3.196) -- (7.9,3.077) -- (7.919,2.958) -- (7.973,2.851) -- (8.058,2.766) -- (8.166,2.712) -- (8.285,2.693) -- cycle;
\node at (8.465,3.077) {\scalebox{1.612}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.902,4.356) -- (4.02,4.375) -- (4.128,4.43) -- (4.213,4.515) -- (4.267,4.622) -- (4.286,4.741) -- (4.267,4.86) -- (4.213,4.967) -- (4.128,5.052) -- (4.02,5.106) -- (3.902,5.125) -- (3.54,5.125) -- (3.421,5.106) -- (3.314,5.052) -- (3.229,4.967) -- (3.174,4.86) -- (3.155,4.741) -- (3.174,4.622) -- (3.229,4.515) -- (3.314,4.43) -- (3.421,4.375) -- (3.54,4.356) -- cycle;
\node at (3.721,4.741) {\scalebox{1.612}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.239,4.676) -- (7.358,4.695) -- (7.465,4.75) -- (7.55,4.835) -- (7.605,4.942) -- (7.624,5.061) -- (7.605,5.18) -- (7.55,5.287) -- (7.465,5.372) -- (7.358,5.426) -- (7.239,5.445) -- (6.877,5.445) -- (6.758,5.426) -- (6.651,5.372) -- (6.566,5.287) -- (6.512,5.18) -- (6.493,5.061) -- (6.512,4.942) -- (6.566,4.835) -- (6.651,4.75) -- (6.758,4.695) -- (6.877,4.676) -- cycle;
\node at (7.058,5.061) {\scalebox{1.612}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.034,1.556) -- (10.153,1.575) -- (10.26,1.629) -- (10.345,1.714) -- (10.4,1.822) -- (10.419,1.94) -- (10.4,2.059) -- (10.345,2.167) -- (10.26,2.252) -- (10.153,2.306) -- (10.034,2.325) -- (9.672,2.325) -- (9.554,2.306) -- (9.446,2.252) -- (9.361,2.167) -- (9.307,2.059) -- (9.288,1.94) -- (9.307,1.822) -- (9.361,1.714) -- (9.446,1.629) -- (9.554,1.575) -- (9.672,1.556) -- cycle;
\node at (9.853,1.94) {\scalebox{1.612}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (5.183,4.165) -- (5.783,4.165) -- (5.783,4.765) -- (5.183,4.765) -- cycle;
\node at (5.483,4.465) {\scalebox{1.612}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.181,6.526) -- (4.781,6.526) -- (4.781,7.126) -- (4.181,7.126) -- cycle;
\node at (4.481,6.826) {\scalebox{1.612}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.972,10.615) -- (5.572,10.615) -- (5.572,11.215) -- (4.972,11.215) -- cycle;
\node at (5.272,10.915) {\scalebox{1.612}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.583,12.04) -- (6.183,12.04) -- (6.183,12.64) -- (5.583,12.64) -- cycle;
\node at (5.883,12.34) {\scalebox{1.612}{\small d}};
\node at (4.451,12.094) {\scalebox{1.612}{\small A}};
\node at (11.767,11.693) {\scalebox{1.612}{\small B}};
\node at (8.211,11.715) {\scalebox{1.612}{\small C}};
\node at (11.72,8.426) {\scalebox{1.612}{\small D}};
\node at (4.462,8.471) {\scalebox{1.612}{\small E}};
\node at (0.765,8.147) {\scalebox{1.612}{\small F}};
\node at (8.376,8.445) {\scalebox{1.612}{\small G}};
\node at (9.031,4.11) {\scalebox{1.612}{\small H}};
\node at (1.385,4.279) {\scalebox{1.612}{\small I}};
\node at (5.14,3.949) {\scalebox{1.612}{\small J}};
\node at (11.372,1.26) {\scalebox{1.612}{\small K}};
\node at (8.941,0.95) {\scalebox{1.612}{\small L}};
\draw[line width=0.5pt] (12.634,10.593) -- (12.634,11.693);
\draw[line width=0.5pt] (12.403,11.308) -- (12.634,11.693) -- (12.865,11.308);
\draw[line width=0.5pt] (12.332,11.143) -- (12.937,11.143);
\node at (12.634,12.078) {\scalebox{1.612}{\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, F, G, J, L & 6 \\ \hline
clairs & C, D, E, 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{H}, 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{J} :}\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{rue du Four ⑬} vers l'ouest jusqu'à Place des Tanneurs \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Haute ⑨}, vers le nord, jusqu'à Place du Vieux-Puits \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Tilleuls ⑧}, vers l'est, jusqu'à Carrefour du Lavoir \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ②}, vers le nord, jusqu'à Placette des Ormes \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Mairie ①}, vers l'est, jusqu'à Carrefour de la Mairie \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{venelle du Chat ④}, vers l'est, jusqu'à Place Sainte-Croix \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Moulin ③}, vers le sud, jusqu'à Place de l'Église \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de l'Église ⑩}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{boulevard du Parc ⑪}, vers le sud, jusqu'à Carrefour Saint-Roch \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Stade ⑫}, vers le sud, jusqu'à Place aux Herbes \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse des Roses ⑮}, vers l'est, jusqu'à Coin du Lavoir \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}
