\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 187}

\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=187}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0187}
\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=0.6067, line join=round, line cap=round]
\draw[line width=0.5pt] (8.405,6.494) -- (8.601,6.429) -- (8.798,6.368) -- (8.997,6.311) -- (9.198,6.259) -- (9.399,6.212) -- (9.602,6.17) -- (9.805,6.132) -- (10.01,6.1) -- (10.215,6.072) -- (10.42,6.049) -- (10.627,6.03) -- (10.833,6.017) -- (11.04,6.008) -- (11.246,6.005);
\draw[line width=0.5pt] (8.405,6.494) -- (8.444,6.316) -- (8.48,6.136) -- (8.511,5.955) -- (8.538,5.773) -- (8.561,5.591) -- (8.579,5.409) -- (8.593,5.226) -- (8.603,5.042) -- (8.609,4.859) -- (8.61,4.675) -- (8.607,4.492) -- (8.599,4.309) -- (8.588,4.126) -- (8.572,3.943);
\draw[line width=0.5pt] (6.093,6.411) -- (6.257,6.391) -- (6.423,6.376) -- (6.588,6.364) -- (6.754,6.357) -- (6.92,6.353) -- (7.086,6.353) -- (7.252,6.357) -- (7.418,6.365) -- (7.584,6.377) -- (7.749,6.393) -- (7.914,6.412) -- (8.078,6.436) -- (8.242,6.463) -- (8.405,6.494);
\draw[line width=0.5pt] (6.093,6.411) -- (6.13,6.208) -- (6.163,6.004) -- (6.19,5.799) -- (6.213,5.594) -- (6.232,5.388) -- (6.245,5.182) -- (6.253,4.976) -- (6.257,4.769) -- (6.256,4.563) -- (6.25,4.357) -- (6.239,4.15) -- (6.223,3.945) -- (6.202,3.739) -- (6.177,3.534);
\draw[line width=0.5pt] (11.246,6.005) -- (11.231,5.874) -- (11.219,5.743) -- (11.209,5.611) -- (11.203,5.479) -- (11.2,5.347) -- (11.201,5.216) -- (11.204,5.084) -- (11.21,4.952) -- (11.22,4.82) -- (11.232,4.689) -- (11.248,4.558) -- (11.266,4.428) -- (11.288,4.298) -- (11.313,4.168);
\draw[line width=0.5pt] (8.572,3.943) -- (8.763,3.989) -- (8.956,4.03) -- (9.15,4.067) -- (9.345,4.099) -- (9.54,4.127) -- (9.736,4.15) -- (9.933,4.169) -- (10.13,4.182) -- (10.327,4.192) -- (10.524,4.196) -- (10.722,4.196) -- (10.919,4.192) -- (11.116,4.182) -- (11.313,4.168);
\draw[line width=0.5pt] (1.756,3.669) -- (1.919,3.635) -- (2.083,3.606) -- (2.248,3.58) -- (2.413,3.558) -- (2.579,3.54) -- (2.746,3.526) -- (2.912,3.515) -- (3.079,3.509) -- (3.246,3.506) -- (3.413,3.508) -- (3.579,3.513) -- (3.746,3.523) -- (3.912,3.536) -- (4.078,3.553);
\draw[line width=0.5pt] (1.756,3.669) -- (1.751,3.492) -- (1.742,3.315) -- (1.729,3.139) -- (1.711,2.962) -- (1.69,2.787) -- (1.664,2.611) -- (1.635,2.436) -- (1.601,2.263) -- (1.563,2.089) -- (1.521,1.917) -- (1.475,1.746) -- (1.425,1.576) -- (1.371,1.408) -- (1.313,1.24);
\draw[line width=0.5pt] (4.078,3.553) -- (4.226,3.529) -- (4.376,3.508) -- (4.525,3.491) -- (4.675,3.477) -- (4.825,3.467) -- (4.976,3.46) -- (5.127,3.457) -- (5.277,3.458) -- (5.428,3.462) -- (5.578,3.469) -- (5.729,3.48) -- (5.878,3.495) -- (6.028,3.513) -- (6.177,3.534);
\draw[line width=0.5pt] (4.078,3.553) -- (4.093,3.42) -- (4.104,3.286) -- (4.112,3.152) -- (4.117,3.018) -- (4.119,2.883) -- (4.118,2.749) -- (4.113,2.615) -- (4.106,2.481) -- (4.095,2.347) -- (4.081,2.213) -- (4.064,2.08) -- (4.044,1.948) -- (4.021,1.815) -- (3.995,1.684);
\draw[line width=0.5pt] (6.177,3.534) -- (6.351,3.537) -- (6.525,3.544) -- (6.699,3.555) -- (6.873,3.57) -- (7.046,3.589) -- (7.219,3.613) -- (7.391,3.64) -- (7.563,3.671) -- (7.733,3.707) -- (7.903,3.746) -- (8.072,3.789) -- (8.24,3.837) -- (8.406,3.888) -- (8.572,3.943);
\draw[line width=0.5pt] (6.177,3.534) -- (6.191,3.418) -- (6.202,3.301) -- (6.21,3.184) -- (6.216,3.067) -- (6.218,2.949) -- (6.218,2.832) -- (6.216,2.714) -- (6.21,2.597) -- (6.202,2.48) -- (6.191,2.363) -- (6.177,2.246) -- (6.161,2.13) -- (6.141,2.014) -- (6.119,1.899);
\draw[line width=0.5pt] (3.995,1.684) -- (4.148,1.676) -- (4.301,1.671) -- (4.454,1.671) -- (4.607,1.674) -- (4.76,1.68) -- (4.913,1.69) -- (5.066,1.704) -- (5.218,1.721) -- (5.37,1.742) -- (5.521,1.766) -- (5.672,1.794) -- (5.822,1.826) -- (5.971,1.861) -- (6.119,1.899);
\draw[line width=0.5pt] (1.313,1.24) -- (1.498,1.301) -- (1.685,1.358) -- (1.873,1.409) -- (2.062,1.457) -- (2.252,1.5) -- (2.443,1.539) -- (2.635,1.573) -- (2.828,1.602) -- (3.022,1.627) -- (3.216,1.648) -- (3.41,1.664) -- (3.605,1.675) -- (3.8,1.682) -- (3.995,1.684);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.565,6.494) -- (8.543,6.574) -- (8.485,6.633) -- (8.405,6.654) -- (8.325,6.633) -- (8.266,6.574) -- (8.245,6.494) -- (8.266,6.414) -- (8.325,6.356) -- (8.405,6.334) -- (8.485,6.356) -- (8.543,6.414) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.253,6.411) -- (6.231,6.491) -- (6.173,6.549) -- (6.093,6.571) -- (6.013,6.549) -- (5.954,6.491) -- (5.933,6.411) -- (5.954,6.331) -- (6.013,6.272) -- (6.093,6.251) -- (6.173,6.272) -- (6.231,6.331) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.406,6.005) -- (11.385,6.085) -- (11.326,6.143) -- (11.246,6.165) -- (11.166,6.143) -- (11.108,6.085) -- (11.086,6.005) -- (11.108,5.925) -- (11.166,5.866) -- (11.246,5.845) -- (11.326,5.866) -- (11.385,5.925) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.473,4.168) -- (11.451,4.248) -- (11.393,4.307) -- (11.313,4.328) -- (11.233,4.307) -- (11.174,4.248) -- (11.153,4.168) -- (11.174,4.088) -- (11.233,4.03) -- (11.313,4.008) -- (11.393,4.03) -- (11.451,4.088) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.732,3.943) -- (8.71,4.023) -- (8.652,4.081) -- (8.572,4.103) -- (8.492,4.081) -- (8.433,4.023) -- (8.412,3.943) -- (8.433,3.863) -- (8.492,3.804) -- (8.572,3.783) -- (8.652,3.804) -- (8.71,3.863) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.916,3.669) -- (1.894,3.749) -- (1.836,3.808) -- (1.756,3.829) -- (1.676,3.808) -- (1.617,3.749) -- (1.596,3.669) -- (1.617,3.589) -- (1.676,3.531) -- (1.756,3.509) -- (1.836,3.531) -- (1.894,3.589) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.238,3.553) -- (4.217,3.633) -- (4.158,3.692) -- (4.078,3.713) -- (3.998,3.692) -- (3.939,3.633) -- (3.918,3.553) -- (3.939,3.473) -- (3.998,3.414) -- (4.078,3.393) -- (4.158,3.414) -- (4.217,3.473) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.337,3.534) -- (6.315,3.614) -- (6.257,3.673) -- (6.177,3.694) -- (6.097,3.673) -- (6.038,3.614) -- (6.017,3.534) -- (6.038,3.454) -- (6.097,3.396) -- (6.177,3.374) -- (6.257,3.396) -- (6.315,3.454) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.279,1.899) -- (6.258,1.979) -- (6.199,2.038) -- (6.119,2.059) -- (6.039,2.038) -- (5.981,1.979) -- (5.959,1.899) -- (5.981,1.819) -- (6.039,1.76) -- (6.119,1.739) -- (6.199,1.76) -- (6.258,1.819) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.155,1.684) -- (4.133,1.764) -- (4.075,1.822) -- (3.995,1.844) -- (3.915,1.822) -- (3.856,1.764) -- (3.835,1.684) -- (3.856,1.604) -- (3.915,1.545) -- (3.995,1.524) -- (4.075,1.545) -- (4.133,1.604) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.473,1.24) -- (1.451,1.32) -- (1.393,1.379) -- (1.313,1.4) -- (1.233,1.379) -- (1.174,1.32) -- (1.153,1.24) -- (1.174,1.16) -- (1.233,1.102) -- (1.313,1.08) -- (1.393,1.102) -- (1.451,1.16) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.872,6.127) -- (9.994,6.146) -- (10.104,6.202) -- (10.191,6.289) -- (10.247,6.399) -- (10.266,6.521) -- (10.247,6.642) -- (10.191,6.752) -- (10.104,6.839) -- (9.994,6.895) -- (9.872,6.914) -- (9.872,6.914) -- (9.75,6.895) -- (9.641,6.839) -- (9.554,6.752) -- (9.498,6.642) -- (9.478,6.521) -- (9.498,6.399) -- (9.554,6.289) -- (9.641,6.202) -- (9.75,6.146) -- (9.872,6.127) -- cycle;
\node at (9.872,6.521) {\scalebox{1.648}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.2,4.806) -- (8.322,4.825) -- (8.432,4.881) -- (8.519,4.968) -- (8.575,5.078) -- (8.594,5.2) -- (8.575,5.322) -- (8.519,5.431) -- (8.432,5.519) -- (8.322,5.574) -- (8.2,5.594) -- (8.2,5.594) -- (8.079,5.574) -- (7.969,5.519) -- (7.882,5.431) -- (7.826,5.322) -- (7.806,5.2) -- (7.826,5.078) -- (7.882,4.968) -- (7.969,4.881) -- (8.079,4.825) -- (8.2,4.806) -- cycle;
\node at (8.2,5.2) {\scalebox{1.648}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.238,6.357) -- (7.36,6.376) -- (7.469,6.432) -- (7.556,6.519) -- (7.612,6.629) -- (7.632,6.751) -- (7.612,6.873) -- (7.556,6.982) -- (7.469,7.07) -- (7.36,7.125) -- (7.238,7.145) -- (7.238,7.145) -- (7.116,7.125) -- (7.006,7.07) -- (6.919,6.982) -- (6.863,6.873) -- (6.844,6.751) -- (6.863,6.629) -- (6.919,6.519) -- (7.006,6.432) -- (7.116,6.376) -- (7.238,6.357) -- cycle;
\node at (7.238,6.751) {\scalebox{1.648}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.647,4.594) -- (6.769,4.613) -- (6.879,4.669) -- (6.966,4.756) -- (7.022,4.866) -- (7.041,4.988) -- (7.022,5.109) -- (6.966,5.219) -- (6.879,5.306) -- (6.769,5.362) -- (6.647,5.381) -- (6.647,5.381) -- (6.525,5.362) -- (6.416,5.306) -- (6.328,5.219) -- (6.272,5.109) -- (6.253,4.988) -- (6.272,4.866) -- (6.328,4.756) -- (6.416,4.669) -- (6.525,4.613) -- (6.647,4.594) -- cycle;
\node at (6.647,4.988) {\scalebox{1.648}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.597,4.704) -- (11.719,4.723) -- (11.829,4.779) -- (11.916,4.866) -- (11.972,4.976) -- (11.991,5.098) -- (11.972,5.22) -- (11.916,5.329) -- (11.829,5.417) -- (11.719,5.473) -- (11.597,5.492) -- (11.597,5.492) -- (11.476,5.473) -- (11.366,5.417) -- (11.279,5.329) -- (11.223,5.22) -- (11.204,5.098) -- (11.223,4.976) -- (11.279,4.866) -- (11.366,4.779) -- (11.476,4.723) -- (11.597,4.704) -- cycle;
\node at (11.597,5.098) {\scalebox{1.648}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.341,3.404) -- (10.463,3.423) -- (10.572,3.479) -- (10.66,3.567) -- (10.716,3.676) -- (10.735,3.798) -- (10.716,3.92) -- (10.66,4.03) -- (10.572,4.117) -- (10.463,4.173) -- (10.341,4.192) -- (10.341,4.192) -- (10.219,4.173) -- (10.109,4.117) -- (10.022,4.03) -- (9.966,3.92) -- (9.947,3.798) -- (9.966,3.676) -- (10.022,3.567) -- (10.109,3.479) -- (10.219,3.423) -- (10.341,3.404) -- cycle;
\node at (10.341,3.798) {\scalebox{1.648}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.786,3.686) -- (2.908,3.706) -- (3.018,3.761) -- (3.105,3.849) -- (3.161,3.958) -- (3.18,4.08) -- (3.161,4.202) -- (3.105,4.312) -- (3.018,4.399) -- (2.908,4.455) -- (2.786,4.474) -- (2.786,4.474) -- (2.665,4.455) -- (2.555,4.399) -- (2.468,4.312) -- (2.412,4.202) -- (2.392,4.08) -- (2.412,3.958) -- (2.468,3.849) -- (2.555,3.761) -- (2.665,3.706) -- (2.786,3.686) -- cycle;
\node at (2.786,4.08) {\scalebox{1.648}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.247,2.113) -- (1.369,2.133) -- (1.479,2.188) -- (1.566,2.276) -- (1.622,2.385) -- (1.641,2.507) -- (1.622,2.629) -- (1.566,2.739) -- (1.479,2.826) -- (1.369,2.882) -- (1.247,2.901) -- (1.247,2.901) -- (1.125,2.882) -- (1.015,2.826) -- (0.928,2.739) -- (0.872,2.629) -- (0.853,2.507) -- (0.872,2.385) -- (0.928,2.276) -- (1.015,2.188) -- (1.125,2.133) -- (1.247,2.113) -- cycle;
\node at (1.247,2.507) {\scalebox{1.648}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.13,3.457) -- (5.252,3.476) -- (5.362,3.532) -- (5.449,3.62) -- (5.505,3.729) -- (5.524,3.851) -- (5.505,3.973) -- (5.449,4.083) -- (5.362,4.17) -- (5.252,4.226) -- (5.13,4.245) -- (5.13,4.245) -- (5.008,4.226) -- (4.899,4.17) -- (4.811,4.083) -- (4.755,3.973) -- (4.736,3.851) -- (4.755,3.729) -- (4.811,3.62) -- (4.899,3.532) -- (5.008,3.476) -- (5.13,3.457) -- cycle;
\node at (5.13,3.851) {\scalebox{1.648}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.854,2.196) -- (4.976,2.216) -- (5.086,2.272) -- (5.173,2.359) -- (5.229,2.468) -- (5.248,2.59) -- (5.229,2.712) -- (5.173,2.822) -- (5.086,2.909) -- (4.976,2.965) -- (4.854,2.984) -- (4.484,2.984) -- (4.362,2.965) -- (4.252,2.909) -- (4.165,2.822) -- (4.109,2.712) -- (4.09,2.59) -- (4.109,2.468) -- (4.165,2.359) -- (4.252,2.272) -- (4.362,2.216) -- (4.484,2.196) -- cycle;
\node at (4.669,2.59) {\scalebox{1.648}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.51,3.634) -- (7.632,3.654) -- (7.742,3.71) -- (7.829,3.797) -- (7.885,3.906) -- (7.904,4.028) -- (7.885,4.15) -- (7.829,4.26) -- (7.742,4.347) -- (7.632,4.403) -- (7.51,4.422) -- (7.139,4.422) -- (7.018,4.403) -- (6.908,4.347) -- (6.821,4.26) -- (6.765,4.15) -- (6.746,4.028) -- (6.765,3.906) -- (6.821,3.797) -- (6.908,3.71) -- (7.018,3.654) -- (7.139,3.634) -- cycle;
\node at (7.325,4.028) {\scalebox{1.648}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.721,2.309) -- (6.842,2.329) -- (6.952,2.384) -- (7.039,2.472) -- (7.095,2.581) -- (7.115,2.703) -- (7.095,2.825) -- (7.039,2.935) -- (6.952,3.022) -- (6.842,3.078) -- (6.721,3.097) -- (6.35,3.097) -- (6.228,3.078) -- (6.118,3.022) -- (6.031,2.935) -- (5.975,2.825) -- (5.956,2.703) -- (5.975,2.581) -- (6.031,2.472) -- (6.118,2.384) -- (6.228,2.329) -- (6.35,2.309) -- cycle;
\node at (6.535,2.703) {\scalebox{1.648}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.142,0.742) -- (5.263,0.761) -- (5.373,0.817) -- (5.46,0.904) -- (5.516,1.014) -- (5.536,1.136) -- (5.516,1.257) -- (5.46,1.367) -- (5.373,1.454) -- (5.263,1.51) -- (5.142,1.53) -- (4.771,1.53) -- (4.649,1.51) -- (4.539,1.454) -- (4.452,1.367) -- (4.396,1.257) -- (4.377,1.136) -- (4.396,1.014) -- (4.452,0.904) -- (4.539,0.817) -- (4.649,0.761) -- (4.771,0.742) -- cycle;
\node at (4.956,1.136) {\scalebox{1.648}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.332,1.992) -- (3.454,2.011) -- (3.564,2.067) -- (3.651,2.155) -- (3.707,2.264) -- (3.726,2.386) -- (3.707,2.508) -- (3.651,2.618) -- (3.564,2.705) -- (3.454,2.761) -- (3.332,2.78) -- (2.961,2.78) -- (2.84,2.761) -- (2.73,2.705) -- (2.643,2.618) -- (2.587,2.508) -- (2.567,2.386) -- (2.587,2.264) -- (2.643,2.155) -- (2.73,2.067) -- (2.84,2.011) -- (2.961,1.992) -- cycle;
\node at (3.147,2.386) {\scalebox{1.648}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.013,0.94) -- (1.613,0.94) -- (1.613,1.54) -- (1.013,1.54) -- cycle;
\node at (1.313,1.24) {\scalebox{1.648}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (9.17,4.372) -- (9.77,4.372) -- (9.77,4.972) -- (9.17,4.972) -- cycle;
\node at (9.47,4.672) {\scalebox{1.648}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.837,1.738) -- (2.437,1.738) -- (2.437,2.338) -- (1.837,2.338) -- cycle;
\node at (2.137,2.038) {\scalebox{1.648}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (6.986,1.685) -- (7.586,1.685) -- (7.586,2.285) -- (6.986,2.285) -- cycle;
\node at (7.286,1.985) {\scalebox{1.648}{\small d}};
\node at (8.795,6.976) {\scalebox{1.648}{\small A}};
\node at (6.047,7.029) {\scalebox{1.648}{\small B}};
\node at (11.816,6.249) {\scalebox{1.648}{\small C}};
\node at (11.932,4.205) {\scalebox{1.648}{\small D}};
\node at (9.191,3.962) {\scalebox{1.648}{\small E}};
\node at (1.137,3.641) {\scalebox{1.648}{\small F}};
\node at (3.464,3.464) {\scalebox{1.648}{\small G}};
\node at (5.998,2.941) {\scalebox{1.648}{\small H}};
\node at (6.07,1.281) {\scalebox{1.648}{\small I}};
\node at (3.547,1.255) {\scalebox{1.648}{\small J}};
\node at (0.765,0.95) {\scalebox{1.648}{\small K}};
\draw[line width=0.5pt] (12.613,5.544) -- (12.613,6.644);
\draw[line width=0.5pt] (12.382,6.259) -- (12.613,6.644) -- (12.844,6.259);
\draw[line width=0.5pt] (12.31,6.094) -- (12.915,6.094);
\node at (12.613,7.029) {\scalebox{1.648}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour K, 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 & ⑧ boulevard du Parc & Ⓐ Carrefour des Tilleuls & Ⓘ Place des Tanneurs \\ \hline
① avenue de la Gare & ⑨ rue du Moulin & Ⓑ Place du Marché & Ⓙ Rond-point du Pont \\ \hline
② allée des Tilleuls & ⑩ rue du Pont & Ⓒ Coin du Lavoir & Ⓚ Place de l'Église \\ \hline
③ rue de la Poste & ⑪ rue de l'Église & Ⓓ Placette des Ormes & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ passage du Marché & ⑫ rue des Écoles & Ⓔ Carrefour de l'Hôpital & \textbf{a} l'école \\ \hline
⑤ quai aux Fleurs & ⑬ chemin du Lavoir & Ⓕ Place aux Herbes & \textbf{b} le cimetière \\ \hline
⑥ rue des Jardins & ⑭ chemin des Vignes & Ⓖ Rond-point de la Poste & \textbf{c} le lavoir \\ \hline
⑦ rue du Puits & \cellcolor{black!32}\bfseries Les carrefours & Ⓗ Carrefour du Lavoir & \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=0.6215, line join=round, line cap=round]
\draw[line width=0.5pt] (8.405,6.652) -- (8.601,6.586) -- (8.798,6.525) -- (8.997,6.468) -- (9.198,6.416) -- (9.399,6.369) -- (9.602,6.327) -- (9.805,6.289) -- (10.01,6.257) -- (10.215,6.229) -- (10.42,6.206) -- (10.627,6.187) -- (10.833,6.174) -- (11.04,6.165) -- (11.246,6.162);
\draw[line width=0.5pt] (8.405,6.652) -- (8.444,6.473) -- (8.48,6.293) -- (8.511,6.112) -- (8.538,5.93) -- (8.561,5.748) -- (8.579,5.566) -- (8.593,5.383) -- (8.603,5.199) -- (8.609,5.016) -- (8.61,4.832) -- (8.607,4.649) -- (8.599,4.466) -- (8.588,4.283) -- (8.572,4.1);
\draw[line width=0.5pt] (6.093,6.568) -- (6.257,6.548) -- (6.423,6.533) -- (6.588,6.521) -- (6.754,6.514) -- (6.92,6.51) -- (7.086,6.51) -- (7.252,6.514) -- (7.418,6.522) -- (7.584,6.534) -- (7.749,6.55) -- (7.914,6.569) -- (8.078,6.593) -- (8.242,6.62) -- (8.405,6.652);
\draw[line width=0.5pt] (6.093,6.568) -- (6.13,6.365) -- (6.163,6.161) -- (6.19,5.956) -- (6.213,5.751) -- (6.232,5.546) -- (6.245,5.339) -- (6.253,5.133) -- (6.257,4.927) -- (6.256,4.72) -- (6.25,4.514) -- (6.239,4.307) -- (6.223,4.102) -- (6.202,3.896) -- (6.177,3.691);
\draw[line width=0.5pt] (11.246,6.162) -- (11.231,6.031) -- (11.219,5.9) -- (11.209,5.768) -- (11.203,5.636) -- (11.2,5.505) -- (11.201,5.373) -- (11.204,5.241) -- (11.21,5.109) -- (11.22,4.977) -- (11.232,4.846) -- (11.248,4.715) -- (11.266,4.585) -- (11.288,4.455) -- (11.313,4.325);
\draw[line width=0.5pt] (8.572,4.1) -- (8.763,4.146) -- (8.956,4.187) -- (9.15,4.224) -- (9.345,4.256) -- (9.54,4.284) -- (9.736,4.307) -- (9.933,4.326) -- (10.13,4.34) -- (10.327,4.349) -- (10.524,4.353) -- (10.722,4.353) -- (10.919,4.349) -- (11.116,4.339) -- (11.313,4.325);
\draw[line width=0.5pt] (1.756,3.826) -- (1.919,3.792) -- (2.083,3.763) -- (2.248,3.737) -- (2.413,3.715) -- (2.579,3.697) -- (2.746,3.683) -- (2.912,3.672) -- (3.079,3.666) -- (3.246,3.663) -- (3.413,3.665) -- (3.579,3.67) -- (3.746,3.68) -- (3.912,3.693) -- (4.078,3.71);
\draw[line width=0.5pt] (1.756,3.826) -- (1.751,3.649) -- (1.742,3.472) -- (1.729,3.296) -- (1.711,3.119) -- (1.69,2.944) -- (1.664,2.768) -- (1.635,2.594) -- (1.601,2.42) -- (1.563,2.246) -- (1.521,2.074) -- (1.475,1.903) -- (1.425,1.733) -- (1.371,1.565) -- (1.313,1.397);
\draw[line width=0.5pt] (4.078,3.71) -- (4.226,3.686) -- (4.376,3.665) -- (4.525,3.648) -- (4.675,3.634) -- (4.825,3.624) -- (4.976,3.617) -- (5.127,3.614) -- (5.277,3.615) -- (5.428,3.619) -- (5.578,3.626) -- (5.729,3.637) -- (5.878,3.652) -- (6.028,3.67) -- (6.177,3.691);
\draw[line width=0.5pt] (4.078,3.71) -- (4.093,3.577) -- (4.104,3.443) -- (4.112,3.309) -- (4.117,3.175) -- (4.119,3.041) -- (4.118,2.906) -- (4.113,2.772) -- (4.106,2.638) -- (4.095,2.504) -- (4.081,2.37) -- (4.064,2.237) -- (4.044,2.105) -- (4.021,1.972) -- (3.995,1.841);
\draw[line width=0.5pt] (6.177,3.691) -- (6.351,3.694) -- (6.525,3.701) -- (6.699,3.712) -- (6.873,3.727) -- (7.046,3.746) -- (7.219,3.77) -- (7.391,3.797) -- (7.563,3.828) -- (7.733,3.864) -- (7.903,3.903) -- (8.072,3.946) -- (8.24,3.994) -- (8.406,4.045) -- (8.572,4.1);
\draw[line width=0.5pt] (6.177,3.691) -- (6.191,3.575) -- (6.202,3.458) -- (6.21,3.341) -- (6.216,3.224) -- (6.218,3.106) -- (6.218,2.989) -- (6.216,2.871) -- (6.21,2.754) -- (6.202,2.637) -- (6.191,2.52) -- (6.177,2.403) -- (6.161,2.287) -- (6.141,2.171) -- (6.119,2.056);
\draw[line width=0.5pt] (3.995,1.841) -- (4.148,1.833) -- (4.301,1.828) -- (4.454,1.828) -- (4.607,1.831) -- (4.76,1.837) -- (4.913,1.847) -- (5.066,1.861) -- (5.218,1.878) -- (5.37,1.899) -- (5.521,1.923) -- (5.672,1.951) -- (5.822,1.983) -- (5.971,2.018) -- (6.119,2.056);
\draw[line width=0.5pt] (1.313,1.397) -- (1.498,1.458) -- (1.685,1.515) -- (1.873,1.567) -- (2.062,1.614) -- (2.252,1.657) -- (2.443,1.696) -- (2.635,1.73) -- (2.828,1.759) -- (3.022,1.784) -- (3.216,1.805) -- (3.41,1.821) -- (3.605,1.832) -- (3.8,1.839) -- (3.995,1.841);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.778,6.652) -- (8.749,6.794) -- (8.669,6.915) -- (8.547,6.996) -- (8.405,7.025) -- (8.262,6.996) -- (8.141,6.915) -- (8.06,6.794) -- (8.031,6.652) -- (8.06,6.509) -- (8.141,6.388) -- (8.262,6.307) -- (8.405,6.278) -- (8.547,6.307) -- (8.669,6.388) -- (8.749,6.509) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.466,6.568) -- (6.437,6.71) -- (6.357,6.832) -- (6.235,6.912) -- (6.093,6.941) -- (5.95,6.912) -- (5.829,6.832) -- (5.748,6.71) -- (5.719,6.568) -- (5.748,6.425) -- (5.829,6.304) -- (5.95,6.223) -- (6.093,6.194) -- (6.235,6.223) -- (6.357,6.304) -- (6.437,6.425) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.62,6.162) -- (11.591,6.305) -- (11.51,6.426) -- (11.389,6.507) -- (11.246,6.535) -- (11.104,6.507) -- (10.982,6.426) -- (10.902,6.305) -- (10.873,6.162) -- (10.902,6.019) -- (10.982,5.898) -- (11.104,5.817) -- (11.246,5.789) -- (11.389,5.817) -- (11.51,5.898) -- (11.591,6.019) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.686,4.325) -- (11.658,4.468) -- (11.577,4.589) -- (11.456,4.67) -- (11.313,4.698) -- (11.17,4.67) -- (11.049,4.589) -- (10.968,4.468) -- (10.94,4.325) -- (10.968,4.182) -- (11.049,4.061) -- (11.17,3.98) -- (11.313,3.952) -- (11.456,3.98) -- (11.577,4.061) -- (11.658,4.182) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.945,4.1) -- (8.916,4.243) -- (8.836,4.364) -- (8.714,4.445) -- (8.572,4.473) -- (8.429,4.445) -- (8.308,4.364) -- (8.227,4.243) -- (8.198,4.1) -- (8.227,3.957) -- (8.308,3.836) -- (8.429,3.755) -- (8.572,3.727) -- (8.714,3.755) -- (8.836,3.836) -- (8.916,3.957) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.129,3.826) -- (2.101,3.969) -- (2.02,4.09) -- (1.899,4.171) -- (1.756,4.199) -- (1.613,4.171) -- (1.492,4.09) -- (1.411,3.969) -- (1.383,3.826) -- (1.411,3.683) -- (1.492,3.562) -- (1.613,3.481) -- (1.756,3.453) -- (1.899,3.481) -- (2.02,3.562) -- (2.101,3.683) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.451,3.71) -- (4.423,3.853) -- (4.342,3.974) -- (4.221,4.055) -- (4.078,4.083) -- (3.935,4.055) -- (3.814,3.974) -- (3.733,3.853) -- (3.705,3.71) -- (3.733,3.567) -- (3.814,3.446) -- (3.935,3.365) -- (4.078,3.337) -- (4.221,3.365) -- (4.342,3.446) -- (4.423,3.567) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.55,3.691) -- (6.522,3.834) -- (6.441,3.955) -- (6.32,4.036) -- (6.177,4.065) -- (6.034,4.036) -- (5.913,3.955) -- (5.832,3.834) -- (5.804,3.691) -- (5.832,3.549) -- (5.913,3.428) -- (6.034,3.347) -- (6.177,3.318) -- (6.32,3.347) -- (6.441,3.428) -- (6.522,3.549) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.493,2.056) -- (6.464,2.199) -- (6.383,2.32) -- (6.262,2.401) -- (6.119,2.429) -- (5.977,2.401) -- (5.856,2.32) -- (5.775,2.199) -- (5.746,2.056) -- (5.775,1.913) -- (5.856,1.792) -- (5.977,1.711) -- (6.119,1.683) -- (6.262,1.711) -- (6.383,1.792) -- (6.464,1.913) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.368,1.841) -- (4.34,1.984) -- (4.259,2.105) -- (4.138,2.186) -- (3.995,2.214) -- (3.852,2.186) -- (3.731,2.105) -- (3.65,1.984) -- (3.621,1.841) -- (3.65,1.698) -- (3.731,1.577) -- (3.852,1.496) -- (3.995,1.468) -- (4.138,1.496) -- (4.259,1.577) -- (4.34,1.698) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.686,1.397) -- (1.658,1.54) -- (1.577,1.661) -- (1.456,1.742) -- (1.313,1.771) -- (1.17,1.742) -- (1.049,1.661) -- (0.968,1.54) -- (0.94,1.397) -- (0.968,1.255) -- (1.049,1.134) -- (1.17,1.053) -- (1.313,1.024) -- (1.456,1.053) -- (1.577,1.134) -- (1.658,1.255) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.871,6.284) -- (9.989,6.303) -- (10.097,6.357) -- (10.182,6.442) -- (10.236,6.55) -- (10.255,6.668) -- (10.236,6.787) -- (10.182,6.894) -- (10.097,6.979) -- (9.989,7.034) -- (9.871,7.053) -- (9.871,7.053) -- (9.752,7.034) -- (9.645,6.979) -- (9.56,6.894) -- (9.505,6.787) -- (9.486,6.668) -- (9.505,6.55) -- (9.56,6.442) -- (9.645,6.357) -- (9.752,6.303) -- (9.871,6.284) -- cycle;
\node at (9.871,6.668) {\scalebox{1.609}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.21,4.973) -- (8.328,4.992) -- (8.436,5.046) -- (8.521,5.131) -- (8.575,5.239) -- (8.594,5.358) -- (8.575,5.476) -- (8.521,5.584) -- (8.436,5.669) -- (8.328,5.723) -- (8.21,5.742) -- (8.21,5.742) -- (8.091,5.723) -- (7.984,5.669) -- (7.899,5.584) -- (7.844,5.476) -- (7.825,5.358) -- (7.844,5.239) -- (7.899,5.131) -- (7.984,5.046) -- (8.091,4.992) -- (8.21,4.973) -- cycle;
\node at (8.21,5.358) {\scalebox{1.609}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.238,6.514) -- (7.357,6.533) -- (7.464,6.587) -- (7.549,6.672) -- (7.604,6.78) -- (7.623,6.899) -- (7.604,7.017) -- (7.549,7.125) -- (7.464,7.21) -- (7.357,7.264) -- (7.238,7.283) -- (7.238,7.283) -- (7.119,7.264) -- (7.012,7.21) -- (6.927,7.125) -- (6.872,7.017) -- (6.854,6.899) -- (6.872,6.78) -- (6.927,6.672) -- (7.012,6.587) -- (7.119,6.533) -- (7.238,6.514) -- cycle;
\node at (7.238,6.899) {\scalebox{1.609}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.638,4.76) -- (6.757,4.779) -- (6.864,4.833) -- (6.949,4.918) -- (7.003,5.025) -- (7.022,5.144) -- (7.003,5.263) -- (6.949,5.37) -- (6.864,5.455) -- (6.757,5.51) -- (6.638,5.529) -- (6.638,5.529) -- (6.519,5.51) -- (6.412,5.455) -- (6.327,5.37) -- (6.272,5.263) -- (6.253,5.144) -- (6.272,5.025) -- (6.327,4.918) -- (6.412,4.833) -- (6.519,4.779) -- (6.638,4.76) -- cycle;
\node at (6.638,5.144) {\scalebox{1.609}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.588,4.87) -- (11.707,4.889) -- (11.814,4.944) -- (11.899,5.029) -- (11.954,5.136) -- (11.973,5.255) -- (11.954,5.373) -- (11.899,5.481) -- (11.814,5.566) -- (11.707,5.62) -- (11.588,5.639) -- (11.588,5.639) -- (11.469,5.62) -- (11.362,5.566) -- (11.277,5.481) -- (11.222,5.373) -- (11.204,5.255) -- (11.222,5.136) -- (11.277,5.029) -- (11.362,4.944) -- (11.469,4.889) -- (11.588,4.87) -- cycle;
\node at (11.588,5.255) {\scalebox{1.609}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.341,3.58) -- (10.459,3.599) -- (10.567,3.653) -- (10.652,3.738) -- (10.706,3.846) -- (10.725,3.964) -- (10.706,4.083) -- (10.652,4.19) -- (10.567,4.276) -- (10.459,4.33) -- (10.341,4.349) -- (10.341,4.349) -- (10.222,4.33) -- (10.115,4.276) -- (10.029,4.19) -- (9.975,4.083) -- (9.956,3.964) -- (9.975,3.846) -- (10.029,3.738) -- (10.115,3.653) -- (10.222,3.599) -- (10.341,3.58) -- cycle;
\node at (10.341,3.964) {\scalebox{1.609}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.785,3.839) -- (2.904,3.858) -- (3.011,3.913) -- (3.096,3.998) -- (3.151,4.105) -- (3.17,4.224) -- (3.151,4.343) -- (3.096,4.45) -- (3.011,4.535) -- (2.904,4.59) -- (2.785,4.609) -- (2.785,4.609) -- (2.667,4.59) -- (2.559,4.535) -- (2.474,4.45) -- (2.42,4.343) -- (2.401,4.224) -- (2.42,4.105) -- (2.474,3.998) -- (2.559,3.913) -- (2.667,3.858) -- (2.785,3.839) -- cycle;
\node at (2.785,4.224) {\scalebox{1.609}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.256,2.278) -- (1.375,2.297) -- (1.482,2.351) -- (1.567,2.436) -- (1.622,2.544) -- (1.641,2.663) -- (1.622,2.781) -- (1.567,2.889) -- (1.482,2.974) -- (1.375,3.028) -- (1.256,3.047) -- (1.256,3.047) -- (1.137,3.028) -- (1.03,2.974) -- (0.945,2.889) -- (0.891,2.781) -- (0.872,2.663) -- (0.891,2.544) -- (0.945,2.436) -- (1.03,2.351) -- (1.137,2.297) -- (1.256,2.278) -- cycle;
\node at (1.256,2.663) {\scalebox{1.609}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.13,3.614) -- (5.249,3.633) -- (5.356,3.688) -- (5.441,3.773) -- (5.496,3.88) -- (5.515,3.999) -- (5.496,4.118) -- (5.441,4.225) -- (5.356,4.31) -- (5.249,4.364) -- (5.13,4.383) -- (5.13,4.383) -- (5.011,4.364) -- (4.904,4.31) -- (4.819,4.225) -- (4.764,4.118) -- (4.746,3.999) -- (4.764,3.88) -- (4.819,3.773) -- (4.904,3.688) -- (5.011,3.633) -- (5.13,3.614) -- cycle;
\node at (5.13,3.999) {\scalebox{1.609}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.679,2.37) -- (4.797,2.389) -- (4.905,2.444) -- (4.99,2.529) -- (5.044,2.636) -- (5.063,2.755) -- (5.044,2.874) -- (4.99,2.981) -- (4.905,3.066) -- (4.797,3.121) -- (4.679,3.139) -- (4.317,3.139) -- (4.198,3.121) -- (4.091,3.066) -- (4.006,2.981) -- (3.951,2.874) -- (3.932,2.755) -- (3.951,2.636) -- (4.006,2.529) -- (4.091,2.444) -- (4.198,2.389) -- (4.317,2.37) -- cycle;
\node at (4.498,2.755) {\scalebox{1.609}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.507,3.791) -- (7.626,3.81) -- (7.733,3.865) -- (7.818,3.95) -- (7.873,4.057) -- (7.892,4.176) -- (7.873,4.295) -- (7.818,4.402) -- (7.733,4.487) -- (7.626,4.542) -- (7.507,4.56) -- (7.145,4.56) -- (7.027,4.542) -- (6.919,4.487) -- (6.834,4.402) -- (6.78,4.295) -- (6.761,4.176) -- (6.78,4.057) -- (6.834,3.95) -- (6.919,3.865) -- (7.027,3.81) -- (7.145,3.791) -- cycle;
\node at (7.326,4.176) {\scalebox{1.609}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.716,2.476) -- (6.835,2.494) -- (6.942,2.549) -- (7.027,2.634) -- (7.082,2.741) -- (7.101,2.86) -- (7.082,2.979) -- (7.027,3.086) -- (6.942,3.171) -- (6.835,3.226) -- (6.716,3.245) -- (6.354,3.245) -- (6.236,3.226) -- (6.128,3.171) -- (6.043,3.086) -- (5.989,2.979) -- (5.97,2.86) -- (5.989,2.741) -- (6.043,2.634) -- (6.128,2.549) -- (6.236,2.494) -- (6.354,2.476) -- cycle;
\node at (6.535,2.86) {\scalebox{1.609}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.136,0.921) -- (5.255,0.94) -- (5.362,0.995) -- (5.447,1.08) -- (5.502,1.187) -- (5.521,1.306) -- (5.502,1.425) -- (5.447,1.532) -- (5.362,1.617) -- (5.255,1.672) -- (5.136,1.69) -- (4.774,1.69) -- (4.656,1.672) -- (4.548,1.617) -- (4.463,1.532) -- (4.409,1.425) -- (4.39,1.306) -- (4.409,1.187) -- (4.463,1.08) -- (4.548,0.995) -- (4.656,0.94) -- (4.774,0.921) -- cycle;
\node at (4.955,1.306) {\scalebox{1.609}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.759,0.6) -- (2.878,0.619) -- (2.985,0.673) -- (3.07,0.759) -- (3.125,0.866) -- (3.144,0.985) -- (3.125,1.103) -- (3.07,1.211) -- (2.985,1.296) -- (2.878,1.35) -- (2.759,1.369) -- (2.397,1.369) -- (2.278,1.35) -- (2.171,1.296) -- (2.086,1.211) -- (2.031,1.103) -- (2.013,0.985) -- (2.031,0.866) -- (2.086,0.759) -- (2.171,0.673) -- (2.278,0.619) -- (2.397,0.6) -- cycle;
\node at (2.578,0.985) {\scalebox{1.609}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.013,1.097) -- (1.613,1.097) -- (1.613,1.697) -- (1.013,1.697) -- cycle;
\node at (1.313,1.397) {\scalebox{1.609}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (9.17,4.53) -- (9.77,4.53) -- (9.77,5.13) -- (9.17,5.13) -- cycle;
\node at (9.47,4.83) {\scalebox{1.609}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.837,1.895) -- (2.437,1.895) -- (2.437,2.495) -- (1.837,2.495) -- cycle;
\node at (2.137,2.195) {\scalebox{1.609}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (6.986,1.842) -- (7.586,1.842) -- (7.586,2.442) -- (6.986,2.442) -- cycle;
\node at (7.286,2.142) {\scalebox{1.609}{\small d}};
\node at (8.795,7.133) {\scalebox{1.609}{\small A}};
\node at (6.047,7.186) {\scalebox{1.609}{\small B}};
\node at (11.816,6.406) {\scalebox{1.609}{\small C}};
\node at (11.932,4.362) {\scalebox{1.609}{\small D}};
\node at (9.191,4.119) {\scalebox{1.609}{\small E}};
\node at (1.137,3.798) {\scalebox{1.609}{\small F}};
\node at (3.464,3.621) {\scalebox{1.609}{\small G}};
\node at (5.998,3.098) {\scalebox{1.609}{\small H}};
\node at (6.07,1.438) {\scalebox{1.609}{\small I}};
\node at (3.547,1.412) {\scalebox{1.609}{\small J}};
\node at (0.765,1.107) {\scalebox{1.609}{\small K}};
\draw[line width=0.5pt] (12.613,5.702) -- (12.613,6.802);
\draw[line width=0.5pt] (12.382,6.417) -- (12.613,6.802) -- (12.844,6.417);
\draw[line width=0.5pt] (12.31,6.252) -- (12.915,6.252);
\node at (12.613,7.187) {\scalebox{1.609}{\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, F, H, J & 5 \\ \hline
clairs & B, C, E, G, I, K & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{6} 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 11 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 5 carrefours sombres et 6 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{K} :}\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{boulevard du Parc ⑧} vers le nord jusqu'à Place aux Herbes \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Puits ⑦}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Pont ⑩}, vers le sud, jusqu'à Rond-point du Pont \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin du Lavoir ⑬}, vers l'est, jusqu'à Place des Tanneurs \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Écoles ⑫}, vers le nord, jusqu'à Carrefour du Lavoir \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{passage du Marché ④}, vers le nord, jusqu'à Place du Marché \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Poste ③}, vers l'est, jusqu'à Carrefour des Tilleuls \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{avenue de la Gare ①}, vers l'est, jusqu'à Coin du Lavoir \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{quai aux Fleurs ⑤}, vers le sud, jusqu'à Placette des Ormes \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Jardins ⑥}, vers l'ouest, jusqu'à Carrefour de l'Hôpital \textbf{(E)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 11 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}
