\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 207}

\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=207}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0207}
\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=0.6109, line join=round, line cap=round]
\draw[line width=0.5pt] (5.192,11.423) -- (5.11,11.211) -- (5.034,10.998) -- (4.963,10.783) -- (4.896,10.566) -- (4.835,10.348) -- (4.779,10.128) -- (4.728,9.907) -- (4.683,9.685) -- (4.642,9.462) -- (4.607,9.238) -- (4.577,9.014) -- (4.553,8.788) -- (4.533,8.563) -- (4.52,8.337);
\draw[line width=0.5pt] (1.221,10.871) -- (1.497,10.954) -- (1.774,11.03) -- (2.053,11.099) -- (2.334,11.162) -- (2.616,11.218) -- (2.899,11.268) -- (3.184,11.311) -- (3.469,11.347) -- (3.755,11.377) -- (4.042,11.399) -- (4.329,11.415) -- (4.617,11.425) -- (4.904,11.427) -- (5.192,11.423);
\draw[line width=0.5pt] (1.221,10.871) -- (1.195,10.647) -- (1.175,10.421) -- (1.16,10.196) -- (1.15,9.97) -- (1.145,9.743) -- (1.146,9.517) -- (1.152,9.291) -- (1.163,9.065) -- (1.18,8.839) -- (1.202,8.614) -- (1.229,8.389) -- (1.262,8.165) -- (1.299,7.942) -- (1.343,7.72);
\draw[line width=0.5pt] (4.52,8.337) -- (4.733,8.306) -- (4.945,8.271) -- (5.156,8.23) -- (5.367,8.185) -- (5.576,8.134) -- (5.784,8.079) -- (5.991,8.019) -- (6.197,7.954) -- (6.4,7.884) -- (6.602,7.809) -- (6.803,7.73) -- (7.001,7.646) -- (7.197,7.557) -- (7.391,7.464);
\draw[line width=0.5pt] (4.52,8.337) -- (4.524,8.1) -- (4.534,7.863) -- (4.55,7.627) -- (4.572,7.391) -- (4.599,7.156) -- (4.631,6.921) -- (4.669,6.687) -- (4.713,6.454) -- (4.762,6.222) -- (4.816,5.991) -- (4.876,5.762) -- (4.941,5.534) -- (5.011,5.308) -- (5.087,5.084);
\draw[line width=0.5pt] (11.221,7.976) -- (11.21,7.728) -- (11.193,7.479) -- (11.17,7.231) -- (11.141,6.984) -- (11.107,6.737) -- (11.066,6.491) -- (11.02,6.246) -- (10.968,6.003) -- (10.91,5.76) -- (10.847,5.519) -- (10.778,5.28) -- (10.703,5.042) -- (10.623,4.806) -- (10.538,4.573);
\draw[line width=0.5pt] (1.343,7.72) -- (1.574,7.729) -- (1.806,7.743) -- (2.037,7.763) -- (2.268,7.789) -- (2.498,7.82) -- (2.728,7.856) -- (2.956,7.897) -- (3.184,7.944) -- (3.41,7.997) -- (3.635,8.054) -- (3.859,8.117) -- (4.081,8.185) -- (4.301,8.258) -- (4.52,8.337);
\draw[line width=0.5pt] (7.391,7.464) -- (7.657,7.542) -- (7.924,7.614) -- (8.194,7.68) -- (8.465,7.739) -- (8.737,7.792) -- (9.011,7.838) -- (9.285,7.878) -- (9.56,7.912) -- (9.836,7.939) -- (10.113,7.959) -- (10.39,7.973) -- (10.667,7.981) -- (10.944,7.982) -- (11.221,7.976);
\draw[line width=0.5pt] (7.391,7.464) -- (7.369,7.312) -- (7.35,7.159) -- (7.335,7.006) -- (7.323,6.852) -- (7.315,6.698) -- (7.311,6.544) -- (7.31,6.39) -- (7.313,6.236) -- (7.319,6.082) -- (7.329,5.929) -- (7.343,5.775) -- (7.36,5.622) -- (7.381,5.47) -- (7.406,5.318);
\draw[line width=0.5pt] (7.406,5.318) -- (7.636,5.299) -- (7.866,5.275) -- (8.095,5.246) -- (8.323,5.211) -- (8.55,5.171) -- (8.777,5.125) -- (9.002,5.074) -- (9.227,5.018) -- (9.449,4.957) -- (9.671,4.89) -- (9.89,4.819) -- (10.108,4.742) -- (10.324,4.66) -- (10.538,4.573);
\draw[line width=0.5pt] (7.406,5.318) -- (7.422,5.086) -- (7.445,4.854) -- (7.472,4.623) -- (7.505,4.393) -- (7.544,4.163) -- (7.588,3.935) -- (7.637,3.707) -- (7.692,3.481) -- (7.752,3.256) -- (7.817,3.032) -- (7.887,2.81) -- (7.963,2.59) -- (8.044,2.372) -- (8.129,2.156);
\draw[line width=0.5pt] (5.087,5.084) -- (5.249,5.126) -- (5.412,5.164) -- (5.575,5.198) -- (5.74,5.228) -- (5.905,5.255) -- (6.071,5.278) -- (6.237,5.296) -- (6.404,5.311) -- (6.57,5.322) -- (6.738,5.329) -- (6.905,5.332) -- (7.072,5.331) -- (7.239,5.327) -- (7.406,5.318);
\draw[line width=0.5pt] (5.087,5.084) -- (5.012,4.838) -- (4.942,4.591) -- (4.878,4.341) -- (4.82,4.091) -- (4.768,3.839) -- (4.722,3.586) -- (4.682,3.331) -- (4.648,3.076) -- (4.62,2.82) -- (4.598,2.564) -- (4.582,2.307) -- (4.572,2.05) -- (4.568,1.793) -- (4.57,1.536);
\draw[line width=0.5pt] (10.538,4.573) -- (10.564,4.383) -- (10.585,4.192) -- (10.602,4.001) -- (10.615,3.809) -- (10.623,3.617) -- (10.626,3.425) -- (10.625,3.233) -- (10.62,3.041) -- (10.61,2.849) -- (10.595,2.658) -- (10.576,2.467) -- (10.553,2.276) -- (10.525,2.087) -- (10.492,1.897);
\draw[line width=0.5pt] (8.129,2.156) -- (8.294,2.112) -- (8.46,2.071) -- (8.626,2.035) -- (8.794,2.003) -- (8.962,1.974) -- (9.131,1.95) -- (9.3,1.929) -- (9.47,1.913) -- (9.64,1.9) -- (9.811,1.892) -- (9.981,1.887) -- (10.152,1.886) -- (10.322,1.89) -- (10.492,1.897);
\draw[line width=0.5pt] (1.711,2.037) -- (1.919,2.033) -- (2.127,2.023) -- (2.335,2.009) -- (2.542,1.99) -- (2.749,1.967) -- (2.955,1.938) -- (3.161,1.904) -- (3.366,1.866) -- (3.57,1.823) -- (3.772,1.775) -- (3.974,1.722) -- (4.174,1.665) -- (4.373,1.603) -- (4.57,1.536);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.352,11.423) -- (5.33,11.503) -- (5.272,11.561) -- (5.192,11.583) -- (5.112,11.561) -- (5.053,11.503) -- (5.032,11.423) -- (5.053,11.343) -- (5.112,11.284) -- (5.192,11.263) -- (5.272,11.284) -- (5.33,11.343) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.381,10.871) -- (1.36,10.951) -- (1.301,11.01) -- (1.221,11.031) -- (1.141,11.01) -- (1.083,10.951) -- (1.061,10.871) -- (1.083,10.791) -- (1.141,10.733) -- (1.221,10.711) -- (1.301,10.733) -- (1.36,10.791) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.68,8.337) -- (4.658,8.417) -- (4.6,8.475) -- (4.52,8.497) -- (4.44,8.475) -- (4.381,8.417) -- (4.36,8.337) -- (4.381,8.257) -- (4.44,8.198) -- (4.52,8.177) -- (4.6,8.198) -- (4.658,8.257) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.381,7.976) -- (11.36,8.056) -- (11.301,8.115) -- (11.221,8.136) -- (11.141,8.115) -- (11.083,8.056) -- (11.061,7.976) -- (11.083,7.896) -- (11.141,7.838) -- (11.221,7.816) -- (11.301,7.838) -- (11.36,7.896) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.503,7.72) -- (1.481,7.8) -- (1.423,7.859) -- (1.343,7.88) -- (1.263,7.859) -- (1.204,7.8) -- (1.183,7.72) -- (1.204,7.64) -- (1.263,7.582) -- (1.343,7.56) -- (1.423,7.582) -- (1.481,7.64) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.551,7.464) -- (7.53,7.544) -- (7.471,7.603) -- (7.391,7.624) -- (7.311,7.603) -- (7.252,7.544) -- (7.231,7.464) -- (7.252,7.384) -- (7.311,7.326) -- (7.391,7.304) -- (7.471,7.326) -- (7.53,7.384) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.566,5.318) -- (7.544,5.398) -- (7.486,5.456) -- (7.406,5.478) -- (7.326,5.456) -- (7.267,5.398) -- (7.246,5.318) -- (7.267,5.238) -- (7.326,5.179) -- (7.406,5.158) -- (7.486,5.179) -- (7.544,5.238) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.247,5.084) -- (5.226,5.164) -- (5.167,5.222) -- (5.087,5.244) -- (5.007,5.222) -- (4.949,5.164) -- (4.927,5.084) -- (4.949,5.004) -- (5.007,4.945) -- (5.087,4.924) -- (5.167,4.945) -- (5.226,5.004) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.698,4.573) -- (10.676,4.653) -- (10.618,4.711) -- (10.538,4.733) -- (10.458,4.711) -- (10.399,4.653) -- (10.378,4.573) -- (10.399,4.493) -- (10.458,4.434) -- (10.538,4.413) -- (10.618,4.434) -- (10.676,4.493) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.289,2.156) -- (8.268,2.236) -- (8.209,2.294) -- (8.129,2.316) -- (8.049,2.294) -- (7.991,2.236) -- (7.969,2.156) -- (7.991,2.076) -- (8.049,2.017) -- (8.129,1.996) -- (8.209,2.017) -- (8.268,2.076) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.871,2.037) -- (1.849,2.117) -- (1.791,2.175) -- (1.711,2.197) -- (1.631,2.175) -- (1.572,2.117) -- (1.551,2.037) -- (1.572,1.957) -- (1.631,1.898) -- (1.711,1.877) -- (1.791,1.898) -- (1.849,1.957) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.652,1.897) -- (10.631,1.977) -- (10.572,2.036) -- (10.492,2.057) -- (10.412,2.036) -- (10.354,1.977) -- (10.332,1.897) -- (10.354,1.817) -- (10.412,1.759) -- (10.492,1.737) -- (10.572,1.759) -- (10.631,1.817) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.73,1.536) -- (4.708,1.616) -- (4.65,1.675) -- (4.57,1.696) -- (4.49,1.675) -- (4.431,1.616) -- (4.41,1.536) -- (4.431,1.456) -- (4.49,1.398) -- (4.57,1.376) -- (4.65,1.398) -- (4.708,1.456) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.113,9.43) -- (5.235,9.449) -- (5.345,9.505) -- (5.432,9.592) -- (5.488,9.702) -- (5.507,9.824) -- (5.488,9.945) -- (5.432,10.055) -- (5.345,10.142) -- (5.235,10.198) -- (5.113,10.218) -- (5.113,10.218) -- (4.992,10.198) -- (4.882,10.142) -- (4.795,10.055) -- (4.739,9.945) -- (4.719,9.824) -- (4.739,9.702) -- (4.795,9.592) -- (4.882,9.505) -- (4.992,9.449) -- (5.113,9.43) -- cycle;
\node at (5.113,9.824) {\scalebox{1.637}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.13,11.307) -- (3.251,11.326) -- (3.361,11.382) -- (3.448,11.469) -- (3.504,11.579) -- (3.523,11.701) -- (3.504,11.823) -- (3.448,11.932) -- (3.361,12.02) -- (3.251,12.076) -- (3.13,12.095) -- (3.13,12.095) -- (3.008,12.076) -- (2.898,12.02) -- (2.811,11.932) -- (2.755,11.823) -- (2.736,11.701) -- (2.755,11.579) -- (2.811,11.469) -- (2.898,11.382) -- (3.008,11.326) -- (3.13,11.307) -- cycle;
\node at (3.13,11.701) {\scalebox{1.637}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.546,8.912) -- (1.667,8.931) -- (1.777,8.987) -- (1.864,9.074) -- (1.92,9.184) -- (1.939,9.306) -- (1.92,9.428) -- (1.864,9.537) -- (1.777,9.624) -- (1.667,9.68) -- (1.546,9.7) -- (1.546,9.7) -- (1.424,9.68) -- (1.314,9.624) -- (1.227,9.537) -- (1.171,9.428) -- (1.152,9.306) -- (1.171,9.184) -- (1.227,9.074) -- (1.314,8.987) -- (1.424,8.931) -- (1.546,8.912) -- cycle;
\node at (1.546,9.306) {\scalebox{1.637}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.106,8.002) -- (6.228,8.021) -- (6.337,8.077) -- (6.424,8.164) -- (6.48,8.274) -- (6.5,8.396) -- (6.48,8.517) -- (6.424,8.627) -- (6.337,8.714) -- (6.228,8.77) -- (6.106,8.79) -- (6.106,8.79) -- (5.984,8.77) -- (5.874,8.714) -- (5.787,8.627) -- (5.731,8.517) -- (5.712,8.396) -- (5.731,8.274) -- (5.787,8.164) -- (5.874,8.077) -- (5.984,8.021) -- (6.106,8.002) -- cycle;
\node at (6.106,8.396) {\scalebox{1.637}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.057,6.361) -- (5.179,6.38) -- (5.289,6.436) -- (5.376,6.523) -- (5.432,6.633) -- (5.451,6.755) -- (5.432,6.876) -- (5.376,6.986) -- (5.289,7.073) -- (5.179,7.129) -- (5.057,7.148) -- (5.057,7.148) -- (4.936,7.129) -- (4.826,7.073) -- (4.739,6.986) -- (4.683,6.876) -- (4.663,6.755) -- (4.683,6.633) -- (4.739,6.523) -- (4.826,6.436) -- (4.936,6.38) -- (5.057,6.361) -- cycle;
\node at (5.057,6.755) {\scalebox{1.637}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.406,5.775) -- (11.528,5.794) -- (11.638,5.85) -- (11.725,5.937) -- (11.781,6.047) -- (11.8,6.169) -- (11.781,6.29) -- (11.725,6.4) -- (11.638,6.487) -- (11.528,6.543) -- (11.406,6.563) -- (11.406,6.563) -- (11.284,6.543) -- (11.174,6.487) -- (11.087,6.4) -- (11.031,6.29) -- (11.012,6.169) -- (11.031,6.047) -- (11.087,5.937) -- (11.174,5.85) -- (11.284,5.794) -- (11.406,5.775) -- cycle;
\node at (11.406,6.169) {\scalebox{1.637}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.032,7.117) -- (3.153,7.136) -- (3.263,7.192) -- (3.35,7.279) -- (3.406,7.389) -- (3.425,7.511) -- (3.406,7.632) -- (3.35,7.742) -- (3.263,7.829) -- (3.153,7.885) -- (3.032,7.905) -- (3.032,7.905) -- (2.91,7.885) -- (2.8,7.829) -- (2.713,7.742) -- (2.657,7.632) -- (2.638,7.511) -- (2.657,7.389) -- (2.713,7.279) -- (2.8,7.192) -- (2.91,7.136) -- (3.032,7.117) -- cycle;
\node at (3.032,7.511) {\scalebox{1.637}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.233,7.875) -- (9.355,7.894) -- (9.464,7.95) -- (9.551,8.037) -- (9.607,8.147) -- (9.627,8.269) -- (9.607,8.39) -- (9.551,8.5) -- (9.464,8.587) -- (9.355,8.643) -- (9.233,8.663) -- (9.233,8.663) -- (9.111,8.643) -- (9.001,8.587) -- (8.914,8.5) -- (8.858,8.39) -- (8.839,8.269) -- (8.858,8.147) -- (8.914,8.037) -- (9.001,7.95) -- (9.111,7.894) -- (9.233,7.875) -- cycle;
\node at (9.233,8.269) {\scalebox{1.637}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.916,5.994) -- (7.038,6.013) -- (7.148,6.069) -- (7.235,6.156) -- (7.291,6.266) -- (7.31,6.388) -- (7.291,6.509) -- (7.235,6.619) -- (7.148,6.706) -- (7.038,6.762) -- (6.916,6.782) -- (6.916,6.782) -- (6.794,6.762) -- (6.684,6.706) -- (6.597,6.619) -- (6.541,6.509) -- (6.522,6.388) -- (6.541,6.266) -- (6.597,6.156) -- (6.684,6.069) -- (6.794,6.013) -- (6.916,5.994) -- cycle;
\node at (6.916,6.388) {\scalebox{1.637}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.279,5.064) -- (9.401,5.083) -- (9.511,5.139) -- (9.598,5.226) -- (9.654,5.336) -- (9.673,5.458) -- (9.654,5.579) -- (9.598,5.689) -- (9.511,5.776) -- (9.401,5.832) -- (9.279,5.851) -- (8.908,5.851) -- (8.787,5.832) -- (8.677,5.776) -- (8.59,5.689) -- (8.534,5.579) -- (8.514,5.458) -- (8.534,5.336) -- (8.59,5.226) -- (8.677,5.139) -- (8.787,5.083) -- (8.908,5.064) -- cycle;
\node at (9.094,5.458) {\scalebox{1.637}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.207,3.401) -- (8.328,3.42) -- (8.438,3.476) -- (8.525,3.563) -- (8.581,3.673) -- (8.6,3.795) -- (8.581,3.917) -- (8.525,4.026) -- (8.438,4.114) -- (8.328,4.17) -- (8.207,4.189) -- (7.836,4.189) -- (7.714,4.17) -- (7.604,4.114) -- (7.517,4.026) -- (7.461,3.917) -- (7.442,3.795) -- (7.461,3.673) -- (7.517,3.563) -- (7.604,3.476) -- (7.714,3.42) -- (7.836,3.401) -- cycle;
\node at (8.021,3.795) {\scalebox{1.637}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.462,4.511) -- (6.584,4.53) -- (6.693,4.586) -- (6.781,4.673) -- (6.836,4.783) -- (6.856,4.905) -- (6.836,5.026) -- (6.781,5.136) -- (6.693,5.223) -- (6.584,5.279) -- (6.462,5.298) -- (6.091,5.298) -- (5.969,5.279) -- (5.86,5.223) -- (5.772,5.136) -- (5.717,5.026) -- (5.697,4.905) -- (5.717,4.783) -- (5.772,4.673) -- (5.86,4.586) -- (5.969,4.53) -- (6.091,4.511) -- cycle;
\node at (6.276,4.905) {\scalebox{1.637}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.257,2.881) -- (5.379,2.9) -- (5.489,2.956) -- (5.576,3.043) -- (5.632,3.153) -- (5.651,3.274) -- (5.632,3.396) -- (5.576,3.506) -- (5.489,3.593) -- (5.379,3.649) -- (5.257,3.668) -- (4.887,3.668) -- (4.765,3.649) -- (4.655,3.593) -- (4.568,3.506) -- (4.512,3.396) -- (4.493,3.274) -- (4.512,3.153) -- (4.568,3.043) -- (4.655,2.956) -- (4.765,2.9) -- (4.887,2.881) -- cycle;
\node at (5.072,3.274) {\scalebox{1.637}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.205,2.833) -- (11.326,2.852) -- (11.436,2.908) -- (11.523,2.995) -- (11.579,3.105) -- (11.598,3.227) -- (11.579,3.348) -- (11.523,3.458) -- (11.436,3.545) -- (11.326,3.601) -- (11.205,3.62) -- (10.834,3.62) -- (10.712,3.601) -- (10.602,3.545) -- (10.515,3.458) -- (10.459,3.348) -- (10.44,3.227) -- (10.459,3.105) -- (10.515,2.995) -- (10.602,2.908) -- (10.712,2.852) -- (10.834,2.833) -- cycle;
\node at (11.019,3.227) {\scalebox{1.637}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.528,1.927) -- (9.65,1.946) -- (9.76,2.002) -- (9.847,2.089) -- (9.903,2.199) -- (9.922,2.321) -- (9.903,2.443) -- (9.847,2.552) -- (9.76,2.639) -- (9.65,2.695) -- (9.528,2.715) -- (9.158,2.715) -- (9.036,2.695) -- (8.926,2.639) -- (8.839,2.552) -- (8.783,2.443) -- (8.764,2.321) -- (8.783,2.199) -- (8.839,2.089) -- (8.926,2.002) -- (9.036,1.946) -- (9.158,1.927) -- cycle;
\node at (9.343,2.321) {\scalebox{1.637}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.414,1.899) -- (3.536,1.918) -- (3.646,1.974) -- (3.733,2.061) -- (3.789,2.171) -- (3.808,2.292) -- (3.789,2.414) -- (3.733,2.524) -- (3.646,2.611) -- (3.536,2.667) -- (3.414,2.686) -- (3.044,2.686) -- (2.922,2.667) -- (2.812,2.611) -- (2.725,2.524) -- (2.669,2.414) -- (2.65,2.292) -- (2.669,2.171) -- (2.725,2.061) -- (2.812,1.974) -- (2.922,1.918) -- (3.044,1.899) -- cycle;
\node at (3.229,2.292) {\scalebox{1.637}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.892,11.123) -- (5.492,11.123) -- (5.492,11.723) -- (4.892,11.723) -- cycle;
\node at (5.192,11.423) {\scalebox{1.637}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (10.049,5.303) -- (10.649,5.303) -- (10.649,5.903) -- (10.049,5.903) -- cycle;
\node at (10.349,5.603) {\scalebox{1.637}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.067,6.6) -- (8.667,6.6) -- (8.667,7.2) -- (8.067,7.2) -- cycle;
\node at (8.367,6.9) {\scalebox{1.637}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.119,8.064) -- (2.719,8.064) -- (2.719,8.664) -- (2.119,8.664) -- cycle;
\node at (2.419,8.364) {\scalebox{1.637}{\small d}};
\node at (5.096,12.035) {\scalebox{1.637}{\small A}};
\node at (0.79,11.316) {\scalebox{1.637}{\small B}};
\node at (4.19,8.862) {\scalebox{1.637}{\small C}};
\node at (11.796,8.21) {\scalebox{1.637}{\small D}};
\node at (0.765,7.946) {\scalebox{1.637}{\small E}};
\node at (7.789,7.94) {\scalebox{1.637}{\small F}};
\node at (7.978,5.08) {\scalebox{1.637}{\small G}};
\node at (4.606,4.693) {\scalebox{1.637}{\small H}};
\node at (11.133,4.399) {\scalebox{1.637}{\small I}};
\node at (8.43,1.614) {\scalebox{1.637}{\small J}};
\node at (1.246,1.627) {\scalebox{1.637}{\small K}};
\node at (10.954,1.483) {\scalebox{1.637}{\small L}};
\node at (4.368,0.95) {\scalebox{1.637}{\small M}};
\draw[line width=0.5pt] (12.521,10.473) -- (12.521,11.573);
\draw[line width=0.5pt] (12.29,11.188) -- (12.521,11.573) -- (12.752,11.188);
\draw[line width=0.5pt] (12.219,11.023) -- (12.824,11.023);
\node at (12.521,11.958) {\scalebox{1.637}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour A, 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 Neuve & Ⓐ Place des Tanneurs & Ⓙ Place de l'Église \\ \hline
① sentier des Prés & ⑩ rue de la Poste & Ⓑ Fontaine Notre-Dame & Ⓚ Placette des Ormes \\ \hline
② chemin du Lavoir & ⑪ rue Pasteur & Ⓒ Carrefour Saint-Roch & Ⓛ Place aux Herbes \\ \hline
③ rue du Puits & ⑫ quai aux Fleurs & Ⓓ Place du Marché & Ⓜ Place du Vieux-Puits \\ \hline
④ rue Haute & ⑬ chemin des Vignes & Ⓔ Place Sainte-Croix & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ allée des Peupliers & ⑭ rue du Stade & Ⓕ Rond-point de la Gare & \textbf{a} l'école \\ \hline
⑥ chemin Creux & ⑮ rue Verte & Ⓖ Carrefour du Moulin & \textbf{b} la mairie \\ \hline
⑦ rue des Écoles & ⑯ impasse Sainte-Anne & Ⓗ Rond-point de la Poste & \textbf{c} le gymnase \\ \hline
⑧ rue des Jardins & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour des Tilleuls & \textbf{d} la boulangerie \\ \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=0.6258, line join=round, line cap=round]
\draw[line width=0.5pt] (5.192,11.423) -- (5.11,11.211) -- (5.034,10.998) -- (4.963,10.783) -- (4.896,10.566) -- (4.835,10.348) -- (4.779,10.128) -- (4.728,9.907) -- (4.683,9.685) -- (4.642,9.462) -- (4.607,9.238) -- (4.577,9.014) -- (4.553,8.788) -- (4.533,8.563) -- (4.52,8.337);
\draw[line width=0.5pt] (1.221,10.871) -- (1.497,10.954) -- (1.774,11.03) -- (2.053,11.099) -- (2.334,11.162) -- (2.616,11.218) -- (2.899,11.268) -- (3.184,11.311) -- (3.469,11.347) -- (3.755,11.377) -- (4.042,11.399) -- (4.329,11.415) -- (4.617,11.425) -- (4.904,11.427) -- (5.192,11.423);
\draw[line width=0.5pt] (1.221,10.871) -- (1.195,10.647) -- (1.175,10.421) -- (1.16,10.196) -- (1.15,9.97) -- (1.145,9.743) -- (1.146,9.517) -- (1.152,9.291) -- (1.163,9.065) -- (1.18,8.839) -- (1.202,8.614) -- (1.229,8.389) -- (1.262,8.165) -- (1.299,7.942) -- (1.343,7.72);
\draw[line width=0.5pt] (4.52,8.337) -- (4.733,8.306) -- (4.945,8.271) -- (5.156,8.23) -- (5.367,8.185) -- (5.576,8.134) -- (5.784,8.079) -- (5.991,8.019) -- (6.197,7.954) -- (6.4,7.884) -- (6.602,7.809) -- (6.803,7.73) -- (7.001,7.646) -- (7.197,7.557) -- (7.391,7.464);
\draw[line width=0.5pt] (4.52,8.337) -- (4.524,8.1) -- (4.534,7.863) -- (4.55,7.627) -- (4.572,7.391) -- (4.599,7.156) -- (4.631,6.921) -- (4.669,6.687) -- (4.713,6.454) -- (4.762,6.222) -- (4.816,5.991) -- (4.876,5.762) -- (4.941,5.534) -- (5.011,5.308) -- (5.087,5.084);
\draw[line width=0.5pt] (11.221,7.976) -- (11.21,7.728) -- (11.193,7.479) -- (11.17,7.231) -- (11.141,6.984) -- (11.107,6.737) -- (11.066,6.491) -- (11.02,6.246) -- (10.968,6.003) -- (10.91,5.76) -- (10.847,5.519) -- (10.778,5.28) -- (10.703,5.042) -- (10.623,4.806) -- (10.538,4.573);
\draw[line width=0.5pt] (1.343,7.72) -- (1.574,7.729) -- (1.806,7.743) -- (2.037,7.763) -- (2.268,7.789) -- (2.498,7.82) -- (2.728,7.856) -- (2.956,7.897) -- (3.184,7.944) -- (3.41,7.997) -- (3.635,8.054) -- (3.859,8.117) -- (4.081,8.185) -- (4.301,8.258) -- (4.52,8.337);
\draw[line width=0.5pt] (7.391,7.464) -- (7.657,7.542) -- (7.924,7.614) -- (8.194,7.68) -- (8.465,7.739) -- (8.737,7.792) -- (9.011,7.838) -- (9.285,7.878) -- (9.56,7.912) -- (9.836,7.939) -- (10.113,7.959) -- (10.39,7.973) -- (10.667,7.981) -- (10.944,7.982) -- (11.221,7.976);
\draw[line width=0.5pt] (7.391,7.464) -- (7.369,7.312) -- (7.35,7.159) -- (7.335,7.006) -- (7.323,6.852) -- (7.315,6.698) -- (7.311,6.544) -- (7.31,6.39) -- (7.313,6.236) -- (7.319,6.082) -- (7.329,5.929) -- (7.343,5.775) -- (7.36,5.622) -- (7.381,5.47) -- (7.406,5.318);
\draw[line width=0.5pt] (7.406,5.318) -- (7.636,5.299) -- (7.866,5.275) -- (8.095,5.246) -- (8.323,5.211) -- (8.55,5.171) -- (8.777,5.125) -- (9.002,5.074) -- (9.227,5.018) -- (9.449,4.957) -- (9.671,4.89) -- (9.89,4.819) -- (10.108,4.742) -- (10.324,4.66) -- (10.538,4.573);
\draw[line width=0.5pt] (7.406,5.318) -- (7.422,5.086) -- (7.445,4.854) -- (7.472,4.623) -- (7.505,4.393) -- (7.544,4.163) -- (7.588,3.935) -- (7.637,3.707) -- (7.692,3.481) -- (7.752,3.256) -- (7.817,3.032) -- (7.887,2.81) -- (7.963,2.59) -- (8.044,2.372) -- (8.129,2.156);
\draw[line width=0.5pt] (5.087,5.084) -- (5.249,5.126) -- (5.412,5.164) -- (5.575,5.198) -- (5.74,5.228) -- (5.905,5.255) -- (6.071,5.278) -- (6.237,5.296) -- (6.404,5.311) -- (6.57,5.322) -- (6.738,5.329) -- (6.905,5.332) -- (7.072,5.331) -- (7.239,5.327) -- (7.406,5.318);
\draw[line width=0.5pt] (5.087,5.084) -- (5.012,4.838) -- (4.942,4.591) -- (4.878,4.341) -- (4.82,4.091) -- (4.768,3.839) -- (4.722,3.586) -- (4.682,3.331) -- (4.648,3.076) -- (4.62,2.82) -- (4.598,2.564) -- (4.582,2.307) -- (4.572,2.05) -- (4.568,1.793) -- (4.57,1.536);
\draw[line width=0.5pt] (10.538,4.573) -- (10.564,4.383) -- (10.585,4.192) -- (10.602,4.001) -- (10.615,3.809) -- (10.623,3.617) -- (10.626,3.425) -- (10.625,3.233) -- (10.62,3.041) -- (10.61,2.849) -- (10.595,2.658) -- (10.576,2.467) -- (10.553,2.276) -- (10.525,2.087) -- (10.492,1.897);
\draw[line width=0.5pt] (8.129,2.156) -- (8.294,2.112) -- (8.46,2.071) -- (8.626,2.035) -- (8.794,2.003) -- (8.962,1.974) -- (9.131,1.95) -- (9.3,1.929) -- (9.47,1.913) -- (9.64,1.9) -- (9.811,1.892) -- (9.981,1.887) -- (10.152,1.886) -- (10.322,1.89) -- (10.492,1.897);
\draw[line width=0.5pt] (1.711,2.037) -- (1.919,2.033) -- (2.127,2.023) -- (2.335,2.009) -- (2.542,1.99) -- (2.749,1.967) -- (2.955,1.938) -- (3.161,1.904) -- (3.366,1.866) -- (3.57,1.823) -- (3.772,1.775) -- (3.974,1.722) -- (4.174,1.665) -- (4.373,1.603) -- (4.57,1.536);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.565,11.423) -- (5.537,11.566) -- (5.456,11.687) -- (5.335,11.768) -- (5.192,11.796) -- (5.049,11.768) -- (4.928,11.687) -- (4.847,11.566) -- (4.819,11.423) -- (4.847,11.28) -- (4.928,11.159) -- (5.049,11.078) -- (5.192,11.05) -- (5.335,11.078) -- (5.456,11.159) -- (5.537,11.28) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.595,10.871) -- (1.566,11.014) -- (1.485,11.135) -- (1.364,11.216) -- (1.221,11.244) -- (1.079,11.216) -- (0.957,11.135) -- (0.877,11.014) -- (0.848,10.871) -- (0.877,10.728) -- (0.957,10.607) -- (1.079,10.526) -- (1.221,10.498) -- (1.364,10.526) -- (1.485,10.607) -- (1.566,10.728) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.893,8.337) -- (4.864,8.48) -- (4.783,8.601) -- (4.662,8.682) -- (4.52,8.71) -- (4.377,8.682) -- (4.256,8.601) -- (4.175,8.48) -- (4.146,8.337) -- (4.175,8.194) -- (4.256,8.073) -- (4.377,7.992) -- (4.52,7.964) -- (4.662,7.992) -- (4.783,8.073) -- (4.864,8.194) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.595,7.976) -- (11.566,8.119) -- (11.485,8.24) -- (11.364,8.321) -- (11.221,8.35) -- (11.079,8.321) -- (10.957,8.24) -- (10.877,8.119) -- (10.848,7.976) -- (10.877,7.834) -- (10.957,7.712) -- (11.079,7.632) -- (11.221,7.603) -- (11.364,7.632) -- (11.485,7.712) -- (11.566,7.834) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.716,7.72) -- (1.687,7.863) -- (1.606,7.984) -- (1.485,8.065) -- (1.343,8.093) -- (1.2,8.065) -- (1.079,7.984) -- (0.998,7.863) -- (0.969,7.72) -- (0.998,7.577) -- (1.079,7.456) -- (1.2,7.375) -- (1.343,7.347) -- (1.485,7.375) -- (1.606,7.456) -- (1.687,7.577) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.764,7.464) -- (7.736,7.607) -- (7.655,7.728) -- (7.534,7.809) -- (7.391,7.837) -- (7.248,7.809) -- (7.127,7.728) -- (7.046,7.607) -- (7.018,7.464) -- (7.046,7.321) -- (7.127,7.2) -- (7.248,7.119) -- (7.391,7.091) -- (7.534,7.119) -- (7.655,7.2) -- (7.736,7.321) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.779,5.318) -- (7.751,5.461) -- (7.67,5.582) -- (7.549,5.663) -- (7.406,5.691) -- (7.263,5.663) -- (7.142,5.582) -- (7.061,5.461) -- (7.033,5.318) -- (7.061,5.175) -- (7.142,5.054) -- (7.263,4.973) -- (7.406,4.945) -- (7.549,4.973) -- (7.67,5.054) -- (7.751,5.175) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.461,5.084) -- (5.432,5.227) -- (5.351,5.348) -- (5.23,5.429) -- (5.087,5.457) -- (4.945,5.429) -- (4.823,5.348) -- (4.743,5.227) -- (4.714,5.084) -- (4.743,4.941) -- (4.823,4.82) -- (4.945,4.739) -- (5.087,4.711) -- (5.23,4.739) -- (5.351,4.82) -- (5.432,4.941) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (10.911,4.573) -- (10.882,4.715) -- (10.801,4.836) -- (10.68,4.917) -- (10.538,4.946) -- (10.395,4.917) -- (10.274,4.836) -- (10.193,4.715) -- (10.164,4.573) -- (10.193,4.43) -- (10.274,4.309) -- (10.395,4.228) -- (10.538,4.199) -- (10.68,4.228) -- (10.801,4.309) -- (10.882,4.43) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.503,2.156) -- (8.474,2.299) -- (8.393,2.42) -- (8.272,2.501) -- (8.129,2.529) -- (7.987,2.501) -- (7.865,2.42) -- (7.785,2.299) -- (7.756,2.156) -- (7.785,2.013) -- (7.865,1.892) -- (7.987,1.811) -- (8.129,1.783) -- (8.272,1.811) -- (8.393,1.892) -- (8.474,2.013) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.084,2.037) -- (2.056,2.18) -- (1.975,2.301) -- (1.854,2.382) -- (1.711,2.41) -- (1.568,2.382) -- (1.447,2.301) -- (1.366,2.18) -- (1.338,2.037) -- (1.366,1.894) -- (1.447,1.773) -- (1.568,1.692) -- (1.711,1.664) -- (1.854,1.692) -- (1.975,1.773) -- (2.056,1.894) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.866,1.897) -- (10.837,2.04) -- (10.756,2.161) -- (10.635,2.242) -- (10.492,2.271) -- (10.35,2.242) -- (10.228,2.161) -- (10.148,2.04) -- (10.119,1.897) -- (10.148,1.755) -- (10.228,1.634) -- (10.35,1.553) -- (10.492,1.524) -- (10.635,1.553) -- (10.756,1.634) -- (10.837,1.755) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.943,1.536) -- (4.915,1.679) -- (4.834,1.8) -- (4.713,1.881) -- (4.57,1.909) -- (4.427,1.881) -- (4.306,1.8) -- (4.225,1.679) -- (4.197,1.536) -- (4.225,1.393) -- (4.306,1.272) -- (4.427,1.191) -- (4.57,1.163) -- (4.713,1.191) -- (4.834,1.272) -- (4.915,1.393) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.104,9.441) -- (5.223,9.46) -- (5.33,9.515) -- (5.415,9.6) -- (5.47,9.707) -- (5.489,9.826) -- (5.47,9.944) -- (5.415,10.052) -- (5.33,10.137) -- (5.223,10.191) -- (5.104,10.21) -- (5.104,10.21) -- (4.985,10.191) -- (4.878,10.137) -- (4.793,10.052) -- (4.738,9.944) -- (4.72,9.826) -- (4.738,9.707) -- (4.793,9.6) -- (4.878,9.515) -- (4.985,9.46) -- (5.104,9.441) -- cycle;
\node at (5.104,9.826) {\scalebox{1.598}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.131,11.307) -- (3.25,11.326) -- (3.357,11.381) -- (3.442,11.466) -- (3.497,11.573) -- (3.515,11.692) -- (3.497,11.81) -- (3.442,11.918) -- (3.357,12.003) -- (3.25,12.057) -- (3.131,12.076) -- (3.131,12.076) -- (3.012,12.057) -- (2.905,12.003) -- (2.82,11.918) -- (2.765,11.81) -- (2.746,11.692) -- (2.765,11.573) -- (2.82,11.466) -- (2.905,11.381) -- (3.012,11.326) -- (3.131,11.307) -- cycle;
\node at (3.131,11.692) {\scalebox{1.598}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.536,8.921) -- (1.655,8.94) -- (1.762,8.994) -- (1.847,9.079) -- (1.902,9.187) -- (1.921,9.305) -- (1.902,9.424) -- (1.847,9.531) -- (1.762,9.617) -- (1.655,9.671) -- (1.536,9.69) -- (1.536,9.69) -- (1.417,9.671) -- (1.31,9.617) -- (1.225,9.531) -- (1.17,9.424) -- (1.152,9.305) -- (1.17,9.187) -- (1.225,9.079) -- (1.31,8.994) -- (1.417,8.94) -- (1.536,8.921) -- cycle;
\node at (1.536,9.305) {\scalebox{1.598}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.103,8.002) -- (6.222,8.021) -- (6.329,8.076) -- (6.414,8.161) -- (6.469,8.268) -- (6.488,8.387) -- (6.469,8.506) -- (6.414,8.613) -- (6.329,8.698) -- (6.222,8.752) -- (6.103,8.771) -- (6.103,8.771) -- (5.984,8.752) -- (5.877,8.698) -- (5.792,8.613) -- (5.737,8.506) -- (5.719,8.387) -- (5.737,8.268) -- (5.792,8.161) -- (5.877,8.076) -- (5.984,8.021) -- (6.103,8.002) -- cycle;
\node at (6.103,8.387) {\scalebox{1.598}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.048,6.368) -- (5.167,6.387) -- (5.274,6.442) -- (5.359,6.527) -- (5.414,6.634) -- (5.433,6.753) -- (5.414,6.872) -- (5.359,6.979) -- (5.274,7.064) -- (5.167,7.119) -- (5.048,7.137) -- (5.048,7.137) -- (4.929,7.119) -- (4.822,7.064) -- (4.737,6.979) -- (4.682,6.872) -- (4.664,6.753) -- (4.682,6.634) -- (4.737,6.527) -- (4.822,6.442) -- (4.929,6.387) -- (5.048,6.368) -- cycle;
\node at (5.048,6.753) {\scalebox{1.598}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.397,5.786) -- (11.516,5.805) -- (11.623,5.859) -- (11.708,5.945) -- (11.763,6.052) -- (11.781,6.171) -- (11.763,6.289) -- (11.708,6.397) -- (11.623,6.482) -- (11.516,6.536) -- (11.397,6.555) -- (11.397,6.555) -- (11.278,6.536) -- (11.171,6.482) -- (11.086,6.397) -- (11.031,6.289) -- (11.012,6.171) -- (11.031,6.052) -- (11.086,5.945) -- (11.171,5.859) -- (11.278,5.805) -- (11.397,5.786) -- cycle;
\node at (11.397,6.171) {\scalebox{1.598}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.03,7.135) -- (3.149,7.154) -- (3.256,7.209) -- (3.341,7.294) -- (3.395,7.401) -- (3.414,7.52) -- (3.395,7.639) -- (3.341,7.746) -- (3.256,7.831) -- (3.149,7.886) -- (3.03,7.904) -- (3.03,7.904) -- (2.911,7.886) -- (2.804,7.831) -- (2.719,7.746) -- (2.664,7.639) -- (2.645,7.52) -- (2.664,7.401) -- (2.719,7.294) -- (2.804,7.209) -- (2.911,7.154) -- (3.03,7.135) -- cycle;
\node at (3.03,7.52) {\scalebox{1.598}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.234,7.875) -- (9.353,7.894) -- (9.46,7.948) -- (9.545,8.033) -- (9.6,8.141) -- (9.619,8.259) -- (9.6,8.378) -- (9.545,8.485) -- (9.46,8.57) -- (9.353,8.625) -- (9.234,8.644) -- (9.234,8.644) -- (9.115,8.625) -- (9.008,8.57) -- (8.923,8.485) -- (8.868,8.378) -- (8.85,8.259) -- (8.868,8.141) -- (8.923,8.033) -- (9.008,7.948) -- (9.115,7.894) -- (9.234,7.875) -- cycle;
\node at (9.234,8.259) {\scalebox{1.598}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.925,6.003) -- (7.044,6.022) -- (7.151,6.077) -- (7.236,6.162) -- (7.291,6.269) -- (7.31,6.388) -- (7.291,6.506) -- (7.236,6.614) -- (7.151,6.699) -- (7.044,6.753) -- (6.925,6.772) -- (6.925,6.772) -- (6.807,6.753) -- (6.699,6.699) -- (6.614,6.614) -- (6.56,6.506) -- (6.541,6.388) -- (6.56,6.269) -- (6.614,6.162) -- (6.699,6.077) -- (6.807,6.022) -- (6.925,6.003) -- cycle;
\node at (6.925,6.388) {\scalebox{1.598}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.272,5.064) -- (9.391,5.083) -- (9.498,5.137) -- (9.583,5.222) -- (9.638,5.33) -- (9.657,5.448) -- (9.638,5.567) -- (9.583,5.674) -- (9.498,5.76) -- (9.391,5.814) -- (9.272,5.833) -- (8.91,5.833) -- (8.792,5.814) -- (8.684,5.76) -- (8.599,5.674) -- (8.545,5.567) -- (8.526,5.448) -- (8.545,5.33) -- (8.599,5.222) -- (8.684,5.137) -- (8.792,5.083) -- (8.91,5.064) -- cycle;
\node at (9.091,5.448) {\scalebox{1.598}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.193,3.408) -- (8.312,3.427) -- (8.419,3.482) -- (8.504,3.567) -- (8.559,3.674) -- (8.577,3.793) -- (8.559,3.912) -- (8.504,4.019) -- (8.419,4.104) -- (8.312,4.159) -- (8.193,4.177) -- (7.831,4.177) -- (7.712,4.159) -- (7.605,4.104) -- (7.52,4.019) -- (7.465,3.912) -- (7.447,3.793) -- (7.465,3.674) -- (7.52,3.567) -- (7.605,3.482) -- (7.712,3.427) -- (7.831,3.408) -- cycle;
\node at (8.012,3.793) {\scalebox{1.598}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.457,4.529) -- (6.575,4.548) -- (6.683,4.603) -- (6.768,4.688) -- (6.822,4.795) -- (6.841,4.914) -- (6.822,5.033) -- (6.768,5.14) -- (6.683,5.225) -- (6.575,5.28) -- (6.457,5.298) -- (6.095,5.298) -- (5.976,5.28) -- (5.869,5.225) -- (5.784,5.14) -- (5.729,5.033) -- (5.71,4.914) -- (5.729,4.795) -- (5.784,4.688) -- (5.869,4.603) -- (5.976,4.548) -- (6.095,4.529) -- cycle;
\node at (6.276,4.914) {\scalebox{1.598}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.244,2.891) -- (5.362,2.91) -- (5.47,2.965) -- (5.555,3.05) -- (5.609,3.157) -- (5.628,3.276) -- (5.609,3.395) -- (5.555,3.502) -- (5.47,3.587) -- (5.362,3.642) -- (5.244,3.66) -- (4.882,3.66) -- (4.763,3.642) -- (4.656,3.587) -- (4.571,3.502) -- (4.516,3.395) -- (4.497,3.276) -- (4.516,3.157) -- (4.571,3.05) -- (4.656,2.965) -- (4.763,2.91) -- (4.882,2.891) -- cycle;
\node at (5.063,3.276) {\scalebox{1.598}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.191,2.842) -- (11.31,2.861) -- (11.417,2.916) -- (11.502,3.001) -- (11.556,3.108) -- (11.575,3.227) -- (11.556,3.345) -- (11.502,3.453) -- (11.417,3.538) -- (11.31,3.592) -- (11.191,3.611) -- (10.829,3.611) -- (10.71,3.592) -- (10.603,3.538) -- (10.518,3.453) -- (10.463,3.345) -- (10.444,3.227) -- (10.463,3.108) -- (10.518,3.001) -- (10.603,2.916) -- (10.71,2.861) -- (10.829,2.842) -- cycle;
\node at (11.01,3.227) {\scalebox{1.598}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.523,1.927) -- (9.642,1.946) -- (9.749,2.0) -- (9.834,2.085) -- (9.889,2.193) -- (9.908,2.311) -- (9.889,2.43) -- (9.834,2.537) -- (9.749,2.623) -- (9.642,2.677) -- (9.523,2.696) -- (9.161,2.696) -- (9.042,2.677) -- (8.935,2.623) -- (8.85,2.537) -- (8.795,2.43) -- (8.777,2.311) -- (8.795,2.193) -- (8.85,2.085) -- (8.935,2.0) -- (9.042,1.946) -- (9.161,1.927) -- cycle;
\node at (9.342,2.311) {\scalebox{1.598}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.408,1.899) -- (3.527,1.918) -- (3.634,1.972) -- (3.719,2.057) -- (3.774,2.164) -- (3.793,2.283) -- (3.774,2.402) -- (3.719,2.509) -- (3.634,2.594) -- (3.527,2.649) -- (3.408,2.668) -- (3.046,2.668) -- (2.928,2.649) -- (2.82,2.594) -- (2.735,2.509) -- (2.681,2.402) -- (2.662,2.283) -- (2.681,2.164) -- (2.735,2.057) -- (2.82,1.972) -- (2.928,1.918) -- (3.046,1.899) -- cycle;
\node at (3.227,2.283) {\scalebox{1.598}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.892,11.123) -- (5.492,11.123) -- (5.492,11.723) -- (4.892,11.723) -- cycle;
\node at (5.192,11.423) {\scalebox{1.598}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (10.049,5.303) -- (10.649,5.303) -- (10.649,5.903) -- (10.049,5.903) -- cycle;
\node at (10.349,5.603) {\scalebox{1.598}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.067,6.6) -- (8.667,6.6) -- (8.667,7.2) -- (8.067,7.2) -- cycle;
\node at (8.367,6.9) {\scalebox{1.598}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.119,8.064) -- (2.719,8.064) -- (2.719,8.664) -- (2.119,8.664) -- cycle;
\node at (2.419,8.364) {\scalebox{1.598}{\small d}};
\node at (5.096,12.035) {\scalebox{1.598}{\small A}};
\node at (0.79,11.316) {\scalebox{1.598}{\small B}};
\node at (4.19,8.862) {\scalebox{1.598}{\small C}};
\node at (11.796,8.21) {\scalebox{1.598}{\small D}};
\node at (0.765,7.946) {\scalebox{1.598}{\small E}};
\node at (7.789,7.94) {\scalebox{1.598}{\small F}};
\node at (7.978,5.08) {\scalebox{1.598}{\small G}};
\node at (4.606,4.693) {\scalebox{1.598}{\small H}};
\node at (11.133,4.399) {\scalebox{1.598}{\small I}};
\node at (8.43,1.614) {\scalebox{1.598}{\small J}};
\node at (1.246,1.627) {\scalebox{1.598}{\small K}};
\node at (10.954,1.483) {\scalebox{1.598}{\small L}};
\node at (4.368,0.95) {\scalebox{1.598}{\small M}};
\draw[line width=0.5pt] (12.521,10.473) -- (12.521,11.573);
\draw[line width=0.5pt] (12.29,11.188) -- (12.521,11.573) -- (12.752,11.188);
\draw[line width=0.5pt] (12.219,11.023) -- (12.824,11.023);
\node at (12.521,11.958) {\scalebox{1.598}{\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, E, F, H, I, J, K & 7 \\ \hline
clairs & B, C, D, G, L, M & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{7} carrefours sombres et \textbf{6} clairs. Il y en a donc \textbf{1 de plus} du côté des sombres.}\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 13 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 7 carrefours sombres et 6 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{A} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend le \textbf{chemin du Lavoir ②} vers l'ouest jusqu'à Fontaine Notre-Dame \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ③}, vers le sud, jusqu'à Place Sainte-Croix \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Écoles ⑦}, vers l'est, jusqu'à Carrefour Saint-Roch \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Haute ④}, vers l'est, jusqu'à Rond-point de la Gare \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Jardins ⑧}, vers l'est, jusqu'à Place du Marché \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin Creux ⑥}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Stade ⑭}, vers le sud, jusqu'à Place aux Herbes \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Verte ⑮}, vers l'ouest, jusqu'à Place de l'Église \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Pasteur ⑪}, vers le nord, jusqu'à Carrefour du Moulin \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{quai aux Fleurs ⑫}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin des Vignes ⑬}, vers le sud, jusqu'à Place du Vieux-Puits \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{impasse Sainte-Anne ⑯}, vers l'ouest, jusqu'à Placette des Ormes \textbf{(K)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 13 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}
