\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 332}

\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=332}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0332}
\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=0.6099, line join=round, line cap=round]
\draw[line width=0.5pt] (4.714,11.044) -- (4.965,10.972) -- (5.218,10.907) -- (5.473,10.847) -- (5.729,10.794) -- (5.986,10.747) -- (6.244,10.705) -- (6.503,10.67) -- (6.763,10.641) -- (7.024,10.617) -- (7.285,10.6) -- (7.546,10.59) -- (7.807,10.585) -- (8.069,10.586) -- (8.33,10.594);
\draw[line width=0.5pt] (2.068,11.016) -- (2.256,11.047) -- (2.443,11.074) -- (2.632,11.096) -- (2.821,11.113) -- (3.01,11.126) -- (3.2,11.135) -- (3.39,11.139) -- (3.58,11.139) -- (3.77,11.134) -- (3.959,11.125) -- (4.149,11.111) -- (4.337,11.093) -- (4.526,11.071) -- (4.714,11.044);
\draw[line width=0.5pt] (8.33,10.594) -- (8.244,10.45) -- (8.161,10.303) -- (8.081,10.155) -- (8.005,10.005) -- (7.933,9.853) -- (7.864,9.699) -- (7.798,9.544) -- (7.737,9.387) -- (7.679,9.229) -- (7.625,9.069) -- (7.574,8.909) -- (7.528,8.747) -- (7.485,8.584) -- (7.446,8.42);
\draw[line width=0.5pt] (7.446,8.42) -- (7.71,8.416) -- (7.974,8.404) -- (8.238,8.387) -- (8.501,8.363) -- (8.764,8.334) -- (9.026,8.298) -- (9.287,8.256) -- (9.547,8.208) -- (9.806,8.153) -- (10.063,8.093) -- (10.319,8.027) -- (10.573,7.954) -- (10.825,7.876) -- (11.076,7.792);
\draw[line width=0.5pt] (7.446,8.42) -- (7.449,8.153) -- (7.459,7.886) -- (7.475,7.619) -- (7.498,7.353) -- (7.526,7.087) -- (7.561,6.822) -- (7.602,6.558) -- (7.649,6.294) -- (7.703,6.032) -- (7.762,5.771) -- (7.828,5.512) -- (7.899,5.255) -- (7.977,4.999) -- (8.061,4.745);
\draw[line width=0.5pt] (11.076,7.792) -- (11.125,7.548) -- (11.169,7.303) -- (11.208,7.057) -- (11.24,6.81) -- (11.267,6.562) -- (11.287,6.314) -- (11.302,6.065) -- (11.311,5.816) -- (11.315,5.567) -- (11.312,5.318) -- (11.303,5.069) -- (11.289,4.821) -- (11.269,4.573) -- (11.243,4.325);
\draw[line width=0.5pt] (2.049,5.301) -- (2.25,5.226) -- (2.453,5.156) -- (2.657,5.09) -- (2.863,5.03) -- (3.07,4.974) -- (3.279,4.923) -- (3.489,4.878) -- (3.699,4.837) -- (3.911,4.8) -- (4.123,4.769) -- (4.336,4.743) -- (4.55,4.722) -- (4.764,4.707) -- (4.978,4.696);
\draw[line width=0.5pt] (2.049,5.301) -- (2.035,5.012) -- (2.015,4.723) -- (1.987,4.435) -- (1.953,4.147) -- (1.912,3.86) -- (1.864,3.574) -- (1.809,3.29) -- (1.748,3.007) -- (1.68,2.725) -- (1.606,2.445) -- (1.525,2.167) -- (1.437,1.89) -- (1.343,1.617) -- (1.243,1.345);
\draw[line width=0.5pt] (8.061,4.745) -- (8.282,4.681) -- (8.504,4.621) -- (8.728,4.567) -- (8.953,4.519) -- (9.179,4.475) -- (9.407,4.437) -- (9.635,4.404) -- (9.863,4.377) -- (10.093,4.354) -- (10.322,4.338) -- (10.552,4.326) -- (10.783,4.321) -- (11.013,4.32) -- (11.243,4.325);
\draw[line width=0.5pt] (8.061,4.745) -- (8.111,4.522) -- (8.155,4.297) -- (8.194,4.071) -- (8.228,3.844) -- (8.256,3.616) -- (8.279,3.388) -- (8.297,3.159) -- (8.309,2.93) -- (8.316,2.7) -- (8.317,2.471) -- (8.314,2.241) -- (8.304,2.012) -- (8.29,1.783) -- (8.269,1.555);
\draw[line width=0.5pt] (4.978,4.696) -- (5.197,4.665) -- (5.417,4.64) -- (5.637,4.621) -- (5.858,4.606) -- (6.079,4.597) -- (6.3,4.592) -- (6.522,4.593) -- (6.743,4.6) -- (6.964,4.611) -- (7.185,4.627) -- (7.405,4.649) -- (7.624,4.676) -- (7.843,4.708) -- (8.061,4.745);
\draw[line width=0.5pt] (4.978,4.696) -- (5.015,4.514) -- (5.048,4.331) -- (5.076,4.147) -- (5.101,3.962) -- (5.12,3.777) -- (5.136,3.592) -- (5.147,3.406) -- (5.154,3.22) -- (5.156,3.034) -- (5.154,2.848) -- (5.148,2.662) -- (5.137,2.477) -- (5.122,2.291) -- (5.102,2.107);
\draw[line width=0.5pt] (11.243,4.325) -- (11.175,4.145) -- (11.111,3.963) -- (11.051,3.78) -- (10.996,3.596) -- (10.945,3.41) -- (10.899,3.223) -- (10.857,3.035) -- (10.819,2.846) -- (10.786,2.656) -- (10.758,2.465) -- (10.733,2.274) -- (10.714,2.082) -- (10.698,1.89) -- (10.688,1.698);
\draw[line width=0.5pt] (5.102,2.107) -- (5.333,2.102) -- (5.563,2.092) -- (5.793,2.077) -- (6.023,2.056) -- (6.252,2.03) -- (6.481,1.998) -- (6.709,1.961) -- (6.936,1.919) -- (7.161,1.872) -- (7.386,1.819) -- (7.609,1.761) -- (7.831,1.697) -- (8.051,1.629) -- (8.269,1.555);
\draw[line width=0.5pt] (8.269,1.555) -- (8.442,1.539) -- (8.616,1.526) -- (8.789,1.518) -- (8.963,1.514) -- (9.137,1.514) -- (9.311,1.519) -- (9.485,1.527) -- (9.658,1.539) -- (9.831,1.555) -- (10.004,1.576) -- (10.176,1.6) -- (10.347,1.629) -- (10.518,1.662) -- (10.688,1.698);
\draw[line width=0.5pt] (1.243,1.345) -- (1.508,1.441) -- (1.775,1.531) -- (2.045,1.615) -- (2.317,1.692) -- (2.59,1.763) -- (2.865,1.827) -- (3.141,1.885) -- (3.419,1.936) -- (3.698,1.981) -- (3.977,2.02) -- (4.258,2.051) -- (4.539,2.076) -- (4.821,2.095) -- (5.102,2.107);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.874,11.044) -- (4.852,11.124) -- (4.794,11.182) -- (4.714,11.204) -- (4.634,11.182) -- (4.575,11.124) -- (4.554,11.044) -- (4.575,10.964) -- (4.634,10.905) -- (4.714,10.884) -- (4.794,10.905) -- (4.852,10.964) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.228,11.016) -- (2.207,11.096) -- (2.148,11.155) -- (2.068,11.176) -- (1.988,11.155) -- (1.93,11.096) -- (1.908,11.016) -- (1.93,10.936) -- (1.988,10.878) -- (2.068,10.856) -- (2.148,10.878) -- (2.207,10.936) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.49,10.594) -- (8.469,10.674) -- (8.41,10.733) -- (8.33,10.754) -- (8.25,10.733) -- (8.191,10.674) -- (8.17,10.594) -- (8.191,10.514) -- (8.25,10.456) -- (8.33,10.434) -- (8.41,10.456) -- (8.469,10.514) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.606,8.42) -- (7.585,8.5) -- (7.526,8.559) -- (7.446,8.58) -- (7.366,8.559) -- (7.308,8.5) -- (7.286,8.42) -- (7.308,8.34) -- (7.366,8.282) -- (7.446,8.26) -- (7.526,8.282) -- (7.585,8.34) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.236,7.792) -- (11.214,7.872) -- (11.156,7.93) -- (11.076,7.952) -- (10.996,7.93) -- (10.937,7.872) -- (10.916,7.792) -- (10.937,7.712) -- (10.996,7.653) -- (11.076,7.632) -- (11.156,7.653) -- (11.214,7.712) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.209,5.301) -- (2.188,5.381) -- (2.129,5.44) -- (2.049,5.461) -- (1.969,5.44) -- (1.911,5.381) -- (1.889,5.301) -- (1.911,5.221) -- (1.969,5.162) -- (2.049,5.141) -- (2.129,5.162) -- (2.188,5.221) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.221,4.745) -- (8.2,4.825) -- (8.141,4.884) -- (8.061,4.905) -- (7.981,4.884) -- (7.922,4.825) -- (7.901,4.745) -- (7.922,4.665) -- (7.981,4.607) -- (8.061,4.585) -- (8.141,4.607) -- (8.2,4.665) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.138,4.696) -- (5.117,4.776) -- (5.058,4.834) -- (4.978,4.856) -- (4.898,4.834) -- (4.839,4.776) -- (4.818,4.696) -- (4.839,4.616) -- (4.898,4.557) -- (4.978,4.536) -- (5.058,4.557) -- (5.117,4.616) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.403,4.325) -- (11.381,4.405) -- (11.323,4.464) -- (11.243,4.485) -- (11.163,4.464) -- (11.104,4.405) -- (11.083,4.325) -- (11.104,4.245) -- (11.163,4.187) -- (11.243,4.165) -- (11.323,4.187) -- (11.381,4.245) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.262,2.107) -- (5.241,2.187) -- (5.182,2.245) -- (5.102,2.267) -- (5.022,2.245) -- (4.964,2.187) -- (4.942,2.107) -- (4.964,2.027) -- (5.022,1.968) -- (5.102,1.947) -- (5.182,1.968) -- (5.241,2.027) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.848,1.698) -- (10.826,1.778) -- (10.768,1.837) -- (10.688,1.858) -- (10.608,1.837) -- (10.549,1.778) -- (10.528,1.698) -- (10.549,1.618) -- (10.608,1.56) -- (10.688,1.538) -- (10.768,1.56) -- (10.826,1.618) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.429,1.555) -- (8.408,1.635) -- (8.349,1.693) -- (8.269,1.715) -- (8.189,1.693) -- (8.131,1.635) -- (8.109,1.555) -- (8.131,1.475) -- (8.189,1.416) -- (8.269,1.395) -- (8.349,1.416) -- (8.408,1.475) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.403,1.345) -- (1.381,1.425) -- (1.323,1.484) -- (1.243,1.505) -- (1.163,1.484) -- (1.104,1.425) -- (1.083,1.345) -- (1.104,1.265) -- (1.163,1.207) -- (1.243,1.185) -- (1.323,1.207) -- (1.381,1.265) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.552,10.667) -- (6.674,10.686) -- (6.783,10.742) -- (6.87,10.829) -- (6.926,10.939) -- (6.946,11.061) -- (6.926,11.182) -- (6.87,11.292) -- (6.783,11.379) -- (6.674,11.435) -- (6.552,11.455) -- (6.552,11.455) -- (6.43,11.435) -- (6.32,11.379) -- (6.233,11.292) -- (6.177,11.182) -- (6.158,11.061) -- (6.177,10.939) -- (6.233,10.829) -- (6.32,10.742) -- (6.43,10.686) -- (6.552,10.667) -- cycle;
\node at (6.552,11.061) {\scalebox{1.64}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.386,11.139) -- (3.507,11.159) -- (3.617,11.214) -- (3.704,11.302) -- (3.76,11.411) -- (3.78,11.533) -- (3.76,11.655) -- (3.704,11.765) -- (3.617,11.852) -- (3.507,11.908) -- (3.386,11.927) -- (3.386,11.927) -- (3.264,11.908) -- (3.154,11.852) -- (3.067,11.765) -- (3.011,11.655) -- (2.992,11.533) -- (3.011,11.411) -- (3.067,11.302) -- (3.154,11.214) -- (3.264,11.159) -- (3.386,11.139) -- cycle;
\node at (3.386,11.533) {\scalebox{1.64}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.354,9.527) -- (7.475,9.546) -- (7.585,9.602) -- (7.672,9.689) -- (7.728,9.799) -- (7.747,9.92) -- (7.728,10.042) -- (7.672,10.152) -- (7.585,10.239) -- (7.475,10.295) -- (7.354,10.314) -- (7.354,10.314) -- (7.232,10.295) -- (7.122,10.239) -- (7.035,10.152) -- (6.979,10.042) -- (6.96,9.92) -- (6.979,9.799) -- (7.035,9.689) -- (7.122,9.602) -- (7.232,9.546) -- (7.354,9.527) -- cycle;
\node at (7.354,9.92) {\scalebox{1.64}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.354,8.25) -- (9.476,8.269) -- (9.586,8.325) -- (9.673,8.412) -- (9.729,8.522) -- (9.748,8.644) -- (9.729,8.766) -- (9.673,8.875) -- (9.586,8.962) -- (9.476,9.018) -- (9.354,9.038) -- (9.354,9.038) -- (9.232,9.018) -- (9.123,8.962) -- (9.035,8.875) -- (8.979,8.766) -- (8.96,8.644) -- (8.979,8.522) -- (9.035,8.412) -- (9.123,8.325) -- (9.232,8.269) -- (9.354,8.25) -- cycle;
\node at (9.354,8.644) {\scalebox{1.64}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.99,6.229) -- (8.112,6.248) -- (8.222,6.304) -- (8.309,6.391) -- (8.365,6.501) -- (8.384,6.623) -- (8.365,6.744) -- (8.309,6.854) -- (8.222,6.941) -- (8.112,6.997) -- (7.99,7.016) -- (7.99,7.016) -- (7.869,6.997) -- (7.759,6.941) -- (7.672,6.854) -- (7.616,6.744) -- (7.597,6.623) -- (7.616,6.501) -- (7.672,6.391) -- (7.759,6.304) -- (7.869,6.248) -- (7.99,6.229) -- cycle;
\node at (7.99,6.623) {\scalebox{1.64}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.696,5.69) -- (11.817,5.71) -- (11.927,5.766) -- (12.014,5.853) -- (12.07,5.963) -- (12.09,6.084) -- (12.07,6.206) -- (12.014,6.316) -- (11.927,6.403) -- (11.817,6.459) -- (11.696,6.478) -- (11.696,6.478) -- (11.574,6.459) -- (11.464,6.403) -- (11.377,6.316) -- (11.321,6.206) -- (11.302,6.084) -- (11.321,5.963) -- (11.377,5.853) -- (11.464,5.766) -- (11.574,5.71) -- (11.696,5.69) -- cycle;
\node at (11.696,6.084) {\scalebox{1.64}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.568,4.869) -- (3.69,4.889) -- (3.8,4.945) -- (3.887,5.032) -- (3.943,5.142) -- (3.962,5.263) -- (3.943,5.385) -- (3.887,5.495) -- (3.8,5.582) -- (3.69,5.638) -- (3.568,5.657) -- (3.568,5.657) -- (3.447,5.638) -- (3.337,5.582) -- (3.25,5.495) -- (3.194,5.385) -- (3.174,5.263) -- (3.194,5.142) -- (3.25,5.032) -- (3.337,4.945) -- (3.447,4.889) -- (3.568,4.869) -- cycle;
\node at (3.568,5.263) {\scalebox{1.64}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.195,2.817) -- (2.317,2.836) -- (2.427,2.892) -- (2.514,2.98) -- (2.57,3.089) -- (2.589,3.211) -- (2.57,3.333) -- (2.514,3.443) -- (2.427,3.53) -- (2.317,3.586) -- (2.195,3.605) -- (2.195,3.605) -- (2.073,3.586) -- (1.964,3.53) -- (1.877,3.443) -- (1.821,3.333) -- (1.801,3.211) -- (1.821,3.089) -- (1.877,2.98) -- (1.964,2.892) -- (2.073,2.836) -- (2.195,2.817) -- cycle;
\node at (2.195,3.211) {\scalebox{1.64}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.686,4.401) -- (9.808,4.42) -- (9.918,4.476) -- (10.005,4.563) -- (10.061,4.673) -- (10.08,4.795) -- (10.061,4.916) -- (10.005,5.026) -- (9.918,5.113) -- (9.808,5.169) -- (9.686,5.188) -- (9.686,5.188) -- (9.564,5.169) -- (9.455,5.113) -- (9.367,5.026) -- (9.312,4.916) -- (9.292,4.795) -- (9.312,4.673) -- (9.367,4.563) -- (9.455,4.476) -- (9.564,4.42) -- (9.686,4.401) -- cycle;
\node at (9.686,4.795) {\scalebox{1.64}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.875,2.791) -- (8.997,2.81) -- (9.107,2.866) -- (9.194,2.953) -- (9.25,3.063) -- (9.269,3.184) -- (9.25,3.306) -- (9.194,3.416) -- (9.107,3.503) -- (8.997,3.559) -- (8.875,3.578) -- (8.505,3.578) -- (8.383,3.559) -- (8.273,3.503) -- (8.186,3.416) -- (8.13,3.306) -- (8.111,3.184) -- (8.13,3.063) -- (8.186,2.953) -- (8.273,2.866) -- (8.383,2.81) -- (8.505,2.791) -- cycle;
\node at (8.69,3.184) {\scalebox{1.64}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.701,4.593) -- (6.822,4.613) -- (6.932,4.669) -- (7.019,4.756) -- (7.075,4.865) -- (7.094,4.987) -- (7.075,5.109) -- (7.019,5.219) -- (6.932,5.306) -- (6.822,5.362) -- (6.701,5.381) -- (6.33,5.381) -- (6.208,5.362) -- (6.098,5.306) -- (6.011,5.219) -- (5.955,5.109) -- (5.936,4.987) -- (5.955,4.865) -- (6.011,4.756) -- (6.098,4.669) -- (6.208,4.613) -- (6.33,4.593) -- cycle;
\node at (6.515,4.987) {\scalebox{1.64}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.726,3.031) -- (5.848,3.051) -- (5.957,3.106) -- (6.044,3.194) -- (6.1,3.303) -- (6.12,3.425) -- (6.1,3.547) -- (6.044,3.657) -- (5.957,3.744) -- (5.848,3.8) -- (5.726,3.819) -- (5.355,3.819) -- (5.233,3.8) -- (5.124,3.744) -- (5.036,3.657) -- (4.98,3.547) -- (4.961,3.425) -- (4.98,3.303) -- (5.036,3.194) -- (5.124,3.106) -- (5.233,3.051) -- (5.355,3.031) -- cycle;
\node at (5.54,3.425) {\scalebox{1.64}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.428,2.559) -- (11.549,2.579) -- (11.659,2.635) -- (11.746,2.722) -- (11.802,2.831) -- (11.822,2.953) -- (11.802,3.075) -- (11.746,3.185) -- (11.659,3.272) -- (11.549,3.328) -- (11.428,3.347) -- (11.057,3.347) -- (10.935,3.328) -- (10.825,3.272) -- (10.738,3.185) -- (10.682,3.075) -- (10.663,2.953) -- (10.682,2.831) -- (10.738,2.722) -- (10.825,2.635) -- (10.935,2.579) -- (11.057,2.559) -- cycle;
\node at (11.242,2.953) {\scalebox{1.64}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.962,1.956) -- (7.083,1.975) -- (7.193,2.031) -- (7.28,2.118) -- (7.336,2.228) -- (7.356,2.349) -- (7.336,2.471) -- (7.28,2.581) -- (7.193,2.668) -- (7.083,2.724) -- (6.962,2.743) -- (6.591,2.743) -- (6.469,2.724) -- (6.359,2.668) -- (6.272,2.581) -- (6.216,2.471) -- (6.197,2.349) -- (6.216,2.228) -- (6.272,2.118) -- (6.359,2.031) -- (6.469,1.975) -- (6.591,1.956) -- cycle;
\node at (6.776,2.349) {\scalebox{1.64}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.647,1.526) -- (9.768,1.545) -- (9.878,1.601) -- (9.965,1.688) -- (10.021,1.798) -- (10.041,1.92) -- (10.021,2.042) -- (9.965,2.152) -- (9.878,2.239) -- (9.768,2.295) -- (9.647,2.314) -- (9.276,2.314) -- (9.154,2.295) -- (9.044,2.239) -- (8.957,2.152) -- (8.901,2.042) -- (8.882,1.92) -- (8.901,1.798) -- (8.957,1.688) -- (9.044,1.601) -- (9.154,1.545) -- (9.276,1.526) -- cycle;
\node at (9.461,1.92) {\scalebox{1.64}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.25,1.878) -- (3.372,1.897) -- (3.482,1.953) -- (3.569,2.04) -- (3.625,2.15) -- (3.644,2.271) -- (3.625,2.393) -- (3.569,2.503) -- (3.482,2.59) -- (3.372,2.646) -- (3.25,2.665) -- (2.88,2.665) -- (2.758,2.646) -- (2.648,2.59) -- (2.561,2.503) -- (2.505,2.393) -- (2.486,2.271) -- (2.505,2.15) -- (2.561,2.04) -- (2.648,1.953) -- (2.758,1.897) -- (2.88,1.878) -- cycle;
\node at (3.065,2.271) {\scalebox{1.64}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (10.776,7.492) -- (11.376,7.492) -- (11.376,8.092) -- (10.776,8.092) -- cycle;
\node at (11.076,7.792) {\scalebox{1.64}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.737,1.643) -- (2.337,1.643) -- (2.337,2.243) -- (1.737,2.243) -- cycle;
\node at (2.037,1.943) {\scalebox{1.64}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (10.619,4.278) -- (11.219,4.278) -- (11.219,4.878) -- (10.619,4.878) -- cycle;
\node at (10.919,4.578) {\scalebox{1.64}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (2.395,11.846) -- (2.995,11.846) -- (2.995,12.446) -- (2.395,12.446) -- cycle;
\node at (2.695,12.146) {\scalebox{1.64}{\small d}};
\node at (4.51,11.629) {\scalebox{1.64}{\small A}};
\node at (1.667,11.489) {\scalebox{1.64}{\small B}};
\node at (8.542,11.177) {\scalebox{1.64}{\small C}};
\node at (7.641,9.009) {\scalebox{1.64}{\small D}};
\node at (11.64,8.048) {\scalebox{1.64}{\small E}};
\node at (1.432,5.241) {\scalebox{1.64}{\small F}};
\node at (8.578,4.403) {\scalebox{1.64}{\small G}};
\node at (4.461,4.354) {\scalebox{1.64}{\small H}};
\node at (11.836,4.146) {\scalebox{1.64}{\small I}};
\node at (4.872,1.531) {\scalebox{1.64}{\small J}};
\node at (11.131,1.264) {\scalebox{1.64}{\small K}};
\node at (8.504,0.981) {\scalebox{1.64}{\small L}};
\node at (0.765,0.95) {\scalebox{1.64}{\small M}};
\draw[line width=0.5pt] (12.543,10.094) -- (12.543,11.194);
\draw[line width=0.5pt] (12.312,10.809) -- (12.543,11.194) -- (12.774,10.809);
\draw[line width=0.5pt] (12.24,10.644) -- (12.845,10.644);
\node at (12.543,11.579) {\scalebox{1.64}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour E, 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 quai aux Fleurs & Ⓐ Place du Vieux-Puits & Ⓙ Carrefour du Lavoir \\ \hline
① passage du Marché & ⑩ chemin Creux & Ⓑ Place Sainte-Croix & Ⓚ Place aux Herbes \\ \hline
② impasse du Vieux-Four & ⑪ rue Neuve & Ⓒ Fontaine Notre-Dame & Ⓛ Rond-point des Vaches \\ \hline
③ avenue de la Gare & ⑫ allée des Peupliers & Ⓓ Carrefour des Tilleuls & Ⓜ Place de l'Église \\ \hline
④ rue du Four & ⑬ rue de la Mairie & Ⓔ Placette des Ormes & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ rue Basse & ⑭ sentier des Prés & Ⓕ Place de la Fontaine & \textbf{a} l'école \\ \hline
⑥ chemin des Vignes & ⑮ rue Verte & Ⓖ Carrefour du Moulin & \textbf{b} la pharmacie \\ \hline
⑦ rue des Lilas & ⑯ allée des Tilleuls & Ⓗ Carrefour Saint-Roch & \textbf{c} la bibliothèque \\ \hline
⑧ venelle du Chat & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Rond-point du Pont & \textbf{d} l'église \\ \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=0.6248, line join=round, line cap=round]
\draw[line width=0.5pt] (4.714,11.044) -- (4.965,10.972) -- (5.218,10.907) -- (5.473,10.847) -- (5.729,10.794) -- (5.986,10.747) -- (6.244,10.705) -- (6.503,10.67) -- (6.763,10.641) -- (7.024,10.617) -- (7.285,10.6) -- (7.546,10.59) -- (7.807,10.585) -- (8.069,10.586) -- (8.33,10.594);
\draw[line width=0.5pt] (2.068,11.016) -- (2.256,11.047) -- (2.443,11.074) -- (2.632,11.096) -- (2.821,11.113) -- (3.01,11.126) -- (3.2,11.135) -- (3.39,11.139) -- (3.58,11.139) -- (3.77,11.134) -- (3.959,11.125) -- (4.149,11.111) -- (4.337,11.093) -- (4.526,11.071) -- (4.714,11.044);
\draw[line width=0.5pt] (8.33,10.594) -- (8.244,10.45) -- (8.161,10.303) -- (8.081,10.155) -- (8.005,10.005) -- (7.933,9.853) -- (7.864,9.699) -- (7.798,9.544) -- (7.737,9.387) -- (7.679,9.229) -- (7.625,9.069) -- (7.574,8.909) -- (7.528,8.747) -- (7.485,8.584) -- (7.446,8.42);
\draw[line width=0.5pt] (7.446,8.42) -- (7.71,8.416) -- (7.974,8.404) -- (8.238,8.387) -- (8.501,8.363) -- (8.764,8.334) -- (9.026,8.298) -- (9.287,8.256) -- (9.547,8.208) -- (9.806,8.153) -- (10.063,8.093) -- (10.319,8.027) -- (10.573,7.954) -- (10.825,7.876) -- (11.076,7.792);
\draw[line width=0.5pt] (7.446,8.42) -- (7.449,8.153) -- (7.459,7.886) -- (7.475,7.619) -- (7.498,7.353) -- (7.526,7.087) -- (7.561,6.822) -- (7.602,6.558) -- (7.649,6.294) -- (7.703,6.032) -- (7.762,5.771) -- (7.828,5.512) -- (7.899,5.255) -- (7.977,4.999) -- (8.061,4.745);
\draw[line width=0.5pt] (11.076,7.792) -- (11.125,7.548) -- (11.169,7.303) -- (11.208,7.057) -- (11.24,6.81) -- (11.267,6.562) -- (11.287,6.314) -- (11.302,6.065) -- (11.311,5.816) -- (11.315,5.567) -- (11.312,5.318) -- (11.303,5.069) -- (11.289,4.821) -- (11.269,4.573) -- (11.243,4.325);
\draw[line width=0.5pt] (2.049,5.301) -- (2.25,5.226) -- (2.453,5.156) -- (2.657,5.09) -- (2.863,5.03) -- (3.07,4.974) -- (3.279,4.923) -- (3.489,4.878) -- (3.699,4.837) -- (3.911,4.8) -- (4.123,4.769) -- (4.336,4.743) -- (4.55,4.722) -- (4.764,4.707) -- (4.978,4.696);
\draw[line width=0.5pt] (2.049,5.301) -- (2.035,5.012) -- (2.015,4.723) -- (1.987,4.435) -- (1.953,4.147) -- (1.912,3.86) -- (1.864,3.574) -- (1.809,3.29) -- (1.748,3.007) -- (1.68,2.725) -- (1.606,2.445) -- (1.525,2.167) -- (1.437,1.89) -- (1.343,1.617) -- (1.243,1.345);
\draw[line width=0.5pt] (8.061,4.745) -- (8.282,4.681) -- (8.504,4.621) -- (8.728,4.567) -- (8.953,4.519) -- (9.179,4.475) -- (9.407,4.437) -- (9.635,4.404) -- (9.863,4.377) -- (10.093,4.354) -- (10.322,4.338) -- (10.552,4.326) -- (10.783,4.321) -- (11.013,4.32) -- (11.243,4.325);
\draw[line width=0.5pt] (8.061,4.745) -- (8.111,4.522) -- (8.155,4.297) -- (8.194,4.071) -- (8.228,3.844) -- (8.256,3.616) -- (8.279,3.388) -- (8.297,3.159) -- (8.309,2.93) -- (8.316,2.7) -- (8.317,2.471) -- (8.314,2.241) -- (8.304,2.012) -- (8.29,1.783) -- (8.269,1.555);
\draw[line width=0.5pt] (4.978,4.696) -- (5.197,4.665) -- (5.417,4.64) -- (5.637,4.621) -- (5.858,4.606) -- (6.079,4.597) -- (6.3,4.592) -- (6.522,4.593) -- (6.743,4.6) -- (6.964,4.611) -- (7.185,4.627) -- (7.405,4.649) -- (7.624,4.676) -- (7.843,4.708) -- (8.061,4.745);
\draw[line width=0.5pt] (4.978,4.696) -- (5.015,4.514) -- (5.048,4.331) -- (5.076,4.147) -- (5.101,3.962) -- (5.12,3.777) -- (5.136,3.592) -- (5.147,3.406) -- (5.154,3.22) -- (5.156,3.034) -- (5.154,2.848) -- (5.148,2.662) -- (5.137,2.477) -- (5.122,2.291) -- (5.102,2.107);
\draw[line width=0.5pt] (11.243,4.325) -- (11.175,4.145) -- (11.111,3.963) -- (11.051,3.78) -- (10.996,3.596) -- (10.945,3.41) -- (10.899,3.223) -- (10.857,3.035) -- (10.819,2.846) -- (10.786,2.656) -- (10.758,2.465) -- (10.733,2.274) -- (10.714,2.082) -- (10.698,1.89) -- (10.688,1.698);
\draw[line width=0.5pt] (5.102,2.107) -- (5.333,2.102) -- (5.563,2.092) -- (5.793,2.077) -- (6.023,2.056) -- (6.252,2.03) -- (6.481,1.998) -- (6.709,1.961) -- (6.936,1.919) -- (7.161,1.872) -- (7.386,1.819) -- (7.609,1.761) -- (7.831,1.697) -- (8.051,1.629) -- (8.269,1.555);
\draw[line width=0.5pt] (8.269,1.555) -- (8.442,1.539) -- (8.616,1.526) -- (8.789,1.518) -- (8.963,1.514) -- (9.137,1.514) -- (9.311,1.519) -- (9.485,1.527) -- (9.658,1.539) -- (9.831,1.555) -- (10.004,1.576) -- (10.176,1.6) -- (10.347,1.629) -- (10.518,1.662) -- (10.688,1.698);
\draw[line width=0.5pt] (1.243,1.345) -- (1.508,1.441) -- (1.775,1.531) -- (2.045,1.615) -- (2.317,1.692) -- (2.59,1.763) -- (2.865,1.827) -- (3.141,1.885) -- (3.419,1.936) -- (3.698,1.981) -- (3.977,2.02) -- (4.258,2.051) -- (4.539,2.076) -- (4.821,2.095) -- (5.102,2.107);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.087,11.044) -- (5.058,11.187) -- (4.977,11.308) -- (4.856,11.389) -- (4.714,11.417) -- (4.571,11.389) -- (4.45,11.308) -- (4.369,11.187) -- (4.34,11.044) -- (4.369,10.901) -- (4.45,10.78) -- (4.571,10.699) -- (4.714,10.671) -- (4.856,10.699) -- (4.977,10.78) -- (5.058,10.901) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.442,11.016) -- (2.413,11.159) -- (2.332,11.28) -- (2.211,11.361) -- (2.068,11.39) -- (1.926,11.361) -- (1.805,11.28) -- (1.724,11.159) -- (1.695,11.016) -- (1.724,10.874) -- (1.805,10.753) -- (1.926,10.672) -- (2.068,10.643) -- (2.211,10.672) -- (2.332,10.753) -- (2.413,10.874) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.703,10.594) -- (8.675,10.737) -- (8.594,10.858) -- (8.473,10.939) -- (8.33,10.967) -- (8.187,10.939) -- (8.066,10.858) -- (7.985,10.737) -- (7.957,10.594) -- (7.985,10.451) -- (8.066,10.33) -- (8.187,10.249) -- (8.33,10.221) -- (8.473,10.249) -- (8.594,10.33) -- (8.675,10.451) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.819,8.42) -- (7.791,8.563) -- (7.71,8.684) -- (7.589,8.765) -- (7.446,8.794) -- (7.303,8.765) -- (7.182,8.684) -- (7.101,8.563) -- (7.073,8.42) -- (7.101,8.278) -- (7.182,8.156) -- (7.303,8.076) -- (7.446,8.047) -- (7.589,8.076) -- (7.71,8.156) -- (7.791,8.278) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.449,7.792) -- (11.42,7.934) -- (11.34,8.055) -- (11.218,8.136) -- (11.076,8.165) -- (10.933,8.136) -- (10.812,8.055) -- (10.731,7.934) -- (10.702,7.792) -- (10.731,7.649) -- (10.812,7.528) -- (10.933,7.447) -- (11.076,7.418) -- (11.218,7.447) -- (11.34,7.528) -- (11.42,7.649) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.422,5.301) -- (2.394,5.444) -- (2.313,5.565) -- (2.192,5.646) -- (2.049,5.674) -- (1.906,5.646) -- (1.785,5.565) -- (1.704,5.444) -- (1.676,5.301) -- (1.704,5.158) -- (1.785,5.037) -- (1.906,4.956) -- (2.049,4.928) -- (2.192,4.956) -- (2.313,5.037) -- (2.394,5.158) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.434,4.745) -- (8.406,4.888) -- (8.325,5.009) -- (8.204,5.09) -- (8.061,5.119) -- (7.918,5.09) -- (7.797,5.009) -- (7.716,4.888) -- (7.688,4.745) -- (7.716,4.603) -- (7.797,4.482) -- (7.918,4.401) -- (8.061,4.372) -- (8.204,4.401) -- (8.325,4.482) -- (8.406,4.603) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.351,4.696) -- (5.323,4.838) -- (5.242,4.96) -- (5.121,5.04) -- (4.978,5.069) -- (4.835,5.04) -- (4.714,4.96) -- (4.633,4.838) -- (4.605,4.696) -- (4.633,4.553) -- (4.714,4.432) -- (4.835,4.351) -- (4.978,4.322) -- (5.121,4.351) -- (5.242,4.432) -- (5.323,4.553) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.616,4.325) -- (11.588,4.468) -- (11.507,4.589) -- (11.386,4.67) -- (11.243,4.698) -- (11.1,4.67) -- (10.979,4.589) -- (10.898,4.468) -- (10.87,4.325) -- (10.898,4.182) -- (10.979,4.061) -- (11.1,3.98) -- (11.243,3.952) -- (11.386,3.98) -- (11.507,4.061) -- (11.588,4.182) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.476,2.107) -- (5.447,2.249) -- (5.366,2.37) -- (5.245,2.451) -- (5.102,2.48) -- (4.96,2.451) -- (4.838,2.37) -- (4.758,2.249) -- (4.729,2.107) -- (4.758,1.964) -- (4.838,1.843) -- (4.96,1.762) -- (5.102,1.733) -- (5.245,1.762) -- (5.366,1.843) -- (5.447,1.964) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.061,1.698) -- (11.033,1.841) -- (10.952,1.962) -- (10.831,2.043) -- (10.688,2.071) -- (10.545,2.043) -- (10.424,1.962) -- (10.343,1.841) -- (10.315,1.698) -- (10.343,1.555) -- (10.424,1.434) -- (10.545,1.353) -- (10.688,1.325) -- (10.831,1.353) -- (10.952,1.434) -- (11.033,1.555) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.643,1.555) -- (8.614,1.698) -- (8.533,1.819) -- (8.412,1.9) -- (8.269,1.928) -- (8.127,1.9) -- (8.006,1.819) -- (7.925,1.698) -- (7.896,1.555) -- (7.925,1.412) -- (8.006,1.291) -- (8.127,1.21) -- (8.269,1.182) -- (8.412,1.21) -- (8.533,1.291) -- (8.614,1.412) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.616,1.345) -- (1.588,1.488) -- (1.507,1.609) -- (1.386,1.69) -- (1.243,1.718) -- (1.1,1.69) -- (0.979,1.609) -- (0.898,1.488) -- (0.87,1.345) -- (0.898,1.202) -- (0.979,1.081) -- (1.1,1.0) -- (1.243,0.972) -- (1.386,1.0) -- (1.507,1.081) -- (1.588,1.202) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.551,10.667) -- (6.669,10.686) -- (6.777,10.74) -- (6.862,10.825) -- (6.916,10.933) -- (6.935,11.051) -- (6.916,11.17) -- (6.862,11.277) -- (6.777,11.362) -- (6.669,11.417) -- (6.551,11.436) -- (6.551,11.436) -- (6.432,11.417) -- (6.325,11.362) -- (6.24,11.277) -- (6.185,11.17) -- (6.166,11.051) -- (6.185,10.933) -- (6.24,10.825) -- (6.325,10.74) -- (6.432,10.686) -- (6.551,10.667) -- cycle;
\node at (6.551,11.051) {\scalebox{1.601}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.386,11.139) -- (3.505,11.158) -- (3.612,11.213) -- (3.697,11.298) -- (3.752,11.405) -- (3.77,11.524) -- (3.752,11.643) -- (3.697,11.75) -- (3.612,11.835) -- (3.505,11.889) -- (3.386,11.908) -- (3.386,11.908) -- (3.267,11.889) -- (3.16,11.835) -- (3.075,11.75) -- (3.02,11.643) -- (3.001,11.524) -- (3.02,11.405) -- (3.075,11.298) -- (3.16,11.213) -- (3.267,11.158) -- (3.386,11.139) -- cycle;
\node at (3.386,11.524) {\scalebox{1.601}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.362,9.098) -- (8.48,9.117) -- (8.588,9.172) -- (8.673,9.257) -- (8.727,9.364) -- (8.746,9.483) -- (8.727,9.602) -- (8.673,9.709) -- (8.588,9.794) -- (8.48,9.848) -- (8.362,9.867) -- (8.362,9.867) -- (8.243,9.848) -- (8.136,9.794) -- (8.05,9.709) -- (7.996,9.602) -- (7.977,9.483) -- (7.996,9.364) -- (8.05,9.257) -- (8.136,9.172) -- (8.243,9.117) -- (8.362,9.098) -- cycle;
\node at (8.362,9.483) {\scalebox{1.601}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.352,8.25) -- (9.471,8.269) -- (9.578,8.323) -- (9.664,8.409) -- (9.718,8.516) -- (9.737,8.635) -- (9.718,8.753) -- (9.664,8.861) -- (9.578,8.946) -- (9.471,9.0) -- (9.352,9.019) -- (9.352,9.019) -- (9.234,9.0) -- (9.126,8.946) -- (9.041,8.861) -- (8.987,8.753) -- (8.968,8.635) -- (8.987,8.516) -- (9.041,8.409) -- (9.126,8.323) -- (9.234,8.269) -- (9.352,8.25) -- cycle;
\node at (9.352,8.635) {\scalebox{1.601}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.981,6.236) -- (8.1,6.255) -- (8.207,6.31) -- (8.292,6.395) -- (8.347,6.502) -- (8.366,6.621) -- (8.347,6.74) -- (8.292,6.847) -- (8.207,6.932) -- (8.1,6.987) -- (7.981,7.006) -- (7.981,7.006) -- (7.862,6.987) -- (7.755,6.932) -- (7.67,6.847) -- (7.616,6.74) -- (7.597,6.621) -- (7.616,6.502) -- (7.67,6.395) -- (7.755,6.31) -- (7.862,6.255) -- (7.981,6.236) -- cycle;
\node at (7.981,6.621) {\scalebox{1.601}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.686,5.699) -- (11.805,5.718) -- (11.912,5.773) -- (11.997,5.858) -- (12.052,5.965) -- (12.071,6.084) -- (12.052,6.203) -- (11.997,6.31) -- (11.912,6.395) -- (11.805,6.45) -- (11.686,6.468) -- (11.686,6.468) -- (11.567,6.45) -- (11.46,6.395) -- (11.375,6.31) -- (11.321,6.203) -- (11.302,6.084) -- (11.321,5.965) -- (11.375,5.858) -- (11.46,5.773) -- (11.567,5.718) -- (11.686,5.699) -- cycle;
\node at (11.686,6.084) {\scalebox{1.601}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.566,4.87) -- (3.685,4.888) -- (3.793,4.943) -- (3.878,5.028) -- (3.932,5.135) -- (3.951,5.254) -- (3.932,5.373) -- (3.878,5.48) -- (3.793,5.565) -- (3.685,5.62) -- (3.566,5.639) -- (3.566,5.639) -- (3.448,5.62) -- (3.34,5.565) -- (3.255,5.48) -- (3.201,5.373) -- (3.182,5.254) -- (3.201,5.135) -- (3.255,5.028) -- (3.34,4.943) -- (3.448,4.888) -- (3.566,4.87) -- cycle;
\node at (3.566,5.254) {\scalebox{1.601}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.186,2.828) -- (2.305,2.847) -- (2.412,2.902) -- (2.497,2.987) -- (2.552,3.094) -- (2.571,3.213) -- (2.552,3.332) -- (2.497,3.439) -- (2.412,3.524) -- (2.305,3.579) -- (2.186,3.597) -- (2.186,3.597) -- (2.067,3.579) -- (1.96,3.524) -- (1.875,3.439) -- (1.82,3.332) -- (1.801,3.213) -- (1.82,3.094) -- (1.875,2.987) -- (1.96,2.902) -- (2.067,2.847) -- (2.186,2.828) -- cycle;
\node at (2.186,3.213) {\scalebox{1.601}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.685,4.401) -- (9.804,4.42) -- (9.911,4.474) -- (9.996,4.559) -- (10.051,4.666) -- (10.069,4.785) -- (10.051,4.904) -- (9.996,5.011) -- (9.911,5.096) -- (9.804,5.151) -- (9.685,5.17) -- (9.685,5.17) -- (9.566,5.151) -- (9.459,5.096) -- (9.374,5.011) -- (9.319,4.904) -- (9.3,4.785) -- (9.319,4.666) -- (9.374,4.559) -- (9.459,4.474) -- (9.566,4.42) -- (9.685,4.401) -- cycle;
\node at (9.685,4.785) {\scalebox{1.601}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.861,2.799) -- (8.98,2.818) -- (9.087,2.873) -- (9.173,2.958) -- (9.227,3.065) -- (9.246,3.184) -- (9.227,3.303) -- (9.173,3.41) -- (9.087,3.495) -- (8.98,3.55) -- (8.861,3.568) -- (8.5,3.568) -- (8.381,3.55) -- (8.274,3.495) -- (8.188,3.41) -- (8.134,3.303) -- (8.115,3.184) -- (8.134,3.065) -- (8.188,2.958) -- (8.274,2.873) -- (8.381,2.818) -- (8.5,2.799) -- cycle;
\node at (8.681,3.184) {\scalebox{1.601}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.696,4.593) -- (6.815,4.612) -- (6.922,4.667) -- (7.007,4.752) -- (7.062,4.859) -- (7.081,4.978) -- (7.062,5.097) -- (7.007,5.204) -- (6.922,5.289) -- (6.815,5.344) -- (6.696,5.362) -- (6.334,5.362) -- (6.216,5.344) -- (6.108,5.289) -- (6.023,5.204) -- (5.969,5.097) -- (5.95,4.978) -- (5.969,4.859) -- (6.023,4.752) -- (6.108,4.667) -- (6.216,4.612) -- (6.334,4.593) -- cycle;
\node at (6.515,4.978) {\scalebox{1.601}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.712,3.04) -- (5.831,3.059) -- (5.938,3.114) -- (6.023,3.199) -- (6.078,3.306) -- (6.097,3.425) -- (6.078,3.544) -- (6.023,3.651) -- (5.938,3.736) -- (5.831,3.79) -- (5.712,3.809) -- (5.35,3.809) -- (5.231,3.79) -- (5.124,3.736) -- (5.039,3.651) -- (4.984,3.544) -- (4.966,3.425) -- (4.984,3.306) -- (5.039,3.199) -- (5.124,3.114) -- (5.231,3.059) -- (5.35,3.04) -- cycle;
\node at (5.531,3.425) {\scalebox{1.601}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.414,2.571) -- (11.533,2.589) -- (11.64,2.644) -- (11.725,2.729) -- (11.78,2.836) -- (11.799,2.955) -- (11.78,3.074) -- (11.725,3.181) -- (11.64,3.266) -- (11.533,3.321) -- (11.414,3.34) -- (11.052,3.34) -- (10.933,3.321) -- (10.826,3.266) -- (10.741,3.181) -- (10.687,3.074) -- (10.668,2.955) -- (10.687,2.836) -- (10.741,2.729) -- (10.826,2.644) -- (10.933,2.589) -- (11.052,2.571) -- cycle;
\node at (11.233,2.955) {\scalebox{1.601}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.956,1.956) -- (7.074,1.974) -- (7.182,2.029) -- (7.267,2.114) -- (7.321,2.221) -- (7.34,2.34) -- (7.321,2.459) -- (7.267,2.566) -- (7.182,2.651) -- (7.074,2.706) -- (6.956,2.725) -- (6.594,2.725) -- (6.475,2.706) -- (6.368,2.651) -- (6.283,2.566) -- (6.228,2.459) -- (6.209,2.34) -- (6.228,2.221) -- (6.283,2.114) -- (6.368,2.029) -- (6.475,1.974) -- (6.594,1.956) -- cycle;
\node at (6.775,2.34) {\scalebox{1.601}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.643,1.526) -- (9.762,1.545) -- (9.869,1.6) -- (9.954,1.685) -- (10.008,1.792) -- (10.027,1.911) -- (10.008,2.029) -- (9.954,2.137) -- (9.869,2.222) -- (9.762,2.276) -- (9.643,2.295) -- (9.281,2.295) -- (9.162,2.276) -- (9.055,2.222) -- (8.97,2.137) -- (8.915,2.029) -- (8.896,1.911) -- (8.915,1.792) -- (8.97,1.685) -- (9.055,1.6) -- (9.162,1.545) -- (9.281,1.526) -- cycle;
\node at (9.462,1.911) {\scalebox{1.601}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.248,1.878) -- (3.367,1.897) -- (3.474,1.951) -- (3.559,2.036) -- (3.613,2.143) -- (3.632,2.262) -- (3.613,2.381) -- (3.559,2.488) -- (3.474,2.573) -- (3.367,2.628) -- (3.248,2.647) -- (2.886,2.647) -- (2.767,2.628) -- (2.66,2.573) -- (2.575,2.488) -- (2.52,2.381) -- (2.501,2.262) -- (2.52,2.143) -- (2.575,2.036) -- (2.66,1.951) -- (2.767,1.897) -- (2.886,1.878) -- cycle;
\node at (3.067,2.262) {\scalebox{1.601}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (10.776,7.492) -- (11.376,7.492) -- (11.376,8.092) -- (10.776,8.092) -- cycle;
\node at (11.076,7.792) {\scalebox{1.601}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.737,1.643) -- (2.337,1.643) -- (2.337,2.243) -- (1.737,2.243) -- cycle;
\node at (2.037,1.943) {\scalebox{1.601}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (10.619,4.278) -- (11.219,4.278) -- (11.219,4.878) -- (10.619,4.878) -- cycle;
\node at (10.919,4.578) {\scalebox{1.601}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (2.395,11.846) -- (2.995,11.846) -- (2.995,12.446) -- (2.395,12.446) -- cycle;
\node at (2.695,12.146) {\scalebox{1.601}{\small d}};
\node at (4.51,11.629) {\scalebox{1.601}{\small A}};
\node at (1.667,11.489) {\scalebox{1.601}{\small B}};
\node at (8.542,11.177) {\scalebox{1.601}{\small C}};
\node at (7.641,9.009) {\scalebox{1.601}{\small D}};
\node at (11.64,8.048) {\scalebox{1.601}{\small E}};
\node at (1.432,5.241) {\scalebox{1.601}{\small F}};
\node at (8.578,4.403) {\scalebox{1.601}{\small G}};
\node at (4.461,4.354) {\scalebox{1.601}{\small H}};
\node at (11.836,4.146) {\scalebox{1.601}{\small I}};
\node at (4.872,1.531) {\scalebox{1.601}{\small J}};
\node at (11.131,1.264) {\scalebox{1.601}{\small K}};
\node at (8.504,0.981) {\scalebox{1.601}{\small L}};
\node at (0.765,0.95) {\scalebox{1.601}{\small M}};
\draw[line width=0.5pt] (12.543,10.094) -- (12.543,11.194);
\draw[line width=0.5pt] (12.312,10.809) -- (12.543,11.194) -- (12.774,10.809);
\draw[line width=0.5pt] (12.24,10.644) -- (12.845,10.644);
\node at (12.543,11.579) {\scalebox{1.601}{\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, D, H, I, L, M & 6 \\ \hline
clairs & B, C, E, F, G, J, K & 7 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{7} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 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 6 carrefours sombres et 7 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{E} :}\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 des Vignes ⑥} vers le sud jusqu'à Rond-point du Pont \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de la Mairie ⑬}, vers le sud, jusqu'à Place aux Herbes \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Verte ⑮}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{sentier des Prés ⑭}, vers l'ouest, jusqu'à Carrefour du Lavoir \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Tilleuls ⑯}, vers l'ouest, jusqu'à Place de l'Église \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{venelle du Chat ⑧}, vers le nord, jusqu'à Place de la Fontaine \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Lilas ⑦}, vers l'est, jusqu'à Carrefour Saint-Roch \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Neuve ⑪}, vers l'est, jusqu'à Carrefour du Moulin \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ⑤}, vers le nord, jusqu'à Carrefour des Tilleuls \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{avenue de la Gare ③}, vers le nord, jusqu'à Fontaine Notre-Dame \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{passage du Marché ①}, vers l'ouest, jusqu'à Place du Vieux-Puits \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{impasse du Vieux-Four ②}, vers l'ouest, jusqu'à Place Sainte-Croix \textbf{(B)}.\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}
