\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 143}

\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=143}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0143}
\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=0.5975, line join=round, line cap=round]
\draw[line width=0.5pt] (6.913,6.932) -- (6.892,6.752) -- (6.875,6.572) -- (6.862,6.392) -- (6.854,6.211) -- (6.85,6.03) -- (6.85,5.849) -- (6.854,5.668) -- (6.862,5.487) -- (6.875,5.307) -- (6.892,5.127) -- (6.913,4.947) -- (6.939,4.768) -- (6.969,4.59) -- (7.003,4.412);
\draw[line width=0.5pt] (11.522,6.594) -- (11.543,6.418) -- (11.559,6.242) -- (11.572,6.065) -- (11.58,5.889) -- (11.584,5.712) -- (11.584,5.535) -- (11.58,5.357) -- (11.572,5.181) -- (11.559,5.004) -- (11.543,4.828) -- (11.522,4.652) -- (11.497,4.477) -- (11.468,4.302) -- (11.435,4.128);
\draw[line width=0.5pt] (4.351,6.343) -- (4.539,6.357) -- (4.726,6.375) -- (4.914,6.398) -- (5.1,6.425) -- (5.286,6.456) -- (5.472,6.492) -- (5.656,6.532) -- (5.84,6.576) -- (6.022,6.625) -- (6.203,6.678) -- (6.383,6.735) -- (6.561,6.797) -- (6.738,6.862) -- (6.913,6.932);
\draw[line width=0.5pt] (4.351,6.343) -- (4.357,6.192) -- (4.36,6.04) -- (4.36,5.888) -- (4.356,5.736) -- (4.348,5.584) -- (4.337,5.432) -- (4.322,5.281) -- (4.304,5.13) -- (4.282,4.979) -- (4.256,4.829) -- (4.227,4.68) -- (4.195,4.531) -- (4.159,4.384) -- (4.119,4.237);
\draw[line width=0.5pt] (7.003,4.412) -- (7.17,4.35) -- (7.339,4.292) -- (7.51,4.239) -- (7.681,4.189) -- (7.854,4.143) -- (8.028,4.101) -- (8.203,4.064) -- (8.378,4.03) -- (8.555,4.001) -- (8.732,3.976) -- (8.909,3.955) -- (9.087,3.938) -- (9.265,3.926) -- (9.444,3.917);
\draw[line width=0.5pt] (7.003,4.412) -- (6.985,4.241) -- (6.962,4.07) -- (6.936,3.899) -- (6.906,3.729) -- (6.872,3.56) -- (6.834,3.392) -- (6.792,3.225) -- (6.746,3.059) -- (6.696,2.894) -- (6.642,2.73) -- (6.585,2.567) -- (6.523,2.406) -- (6.458,2.246) -- (6.389,2.088);
\draw[line width=0.5pt] (4.119,4.237) -- (4.326,4.218) -- (4.532,4.203) -- (4.739,4.194) -- (4.946,4.19) -- (5.154,4.19) -- (5.361,4.195) -- (5.568,4.205) -- (5.775,4.22) -- (5.981,4.24) -- (6.187,4.265) -- (6.392,4.295) -- (6.597,4.329) -- (6.8,4.368) -- (7.003,4.412);
\draw[line width=0.5pt] (4.119,4.237) -- (4.156,4.069) -- (4.189,3.9) -- (4.218,3.73) -- (4.242,3.559) -- (4.263,3.388) -- (4.28,3.217) -- (4.292,3.045) -- (4.301,2.873) -- (4.306,2.7) -- (4.306,2.528) -- (4.303,2.356) -- (4.295,2.184) -- (4.284,2.012) -- (4.268,1.841);
\draw[line width=0.5pt] (9.444,3.917) -- (9.587,3.91) -- (9.731,3.907) -- (9.874,3.907) -- (10.018,3.91) -- (10.161,3.917) -- (10.305,3.927) -- (10.448,3.941) -- (10.591,3.958) -- (10.733,3.978) -- (10.875,4.001) -- (11.016,4.028) -- (11.156,4.058) -- (11.296,4.092) -- (11.435,4.128);
\draw[line width=0.5pt] (9.444,3.917) -- (9.429,3.733) -- (9.41,3.548) -- (9.386,3.364) -- (9.359,3.181) -- (9.326,2.998) -- (9.29,2.816) -- (9.249,2.635) -- (9.205,2.455) -- (9.155,2.277) -- (9.102,2.099) -- (9.045,1.923) -- (8.983,1.748) -- (8.917,1.574) -- (8.848,1.403);
\draw[line width=0.5pt] (1.804,3.881) -- (1.972,3.881) -- (2.14,3.885) -- (2.308,3.893) -- (2.476,3.905) -- (2.643,3.92) -- (2.81,3.94) -- (2.976,3.964) -- (3.142,3.991) -- (3.307,4.022) -- (3.472,4.058) -- (3.635,4.097) -- (3.798,4.14) -- (3.959,4.186) -- (4.119,4.237);
\draw[line width=0.5pt] (1.804,3.881) -- (1.809,3.716) -- (1.81,3.55) -- (1.808,3.385) -- (1.801,3.219) -- (1.79,3.054) -- (1.776,2.889) -- (1.758,2.724) -- (1.735,2.56) -- (1.709,2.396) -- (1.68,2.233) -- (1.646,2.071) -- (1.608,1.909) -- (1.567,1.749) -- (1.522,1.59);
\draw[line width=0.5pt] (6.389,2.088) -- (6.556,2.013) -- (6.725,1.941) -- (6.895,1.874) -- (7.067,1.81) -- (7.24,1.751) -- (7.414,1.696) -- (7.59,1.644) -- (7.767,1.597) -- (7.945,1.554) -- (8.124,1.515) -- (8.304,1.481) -- (8.485,1.451) -- (8.666,1.425) -- (8.848,1.403);
\draw[line width=0.5pt] (4.268,1.841) -- (4.416,1.881) -- (4.564,1.919) -- (4.714,1.952) -- (4.864,1.983) -- (5.015,2.009) -- (5.166,2.032) -- (5.319,2.052) -- (5.471,2.068) -- (5.624,2.08) -- (5.777,2.089) -- (5.93,2.095) -- (6.083,2.096) -- (6.236,2.094) -- (6.389,2.088);
\draw[line width=0.5pt] (1.522,1.59) -- (1.719,1.578) -- (1.917,1.57) -- (2.115,1.567) -- (2.313,1.569) -- (2.51,1.575) -- (2.708,1.586) -- (2.905,1.602) -- (3.102,1.622) -- (3.299,1.647) -- (3.494,1.677) -- (3.689,1.711) -- (3.883,1.75) -- (4.076,1.793) -- (4.268,1.841);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.073,6.932) -- (7.052,7.012) -- (6.993,7.071) -- (6.913,7.092) -- (6.833,7.071) -- (6.775,7.012) -- (6.753,6.932) -- (6.775,6.852) -- (6.833,6.793) -- (6.913,6.772) -- (6.993,6.793) -- (7.052,6.852) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.682,6.594) -- (11.661,6.674) -- (11.602,6.732) -- (11.522,6.754) -- (11.442,6.732) -- (11.384,6.674) -- (11.362,6.594) -- (11.384,6.514) -- (11.442,6.455) -- (11.522,6.434) -- (11.602,6.455) -- (11.661,6.514) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.511,6.343) -- (4.489,6.423) -- (4.431,6.482) -- (4.351,6.503) -- (4.271,6.482) -- (4.212,6.423) -- (4.191,6.343) -- (4.212,6.263) -- (4.271,6.205) -- (4.351,6.183) -- (4.431,6.205) -- (4.489,6.263) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.163,4.412) -- (7.141,4.492) -- (7.083,4.551) -- (7.003,4.572) -- (6.923,4.551) -- (6.864,4.492) -- (6.843,4.412) -- (6.864,4.332) -- (6.923,4.273) -- (7.003,4.252) -- (7.083,4.273) -- (7.141,4.332) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.279,4.237) -- (4.258,4.317) -- (4.199,4.375) -- (4.119,4.397) -- (4.039,4.375) -- (3.981,4.317) -- (3.959,4.237) -- (3.981,4.157) -- (4.039,4.098) -- (4.119,4.077) -- (4.199,4.098) -- (4.258,4.157) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.595,4.128) -- (11.573,4.208) -- (11.515,4.267) -- (11.435,4.288) -- (11.355,4.267) -- (11.296,4.208) -- (11.275,4.128) -- (11.296,4.048) -- (11.355,3.99) -- (11.435,3.968) -- (11.515,3.99) -- (11.573,4.048) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.604,3.917) -- (9.582,3.997) -- (9.524,4.056) -- (9.444,4.077) -- (9.364,4.056) -- (9.305,3.997) -- (9.284,3.917) -- (9.305,3.837) -- (9.364,3.779) -- (9.444,3.757) -- (9.524,3.779) -- (9.582,3.837) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.964,3.881) -- (1.943,3.961) -- (1.884,4.02) -- (1.804,4.041) -- (1.724,4.02) -- (1.665,3.961) -- (1.644,3.881) -- (1.665,3.801) -- (1.724,3.743) -- (1.804,3.721) -- (1.884,3.743) -- (1.943,3.801) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.549,2.088) -- (6.528,2.168) -- (6.469,2.227) -- (6.389,2.248) -- (6.309,2.227) -- (6.251,2.168) -- (6.229,2.088) -- (6.251,2.008) -- (6.309,1.95) -- (6.389,1.928) -- (6.469,1.95) -- (6.528,2.008) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.428,1.841) -- (4.407,1.921) -- (4.348,1.979) -- (4.268,2.001) -- (4.188,1.979) -- (4.129,1.921) -- (4.108,1.841) -- (4.129,1.761) -- (4.188,1.702) -- (4.268,1.681) -- (4.348,1.702) -- (4.407,1.761) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.682,1.59) -- (1.661,1.67) -- (1.602,1.728) -- (1.522,1.75) -- (1.442,1.728) -- (1.384,1.67) -- (1.362,1.59) -- (1.384,1.51) -- (1.442,1.451) -- (1.522,1.43) -- (1.602,1.451) -- (1.661,1.51) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.008,1.403) -- (8.986,1.483) -- (8.928,1.541) -- (8.848,1.563) -- (8.768,1.541) -- (8.709,1.483) -- (8.688,1.403) -- (8.709,1.323) -- (8.768,1.264) -- (8.848,1.243) -- (8.928,1.264) -- (8.986,1.323) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.248,5.288) -- (7.369,5.308) -- (7.479,5.364) -- (7.566,5.451) -- (7.622,5.561) -- (7.641,5.682) -- (7.622,5.804) -- (7.566,5.914) -- (7.479,6.001) -- (7.369,6.057) -- (7.248,6.076) -- (7.248,6.076) -- (7.126,6.057) -- (7.016,6.001) -- (6.929,5.914) -- (6.873,5.804) -- (6.854,5.682) -- (6.873,5.561) -- (6.929,5.451) -- (7.016,5.364) -- (7.126,5.308) -- (7.248,5.288) -- cycle;
\node at (7.248,5.682) {\scalebox{1.674}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.974,4.95) -- (12.095,4.969) -- (12.205,5.025) -- (12.292,5.112) -- (12.348,5.222) -- (12.368,5.344) -- (12.348,5.465) -- (12.292,5.575) -- (12.205,5.662) -- (12.095,5.718) -- (11.974,5.737) -- (11.974,5.737) -- (11.852,5.718) -- (11.742,5.662) -- (11.655,5.575) -- (11.599,5.465) -- (11.58,5.344) -- (11.599,5.222) -- (11.655,5.112) -- (11.742,5.025) -- (11.852,4.969) -- (11.974,4.95) -- cycle;
\node at (11.974,5.344) {\scalebox{1.674}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.568,6.522) -- (5.69,6.541) -- (5.8,6.597) -- (5.887,6.684) -- (5.943,6.794) -- (5.962,6.916) -- (5.943,7.038) -- (5.887,7.147) -- (5.8,7.235) -- (5.69,7.29) -- (5.568,7.31) -- (5.568,7.31) -- (5.446,7.29) -- (5.336,7.235) -- (5.249,7.147) -- (5.193,7.038) -- (5.174,6.916) -- (5.193,6.794) -- (5.249,6.684) -- (5.336,6.597) -- (5.446,6.541) -- (5.568,6.522) -- cycle;
\node at (5.568,6.916) {\scalebox{1.674}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.713,4.844) -- (4.835,4.863) -- (4.945,4.919) -- (5.032,5.006) -- (5.088,5.116) -- (5.107,5.238) -- (5.088,5.359) -- (5.032,5.469) -- (4.945,5.556) -- (4.835,5.612) -- (4.713,5.631) -- (4.713,5.631) -- (4.592,5.612) -- (4.482,5.556) -- (4.395,5.469) -- (4.339,5.359) -- (4.32,5.238) -- (4.339,5.116) -- (4.395,5.006) -- (4.482,4.919) -- (4.592,4.863) -- (4.713,4.844) -- cycle;
\node at (4.713,5.238) {\scalebox{1.674}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.281,4.056) -- (8.403,4.075) -- (8.512,4.131) -- (8.6,4.218) -- (8.656,4.328) -- (8.675,4.45) -- (8.656,4.572) -- (8.6,4.681) -- (8.512,4.769) -- (8.403,4.825) -- (8.281,4.844) -- (8.281,4.844) -- (8.159,4.825) -- (8.049,4.769) -- (7.962,4.681) -- (7.906,4.572) -- (7.887,4.45) -- (7.906,4.328) -- (7.962,4.218) -- (8.049,4.131) -- (8.159,4.075) -- (8.281,4.056) -- cycle;
\node at (8.281,4.45) {\scalebox{1.674}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.173,2.731) -- (7.294,2.75) -- (7.404,2.806) -- (7.491,2.893) -- (7.547,3.003) -- (7.567,3.124) -- (7.547,3.246) -- (7.491,3.356) -- (7.404,3.443) -- (7.294,3.499) -- (7.173,3.518) -- (7.173,3.518) -- (7.051,3.499) -- (6.941,3.443) -- (6.854,3.356) -- (6.798,3.246) -- (6.779,3.124) -- (6.798,3.003) -- (6.854,2.893) -- (6.941,2.806) -- (7.051,2.75) -- (7.173,2.731) -- cycle;
\node at (7.173,3.124) {\scalebox{1.674}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.544,4.205) -- (5.666,4.224) -- (5.776,4.28) -- (5.863,4.367) -- (5.919,4.477) -- (5.938,4.599) -- (5.919,4.72) -- (5.863,4.83) -- (5.776,4.917) -- (5.666,4.973) -- (5.544,4.993) -- (5.544,4.993) -- (5.423,4.973) -- (5.313,4.917) -- (5.226,4.83) -- (5.17,4.72) -- (5.15,4.599) -- (5.17,4.477) -- (5.226,4.367) -- (5.313,4.28) -- (5.423,4.224) -- (5.544,4.205) -- cycle;
\node at (5.544,4.599) {\scalebox{1.674}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.686,2.675) -- (4.807,2.695) -- (4.917,2.751) -- (5.004,2.838) -- (5.06,2.948) -- (5.08,3.069) -- (5.06,3.191) -- (5.004,3.301) -- (4.917,3.388) -- (4.807,3.444) -- (4.686,3.463) -- (4.686,3.463) -- (4.564,3.444) -- (4.454,3.388) -- (4.367,3.301) -- (4.311,3.191) -- (4.292,3.069) -- (4.311,2.948) -- (4.367,2.838) -- (4.454,2.751) -- (4.564,2.695) -- (4.686,2.675) -- cycle;
\node at (4.686,3.069) {\scalebox{1.674}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.817,3.034) -- (10.939,3.053) -- (11.049,3.109) -- (11.136,3.197) -- (11.192,3.306) -- (11.211,3.428) -- (11.192,3.55) -- (11.136,3.66) -- (11.049,3.747) -- (10.939,3.803) -- (10.817,3.822) -- (10.817,3.822) -- (10.696,3.803) -- (10.586,3.747) -- (10.499,3.66) -- (10.443,3.55) -- (10.424,3.428) -- (10.443,3.306) -- (10.499,3.197) -- (10.586,3.109) -- (10.696,3.053) -- (10.817,3.034) -- cycle;
\node at (10.817,3.428) {\scalebox{1.674}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.818,2.151) -- (9.94,2.17) -- (10.05,2.226) -- (10.137,2.313) -- (10.193,2.423) -- (10.212,2.545) -- (10.193,2.666) -- (10.137,2.776) -- (10.05,2.863) -- (9.94,2.919) -- (9.818,2.938) -- (9.447,2.938) -- (9.326,2.919) -- (9.216,2.863) -- (9.129,2.776) -- (9.073,2.666) -- (9.053,2.545) -- (9.073,2.423) -- (9.129,2.313) -- (9.216,2.226) -- (9.326,2.17) -- (9.447,2.151) -- cycle;
\node at (9.633,2.545) {\scalebox{1.674}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.067,2.995) -- (3.188,3.014) -- (3.298,3.07) -- (3.385,3.157) -- (3.441,3.267) -- (3.461,3.388) -- (3.441,3.51) -- (3.385,3.62) -- (3.298,3.707) -- (3.188,3.763) -- (3.067,3.782) -- (2.696,3.782) -- (2.574,3.763) -- (2.465,3.707) -- (2.377,3.62) -- (2.321,3.51) -- (2.302,3.388) -- (2.321,3.267) -- (2.377,3.157) -- (2.465,3.07) -- (2.574,3.014) -- (2.696,2.995) -- cycle;
\node at (2.881,3.388) {\scalebox{1.674}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.552,2.378) -- (1.674,2.397) -- (1.784,2.453) -- (1.871,2.54) -- (1.927,2.65) -- (1.946,2.772) -- (1.927,2.894) -- (1.871,3.003) -- (1.784,3.091) -- (1.674,3.147) -- (1.552,3.166) -- (1.181,3.166) -- (1.06,3.147) -- (0.95,3.091) -- (0.863,3.003) -- (0.807,2.894) -- (0.787,2.772) -- (0.807,2.65) -- (0.863,2.54) -- (0.95,2.453) -- (1.06,2.397) -- (1.181,2.378) -- cycle;
\node at (1.367,2.772) {\scalebox{1.674}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.881,1.63) -- (8.003,1.649) -- (8.113,1.705) -- (8.2,1.792) -- (8.256,1.902) -- (8.275,2.024) -- (8.256,2.145) -- (8.2,2.255) -- (8.113,2.342) -- (8.003,2.398) -- (7.881,2.418) -- (7.511,2.418) -- (7.389,2.398) -- (7.279,2.342) -- (7.192,2.255) -- (7.136,2.145) -- (7.117,2.024) -- (7.136,1.902) -- (7.192,1.792) -- (7.279,1.705) -- (7.389,1.649) -- (7.511,1.63) -- cycle;
\node at (7.696,2.024) {\scalebox{1.674}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.455,0.904) -- (5.577,0.924) -- (5.687,0.98) -- (5.774,1.067) -- (5.83,1.177) -- (5.849,1.298) -- (5.83,1.42) -- (5.774,1.53) -- (5.687,1.617) -- (5.577,1.673) -- (5.455,1.692) -- (5.084,1.692) -- (4.963,1.673) -- (4.853,1.617) -- (4.766,1.53) -- (4.71,1.42) -- (4.69,1.298) -- (4.71,1.177) -- (4.766,1.067) -- (4.853,0.98) -- (4.963,0.924) -- (5.084,0.904) -- cycle;
\node at (5.27,1.298) {\scalebox{1.674}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.055,1.6) -- (3.177,1.62) -- (3.286,1.676) -- (3.374,1.763) -- (3.43,1.872) -- (3.449,1.994) -- (3.43,2.116) -- (3.374,2.226) -- (3.286,2.313) -- (3.177,2.369) -- (3.055,2.388) -- (2.684,2.388) -- (2.562,2.369) -- (2.453,2.313) -- (2.365,2.226) -- (2.31,2.116) -- (2.29,1.994) -- (2.31,1.872) -- (2.365,1.763) -- (2.453,1.676) -- (2.562,1.62) -- (2.684,1.6) -- cycle;
\node at (2.87,1.994) {\scalebox{1.674}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.504,3.581) -- (2.104,3.581) -- (2.104,4.181) -- (1.504,4.181) -- cycle;
\node at (1.804,3.881) {\scalebox{1.674}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.6,4.393) -- (1.2,4.393) -- (1.2,4.993) -- (0.6,4.993) -- cycle;
\node at (0.9,4.693) {\scalebox{1.674}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.325,6.226) -- (7.925,6.226) -- (7.925,6.826) -- (7.325,6.826) -- cycle;
\node at (7.625,6.526) {\scalebox{1.674}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.08,2.628) -- (8.68,2.628) -- (8.68,3.228) -- (8.08,3.228) -- cycle;
\node at (8.38,2.928) {\scalebox{1.674}{\small d}};
\node at (7.005,7.545) {\scalebox{1.674}{\small A}};
\node at (12.071,6.881) {\scalebox{1.674}{\small B}};
\node at (3.94,6.808) {\scalebox{1.674}{\small C}};
\node at (7.47,4.819) {\scalebox{1.674}{\small D}};
\node at (3.504,4.313) {\scalebox{1.674}{\small E}};
\node at (12.054,4.151) {\scalebox{1.674}{\small F}};
\node at (10.064,3.911) {\scalebox{1.674}{\small G}};
\node at (1.184,3.872) {\scalebox{1.674}{\small H}};
\node at (6.363,1.469) {\scalebox{1.674}{\small I}};
\node at (3.82,1.412) {\scalebox{1.674}{\small J}};
\node at (0.962,1.323) {\scalebox{1.674}{\small K}};
\node at (9.271,0.95) {\scalebox{1.674}{\small L}};
\draw[line width=0.5pt] (12.822,5.982) -- (12.822,7.082);
\draw[line width=0.5pt] (12.591,6.697) -- (12.822,7.082) -- (13.053,6.697);
\draw[line width=0.5pt] (12.52,6.532) -- (13.125,6.532);
\node at (12.822,7.467) {\scalebox{1.674}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour H, 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 avenue de la Gare & Ⓑ Placette des Ormes & Ⓚ Place de l'Église \\ \hline
① rue du Puits & ⑩ rue Verte & Ⓒ Place aux Herbes & Ⓛ Place des Tanneurs \\ \hline
② impasse du Vieux-Four & ⑪ rue Basse & Ⓓ Rond-point du Stade & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ sentier des Prés & ⑫ rue du Moulin & Ⓔ Rond-point du Pont & \textbf{a} l'école \\ \hline
④ chemin de Ronde & ⑬ rue Neuve & Ⓕ Place du Vieux-Puits & \textbf{b} le stade \\ \hline
⑤ rue du Stade & ⑭ passage du Marché & Ⓖ Carrefour du Moulin & \textbf{c} le cimetière \\ \hline
⑥ rue des Écoles & ⑮ rue Pasteur & Ⓗ Fontaine Notre-Dame & \textbf{d} la bibliothèque \\ \hline
⑦ allée des Tilleuls & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour Saint-Roch &  \\ \hline
⑧ rue Haute & Ⓐ Place Sainte-Croix & Ⓙ Rond-point des Vaches &  \\ \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=0.612, line join=round, line cap=round]
\draw[line width=0.5pt] (6.913,6.932) -- (6.892,6.752) -- (6.875,6.572) -- (6.862,6.392) -- (6.854,6.211) -- (6.85,6.03) -- (6.85,5.849) -- (6.854,5.668) -- (6.862,5.487) -- (6.875,5.307) -- (6.892,5.127) -- (6.913,4.947) -- (6.939,4.768) -- (6.969,4.59) -- (7.003,4.412);
\draw[line width=0.5pt] (11.522,6.594) -- (11.543,6.418) -- (11.559,6.242) -- (11.572,6.065) -- (11.58,5.889) -- (11.584,5.712) -- (11.584,5.535) -- (11.58,5.357) -- (11.572,5.181) -- (11.559,5.004) -- (11.543,4.828) -- (11.522,4.652) -- (11.497,4.477) -- (11.468,4.302) -- (11.435,4.128);
\draw[line width=0.5pt] (4.351,6.343) -- (4.539,6.357) -- (4.726,6.375) -- (4.914,6.398) -- (5.1,6.425) -- (5.286,6.456) -- (5.472,6.492) -- (5.656,6.532) -- (5.84,6.576) -- (6.022,6.625) -- (6.203,6.678) -- (6.383,6.735) -- (6.561,6.797) -- (6.738,6.862) -- (6.913,6.932);
\draw[line width=0.5pt] (4.351,6.343) -- (4.357,6.192) -- (4.36,6.04) -- (4.36,5.888) -- (4.356,5.736) -- (4.348,5.584) -- (4.337,5.432) -- (4.322,5.281) -- (4.304,5.13) -- (4.282,4.979) -- (4.256,4.829) -- (4.227,4.68) -- (4.195,4.531) -- (4.159,4.384) -- (4.119,4.237);
\draw[line width=0.5pt] (7.003,4.412) -- (7.17,4.35) -- (7.339,4.292) -- (7.51,4.239) -- (7.681,4.189) -- (7.854,4.143) -- (8.028,4.101) -- (8.203,4.064) -- (8.378,4.03) -- (8.555,4.001) -- (8.732,3.976) -- (8.909,3.955) -- (9.087,3.938) -- (9.265,3.926) -- (9.444,3.917);
\draw[line width=0.5pt] (7.003,4.412) -- (6.985,4.241) -- (6.962,4.07) -- (6.936,3.899) -- (6.906,3.729) -- (6.872,3.56) -- (6.834,3.392) -- (6.792,3.225) -- (6.746,3.059) -- (6.696,2.894) -- (6.642,2.73) -- (6.585,2.567) -- (6.523,2.406) -- (6.458,2.246) -- (6.389,2.088);
\draw[line width=0.5pt] (4.119,4.237) -- (4.326,4.218) -- (4.532,4.203) -- (4.739,4.194) -- (4.946,4.19) -- (5.154,4.19) -- (5.361,4.195) -- (5.568,4.205) -- (5.775,4.22) -- (5.981,4.24) -- (6.187,4.265) -- (6.392,4.295) -- (6.597,4.329) -- (6.8,4.368) -- (7.003,4.412);
\draw[line width=0.5pt] (4.119,4.237) -- (4.156,4.069) -- (4.189,3.9) -- (4.218,3.73) -- (4.242,3.559) -- (4.263,3.388) -- (4.28,3.217) -- (4.292,3.045) -- (4.301,2.873) -- (4.306,2.7) -- (4.306,2.528) -- (4.303,2.356) -- (4.295,2.184) -- (4.284,2.012) -- (4.268,1.841);
\draw[line width=0.5pt] (9.444,3.917) -- (9.587,3.91) -- (9.731,3.907) -- (9.874,3.907) -- (10.018,3.91) -- (10.161,3.917) -- (10.305,3.927) -- (10.448,3.941) -- (10.591,3.958) -- (10.733,3.978) -- (10.875,4.001) -- (11.016,4.028) -- (11.156,4.058) -- (11.296,4.092) -- (11.435,4.128);
\draw[line width=0.5pt] (9.444,3.917) -- (9.429,3.733) -- (9.41,3.548) -- (9.386,3.364) -- (9.359,3.181) -- (9.326,2.998) -- (9.29,2.816) -- (9.249,2.635) -- (9.205,2.455) -- (9.155,2.277) -- (9.102,2.099) -- (9.045,1.923) -- (8.983,1.748) -- (8.917,1.574) -- (8.848,1.403);
\draw[line width=0.5pt] (1.804,3.881) -- (1.972,3.881) -- (2.14,3.885) -- (2.308,3.893) -- (2.476,3.905) -- (2.643,3.92) -- (2.81,3.94) -- (2.976,3.964) -- (3.142,3.991) -- (3.307,4.022) -- (3.472,4.058) -- (3.635,4.097) -- (3.798,4.14) -- (3.959,4.186) -- (4.119,4.237);
\draw[line width=0.5pt] (1.804,3.881) -- (1.809,3.716) -- (1.81,3.55) -- (1.808,3.385) -- (1.801,3.219) -- (1.79,3.054) -- (1.776,2.889) -- (1.758,2.724) -- (1.735,2.56) -- (1.709,2.396) -- (1.68,2.233) -- (1.646,2.071) -- (1.608,1.909) -- (1.567,1.749) -- (1.522,1.59);
\draw[line width=0.5pt] (6.389,2.088) -- (6.556,2.013) -- (6.725,1.941) -- (6.895,1.874) -- (7.067,1.81) -- (7.24,1.751) -- (7.414,1.696) -- (7.59,1.644) -- (7.767,1.597) -- (7.945,1.554) -- (8.124,1.515) -- (8.304,1.481) -- (8.485,1.451) -- (8.666,1.425) -- (8.848,1.403);
\draw[line width=0.5pt] (4.268,1.841) -- (4.416,1.881) -- (4.564,1.919) -- (4.714,1.952) -- (4.864,1.983) -- (5.015,2.009) -- (5.166,2.032) -- (5.319,2.052) -- (5.471,2.068) -- (5.624,2.08) -- (5.777,2.089) -- (5.93,2.095) -- (6.083,2.096) -- (6.236,2.094) -- (6.389,2.088);
\draw[line width=0.5pt] (1.522,1.59) -- (1.719,1.578) -- (1.917,1.57) -- (2.115,1.567) -- (2.313,1.569) -- (2.51,1.575) -- (2.708,1.586) -- (2.905,1.602) -- (3.102,1.622) -- (3.299,1.647) -- (3.494,1.677) -- (3.689,1.711) -- (3.883,1.75) -- (4.076,1.793) -- (4.268,1.841);
\filldraw[fill=white, draw=black, line width=0.5pt] (7.286,6.932) -- (7.258,7.075) -- (7.177,7.196) -- (7.056,7.277) -- (6.913,7.305) -- (6.77,7.277) -- (6.649,7.196) -- (6.568,7.075) -- (6.54,6.932) -- (6.568,6.789) -- (6.649,6.668) -- (6.77,6.587) -- (6.913,6.559) -- (7.056,6.587) -- (7.177,6.668) -- (7.258,6.789) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.895,6.594) -- (11.867,6.737) -- (11.786,6.858) -- (11.665,6.939) -- (11.522,6.967) -- (11.379,6.939) -- (11.258,6.858) -- (11.177,6.737) -- (11.149,6.594) -- (11.177,6.451) -- (11.258,6.33) -- (11.379,6.249) -- (11.522,6.221) -- (11.665,6.249) -- (11.786,6.33) -- (11.867,6.451) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.724,6.343) -- (4.696,6.486) -- (4.615,6.607) -- (4.494,6.688) -- (4.351,6.717) -- (4.208,6.688) -- (4.087,6.607) -- (4.006,6.486) -- (3.978,6.343) -- (4.006,6.201) -- (4.087,6.079) -- (4.208,5.999) -- (4.351,5.97) -- (4.494,5.999) -- (4.615,6.079) -- (4.696,6.201) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.376,4.412) -- (7.347,4.555) -- (7.266,4.676) -- (7.145,4.757) -- (7.003,4.785) -- (6.86,4.757) -- (6.739,4.676) -- (6.658,4.555) -- (6.629,4.412) -- (6.658,4.269) -- (6.739,4.148) -- (6.86,4.067) -- (7.003,4.039) -- (7.145,4.067) -- (7.266,4.148) -- (7.347,4.269) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.493,4.237) -- (4.464,4.38) -- (4.383,4.501) -- (4.262,4.582) -- (4.119,4.61) -- (3.976,4.582) -- (3.855,4.501) -- (3.774,4.38) -- (3.746,4.237) -- (3.774,4.094) -- (3.855,3.973) -- (3.976,3.892) -- (4.119,3.864) -- (4.262,3.892) -- (4.383,3.973) -- (4.464,4.094) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.808,4.128) -- (11.779,4.271) -- (11.699,4.392) -- (11.577,4.473) -- (11.435,4.502) -- (11.292,4.473) -- (11.171,4.392) -- (11.09,4.271) -- (11.061,4.128) -- (11.09,3.986) -- (11.171,3.864) -- (11.292,3.784) -- (11.435,3.755) -- (11.577,3.784) -- (11.699,3.864) -- (11.779,3.986) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.817,3.917) -- (9.789,4.06) -- (9.708,4.181) -- (9.587,4.262) -- (9.444,4.29) -- (9.301,4.262) -- (9.18,4.181) -- (9.099,4.06) -- (9.071,3.917) -- (9.099,3.774) -- (9.18,3.653) -- (9.301,3.572) -- (9.444,3.544) -- (9.587,3.572) -- (9.708,3.653) -- (9.789,3.774) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.177,3.881) -- (2.149,4.024) -- (2.068,4.145) -- (1.947,4.226) -- (1.804,4.254) -- (1.661,4.226) -- (1.54,4.145) -- (1.459,4.024) -- (1.431,3.881) -- (1.459,3.738) -- (1.54,3.617) -- (1.661,3.536) -- (1.804,3.508) -- (1.947,3.536) -- (2.068,3.617) -- (2.149,3.738) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.763,2.088) -- (6.734,2.231) -- (6.653,2.352) -- (6.532,2.433) -- (6.389,2.462) -- (6.247,2.433) -- (6.126,2.352) -- (6.045,2.231) -- (6.016,2.088) -- (6.045,1.946) -- (6.126,1.825) -- (6.247,1.744) -- (6.389,1.715) -- (6.532,1.744) -- (6.653,1.825) -- (6.734,1.946) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.641,1.841) -- (4.613,1.983) -- (4.532,2.105) -- (4.411,2.185) -- (4.268,2.214) -- (4.125,2.185) -- (4.004,2.105) -- (3.923,1.983) -- (3.895,1.841) -- (3.923,1.698) -- (4.004,1.577) -- (4.125,1.496) -- (4.268,1.467) -- (4.411,1.496) -- (4.532,1.577) -- (4.613,1.698) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.895,1.59) -- (1.867,1.733) -- (1.786,1.854) -- (1.665,1.935) -- (1.522,1.963) -- (1.379,1.935) -- (1.258,1.854) -- (1.177,1.733) -- (1.149,1.59) -- (1.177,1.447) -- (1.258,1.326) -- (1.379,1.245) -- (1.522,1.217) -- (1.665,1.245) -- (1.786,1.326) -- (1.867,1.447) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.221,1.403) -- (9.192,1.546) -- (9.112,1.667) -- (8.991,1.748) -- (8.848,1.776) -- (8.705,1.748) -- (8.584,1.667) -- (8.503,1.546) -- (8.474,1.403) -- (8.503,1.26) -- (8.584,1.139) -- (8.705,1.058) -- (8.848,1.03) -- (8.991,1.058) -- (9.112,1.139) -- (9.192,1.26) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.238,5.297) -- (7.357,5.316) -- (7.464,5.371) -- (7.549,5.456) -- (7.604,5.563) -- (7.623,5.682) -- (7.604,5.801) -- (7.549,5.908) -- (7.464,5.993) -- (7.357,6.048) -- (7.238,6.066) -- (7.238,6.066) -- (7.119,6.048) -- (7.012,5.993) -- (6.927,5.908) -- (6.872,5.801) -- (6.854,5.682) -- (6.872,5.563) -- (6.927,5.456) -- (7.012,5.371) -- (7.119,5.316) -- (7.238,5.297) -- cycle;
\node at (7.238,5.682) {\scalebox{1.634}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.964,4.959) -- (12.083,4.978) -- (12.19,5.033) -- (12.275,5.118) -- (12.33,5.225) -- (12.349,5.344) -- (12.33,5.463) -- (12.275,5.57) -- (12.19,5.655) -- (12.083,5.71) -- (11.964,5.728) -- (11.964,5.728) -- (11.846,5.71) -- (11.738,5.655) -- (11.653,5.57) -- (11.599,5.463) -- (11.58,5.344) -- (11.599,5.225) -- (11.653,5.118) -- (11.738,5.033) -- (11.846,4.978) -- (11.964,4.959) -- cycle;
\node at (11.964,5.344) {\scalebox{1.634}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.57,6.522) -- (5.689,6.541) -- (5.796,6.596) -- (5.881,6.681) -- (5.936,6.788) -- (5.955,6.907) -- (5.936,7.026) -- (5.881,7.133) -- (5.796,7.218) -- (5.689,7.272) -- (5.57,7.291) -- (5.57,7.291) -- (5.451,7.272) -- (5.344,7.218) -- (5.259,7.133) -- (5.204,7.026) -- (5.186,6.907) -- (5.204,6.788) -- (5.259,6.681) -- (5.344,6.596) -- (5.451,6.541) -- (5.57,6.522) -- cycle;
\node at (5.57,6.907) {\scalebox{1.634}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.704,4.854) -- (4.823,4.873) -- (4.93,4.927) -- (5.015,5.013) -- (5.07,5.12) -- (5.089,5.239) -- (5.07,5.357) -- (5.015,5.465) -- (4.93,5.55) -- (4.823,5.604) -- (4.704,5.623) -- (4.704,5.623) -- (4.585,5.604) -- (4.478,5.55) -- (4.393,5.465) -- (4.338,5.357) -- (4.32,5.239) -- (4.338,5.12) -- (4.393,5.013) -- (4.478,4.927) -- (4.585,4.873) -- (4.704,4.854) -- cycle;
\node at (4.704,5.239) {\scalebox{1.634}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.279,4.056) -- (8.398,4.075) -- (8.505,4.13) -- (8.59,4.215) -- (8.645,4.322) -- (8.664,4.441) -- (8.645,4.56) -- (8.59,4.667) -- (8.505,4.752) -- (8.398,4.806) -- (8.279,4.825) -- (8.279,4.825) -- (8.16,4.806) -- (8.053,4.752) -- (7.968,4.667) -- (7.913,4.56) -- (7.895,4.441) -- (7.913,4.322) -- (7.968,4.215) -- (8.053,4.13) -- (8.16,4.075) -- (8.279,4.056) -- cycle;
\node at (8.279,4.441) {\scalebox{1.634}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.164,2.742) -- (7.282,2.761) -- (7.39,2.816) -- (7.475,2.901) -- (7.529,3.008) -- (7.548,3.127) -- (7.529,3.246) -- (7.475,3.353) -- (7.39,3.438) -- (7.282,3.493) -- (7.164,3.511) -- (7.164,3.511) -- (7.045,3.493) -- (6.938,3.438) -- (6.853,3.353) -- (6.798,3.246) -- (6.779,3.127) -- (6.798,3.008) -- (6.853,2.901) -- (6.938,2.816) -- (7.045,2.761) -- (7.164,2.742) -- cycle;
\node at (7.164,3.127) {\scalebox{1.634}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.545,4.205) -- (5.664,4.224) -- (5.771,4.278) -- (5.856,4.363) -- (5.911,4.47) -- (5.929,4.589) -- (5.911,4.708) -- (5.856,4.815) -- (5.771,4.9) -- (5.664,4.955) -- (5.545,4.974) -- (5.545,4.974) -- (5.426,4.955) -- (5.319,4.9) -- (5.234,4.815) -- (5.179,4.708) -- (5.16,4.589) -- (5.179,4.47) -- (5.234,4.363) -- (5.319,4.278) -- (5.426,4.224) -- (5.545,4.205) -- cycle;
\node at (5.545,4.589) {\scalebox{1.634}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.676,2.684) -- (4.795,2.703) -- (4.902,2.758) -- (4.987,2.843) -- (5.042,2.95) -- (5.061,3.069) -- (5.042,3.188) -- (4.987,3.295) -- (4.902,3.38) -- (4.795,3.434) -- (4.676,3.453) -- (4.676,3.453) -- (4.557,3.434) -- (4.45,3.38) -- (4.365,3.295) -- (4.311,3.188) -- (4.292,3.069) -- (4.311,2.95) -- (4.365,2.843) -- (4.45,2.758) -- (4.557,2.703) -- (4.676,2.684) -- cycle;
\node at (4.676,3.069) {\scalebox{1.634}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.815,3.057) -- (10.934,3.075) -- (11.041,3.13) -- (11.127,3.215) -- (11.181,3.322) -- (11.2,3.441) -- (11.181,3.56) -- (11.127,3.667) -- (11.041,3.752) -- (10.934,3.807) -- (10.815,3.826) -- (10.815,3.826) -- (10.697,3.807) -- (10.589,3.752) -- (10.504,3.667) -- (10.45,3.56) -- (10.431,3.441) -- (10.45,3.322) -- (10.504,3.215) -- (10.589,3.13) -- (10.697,3.075) -- (10.815,3.057) -- cycle;
\node at (10.815,3.441) {\scalebox{1.634}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.805,2.162) -- (9.923,2.181) -- (10.031,2.236) -- (10.116,2.321) -- (10.17,2.428) -- (10.189,2.547) -- (10.17,2.666) -- (10.116,2.773) -- (10.031,2.858) -- (9.923,2.912) -- (9.805,2.931) -- (9.443,2.931) -- (9.324,2.912) -- (9.217,2.858) -- (9.132,2.773) -- (9.077,2.666) -- (9.058,2.547) -- (9.077,2.428) -- (9.132,2.321) -- (9.217,2.236) -- (9.324,2.181) -- (9.443,2.162) -- cycle;
\node at (9.624,2.547) {\scalebox{1.634}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.061,3.017) -- (3.179,3.036) -- (3.287,3.09) -- (3.372,3.176) -- (3.426,3.283) -- (3.445,3.402) -- (3.426,3.52) -- (3.372,3.628) -- (3.287,3.713) -- (3.179,3.767) -- (3.061,3.786) -- (2.699,3.786) -- (2.58,3.767) -- (2.473,3.713) -- (2.388,3.628) -- (2.333,3.52) -- (2.314,3.402) -- (2.333,3.283) -- (2.388,3.176) -- (2.473,3.09) -- (2.58,3.036) -- (2.699,3.017) -- cycle;
\node at (2.88,3.402) {\scalebox{1.634}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.557,2.386) -- (1.676,2.405) -- (1.783,2.46) -- (1.868,2.545) -- (1.923,2.652) -- (1.941,2.771) -- (1.923,2.89) -- (1.868,2.997) -- (1.783,3.082) -- (1.676,3.137) -- (1.557,3.155) -- (1.195,3.155) -- (1.076,3.137) -- (0.969,3.082) -- (0.884,2.997) -- (0.829,2.89) -- (0.81,2.771) -- (0.829,2.652) -- (0.884,2.545) -- (0.969,2.46) -- (1.076,2.405) -- (1.195,2.386) -- cycle;
\node at (1.376,2.771) {\scalebox{1.634}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.875,1.63) -- (7.993,1.649) -- (8.101,1.704) -- (8.186,1.789) -- (8.24,1.896) -- (8.259,2.015) -- (8.24,2.133) -- (8.186,2.241) -- (8.101,2.326) -- (7.993,2.38) -- (7.875,2.399) -- (7.513,2.399) -- (7.394,2.38) -- (7.287,2.326) -- (7.202,2.241) -- (7.147,2.133) -- (7.128,2.015) -- (7.147,1.896) -- (7.202,1.789) -- (7.287,1.704) -- (7.394,1.649) -- (7.513,1.63) -- cycle;
\node at (7.694,2.015) {\scalebox{1.634}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.448,0.931) -- (5.567,0.95) -- (5.674,1.005) -- (5.759,1.09) -- (5.814,1.197) -- (5.833,1.316) -- (5.814,1.435) -- (5.759,1.542) -- (5.674,1.627) -- (5.567,1.681) -- (5.448,1.7) -- (5.086,1.7) -- (4.967,1.681) -- (4.86,1.627) -- (4.775,1.542) -- (4.721,1.435) -- (4.702,1.316) -- (4.721,1.197) -- (4.775,1.09) -- (4.86,1.005) -- (4.967,0.95) -- (5.086,0.931) -- cycle;
\node at (5.267,1.316) {\scalebox{1.634}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.051,1.6) -- (3.17,1.619) -- (3.277,1.674) -- (3.362,1.759) -- (3.417,1.866) -- (3.436,1.985) -- (3.417,2.104) -- (3.362,2.211) -- (3.277,2.296) -- (3.17,2.351) -- (3.051,2.369) -- (2.689,2.369) -- (2.571,2.351) -- (2.463,2.296) -- (2.378,2.211) -- (2.324,2.104) -- (2.305,1.985) -- (2.324,1.866) -- (2.378,1.759) -- (2.463,1.674) -- (2.571,1.619) -- (2.689,1.6) -- cycle;
\node at (2.87,1.985) {\scalebox{1.634}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.504,3.581) -- (2.104,3.581) -- (2.104,4.181) -- (1.504,4.181) -- cycle;
\node at (1.804,3.881) {\scalebox{1.634}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.6,4.393) -- (1.2,4.393) -- (1.2,4.993) -- (0.6,4.993) -- cycle;
\node at (0.9,4.693) {\scalebox{1.634}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.325,6.226) -- (7.925,6.226) -- (7.925,6.826) -- (7.325,6.826) -- cycle;
\node at (7.625,6.526) {\scalebox{1.634}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.08,2.628) -- (8.68,2.628) -- (8.68,3.228) -- (8.08,3.228) -- cycle;
\node at (8.38,2.928) {\scalebox{1.634}{\small d}};
\node at (7.005,7.545) {\scalebox{1.634}{\small A}};
\node at (12.071,6.881) {\scalebox{1.634}{\small B}};
\node at (3.94,6.808) {\scalebox{1.634}{\small C}};
\node at (7.47,4.819) {\scalebox{1.634}{\small D}};
\node at (3.504,4.313) {\scalebox{1.634}{\small E}};
\node at (12.054,4.151) {\scalebox{1.634}{\small F}};
\node at (10.064,3.911) {\scalebox{1.634}{\small G}};
\node at (1.184,3.872) {\scalebox{1.634}{\small H}};
\node at (6.363,1.469) {\scalebox{1.634}{\small I}};
\node at (3.82,1.412) {\scalebox{1.634}{\small J}};
\node at (0.962,1.323) {\scalebox{1.634}{\small K}};
\node at (9.271,0.95) {\scalebox{1.634}{\small L}};
\draw[line width=0.5pt] (12.822,5.982) -- (12.822,7.082);
\draw[line width=0.5pt] (12.591,6.697) -- (12.822,7.082) -- (13.053,6.697);
\draw[line width=0.5pt] (12.52,6.532) -- (13.125,6.532);
\node at (12.822,7.467) {\scalebox{1.634}{\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 & C, D, F, H, J, L & 6 \\ \hline
clairs & A, B, E, G, 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{F}, 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{H} :}\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 Moulin ⑫} vers le sud jusqu'à Place de l'Église \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Pasteur ⑮}, vers l'est, jusqu'à Rond-point des Vaches \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Haute ⑧}, vers le nord, jusqu'à Rond-point du Pont \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin de Ronde ④}, vers le nord, jusqu'à Place aux Herbes \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ③}, vers l'est, jusqu'à Place Sainte-Croix \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Puits ①}, vers le sud, jusqu'à Rond-point du Stade \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Écoles ⑥}, vers le sud, jusqu'à Carrefour Saint-Roch \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Neuve ⑬}, vers l'est, jusqu'à Place des Tanneurs \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Verte ⑩}, vers le nord, jusqu'à Carrefour du Moulin \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{avenue de la Gare ⑨}, vers l'est, jusqu'à Place du Vieux-Puits \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse du Vieux-Four ②}, vers le nord, jusqu'à Placette des Ormes \textbf{(B)}.\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}
