\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 232}

\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=232}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0232}
\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=0.5894, line join=round, line cap=round]
\draw[line width=0.5pt] (4.255,6.662) -- (4.442,6.59) -- (4.631,6.523) -- (4.821,6.46) -- (5.012,6.402) -- (5.205,6.348) -- (5.399,6.299) -- (5.595,6.254) -- (5.791,6.214) -- (5.988,6.179) -- (6.186,6.148) -- (6.384,6.122) -- (6.584,6.1) -- (6.783,6.084) -- (6.983,6.072);
\draw[line width=0.5pt] (6.983,6.072) -- (7.148,6.093) -- (7.314,6.11) -- (7.48,6.123) -- (7.647,6.133) -- (7.813,6.138) -- (7.98,6.139) -- (8.147,6.137) -- (8.314,6.131) -- (8.48,6.121) -- (8.647,6.106) -- (8.812,6.088) -- (8.978,6.066) -- (9.143,6.041) -- (9.307,6.011);
\draw[line width=0.5pt] (6.983,6.072) -- (6.941,5.914) -- (6.904,5.755) -- (6.87,5.595) -- (6.84,5.434) -- (6.814,5.272) -- (6.792,5.11) -- (6.773,4.948) -- (6.758,4.785) -- (6.747,4.622) -- (6.74,4.458) -- (6.737,4.295) -- (6.737,4.131) -- (6.742,3.968) -- (6.75,3.805);
\draw[line width=0.5pt] (11.44,6.041) -- (11.479,5.892) -- (11.514,5.743) -- (11.545,5.592) -- (11.573,5.441) -- (11.597,5.29) -- (11.618,5.138) -- (11.635,4.985) -- (11.649,4.832) -- (11.659,4.679) -- (11.665,4.525) -- (11.668,4.371) -- (11.667,4.218) -- (11.662,4.065) -- (11.654,3.911);
\draw[line width=0.5pt] (2.182,6.013) -- (2.336,6.036) -- (2.49,6.063) -- (2.642,6.094) -- (2.794,6.128) -- (2.946,6.166) -- (3.096,6.207) -- (3.246,6.252) -- (3.394,6.3) -- (3.541,6.352) -- (3.687,6.407) -- (3.831,6.466) -- (3.974,6.528) -- (4.116,6.593) -- (4.255,6.662);
\draw[line width=0.5pt] (2.182,6.013) -- (2.197,5.863) -- (2.208,5.712) -- (2.216,5.561) -- (2.22,5.41) -- (2.221,5.258) -- (2.218,5.107) -- (2.211,4.956) -- (2.201,4.805) -- (2.187,4.654) -- (2.17,4.504) -- (2.149,4.354) -- (2.125,4.205) -- (2.098,4.056) -- (2.066,3.908);
\draw[line width=0.5pt] (9.307,6.011) -- (9.458,5.99) -- (9.61,5.972) -- (9.763,5.958) -- (9.915,5.947) -- (10.068,5.941) -- (10.221,5.937) -- (10.375,5.938) -- (10.528,5.942) -- (10.681,5.949) -- (10.833,5.96) -- (10.986,5.975) -- (11.138,5.993) -- (11.289,6.015) -- (11.44,6.041);
\draw[line width=0.5pt] (9.307,6.011) -- (9.324,5.885) -- (9.339,5.758) -- (9.351,5.631) -- (9.359,5.504) -- (9.365,5.377) -- (9.368,5.249) -- (9.367,5.122) -- (9.364,4.995) -- (9.358,4.867) -- (9.349,4.74) -- (9.336,4.613) -- (9.321,4.487) -- (9.303,4.361) -- (9.282,4.235);
\draw[line width=0.5pt] (9.282,4.235) -- (9.446,4.186) -- (9.612,4.141) -- (9.779,4.1) -- (9.947,4.063) -- (10.115,4.03) -- (10.285,4.001) -- (10.455,3.975) -- (10.625,3.954) -- (10.796,3.937) -- (10.967,3.924) -- (11.139,3.914) -- (11.311,3.909) -- (11.483,3.908) -- (11.654,3.911);
\draw[line width=0.5pt] (9.282,4.235) -- (9.31,4.05) -- (9.334,3.863) -- (9.354,3.677) -- (9.369,3.489) -- (9.38,3.302) -- (9.387,3.114) -- (9.389,2.926) -- (9.386,2.738) -- (9.38,2.55) -- (9.368,2.363) -- (9.353,2.176) -- (9.333,1.989) -- (9.308,1.803) -- (9.28,1.617);
\draw[line width=0.5pt] (11.654,3.911) -- (11.631,3.736) -- (11.612,3.56) -- (11.597,3.383) -- (11.587,3.206) -- (11.58,3.029) -- (11.578,2.852) -- (11.58,2.675) -- (11.586,2.498) -- (11.596,2.321) -- (11.61,2.144) -- (11.629,1.968) -- (11.651,1.792) -- (11.678,1.617) -- (11.709,1.443);
\draw[line width=0.5pt] (2.066,3.908) -- (2.063,3.759) -- (2.057,3.609) -- (2.047,3.46) -- (2.034,3.311) -- (2.017,3.162) -- (1.996,3.014) -- (1.972,2.866) -- (1.945,2.719) -- (1.914,2.573) -- (1.88,2.427) -- (1.842,2.282) -- (1.801,2.138) -- (1.757,1.995) -- (1.709,1.854);
\draw[line width=0.5pt] (6.75,3.805) -- (6.934,3.808) -- (7.118,3.815) -- (7.302,3.826) -- (7.486,3.842) -- (7.669,3.862) -- (7.852,3.887) -- (8.034,3.916) -- (8.215,3.949) -- (8.396,3.986) -- (8.575,4.027) -- (8.754,4.073) -- (8.931,4.123) -- (9.107,4.177) -- (9.282,4.235);
\draw[line width=0.5pt] (6.75,3.805) -- (6.802,3.651) -- (6.85,3.495) -- (6.895,3.339) -- (6.936,3.181) -- (6.973,3.023) -- (7.007,2.863) -- (7.036,2.703) -- (7.062,2.543) -- (7.085,2.381) -- (7.103,2.219) -- (7.118,2.057) -- (7.128,1.895) -- (7.135,1.732) -- (7.138,1.57);
\draw[line width=0.5pt] (9.28,1.617) -- (9.45,1.578) -- (9.621,1.543) -- (9.793,1.513) -- (9.966,1.486) -- (10.139,1.463) -- (10.313,1.445) -- (10.487,1.43) -- (10.662,1.42) -- (10.836,1.413) -- (11.011,1.411) -- (11.186,1.413) -- (11.36,1.419) -- (11.535,1.429) -- (11.709,1.443);
\draw[line width=0.5pt] (7.138,1.57) -- (7.289,1.597) -- (7.441,1.62) -- (7.594,1.639) -- (7.747,1.655) -- (7.9,1.668) -- (8.053,1.677) -- (8.207,1.682) -- (8.361,1.683) -- (8.514,1.681) -- (8.668,1.676) -- (8.821,1.667) -- (8.975,1.654) -- (9.127,1.637) -- (9.28,1.617);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.415,6.662) -- (4.394,6.742) -- (4.335,6.801) -- (4.255,6.822) -- (4.175,6.801) -- (4.117,6.742) -- (4.095,6.662) -- (4.117,6.582) -- (4.175,6.523) -- (4.255,6.502) -- (4.335,6.523) -- (4.394,6.582) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.143,6.072) -- (7.121,6.152) -- (7.063,6.21) -- (6.983,6.232) -- (6.903,6.21) -- (6.844,6.152) -- (6.823,6.072) -- (6.844,5.992) -- (6.903,5.933) -- (6.983,5.912) -- (7.063,5.933) -- (7.121,5.992) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.6,6.041) -- (11.579,6.121) -- (11.52,6.179) -- (11.44,6.201) -- (11.36,6.179) -- (11.302,6.121) -- (11.28,6.041) -- (11.302,5.961) -- (11.36,5.902) -- (11.44,5.881) -- (11.52,5.902) -- (11.579,5.961) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.342,6.013) -- (2.321,6.093) -- (2.262,6.152) -- (2.182,6.173) -- (2.102,6.152) -- (2.043,6.093) -- (2.022,6.013) -- (2.043,5.933) -- (2.102,5.874) -- (2.182,5.853) -- (2.262,5.874) -- (2.321,5.933) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.467,6.011) -- (9.445,6.091) -- (9.387,6.149) -- (9.307,6.171) -- (9.227,6.149) -- (9.168,6.091) -- (9.147,6.011) -- (9.168,5.931) -- (9.227,5.872) -- (9.307,5.851) -- (9.387,5.872) -- (9.445,5.931) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.442,4.235) -- (9.42,4.315) -- (9.362,4.374) -- (9.282,4.395) -- (9.202,4.374) -- (9.143,4.315) -- (9.122,4.235) -- (9.143,4.155) -- (9.202,4.097) -- (9.282,4.075) -- (9.362,4.097) -- (9.42,4.155) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.814,3.911) -- (11.793,3.991) -- (11.734,4.05) -- (11.654,4.071) -- (11.574,4.05) -- (11.516,3.991) -- (11.494,3.911) -- (11.516,3.831) -- (11.574,3.773) -- (11.654,3.751) -- (11.734,3.773) -- (11.793,3.831) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.226,3.908) -- (2.205,3.988) -- (2.146,4.047) -- (2.066,4.068) -- (1.986,4.047) -- (1.928,3.988) -- (1.906,3.908) -- (1.928,3.828) -- (1.986,3.77) -- (2.066,3.748) -- (2.146,3.77) -- (2.205,3.828) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.91,3.805) -- (6.889,3.885) -- (6.83,3.944) -- (6.75,3.965) -- (6.67,3.944) -- (6.612,3.885) -- (6.59,3.805) -- (6.612,3.725) -- (6.67,3.666) -- (6.75,3.645) -- (6.83,3.666) -- (6.889,3.725) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.869,1.854) -- (1.847,1.934) -- (1.789,1.992) -- (1.709,2.014) -- (1.629,1.992) -- (1.57,1.934) -- (1.549,1.854) -- (1.57,1.774) -- (1.629,1.715) -- (1.709,1.694) -- (1.789,1.715) -- (1.847,1.774) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.44,1.617) -- (9.418,1.697) -- (9.36,1.756) -- (9.28,1.777) -- (9.2,1.756) -- (9.141,1.697) -- (9.12,1.617) -- (9.141,1.537) -- (9.2,1.479) -- (9.28,1.457) -- (9.36,1.479) -- (9.418,1.537) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.298,1.57) -- (7.277,1.65) -- (7.218,1.708) -- (7.138,1.73) -- (7.058,1.708) -- (7.0,1.65) -- (6.978,1.57) -- (7.0,1.49) -- (7.058,1.431) -- (7.138,1.41) -- (7.218,1.431) -- (7.277,1.49) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.869,1.443) -- (11.847,1.523) -- (11.789,1.582) -- (11.709,1.603) -- (11.629,1.582) -- (11.57,1.523) -- (11.549,1.443) -- (11.57,1.363) -- (11.629,1.305) -- (11.709,1.283) -- (11.789,1.305) -- (11.847,1.363) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.678,6.245) -- (5.8,6.265) -- (5.91,6.321) -- (5.997,6.408) -- (6.053,6.518) -- (6.072,6.639) -- (6.053,6.761) -- (5.997,6.871) -- (5.91,6.958) -- (5.8,7.014) -- (5.678,7.033) -- (5.678,7.033) -- (5.556,7.014) -- (5.447,6.958) -- (5.359,6.871) -- (5.303,6.761) -- (5.284,6.639) -- (5.303,6.518) -- (5.359,6.408) -- (5.447,6.321) -- (5.556,6.265) -- (5.678,6.245) -- cycle;
\node at (5.678,6.639) {\scalebox{1.697}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.157,6.137) -- (8.279,6.156) -- (8.389,6.212) -- (8.476,6.299) -- (8.532,6.409) -- (8.551,6.531) -- (8.532,6.653) -- (8.476,6.762) -- (8.389,6.85) -- (8.279,6.905) -- (8.157,6.925) -- (8.157,6.925) -- (8.036,6.905) -- (7.926,6.85) -- (7.839,6.762) -- (7.783,6.653) -- (7.764,6.531) -- (7.783,6.409) -- (7.839,6.299) -- (7.926,6.212) -- (8.036,6.156) -- (8.157,6.137) -- cycle;
\node at (8.157,6.531) {\scalebox{1.697}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.165,4.514) -- (7.287,4.533) -- (7.396,4.589) -- (7.483,4.676) -- (7.539,4.786) -- (7.559,4.908) -- (7.539,5.029) -- (7.483,5.139) -- (7.396,5.226) -- (7.287,5.282) -- (7.165,5.302) -- (7.165,5.302) -- (7.043,5.282) -- (6.933,5.226) -- (6.846,5.139) -- (6.79,5.029) -- (6.771,4.908) -- (6.79,4.786) -- (6.846,4.676) -- (6.933,4.589) -- (7.043,4.533) -- (7.165,4.514) -- cycle;
\node at (7.165,4.908) {\scalebox{1.697}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.243,4.552) -- (11.365,4.571) -- (11.475,4.627) -- (11.562,4.714) -- (11.618,4.824) -- (11.637,4.945) -- (11.618,5.067) -- (11.562,5.177) -- (11.475,5.264) -- (11.365,5.32) -- (11.243,5.339) -- (11.243,5.339) -- (11.121,5.32) -- (11.012,5.264) -- (10.924,5.177) -- (10.868,5.067) -- (10.849,4.945) -- (10.868,4.824) -- (10.924,4.714) -- (11.012,4.627) -- (11.121,4.571) -- (11.243,4.552) -- cycle;
\node at (11.243,4.945) {\scalebox{1.697}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.363,5.482) -- (3.485,5.501) -- (3.595,5.557) -- (3.682,5.645) -- (3.738,5.754) -- (3.757,5.876) -- (3.738,5.998) -- (3.682,6.108) -- (3.595,6.195) -- (3.485,6.251) -- (3.363,6.27) -- (3.363,6.27) -- (3.241,6.251) -- (3.132,6.195) -- (3.045,6.108) -- (2.989,5.998) -- (2.969,5.876) -- (2.989,5.754) -- (3.045,5.645) -- (3.132,5.557) -- (3.241,5.501) -- (3.363,5.482) -- cycle;
\node at (3.363,5.876) {\scalebox{1.697}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.604,4.54) -- (2.726,4.56) -- (2.836,4.615) -- (2.923,4.703) -- (2.979,4.812) -- (2.998,4.934) -- (2.979,5.056) -- (2.923,5.166) -- (2.836,5.253) -- (2.726,5.309) -- (2.604,5.328) -- (2.604,5.328) -- (2.483,5.309) -- (2.373,5.253) -- (2.286,5.166) -- (2.23,5.056) -- (2.21,4.934) -- (2.23,4.812) -- (2.286,4.703) -- (2.373,4.615) -- (2.483,4.56) -- (2.604,4.54) -- cycle;
\node at (2.604,4.934) {\scalebox{1.697}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.38,5.15) -- (10.502,5.169) -- (10.612,5.225) -- (10.699,5.312) -- (10.755,5.422) -- (10.774,5.544) -- (10.755,5.666) -- (10.699,5.775) -- (10.612,5.863) -- (10.502,5.919) -- (10.38,5.938) -- (10.38,5.938) -- (10.258,5.919) -- (10.149,5.863) -- (10.061,5.775) -- (10.006,5.666) -- (9.986,5.544) -- (10.006,5.422) -- (10.061,5.312) -- (10.149,5.225) -- (10.258,5.169) -- (10.38,5.15) -- cycle;
\node at (10.38,5.544) {\scalebox{1.697}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.973,4.734) -- (9.095,4.753) -- (9.205,4.809) -- (9.292,4.896) -- (9.348,5.006) -- (9.367,5.127) -- (9.348,5.249) -- (9.292,5.359) -- (9.205,5.446) -- (9.095,5.502) -- (8.973,5.521) -- (8.973,5.521) -- (8.852,5.502) -- (8.742,5.446) -- (8.655,5.359) -- (8.599,5.249) -- (8.58,5.127) -- (8.599,5.006) -- (8.655,4.896) -- (8.742,4.809) -- (8.852,4.753) -- (8.973,4.734) -- cycle;
\node at (8.973,5.127) {\scalebox{1.697}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.761,3.151) -- (10.883,3.17) -- (10.993,3.226) -- (11.08,3.313) -- (11.136,3.423) -- (11.155,3.545) -- (11.136,3.666) -- (11.08,3.776) -- (10.993,3.863) -- (10.883,3.919) -- (10.761,3.938) -- (10.761,3.938) -- (10.64,3.919) -- (10.53,3.863) -- (10.443,3.776) -- (10.387,3.666) -- (10.367,3.545) -- (10.387,3.423) -- (10.443,3.313) -- (10.53,3.226) -- (10.64,3.17) -- (10.761,3.151) -- cycle;
\node at (10.761,3.545) {\scalebox{1.697}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.968,2.532) -- (10.09,2.551) -- (10.199,2.607) -- (10.287,2.694) -- (10.343,2.804) -- (10.362,2.926) -- (10.343,3.048) -- (10.287,3.157) -- (10.199,3.244) -- (10.09,3.3) -- (9.968,3.32) -- (9.597,3.32) -- (9.475,3.3) -- (9.366,3.244) -- (9.278,3.157) -- (9.223,3.048) -- (9.203,2.926) -- (9.223,2.804) -- (9.278,2.694) -- (9.366,2.607) -- (9.475,2.551) -- (9.597,2.532) -- cycle;
\node at (9.783,2.926) {\scalebox{1.697}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (12.159,2.29) -- (12.281,2.309) -- (12.39,2.365) -- (12.478,2.452) -- (12.534,2.562) -- (12.553,2.684) -- (12.534,2.805) -- (12.478,2.915) -- (12.39,3.002) -- (12.281,3.058) -- (12.159,3.078) -- (11.788,3.078) -- (11.666,3.058) -- (11.557,3.002) -- (11.47,2.915) -- (11.414,2.805) -- (11.394,2.684) -- (11.414,2.562) -- (11.47,2.452) -- (11.557,2.365) -- (11.666,2.309) -- (11.788,2.29) -- cycle;
\node at (11.974,2.684) {\scalebox{1.697}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.546,2.405) -- (2.668,2.424) -- (2.777,2.48) -- (2.864,2.567) -- (2.92,2.677) -- (2.94,2.799) -- (2.92,2.92) -- (2.864,3.03) -- (2.777,3.117) -- (2.668,3.173) -- (2.546,3.193) -- (2.175,3.193) -- (2.053,3.173) -- (1.944,3.117) -- (1.856,3.03) -- (1.8,2.92) -- (1.781,2.799) -- (1.8,2.677) -- (1.856,2.567) -- (1.944,2.48) -- (2.053,2.424) -- (2.175,2.405) -- cycle;
\node at (2.36,2.799) {\scalebox{1.697}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.285,3.133) -- (8.407,3.153) -- (8.517,3.209) -- (8.604,3.296) -- (8.66,3.406) -- (8.679,3.527) -- (8.66,3.649) -- (8.604,3.759) -- (8.517,3.846) -- (8.407,3.902) -- (8.285,3.921) -- (7.914,3.921) -- (7.793,3.902) -- (7.683,3.846) -- (7.596,3.759) -- (7.54,3.649) -- (7.52,3.527) -- (7.54,3.406) -- (7.596,3.296) -- (7.683,3.209) -- (7.793,3.153) -- (7.914,3.133) -- cycle;
\node at (8.1,3.527) {\scalebox{1.697}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.534,1.895) -- (6.656,1.914) -- (6.766,1.97) -- (6.853,2.057) -- (6.909,2.167) -- (6.928,2.289) -- (6.909,2.411) -- (6.853,2.52) -- (6.766,2.608) -- (6.656,2.663) -- (6.534,2.683) -- (6.164,2.683) -- (6.042,2.663) -- (5.932,2.608) -- (5.845,2.52) -- (5.789,2.411) -- (5.77,2.289) -- (5.789,2.167) -- (5.845,2.057) -- (5.932,1.97) -- (6.042,1.914) -- (6.164,1.895) -- cycle;
\node at (6.349,2.289) {\scalebox{1.697}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.701,1.429) -- (10.822,1.448) -- (10.932,1.504) -- (11.019,1.591) -- (11.075,1.701) -- (11.094,1.823) -- (11.075,1.945) -- (11.019,2.054) -- (10.932,2.142) -- (10.822,2.198) -- (10.701,2.217) -- (10.33,2.217) -- (10.208,2.198) -- (10.098,2.142) -- (10.011,2.054) -- (9.955,1.945) -- (9.936,1.823) -- (9.955,1.701) -- (10.011,1.591) -- (10.098,1.504) -- (10.208,1.448) -- (10.33,1.429) -- cycle;
\node at (10.515,1.823) {\scalebox{1.697}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.384,1.682) -- (8.505,1.701) -- (8.615,1.757) -- (8.702,1.844) -- (8.758,1.954) -- (8.777,2.076) -- (8.758,2.197) -- (8.702,2.307) -- (8.615,2.394) -- (8.505,2.45) -- (8.384,2.47) -- (8.013,2.47) -- (7.891,2.45) -- (7.781,2.394) -- (7.694,2.307) -- (7.638,2.197) -- (7.619,2.076) -- (7.638,1.954) -- (7.694,1.844) -- (7.781,1.757) -- (7.891,1.701) -- (8.013,1.682) -- cycle;
\node at (8.198,2.076) {\scalebox{1.697}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (11.409,1.143) -- (12.009,1.143) -- (12.009,1.743) -- (11.409,1.743) -- cycle;
\node at (11.709,1.443) {\scalebox{1.697}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.407,5.233) -- (2.007,5.233) -- (2.007,5.833) -- (1.407,5.833) -- cycle;
\node at (1.707,5.533) {\scalebox{1.697}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (0.6,2.438) -- (1.2,2.438) -- (1.2,3.038) -- (0.6,3.038) -- cycle;
\node at (0.9,2.738) {\scalebox{1.697}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (12.374,5.209) -- (12.974,5.209) -- (12.974,5.809) -- (12.374,5.809) -- cycle;
\node at (12.674,5.509) {\scalebox{1.697}{\small d}};
\node at (3.788,7.069) {\scalebox{1.697}{\small A}};
\node at (6.911,6.688) {\scalebox{1.697}{\small B}};
\node at (12.003,6.301) {\scalebox{1.697}{\small C}};
\node at (1.603,6.235) {\scalebox{1.697}{\small D}};
\node at (9.763,6.43) {\scalebox{1.697}{\small E}};
\node at (9.9,4.279) {\scalebox{1.697}{\small F}};
\node at (12.274,3.887) {\scalebox{1.697}{\small G}};
\node at (1.447,3.886) {\scalebox{1.697}{\small H}};
\node at (6.222,3.481) {\scalebox{1.697}{\small I}};
\node at (1.134,1.621) {\scalebox{1.697}{\small J}};
\node at (9.678,1.142) {\scalebox{1.697}{\small K}};
\node at (7.12,0.95) {\scalebox{1.697}{\small L}};
\node at (12.243,1.129) {\scalebox{1.697}{\small M}};
\draw[line width=0.5pt] (13.009,5.712) -- (13.009,6.812);
\draw[line width=0.5pt] (12.778,6.427) -- (13.009,6.812) -- (13.24,6.427);
\draw[line width=0.5pt] (12.706,6.262) -- (13.311,6.262);
\node at (13.009,7.197) {\scalebox{1.697}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour M, 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 boulevard du Parc & Ⓐ Place du Vieux-Puits & Ⓙ Place de la Fontaine \\ \hline
① chemin Creux & ⑩ rue du Puits & Ⓑ Carrefour de l'Hôpital & Ⓚ Carrefour Saint-Roch \\ \hline
② rue des Jardins & ⑪ rue des Lilas & Ⓒ Placette des Ormes & Ⓛ Place Sainte-Croix \\ \hline
③ rue Pasteur & ⑫ impasse des Roses & Ⓓ Place du Marché & Ⓜ Coin du Lavoir \\ \hline
④ rue du Four & ⑬ rue Basse & Ⓔ Rond-point des Vaches & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ chemin des Vignes & ⑭ rue du Moulin & Ⓕ Rond-point de la Gare & \textbf{a} l'école \\ \hline
⑥ rue de l'Église & ⑮ allée des Peupliers & Ⓖ Carrefour des Tilleuls & \textbf{b} la boulangerie \\ \hline
⑦ rue des Écoles & ⑯ rue Neuve & Ⓗ Place des Tanneurs & \textbf{c} la salle des fêtes \\ \hline
⑧ venelle du Chat & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour du Lavoir & \textbf{d} la pharmacie \\ \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=0.6038, line join=round, line cap=round]
\draw[line width=0.5pt] (4.255,6.662) -- (4.442,6.59) -- (4.631,6.523) -- (4.821,6.46) -- (5.012,6.402) -- (5.205,6.348) -- (5.399,6.299) -- (5.595,6.254) -- (5.791,6.214) -- (5.988,6.179) -- (6.186,6.148) -- (6.384,6.122) -- (6.584,6.1) -- (6.783,6.084) -- (6.983,6.072);
\draw[line width=0.5pt] (6.983,6.072) -- (7.148,6.093) -- (7.314,6.11) -- (7.48,6.123) -- (7.647,6.133) -- (7.813,6.138) -- (7.98,6.139) -- (8.147,6.137) -- (8.314,6.131) -- (8.48,6.121) -- (8.647,6.106) -- (8.812,6.088) -- (8.978,6.066) -- (9.143,6.041) -- (9.307,6.011);
\draw[line width=0.5pt] (6.983,6.072) -- (6.941,5.914) -- (6.904,5.755) -- (6.87,5.595) -- (6.84,5.434) -- (6.814,5.272) -- (6.792,5.11) -- (6.773,4.948) -- (6.758,4.785) -- (6.747,4.622) -- (6.74,4.458) -- (6.737,4.295) -- (6.737,4.131) -- (6.742,3.968) -- (6.75,3.805);
\draw[line width=0.5pt] (11.44,6.041) -- (11.479,5.892) -- (11.514,5.743) -- (11.545,5.592) -- (11.573,5.441) -- (11.597,5.29) -- (11.618,5.138) -- (11.635,4.985) -- (11.649,4.832) -- (11.659,4.679) -- (11.665,4.525) -- (11.668,4.371) -- (11.667,4.218) -- (11.662,4.065) -- (11.654,3.911);
\draw[line width=0.5pt] (2.182,6.013) -- (2.336,6.036) -- (2.49,6.063) -- (2.642,6.094) -- (2.794,6.128) -- (2.946,6.166) -- (3.096,6.207) -- (3.246,6.252) -- (3.394,6.3) -- (3.541,6.352) -- (3.687,6.407) -- (3.831,6.466) -- (3.974,6.528) -- (4.116,6.593) -- (4.255,6.662);
\draw[line width=0.5pt] (2.182,6.013) -- (2.197,5.863) -- (2.208,5.712) -- (2.216,5.561) -- (2.22,5.41) -- (2.221,5.258) -- (2.218,5.107) -- (2.211,4.956) -- (2.201,4.805) -- (2.187,4.654) -- (2.17,4.504) -- (2.149,4.354) -- (2.125,4.205) -- (2.098,4.056) -- (2.066,3.908);
\draw[line width=0.5pt] (9.307,6.011) -- (9.458,5.99) -- (9.61,5.972) -- (9.763,5.958) -- (9.915,5.947) -- (10.068,5.941) -- (10.221,5.937) -- (10.375,5.938) -- (10.528,5.942) -- (10.681,5.949) -- (10.833,5.96) -- (10.986,5.975) -- (11.138,5.993) -- (11.289,6.015) -- (11.44,6.041);
\draw[line width=0.5pt] (9.307,6.011) -- (9.324,5.885) -- (9.339,5.758) -- (9.351,5.631) -- (9.359,5.504) -- (9.365,5.377) -- (9.368,5.249) -- (9.367,5.122) -- (9.364,4.995) -- (9.358,4.867) -- (9.349,4.74) -- (9.336,4.613) -- (9.321,4.487) -- (9.303,4.361) -- (9.282,4.235);
\draw[line width=0.5pt] (9.282,4.235) -- (9.446,4.186) -- (9.612,4.141) -- (9.779,4.1) -- (9.947,4.063) -- (10.115,4.03) -- (10.285,4.001) -- (10.455,3.975) -- (10.625,3.954) -- (10.796,3.937) -- (10.967,3.924) -- (11.139,3.914) -- (11.311,3.909) -- (11.483,3.908) -- (11.654,3.911);
\draw[line width=0.5pt] (9.282,4.235) -- (9.31,4.05) -- (9.334,3.863) -- (9.354,3.677) -- (9.369,3.489) -- (9.38,3.302) -- (9.387,3.114) -- (9.389,2.926) -- (9.386,2.738) -- (9.38,2.55) -- (9.368,2.363) -- (9.353,2.176) -- (9.333,1.989) -- (9.308,1.803) -- (9.28,1.617);
\draw[line width=0.5pt] (11.654,3.911) -- (11.631,3.736) -- (11.612,3.56) -- (11.597,3.383) -- (11.587,3.206) -- (11.58,3.029) -- (11.578,2.852) -- (11.58,2.675) -- (11.586,2.498) -- (11.596,2.321) -- (11.61,2.144) -- (11.629,1.968) -- (11.651,1.792) -- (11.678,1.617) -- (11.709,1.443);
\draw[line width=0.5pt] (2.066,3.908) -- (2.063,3.759) -- (2.057,3.609) -- (2.047,3.46) -- (2.034,3.311) -- (2.017,3.162) -- (1.996,3.014) -- (1.972,2.866) -- (1.945,2.719) -- (1.914,2.573) -- (1.88,2.427) -- (1.842,2.282) -- (1.801,2.138) -- (1.757,1.995) -- (1.709,1.854);
\draw[line width=0.5pt] (6.75,3.805) -- (6.934,3.808) -- (7.118,3.815) -- (7.302,3.826) -- (7.486,3.842) -- (7.669,3.862) -- (7.852,3.887) -- (8.034,3.916) -- (8.215,3.949) -- (8.396,3.986) -- (8.575,4.027) -- (8.754,4.073) -- (8.931,4.123) -- (9.107,4.177) -- (9.282,4.235);
\draw[line width=0.5pt] (6.75,3.805) -- (6.802,3.651) -- (6.85,3.495) -- (6.895,3.339) -- (6.936,3.181) -- (6.973,3.023) -- (7.007,2.863) -- (7.036,2.703) -- (7.062,2.543) -- (7.085,2.381) -- (7.103,2.219) -- (7.118,2.057) -- (7.128,1.895) -- (7.135,1.732) -- (7.138,1.57);
\draw[line width=0.5pt] (9.28,1.617) -- (9.45,1.578) -- (9.621,1.543) -- (9.793,1.513) -- (9.966,1.486) -- (10.139,1.463) -- (10.313,1.445) -- (10.487,1.43) -- (10.662,1.42) -- (10.836,1.413) -- (11.011,1.411) -- (11.186,1.413) -- (11.36,1.419) -- (11.535,1.429) -- (11.709,1.443);
\draw[line width=0.5pt] (7.138,1.57) -- (7.289,1.597) -- (7.441,1.62) -- (7.594,1.639) -- (7.747,1.655) -- (7.9,1.668) -- (8.053,1.677) -- (8.207,1.682) -- (8.361,1.683) -- (8.514,1.681) -- (8.668,1.676) -- (8.821,1.667) -- (8.975,1.654) -- (9.127,1.637) -- (9.28,1.617);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.629,6.662) -- (4.6,6.805) -- (4.519,6.926) -- (4.398,7.007) -- (4.255,7.035) -- (4.113,7.007) -- (3.992,6.926) -- (3.911,6.805) -- (3.882,6.662) -- (3.911,6.519) -- (3.992,6.398) -- (4.113,6.317) -- (4.255,6.289) -- (4.398,6.317) -- (4.519,6.398) -- (4.6,6.519) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.356,6.072) -- (7.327,6.214) -- (7.247,6.336) -- (7.125,6.416) -- (6.983,6.445) -- (6.84,6.416) -- (6.719,6.336) -- (6.638,6.214) -- (6.609,6.072) -- (6.638,5.929) -- (6.719,5.808) -- (6.84,5.727) -- (6.983,5.698) -- (7.125,5.727) -- (7.247,5.808) -- (7.327,5.929) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.813,6.041) -- (11.785,6.184) -- (11.704,6.305) -- (11.583,6.386) -- (11.44,6.414) -- (11.297,6.386) -- (11.176,6.305) -- (11.095,6.184) -- (11.067,6.041) -- (11.095,5.898) -- (11.176,5.777) -- (11.297,5.696) -- (11.44,5.668) -- (11.583,5.696) -- (11.704,5.777) -- (11.785,5.898) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.555,6.013) -- (2.527,6.156) -- (2.446,6.277) -- (2.325,6.358) -- (2.182,6.386) -- (2.039,6.358) -- (1.918,6.277) -- (1.837,6.156) -- (1.809,6.013) -- (1.837,5.87) -- (1.918,5.749) -- (2.039,5.668) -- (2.182,5.64) -- (2.325,5.668) -- (2.446,5.749) -- (2.527,5.87) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.68,6.011) -- (9.651,6.154) -- (9.57,6.275) -- (9.449,6.356) -- (9.307,6.384) -- (9.164,6.356) -- (9.043,6.275) -- (8.962,6.154) -- (8.933,6.011) -- (8.962,5.868) -- (9.043,5.747) -- (9.164,5.666) -- (9.307,5.638) -- (9.449,5.666) -- (9.57,5.747) -- (9.651,5.868) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.655,4.235) -- (9.626,4.378) -- (9.546,4.499) -- (9.424,4.58) -- (9.282,4.608) -- (9.139,4.58) -- (9.018,4.499) -- (8.937,4.378) -- (8.908,4.235) -- (8.937,4.092) -- (9.018,3.971) -- (9.139,3.89) -- (9.282,3.862) -- (9.424,3.89) -- (9.546,3.971) -- (9.626,4.092) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (12.027,3.911) -- (11.999,4.054) -- (11.918,4.175) -- (11.797,4.256) -- (11.654,4.284) -- (11.511,4.256) -- (11.39,4.175) -- (11.309,4.054) -- (11.281,3.911) -- (11.309,3.768) -- (11.39,3.647) -- (11.511,3.566) -- (11.654,3.538) -- (11.797,3.566) -- (11.918,3.647) -- (11.999,3.768) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.44,3.908) -- (2.411,4.051) -- (2.33,4.172) -- (2.209,4.253) -- (2.066,4.281) -- (1.923,4.253) -- (1.802,4.172) -- (1.721,4.051) -- (1.693,3.908) -- (1.721,3.765) -- (1.802,3.644) -- (1.923,3.563) -- (2.066,3.535) -- (2.209,3.563) -- (2.33,3.644) -- (2.411,3.765) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.123,3.805) -- (7.095,3.948) -- (7.014,4.069) -- (6.893,4.15) -- (6.75,4.178) -- (6.607,4.15) -- (6.486,4.069) -- (6.405,3.948) -- (6.377,3.805) -- (6.405,3.662) -- (6.486,3.541) -- (6.607,3.46) -- (6.75,3.432) -- (6.893,3.46) -- (7.014,3.541) -- (7.095,3.662) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.082,1.854) -- (2.054,1.997) -- (1.973,2.118) -- (1.852,2.199) -- (1.709,2.227) -- (1.566,2.199) -- (1.445,2.118) -- (1.364,1.997) -- (1.336,1.854) -- (1.364,1.711) -- (1.445,1.59) -- (1.566,1.509) -- (1.709,1.481) -- (1.852,1.509) -- (1.973,1.59) -- (2.054,1.711) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.653,1.617) -- (9.624,1.76) -- (9.543,1.881) -- (9.422,1.962) -- (9.28,1.99) -- (9.137,1.962) -- (9.016,1.881) -- (8.935,1.76) -- (8.906,1.617) -- (8.935,1.474) -- (9.016,1.353) -- (9.137,1.272) -- (9.28,1.244) -- (9.422,1.272) -- (9.543,1.353) -- (9.624,1.474) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.511,1.57) -- (7.483,1.713) -- (7.402,1.834) -- (7.281,1.915) -- (7.138,1.943) -- (6.995,1.915) -- (6.874,1.834) -- (6.793,1.713) -- (6.765,1.57) -- (6.793,1.427) -- (6.874,1.306) -- (6.995,1.225) -- (7.138,1.197) -- (7.281,1.225) -- (7.402,1.306) -- (7.483,1.427) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (12.082,1.443) -- (12.054,1.586) -- (11.973,1.707) -- (11.852,1.788) -- (11.709,1.816) -- (11.566,1.788) -- (11.445,1.707) -- (11.364,1.586) -- (11.336,1.443) -- (11.364,1.3) -- (11.445,1.179) -- (11.566,1.098) -- (11.709,1.07) -- (11.852,1.098) -- (11.973,1.179) -- (12.054,1.3) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.676,6.246) -- (5.795,6.264) -- (5.902,6.319) -- (5.987,6.404) -- (6.042,6.511) -- (6.061,6.63) -- (6.042,6.749) -- (5.987,6.856) -- (5.902,6.941) -- (5.795,6.996) -- (5.676,7.015) -- (5.676,7.015) -- (5.557,6.996) -- (5.45,6.941) -- (5.365,6.856) -- (5.31,6.749) -- (5.292,6.63) -- (5.31,6.511) -- (5.365,6.404) -- (5.45,6.319) -- (5.557,6.264) -- (5.676,6.246) -- cycle;
\node at (5.676,6.63) {\scalebox{1.656}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.157,6.137) -- (8.276,6.156) -- (8.383,6.21) -- (8.468,6.295) -- (8.523,6.403) -- (8.542,6.521) -- (8.523,6.64) -- (8.468,6.747) -- (8.383,6.833) -- (8.276,6.887) -- (8.157,6.906) -- (8.157,6.906) -- (8.038,6.887) -- (7.931,6.833) -- (7.846,6.747) -- (7.791,6.64) -- (7.773,6.521) -- (7.791,6.403) -- (7.846,6.295) -- (7.931,6.21) -- (8.038,6.156) -- (8.157,6.137) -- cycle;
\node at (8.157,6.521) {\scalebox{1.656}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.155,4.524) -- (7.274,4.543) -- (7.381,4.598) -- (7.467,4.683) -- (7.521,4.79) -- (7.54,4.909) -- (7.521,5.027) -- (7.467,5.135) -- (7.381,5.22) -- (7.274,5.274) -- (7.155,5.293) -- (7.155,5.293) -- (7.037,5.274) -- (6.929,5.22) -- (6.844,5.135) -- (6.79,5.027) -- (6.771,4.909) -- (6.79,4.79) -- (6.844,4.683) -- (6.929,4.598) -- (7.037,4.543) -- (7.155,4.524) -- cycle;
\node at (7.155,4.909) {\scalebox{1.656}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.252,4.562) -- (11.371,4.581) -- (11.478,4.635) -- (11.564,4.72) -- (11.618,4.828) -- (11.637,4.946) -- (11.618,5.065) -- (11.564,5.172) -- (11.478,5.257) -- (11.371,5.312) -- (11.252,5.331) -- (11.252,5.331) -- (11.134,5.312) -- (11.026,5.257) -- (10.941,5.172) -- (10.887,5.065) -- (10.868,4.946) -- (10.887,4.828) -- (10.941,4.72) -- (11.026,4.635) -- (11.134,4.581) -- (11.252,4.562) -- cycle;
\node at (11.252,4.946) {\scalebox{1.656}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.36,5.5) -- (3.479,5.519) -- (3.586,5.574) -- (3.671,5.659) -- (3.726,5.766) -- (3.745,5.885) -- (3.726,6.004) -- (3.671,6.111) -- (3.586,6.196) -- (3.479,6.251) -- (3.36,6.27) -- (3.36,6.27) -- (3.242,6.251) -- (3.134,6.196) -- (3.049,6.111) -- (2.995,6.004) -- (2.976,5.885) -- (2.995,5.766) -- (3.049,5.659) -- (3.134,5.574) -- (3.242,5.519) -- (3.36,5.5) -- cycle;
\node at (3.36,5.885) {\scalebox{1.656}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.595,4.55) -- (2.714,4.569) -- (2.821,4.624) -- (2.906,4.709) -- (2.961,4.816) -- (2.979,4.935) -- (2.961,5.053) -- (2.906,5.161) -- (2.821,5.246) -- (2.714,5.3) -- (2.595,5.319) -- (2.595,5.319) -- (2.476,5.3) -- (2.369,5.246) -- (2.284,5.161) -- (2.229,5.053) -- (2.21,4.935) -- (2.229,4.816) -- (2.284,4.709) -- (2.369,4.624) -- (2.476,4.569) -- (2.595,4.55) -- cycle;
\node at (2.595,4.935) {\scalebox{1.656}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.38,5.169) -- (10.499,5.188) -- (10.606,5.242) -- (10.691,5.327) -- (10.746,5.434) -- (10.765,5.553) -- (10.746,5.672) -- (10.691,5.779) -- (10.606,5.864) -- (10.499,5.919) -- (10.38,5.938) -- (10.38,5.938) -- (10.261,5.919) -- (10.154,5.864) -- (10.069,5.779) -- (10.014,5.672) -- (9.995,5.553) -- (10.014,5.434) -- (10.069,5.327) -- (10.154,5.242) -- (10.261,5.188) -- (10.38,5.169) -- cycle;
\node at (10.38,5.553) {\scalebox{1.656}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.983,4.743) -- (9.102,4.762) -- (9.209,4.816) -- (9.294,4.901) -- (9.349,5.009) -- (9.367,5.127) -- (9.349,5.246) -- (9.294,5.353) -- (9.209,5.438) -- (9.102,5.493) -- (8.983,5.512) -- (8.983,5.512) -- (8.864,5.493) -- (8.757,5.438) -- (8.672,5.353) -- (8.617,5.246) -- (8.598,5.127) -- (8.617,5.009) -- (8.672,4.901) -- (8.757,4.816) -- (8.864,4.762) -- (8.983,4.743) -- cycle;
\node at (8.983,5.127) {\scalebox{1.656}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.762,3.169) -- (10.881,3.188) -- (10.988,3.243) -- (11.073,3.328) -- (11.128,3.435) -- (11.147,3.554) -- (11.128,3.673) -- (11.073,3.78) -- (10.988,3.865) -- (10.881,3.92) -- (10.762,3.938) -- (10.762,3.938) -- (10.643,3.92) -- (10.536,3.865) -- (10.451,3.78) -- (10.396,3.673) -- (10.378,3.554) -- (10.396,3.435) -- (10.451,3.328) -- (10.536,3.243) -- (10.643,3.188) -- (10.762,3.169) -- cycle;
\node at (10.762,3.554) {\scalebox{1.656}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.954,2.541) -- (10.073,2.56) -- (10.18,2.615) -- (10.265,2.7) -- (10.32,2.807) -- (10.339,2.926) -- (10.32,3.045) -- (10.265,3.152) -- (10.18,3.237) -- (10.073,3.292) -- (9.954,3.31) -- (9.592,3.31) -- (9.473,3.292) -- (9.366,3.237) -- (9.281,3.152) -- (9.226,3.045) -- (9.208,2.926) -- (9.226,2.807) -- (9.281,2.7) -- (9.366,2.615) -- (9.473,2.56) -- (9.592,2.541) -- cycle;
\node at (9.773,2.926) {\scalebox{1.656}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (12.145,2.299) -- (12.264,2.318) -- (12.371,2.372) -- (12.456,2.457) -- (12.511,2.565) -- (12.53,2.683) -- (12.511,2.802) -- (12.456,2.91) -- (12.371,2.995) -- (12.264,3.049) -- (12.145,3.068) -- (11.783,3.068) -- (11.664,3.049) -- (11.557,2.995) -- (11.472,2.91) -- (11.418,2.802) -- (11.399,2.683) -- (11.418,2.565) -- (11.472,2.457) -- (11.557,2.372) -- (11.664,2.318) -- (11.783,2.299) -- cycle;
\node at (11.964,2.683) {\scalebox{1.656}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.532,2.416) -- (2.651,2.435) -- (2.758,2.489) -- (2.843,2.574) -- (2.898,2.681) -- (2.917,2.8) -- (2.898,2.919) -- (2.843,3.026) -- (2.758,3.111) -- (2.651,3.166) -- (2.532,3.185) -- (2.17,3.185) -- (2.051,3.166) -- (1.944,3.111) -- (1.859,3.026) -- (1.805,2.919) -- (1.786,2.8) -- (1.805,2.681) -- (1.859,2.574) -- (1.944,2.489) -- (2.051,2.435) -- (2.17,2.416) -- cycle;
\node at (2.351,2.8) {\scalebox{1.656}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.279,3.152) -- (8.398,3.171) -- (8.505,3.225) -- (8.59,3.311) -- (8.645,3.418) -- (8.664,3.537) -- (8.645,3.655) -- (8.59,3.763) -- (8.505,3.848) -- (8.398,3.902) -- (8.279,3.921) -- (7.917,3.921) -- (7.798,3.902) -- (7.691,3.848) -- (7.606,3.763) -- (7.551,3.655) -- (7.533,3.537) -- (7.551,3.418) -- (7.606,3.311) -- (7.691,3.225) -- (7.798,3.171) -- (7.917,3.152) -- cycle;
\node at (8.098,3.537) {\scalebox{1.656}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.547,1.907) -- (6.666,1.925) -- (6.773,1.98) -- (6.858,2.065) -- (6.913,2.172) -- (6.932,2.291) -- (6.913,2.41) -- (6.858,2.517) -- (6.773,2.602) -- (6.666,2.657) -- (6.547,2.676) -- (6.185,2.676) -- (6.067,2.657) -- (5.959,2.602) -- (5.874,2.517) -- (5.82,2.41) -- (5.801,2.291) -- (5.82,2.172) -- (5.874,2.065) -- (5.959,1.98) -- (6.067,1.925) -- (6.185,1.907) -- cycle;
\node at (6.366,2.291) {\scalebox{1.656}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.696,1.429) -- (10.814,1.448) -- (10.922,1.502) -- (11.007,1.588) -- (11.061,1.695) -- (11.08,1.814) -- (11.061,1.932) -- (11.007,2.04) -- (10.922,2.125) -- (10.814,2.179) -- (10.696,2.198) -- (10.334,2.198) -- (10.215,2.179) -- (10.108,2.125) -- (10.023,2.04) -- (9.968,1.932) -- (9.949,1.814) -- (9.968,1.695) -- (10.023,1.588) -- (10.108,1.502) -- (10.215,1.448) -- (10.334,1.429) -- cycle;
\node at (10.515,1.814) {\scalebox{1.656}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.379,1.682) -- (8.498,1.701) -- (8.605,1.755) -- (8.69,1.84) -- (8.745,1.947) -- (8.764,2.066) -- (8.745,2.185) -- (8.69,2.292) -- (8.605,2.377) -- (8.498,2.432) -- (8.379,2.451) -- (8.017,2.451) -- (7.899,2.432) -- (7.791,2.377) -- (7.706,2.292) -- (7.652,2.185) -- (7.633,2.066) -- (7.652,1.947) -- (7.706,1.84) -- (7.791,1.755) -- (7.899,1.701) -- (8.017,1.682) -- cycle;
\node at (8.198,2.066) {\scalebox{1.656}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (11.409,1.143) -- (12.009,1.143) -- (12.009,1.743) -- (11.409,1.743) -- cycle;
\node at (11.709,1.443) {\scalebox{1.656}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.407,5.233) -- (2.007,5.233) -- (2.007,5.833) -- (1.407,5.833) -- cycle;
\node at (1.707,5.533) {\scalebox{1.656}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (0.6,2.438) -- (1.2,2.438) -- (1.2,3.038) -- (0.6,3.038) -- cycle;
\node at (0.9,2.738) {\scalebox{1.656}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (12.374,5.209) -- (12.974,5.209) -- (12.974,5.809) -- (12.374,5.809) -- cycle;
\node at (12.674,5.509) {\scalebox{1.656}{\small d}};
\node at (3.788,7.069) {\scalebox{1.656}{\small A}};
\node at (6.911,6.688) {\scalebox{1.656}{\small B}};
\node at (12.003,6.301) {\scalebox{1.656}{\small C}};
\node at (1.603,6.235) {\scalebox{1.656}{\small D}};
\node at (9.763,6.43) {\scalebox{1.656}{\small E}};
\node at (9.9,4.279) {\scalebox{1.656}{\small F}};
\node at (12.274,3.887) {\scalebox{1.656}{\small G}};
\node at (1.447,3.886) {\scalebox{1.656}{\small H}};
\node at (6.222,3.481) {\scalebox{1.656}{\small I}};
\node at (1.134,1.621) {\scalebox{1.656}{\small J}};
\node at (9.678,1.142) {\scalebox{1.656}{\small K}};
\node at (7.12,0.95) {\scalebox{1.656}{\small L}};
\node at (12.243,1.129) {\scalebox{1.656}{\small M}};
\draw[line width=0.5pt] (13.009,5.712) -- (13.009,6.812);
\draw[line width=0.5pt] (12.778,6.427) -- (13.009,6.812) -- (13.24,6.427);
\draw[line width=0.5pt] (12.706,6.262) -- (13.311,6.262);
\node at (13.009,7.197) {\scalebox{1.656}{\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, G, H, I, K & 6 \\ \hline
clairs & B, C, D, F, J, L, M & 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{M} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{rue des Lilas ⑪} vers le nord jusqu'à Carrefour des Tilleuls \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Four ④}, vers le nord, jusqu'à Placette des Ormes \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Écoles ⑦}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{venelle du Chat ⑧}, vers le sud, jusqu'à Rond-point de la Gare \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Puits ⑩}, vers le sud, jusqu'à Carrefour Saint-Roch \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Neuve ⑯}, vers l'ouest, jusqu'à Place Sainte-Croix \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Moulin ⑭}, vers le nord, jusqu'à Carrefour du Lavoir \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Pasteur ③}, vers le nord, jusqu'à Carrefour de l'Hôpital \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin Creux ①}, vers l'ouest, jusqu'à Place du Vieux-Puits \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin des Vignes ⑤}, vers l'ouest, jusqu'à Place du Marché \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de l'Église ⑥}, vers le sud, jusqu'à Place des Tanneurs \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{impasse des Roses ⑫}, vers le sud, jusqu'à Place de la Fontaine \textbf{(J)}.\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}
