\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 391}

\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=391}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0391}
\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=0.604, line join=round, line cap=round]
\draw[line width=0.5pt] (5.169,8.432) -- (5.337,8.377) -- (5.507,8.326) -- (5.677,8.279) -- (5.849,8.236) -- (6.021,8.197) -- (6.195,8.162) -- (6.369,8.132) -- (6.544,8.105) -- (6.72,8.083) -- (6.896,8.064) -- (7.072,8.05) -- (7.248,8.04) -- (7.425,8.034) -- (7.602,8.032);
\draw[line width=0.5pt] (5.169,8.432) -- (5.141,8.225) -- (5.108,8.019) -- (5.071,7.814) -- (5.028,7.609) -- (4.981,7.406) -- (4.928,7.204) -- (4.871,7.003) -- (4.81,6.803) -- (4.744,6.605) -- (4.673,6.409) -- (4.597,6.214) -- (4.517,6.021) -- (4.433,5.83) -- (4.343,5.642);
\draw[line width=0.5pt] (7.602,8.032) -- (7.578,7.865) -- (7.559,7.697) -- (7.543,7.528) -- (7.532,7.36) -- (7.524,7.19) -- (7.52,7.021) -- (7.521,6.852) -- (7.525,6.683) -- (7.533,6.513) -- (7.546,6.345) -- (7.562,6.176) -- (7.582,6.008) -- (7.606,5.841) -- (7.634,5.674);
\draw[line width=0.5pt] (7.634,5.674) -- (7.869,5.708) -- (8.104,5.737) -- (8.34,5.76) -- (8.576,5.777) -- (8.813,5.789) -- (9.05,5.795) -- (9.287,5.796) -- (9.524,5.791) -- (9.761,5.781) -- (9.997,5.765) -- (10.233,5.743) -- (10.469,5.716) -- (10.703,5.683) -- (10.937,5.645);
\draw[line width=0.5pt] (7.634,5.674) -- (7.651,5.365) -- (7.675,5.057) -- (7.706,4.749) -- (7.745,4.442) -- (7.791,4.136) -- (7.844,3.831) -- (7.904,3.527) -- (7.971,3.225) -- (8.046,2.924) -- (8.127,2.626) -- (8.215,2.329) -- (8.311,2.035) -- (8.413,1.743) -- (8.522,1.454);
\draw[line width=0.5pt] (10.937,5.645) -- (11.013,5.359) -- (11.082,5.07) -- (11.144,4.781) -- (11.2,4.489) -- (11.248,4.197) -- (11.29,3.903) -- (11.325,3.608) -- (11.352,3.313) -- (11.373,3.017) -- (11.387,2.72) -- (11.394,2.424) -- (11.394,2.127) -- (11.387,1.831) -- (11.373,1.535);
\draw[line width=0.5pt] (4.343,5.642) -- (4.577,5.608) -- (4.811,5.58) -- (5.046,5.557) -- (5.282,5.54) -- (5.518,5.528) -- (5.754,5.522) -- (5.99,5.522) -- (6.226,5.527) -- (6.462,5.538) -- (6.698,5.554) -- (6.933,5.576) -- (7.168,5.603) -- (7.401,5.636) -- (7.634,5.674);
\draw[line width=0.5pt] (4.343,5.642) -- (4.397,5.356) -- (4.443,5.07) -- (4.482,4.782) -- (4.515,4.493) -- (4.541,4.204) -- (4.56,3.914) -- (4.572,3.623) -- (4.578,3.333) -- (4.576,3.042) -- (4.568,2.751) -- (4.553,2.461) -- (4.531,2.172) -- (4.503,1.883) -- (4.467,1.595);
\draw[line width=0.5pt] (1.373,5.499) -- (1.585,5.477) -- (1.798,5.46) -- (2.011,5.447) -- (2.224,5.44) -- (2.437,5.438) -- (2.651,5.44) -- (2.864,5.448) -- (3.077,5.461) -- (3.29,5.478) -- (3.502,5.501) -- (3.714,5.529) -- (3.925,5.561) -- (4.135,5.599) -- (4.343,5.642);
\draw[line width=0.5pt] (1.373,5.499) -- (1.447,5.231) -- (1.514,4.962) -- (1.574,4.69) -- (1.629,4.417) -- (1.677,4.143) -- (1.718,3.868) -- (1.753,3.592) -- (1.781,3.315) -- (1.803,3.038) -- (1.819,2.76) -- (1.828,2.482) -- (1.83,2.204) -- (1.826,1.926) -- (1.815,1.648);
\draw[line width=0.5pt] (1.815,1.648) -- (2.002,1.615) -- (2.19,1.587) -- (2.379,1.563) -- (2.569,1.544) -- (2.758,1.529) -- (2.948,1.518) -- (3.139,1.512) -- (3.329,1.51) -- (3.52,1.513) -- (3.71,1.521) -- (3.9,1.532) -- (4.089,1.549) -- (4.278,1.569) -- (4.467,1.595);
\draw[line width=0.5pt] (4.467,1.595) -- (4.756,1.629) -- (5.046,1.656) -- (5.336,1.677) -- (5.627,1.691) -- (5.918,1.698) -- (6.209,1.698) -- (6.5,1.691) -- (6.791,1.678) -- (7.082,1.657) -- (7.372,1.63) -- (7.661,1.596) -- (7.949,1.556) -- (8.236,1.508) -- (8.522,1.454);
\draw[line width=0.5pt] (8.522,1.454) -- (8.725,1.428) -- (8.929,1.408) -- (9.133,1.392) -- (9.337,1.381) -- (9.542,1.375) -- (9.746,1.373) -- (9.951,1.377) -- (10.156,1.385) -- (10.36,1.398) -- (10.564,1.416) -- (10.767,1.438) -- (10.97,1.466) -- (11.172,1.498) -- (11.373,1.535);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.329,8.432) -- (5.308,8.512) -- (5.249,8.57) -- (5.169,8.592) -- (5.089,8.57) -- (5.031,8.512) -- (5.009,8.432) -- (5.031,8.352) -- (5.089,8.293) -- (5.169,8.272) -- (5.249,8.293) -- (5.308,8.352) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.762,8.032) -- (7.74,8.112) -- (7.682,8.171) -- (7.602,8.192) -- (7.522,8.171) -- (7.463,8.112) -- (7.442,8.032) -- (7.463,7.952) -- (7.522,7.894) -- (7.602,7.872) -- (7.682,7.894) -- (7.74,7.952) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.794,5.674) -- (7.773,5.754) -- (7.714,5.812) -- (7.634,5.834) -- (7.554,5.812) -- (7.496,5.754) -- (7.474,5.674) -- (7.496,5.594) -- (7.554,5.535) -- (7.634,5.514) -- (7.714,5.535) -- (7.773,5.594) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.097,5.645) -- (11.075,5.725) -- (11.017,5.784) -- (10.937,5.805) -- (10.857,5.784) -- (10.798,5.725) -- (10.777,5.645) -- (10.798,5.565) -- (10.857,5.507) -- (10.937,5.485) -- (11.017,5.507) -- (11.075,5.565) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.503,5.642) -- (4.482,5.722) -- (4.423,5.78) -- (4.343,5.802) -- (4.263,5.78) -- (4.205,5.722) -- (4.183,5.642) -- (4.205,5.562) -- (4.263,5.503) -- (4.343,5.482) -- (4.423,5.503) -- (4.482,5.562) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.533,5.499) -- (1.512,5.579) -- (1.453,5.638) -- (1.373,5.659) -- (1.293,5.638) -- (1.235,5.579) -- (1.213,5.499) -- (1.235,5.419) -- (1.293,5.361) -- (1.373,5.339) -- (1.453,5.361) -- (1.512,5.419) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.975,1.648) -- (1.954,1.728) -- (1.895,1.787) -- (1.815,1.808) -- (1.735,1.787) -- (1.676,1.728) -- (1.655,1.648) -- (1.676,1.568) -- (1.735,1.51) -- (1.815,1.488) -- (1.895,1.51) -- (1.954,1.568) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.627,1.595) -- (4.606,1.675) -- (4.547,1.733) -- (4.467,1.755) -- (4.387,1.733) -- (4.328,1.675) -- (4.307,1.595) -- (4.328,1.515) -- (4.387,1.456) -- (4.467,1.435) -- (4.547,1.456) -- (4.606,1.515) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.533,1.535) -- (11.512,1.615) -- (11.453,1.674) -- (11.373,1.695) -- (11.293,1.674) -- (11.235,1.615) -- (11.213,1.535) -- (11.235,1.455) -- (11.293,1.396) -- (11.373,1.375) -- (11.453,1.396) -- (11.512,1.455) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.682,1.454) -- (8.661,1.534) -- (8.602,1.592) -- (8.522,1.614) -- (8.442,1.592) -- (8.384,1.534) -- (8.362,1.454) -- (8.384,1.374) -- (8.442,1.315) -- (8.522,1.294) -- (8.602,1.315) -- (8.661,1.374) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.305,7.349) -- (6.427,7.368) -- (6.537,7.424) -- (6.624,7.511) -- (6.68,7.621) -- (6.699,7.743) -- (6.68,7.865) -- (6.624,7.975) -- (6.537,8.062) -- (6.427,8.118) -- (6.305,8.137) -- (6.305,8.137) -- (6.184,8.118) -- (6.074,8.062) -- (5.987,7.975) -- (5.931,7.865) -- (5.911,7.743) -- (5.931,7.621) -- (5.987,7.511) -- (6.074,7.424) -- (6.184,7.368) -- (6.305,7.349) -- cycle;
\node at (6.305,7.743) {\scalebox{1.656}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.249,6.497) -- (5.371,6.516) -- (5.481,6.572) -- (5.568,6.659) -- (5.624,6.769) -- (5.643,6.891) -- (5.624,7.012) -- (5.568,7.122) -- (5.481,7.209) -- (5.371,7.265) -- (5.249,7.285) -- (5.249,7.285) -- (5.127,7.265) -- (5.018,7.209) -- (4.931,7.122) -- (4.875,7.012) -- (4.855,6.891) -- (4.875,6.769) -- (4.931,6.659) -- (5.018,6.572) -- (5.127,6.516) -- (5.249,6.497) -- cycle;
\node at (5.249,6.891) {\scalebox{1.656}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.127,6.453) -- (7.249,6.472) -- (7.358,6.528) -- (7.446,6.615) -- (7.502,6.725) -- (7.521,6.846) -- (7.502,6.968) -- (7.446,7.078) -- (7.358,7.165) -- (7.249,7.221) -- (7.127,7.24) -- (7.127,7.24) -- (7.005,7.221) -- (6.895,7.165) -- (6.808,7.078) -- (6.752,6.968) -- (6.733,6.846) -- (6.752,6.725) -- (6.808,6.615) -- (6.895,6.528) -- (7.005,6.472) -- (7.127,6.453) -- cycle;
\node at (7.127,6.846) {\scalebox{1.656}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.29,5.796) -- (9.412,5.815) -- (9.522,5.871) -- (9.609,5.958) -- (9.665,6.068) -- (9.684,6.19) -- (9.665,6.311) -- (9.609,6.421) -- (9.522,6.508) -- (9.412,6.564) -- (9.29,6.584) -- (9.29,6.584) -- (9.168,6.564) -- (9.059,6.508) -- (8.971,6.421) -- (8.916,6.311) -- (8.896,6.19) -- (8.916,6.068) -- (8.971,5.958) -- (9.059,5.871) -- (9.168,5.815) -- (9.29,5.796) -- cycle;
\node at (9.29,6.19) {\scalebox{1.656}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.29,3.214) -- (8.411,3.234) -- (8.521,3.29) -- (8.608,3.377) -- (8.664,3.487) -- (8.683,3.608) -- (8.664,3.73) -- (8.608,3.84) -- (8.521,3.927) -- (8.411,3.983) -- (8.29,4.002) -- (8.29,4.002) -- (8.168,3.983) -- (8.058,3.927) -- (7.971,3.84) -- (7.915,3.73) -- (7.896,3.608) -- (7.915,3.487) -- (7.971,3.377) -- (8.058,3.29) -- (8.168,3.234) -- (8.29,3.214) -- cycle;
\node at (8.29,3.608) {\scalebox{1.656}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.716,3.256) -- (11.838,3.275) -- (11.948,3.331) -- (12.035,3.418) -- (12.091,3.528) -- (12.11,3.65) -- (12.091,3.771) -- (12.035,3.881) -- (11.948,3.968) -- (11.838,4.024) -- (11.716,4.044) -- (11.716,4.044) -- (11.595,4.024) -- (11.485,3.968) -- (11.398,3.881) -- (11.342,3.771) -- (11.322,3.65) -- (11.342,3.528) -- (11.398,3.418) -- (11.485,3.331) -- (11.595,3.275) -- (11.716,3.256) -- cycle;
\node at (11.716,3.65) {\scalebox{1.656}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.986,5.522) -- (6.108,5.541) -- (6.218,5.597) -- (6.305,5.684) -- (6.361,5.794) -- (6.38,5.916) -- (6.361,6.038) -- (6.305,6.147) -- (6.218,6.235) -- (6.108,6.291) -- (5.986,6.31) -- (5.986,6.31) -- (5.865,6.291) -- (5.755,6.235) -- (5.668,6.147) -- (5.612,6.038) -- (5.592,5.916) -- (5.612,5.794) -- (5.668,5.684) -- (5.755,5.597) -- (5.865,5.541) -- (5.986,5.522) -- cycle;
\node at (5.986,5.916) {\scalebox{1.656}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.966,3.241) -- (5.088,3.261) -- (5.197,3.317) -- (5.285,3.404) -- (5.341,3.513) -- (5.36,3.635) -- (5.341,3.757) -- (5.285,3.867) -- (5.197,3.954) -- (5.088,4.01) -- (4.966,4.029) -- (4.966,4.029) -- (4.844,4.01) -- (4.734,3.954) -- (4.647,3.867) -- (4.591,3.757) -- (4.572,3.635) -- (4.591,3.513) -- (4.647,3.404) -- (4.734,3.317) -- (4.844,3.261) -- (4.966,3.241) -- cycle;
\node at (4.966,3.635) {\scalebox{1.656}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.845,5.448) -- (2.967,5.467) -- (3.077,5.523) -- (3.164,5.61) -- (3.22,5.72) -- (3.239,5.841) -- (3.22,5.963) -- (3.164,6.073) -- (3.077,6.16) -- (2.967,6.216) -- (2.845,6.235) -- (2.845,6.235) -- (2.724,6.216) -- (2.614,6.16) -- (2.527,6.073) -- (2.471,5.963) -- (2.451,5.841) -- (2.471,5.72) -- (2.527,5.61) -- (2.614,5.523) -- (2.724,5.467) -- (2.845,5.448) -- cycle;
\node at (2.845,5.841) {\scalebox{1.656}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.33,3.243) -- (2.451,3.262) -- (2.561,3.318) -- (2.648,3.405) -- (2.704,3.515) -- (2.723,3.637) -- (2.704,3.759) -- (2.648,3.868) -- (2.561,3.956) -- (2.451,4.012) -- (2.33,4.031) -- (1.959,4.031) -- (1.837,4.012) -- (1.727,3.956) -- (1.64,3.868) -- (1.584,3.759) -- (1.565,3.637) -- (1.584,3.515) -- (1.64,3.405) -- (1.727,3.318) -- (1.837,3.262) -- (1.959,3.243) -- cycle;
\node at (2.144,3.637) {\scalebox{1.656}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.332,1.512) -- (3.454,1.531) -- (3.564,1.587) -- (3.651,1.674) -- (3.707,1.784) -- (3.726,1.906) -- (3.707,2.028) -- (3.651,2.137) -- (3.564,2.224) -- (3.454,2.28) -- (3.332,2.3) -- (2.961,2.3) -- (2.84,2.28) -- (2.73,2.224) -- (2.643,2.137) -- (2.587,2.028) -- (2.567,1.906) -- (2.587,1.784) -- (2.643,1.674) -- (2.73,1.587) -- (2.84,1.531) -- (2.961,1.512) -- cycle;
\node at (3.147,1.906) {\scalebox{1.656}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.699,1.691) -- (6.821,1.71) -- (6.931,1.766) -- (7.018,1.854) -- (7.074,1.963) -- (7.093,2.085) -- (7.074,2.207) -- (7.018,2.317) -- (6.931,2.404) -- (6.821,2.46) -- (6.699,2.479) -- (6.329,2.479) -- (6.207,2.46) -- (6.097,2.404) -- (6.01,2.317) -- (5.954,2.207) -- (5.935,2.085) -- (5.954,1.963) -- (6.01,1.854) -- (6.097,1.766) -- (6.207,1.71) -- (6.329,1.691) -- cycle;
\node at (6.514,2.085) {\scalebox{1.656}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.125,1.377) -- (10.247,1.396) -- (10.357,1.452) -- (10.444,1.539) -- (10.5,1.649) -- (10.519,1.77) -- (10.5,1.892) -- (10.444,2.002) -- (10.357,2.089) -- (10.247,2.145) -- (10.125,2.164) -- (9.754,2.164) -- (9.633,2.145) -- (9.523,2.089) -- (9.436,2.002) -- (9.38,1.892) -- (9.36,1.77) -- (9.38,1.649) -- (9.436,1.539) -- (9.523,1.452) -- (9.633,1.396) -- (9.754,1.377) -- cycle;
\node at (9.94,1.77) {\scalebox{1.656}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.302,7.732) -- (7.902,7.732) -- (7.902,8.332) -- (7.302,8.332) -- cycle;
\node at (7.602,8.032) {\scalebox{1.656}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (5.811,8.985) -- (6.411,8.985) -- (6.411,9.585) -- (5.811,9.585) -- cycle;
\node at (6.111,9.285) {\scalebox{1.656}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.376,6.692) -- (9.976,6.692) -- (9.976,7.292) -- (9.376,7.292) -- cycle;
\node at (9.676,6.992) {\scalebox{1.656}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (0.822,3.856) -- (1.422,3.856) -- (1.422,4.456) -- (0.822,4.456) -- cycle;
\node at (1.122,4.156) {\scalebox{1.656}{\small d}};
\node at (4.994,9.026) {\scalebox{1.656}{\small A}};
\node at (7.814,8.615) {\scalebox{1.656}{\small B}};
\node at (8.099,6.085) {\scalebox{1.656}{\small C}};
\node at (11.539,5.793) {\scalebox{1.656}{\small D}};
\node at (3.805,5.948) {\scalebox{1.656}{\small E}};
\node at (0.765,5.62) {\scalebox{1.656}{\small F}};
\node at (1.292,1.315) {\scalebox{1.656}{\small G}};
\node at (4.134,1.071) {\scalebox{1.656}{\small H}};
\node at (11.907,1.22) {\scalebox{1.656}{\small I}};
\node at (8.884,0.95) {\scalebox{1.656}{\small J}};
\draw[line width=0.5pt] (12.673,7.482) -- (12.673,8.582);
\draw[line width=0.5pt] (12.442,8.197) -- (12.673,8.582) -- (12.904,8.197);
\draw[line width=0.5pt] (12.371,8.032) -- (12.976,8.032);
\node at (12.673,8.967) {\scalebox{1.656}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour B, 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 rue du Four & Ⓑ Fontaine Notre-Dame & Ⓙ Carrefour de l'Hôpital \\ \hline
① sentier des Prés & ⑨ rue Basse & Ⓒ Rond-point des Vaches & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
② rue du Puits & ⑩ rue Neuve & Ⓓ Placette des Ormes & \textbf{a} l'école \\ \hline
③ rue Haute & ⑪ chemin des Vignes & Ⓔ Rond-point de la Gare & \textbf{b} la salle des fêtes \\ \hline
④ allée des Tilleuls & ⑫ rue de l'Église & Ⓕ Coin du Lavoir & \textbf{c} le marché couvert \\ \hline
⑤ boulevard du Parc & ⑬ allée des Peupliers & Ⓖ Place des Tanneurs & \textbf{d} la pharmacie \\ \hline
⑥ rue des Lilas & \cellcolor{black!32}\bfseries Les carrefours & Ⓗ Rond-point du Stade &  \\ \hline
⑦ rue des Jardins & Ⓐ Place de l'Église & Ⓘ Place Sainte-Croix &  \\ \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=0.6188, line join=round, line cap=round]
\draw[line width=0.5pt] (5.169,8.432) -- (5.337,8.377) -- (5.507,8.326) -- (5.677,8.279) -- (5.849,8.236) -- (6.021,8.197) -- (6.195,8.162) -- (6.369,8.132) -- (6.544,8.105) -- (6.72,8.083) -- (6.896,8.064) -- (7.072,8.05) -- (7.248,8.04) -- (7.425,8.034) -- (7.602,8.032);
\draw[line width=0.5pt] (5.169,8.432) -- (5.141,8.225) -- (5.108,8.019) -- (5.071,7.814) -- (5.028,7.609) -- (4.981,7.406) -- (4.928,7.204) -- (4.871,7.003) -- (4.81,6.803) -- (4.744,6.605) -- (4.673,6.409) -- (4.597,6.214) -- (4.517,6.021) -- (4.433,5.83) -- (4.343,5.642);
\draw[line width=0.5pt] (7.602,8.032) -- (7.578,7.865) -- (7.559,7.697) -- (7.543,7.528) -- (7.532,7.36) -- (7.524,7.19) -- (7.52,7.021) -- (7.521,6.852) -- (7.525,6.683) -- (7.533,6.513) -- (7.546,6.345) -- (7.562,6.176) -- (7.582,6.008) -- (7.606,5.841) -- (7.634,5.674);
\draw[line width=0.5pt] (7.634,5.674) -- (7.869,5.708) -- (8.104,5.737) -- (8.34,5.76) -- (8.576,5.777) -- (8.813,5.789) -- (9.05,5.795) -- (9.287,5.796) -- (9.524,5.791) -- (9.761,5.781) -- (9.997,5.765) -- (10.233,5.743) -- (10.469,5.716) -- (10.703,5.683) -- (10.937,5.645);
\draw[line width=0.5pt] (7.634,5.674) -- (7.651,5.365) -- (7.675,5.057) -- (7.706,4.749) -- (7.745,4.442) -- (7.791,4.136) -- (7.844,3.831) -- (7.904,3.527) -- (7.971,3.225) -- (8.046,2.924) -- (8.127,2.626) -- (8.215,2.329) -- (8.311,2.035) -- (8.413,1.743) -- (8.522,1.454);
\draw[line width=0.5pt] (10.937,5.645) -- (11.013,5.359) -- (11.082,5.07) -- (11.144,4.781) -- (11.2,4.489) -- (11.248,4.197) -- (11.29,3.903) -- (11.325,3.608) -- (11.352,3.313) -- (11.373,3.017) -- (11.387,2.72) -- (11.394,2.424) -- (11.394,2.127) -- (11.387,1.831) -- (11.373,1.535);
\draw[line width=0.5pt] (4.343,5.642) -- (4.577,5.608) -- (4.811,5.58) -- (5.046,5.557) -- (5.282,5.54) -- (5.518,5.528) -- (5.754,5.522) -- (5.99,5.522) -- (6.226,5.527) -- (6.462,5.538) -- (6.698,5.554) -- (6.933,5.576) -- (7.168,5.603) -- (7.401,5.636) -- (7.634,5.674);
\draw[line width=0.5pt] (4.343,5.642) -- (4.397,5.356) -- (4.443,5.07) -- (4.482,4.782) -- (4.515,4.493) -- (4.541,4.204) -- (4.56,3.914) -- (4.572,3.623) -- (4.578,3.333) -- (4.576,3.042) -- (4.568,2.751) -- (4.553,2.461) -- (4.531,2.172) -- (4.503,1.883) -- (4.467,1.595);
\draw[line width=0.5pt] (1.373,5.499) -- (1.585,5.477) -- (1.798,5.46) -- (2.011,5.447) -- (2.224,5.44) -- (2.437,5.438) -- (2.651,5.44) -- (2.864,5.448) -- (3.077,5.461) -- (3.29,5.478) -- (3.502,5.501) -- (3.714,5.529) -- (3.925,5.561) -- (4.135,5.599) -- (4.343,5.642);
\draw[line width=0.5pt] (1.373,5.499) -- (1.447,5.231) -- (1.514,4.962) -- (1.574,4.69) -- (1.629,4.417) -- (1.677,4.143) -- (1.718,3.868) -- (1.753,3.592) -- (1.781,3.315) -- (1.803,3.038) -- (1.819,2.76) -- (1.828,2.482) -- (1.83,2.204) -- (1.826,1.926) -- (1.815,1.648);
\draw[line width=0.5pt] (1.815,1.648) -- (2.002,1.615) -- (2.19,1.587) -- (2.379,1.563) -- (2.569,1.544) -- (2.758,1.529) -- (2.948,1.518) -- (3.139,1.512) -- (3.329,1.51) -- (3.52,1.513) -- (3.71,1.521) -- (3.9,1.532) -- (4.089,1.549) -- (4.278,1.569) -- (4.467,1.595);
\draw[line width=0.5pt] (4.467,1.595) -- (4.756,1.629) -- (5.046,1.656) -- (5.336,1.677) -- (5.627,1.691) -- (5.918,1.698) -- (6.209,1.698) -- (6.5,1.691) -- (6.791,1.678) -- (7.082,1.657) -- (7.372,1.63) -- (7.661,1.596) -- (7.949,1.556) -- (8.236,1.508) -- (8.522,1.454);
\draw[line width=0.5pt] (8.522,1.454) -- (8.725,1.428) -- (8.929,1.408) -- (9.133,1.392) -- (9.337,1.381) -- (9.542,1.375) -- (9.746,1.373) -- (9.951,1.377) -- (10.156,1.385) -- (10.36,1.398) -- (10.564,1.416) -- (10.767,1.438) -- (10.97,1.466) -- (11.172,1.498) -- (11.373,1.535);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.543,8.432) -- (5.514,8.574) -- (5.433,8.696) -- (5.312,8.776) -- (5.169,8.805) -- (5.027,8.776) -- (4.905,8.696) -- (4.825,8.574) -- (4.796,8.432) -- (4.825,8.289) -- (4.905,8.168) -- (5.027,8.087) -- (5.169,8.058) -- (5.312,8.087) -- (5.433,8.168) -- (5.514,8.289) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.975,8.032) -- (7.947,8.175) -- (7.866,8.296) -- (7.745,8.377) -- (7.602,8.406) -- (7.459,8.377) -- (7.338,8.296) -- (7.257,8.175) -- (7.229,8.032) -- (7.257,7.89) -- (7.338,7.769) -- (7.459,7.688) -- (7.602,7.659) -- (7.745,7.688) -- (7.866,7.769) -- (7.947,7.89) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.007,5.674) -- (7.979,5.817) -- (7.898,5.938) -- (7.777,6.019) -- (7.634,6.047) -- (7.491,6.019) -- (7.37,5.938) -- (7.289,5.817) -- (7.261,5.674) -- (7.289,5.531) -- (7.37,5.41) -- (7.491,5.329) -- (7.634,5.301) -- (7.777,5.329) -- (7.898,5.41) -- (7.979,5.531) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.31,5.645) -- (11.282,5.788) -- (11.201,5.909) -- (11.08,5.99) -- (10.937,6.018) -- (10.794,5.99) -- (10.673,5.909) -- (10.592,5.788) -- (10.564,5.645) -- (10.592,5.502) -- (10.673,5.381) -- (10.794,5.3) -- (10.937,5.272) -- (11.08,5.3) -- (11.201,5.381) -- (11.282,5.502) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.717,5.642) -- (4.688,5.784) -- (4.607,5.906) -- (4.486,5.986) -- (4.343,6.015) -- (4.201,5.986) -- (4.08,5.906) -- (3.999,5.784) -- (3.97,5.642) -- (3.999,5.499) -- (4.08,5.378) -- (4.201,5.297) -- (4.343,5.268) -- (4.486,5.297) -- (4.607,5.378) -- (4.688,5.499) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.746,5.499) -- (1.718,5.642) -- (1.637,5.763) -- (1.516,5.844) -- (1.373,5.873) -- (1.23,5.844) -- (1.109,5.763) -- (1.028,5.642) -- (1.0,5.499) -- (1.028,5.357) -- (1.109,5.236) -- (1.23,5.155) -- (1.373,5.126) -- (1.516,5.155) -- (1.637,5.236) -- (1.718,5.357) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.188,1.648) -- (2.16,1.791) -- (2.079,1.912) -- (1.958,1.993) -- (1.815,2.021) -- (1.672,1.993) -- (1.551,1.912) -- (1.47,1.791) -- (1.442,1.648) -- (1.47,1.505) -- (1.551,1.384) -- (1.672,1.303) -- (1.815,1.275) -- (1.958,1.303) -- (2.079,1.384) -- (2.16,1.505) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.84,1.595) -- (4.812,1.737) -- (4.731,1.858) -- (4.61,1.939) -- (4.467,1.968) -- (4.324,1.939) -- (4.203,1.858) -- (4.122,1.737) -- (4.094,1.595) -- (4.122,1.452) -- (4.203,1.331) -- (4.324,1.25) -- (4.467,1.221) -- (4.61,1.25) -- (4.731,1.331) -- (4.812,1.452) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.746,1.535) -- (11.718,1.678) -- (11.637,1.799) -- (11.516,1.88) -- (11.373,1.908) -- (11.23,1.88) -- (11.109,1.799) -- (11.028,1.678) -- (11.0,1.535) -- (11.028,1.392) -- (11.109,1.271) -- (11.23,1.19) -- (11.373,1.162) -- (11.516,1.19) -- (11.637,1.271) -- (11.718,1.392) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.895,1.454) -- (8.867,1.596) -- (8.786,1.718) -- (8.665,1.798) -- (8.522,1.827) -- (8.379,1.798) -- (8.258,1.718) -- (8.177,1.596) -- (8.149,1.454) -- (8.177,1.311) -- (8.258,1.19) -- (8.379,1.109) -- (8.522,1.08) -- (8.665,1.109) -- (8.786,1.19) -- (8.867,1.311) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.307,7.368) -- (6.426,7.387) -- (6.533,7.441) -- (6.618,7.526) -- (6.673,7.633) -- (6.691,7.752) -- (6.673,7.871) -- (6.618,7.978) -- (6.533,8.063) -- (6.426,8.118) -- (6.307,8.137) -- (6.307,8.137) -- (6.188,8.118) -- (6.081,8.063) -- (5.996,7.978) -- (5.941,7.871) -- (5.922,7.752) -- (5.941,7.633) -- (5.996,7.526) -- (6.081,7.441) -- (6.188,7.387) -- (6.307,7.368) -- cycle;
\node at (6.307,7.752) {\scalebox{1.616}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.24,6.509) -- (5.359,6.528) -- (5.466,6.582) -- (5.551,6.667) -- (5.606,6.775) -- (5.625,6.893) -- (5.606,7.012) -- (5.551,7.119) -- (5.466,7.205) -- (5.359,7.259) -- (5.24,7.278) -- (5.24,7.278) -- (5.121,7.259) -- (5.014,7.205) -- (4.929,7.119) -- (4.874,7.012) -- (4.856,6.893) -- (4.874,6.775) -- (4.929,6.667) -- (5.014,6.582) -- (5.121,6.528) -- (5.24,6.509) -- cycle;
\node at (5.24,6.893) {\scalebox{1.616}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.136,6.462) -- (7.255,6.481) -- (7.362,6.535) -- (7.447,6.621) -- (7.502,6.728) -- (7.521,6.847) -- (7.502,6.965) -- (7.447,7.073) -- (7.362,7.158) -- (7.255,7.212) -- (7.136,7.231) -- (7.136,7.231) -- (7.017,7.212) -- (6.91,7.158) -- (6.825,7.073) -- (6.771,6.965) -- (6.752,6.847) -- (6.771,6.728) -- (6.825,6.621) -- (6.91,6.535) -- (7.017,6.481) -- (7.136,6.462) -- cycle;
\node at (7.136,6.847) {\scalebox{1.616}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.29,5.796) -- (9.409,5.815) -- (9.516,5.869) -- (9.601,5.954) -- (9.656,6.062) -- (9.675,6.18) -- (9.656,6.299) -- (9.601,6.406) -- (9.516,6.491) -- (9.409,6.546) -- (9.29,6.565) -- (9.29,6.565) -- (9.171,6.546) -- (9.064,6.491) -- (8.979,6.406) -- (8.924,6.299) -- (8.906,6.18) -- (8.924,6.062) -- (8.979,5.954) -- (9.064,5.869) -- (9.171,5.815) -- (9.29,5.796) -- cycle;
\node at (9.29,6.18) {\scalebox{1.616}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.28,3.222) -- (8.399,3.241) -- (8.506,3.295) -- (8.591,3.38) -- (8.646,3.487) -- (8.665,3.606) -- (8.646,3.725) -- (8.591,3.832) -- (8.506,3.917) -- (8.399,3.972) -- (8.28,3.991) -- (8.28,3.991) -- (8.162,3.972) -- (8.054,3.917) -- (7.969,3.832) -- (7.915,3.725) -- (7.896,3.606) -- (7.915,3.487) -- (7.969,3.38) -- (8.054,3.295) -- (8.162,3.241) -- (8.28,3.222) -- cycle;
\node at (8.28,3.606) {\scalebox{1.616}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.707,3.264) -- (11.826,3.283) -- (11.933,3.338) -- (12.018,3.423) -- (12.073,3.53) -- (12.091,3.649) -- (12.073,3.768) -- (12.018,3.875) -- (11.933,3.96) -- (11.826,4.014) -- (11.707,4.033) -- (11.707,4.033) -- (11.588,4.014) -- (11.481,3.96) -- (11.396,3.875) -- (11.341,3.768) -- (11.322,3.649) -- (11.341,3.53) -- (11.396,3.423) -- (11.481,3.338) -- (11.588,3.283) -- (11.707,3.264) -- cycle;
\node at (11.707,3.649) {\scalebox{1.616}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.986,5.522) -- (6.105,5.541) -- (6.212,5.595) -- (6.297,5.681) -- (6.352,5.788) -- (6.371,5.907) -- (6.352,6.025) -- (6.297,6.133) -- (6.212,6.218) -- (6.105,6.272) -- (5.986,6.291) -- (5.986,6.291) -- (5.868,6.272) -- (5.76,6.218) -- (5.675,6.133) -- (5.621,6.025) -- (5.602,5.907) -- (5.621,5.788) -- (5.675,5.681) -- (5.76,5.595) -- (5.868,5.541) -- (5.986,5.522) -- cycle;
\node at (5.986,5.907) {\scalebox{1.616}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.957,3.25) -- (5.075,3.269) -- (5.183,3.324) -- (5.268,3.409) -- (5.322,3.516) -- (5.341,3.635) -- (5.322,3.754) -- (5.268,3.861) -- (5.183,3.946) -- (5.075,4.001) -- (4.957,4.019) -- (4.957,4.019) -- (4.838,4.001) -- (4.731,3.946) -- (4.645,3.861) -- (4.591,3.754) -- (4.572,3.635) -- (4.591,3.516) -- (4.645,3.409) -- (4.731,3.324) -- (4.838,3.269) -- (4.957,3.25) -- cycle;
\node at (4.957,3.635) {\scalebox{1.616}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.846,5.448) -- (2.965,5.466) -- (3.072,5.521) -- (3.157,5.606) -- (3.211,5.713) -- (3.23,5.832) -- (3.211,5.951) -- (3.157,6.058) -- (3.072,6.143) -- (2.965,6.198) -- (2.846,6.217) -- (2.846,6.217) -- (2.727,6.198) -- (2.62,6.143) -- (2.535,6.058) -- (2.48,5.951) -- (2.461,5.832) -- (2.48,5.713) -- (2.535,5.606) -- (2.62,5.521) -- (2.727,5.466) -- (2.846,5.448) -- cycle;
\node at (2.846,5.832) {\scalebox{1.616}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.316,3.251) -- (2.435,3.27) -- (2.542,3.325) -- (2.627,3.41) -- (2.682,3.517) -- (2.7,3.636) -- (2.682,3.755) -- (2.627,3.862) -- (2.542,3.947) -- (2.435,4.002) -- (2.316,4.02) -- (1.954,4.02) -- (1.835,4.002) -- (1.728,3.947) -- (1.643,3.862) -- (1.588,3.755) -- (1.569,3.636) -- (1.588,3.517) -- (1.643,3.41) -- (1.728,3.325) -- (1.835,3.27) -- (1.954,3.251) -- cycle;
\node at (2.135,3.636) {\scalebox{1.616}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.327,1.512) -- (3.446,1.531) -- (3.553,1.585) -- (3.639,1.67) -- (3.693,1.778) -- (3.712,1.896) -- (3.693,2.015) -- (3.639,2.122) -- (3.553,2.207) -- (3.446,2.262) -- (3.327,2.281) -- (2.966,2.281) -- (2.847,2.262) -- (2.74,2.207) -- (2.654,2.122) -- (2.6,2.015) -- (2.581,1.896) -- (2.6,1.778) -- (2.654,1.67) -- (2.74,1.585) -- (2.847,1.531) -- (2.966,1.512) -- cycle;
\node at (3.147,1.896) {\scalebox{1.616}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.695,1.691) -- (6.813,1.71) -- (6.921,1.765) -- (7.006,1.85) -- (7.06,1.957) -- (7.079,2.076) -- (7.06,2.194) -- (7.006,2.302) -- (6.921,2.387) -- (6.813,2.441) -- (6.695,2.46) -- (6.333,2.46) -- (6.214,2.441) -- (6.107,2.387) -- (6.022,2.302) -- (5.967,2.194) -- (5.948,2.076) -- (5.967,1.957) -- (6.022,1.85) -- (6.107,1.765) -- (6.214,1.71) -- (6.333,1.691) -- cycle;
\node at (6.514,2.076) {\scalebox{1.616}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.121,1.377) -- (10.24,1.395) -- (10.347,1.45) -- (10.432,1.535) -- (10.487,1.642) -- (10.505,1.761) -- (10.487,1.88) -- (10.432,1.987) -- (10.347,2.072) -- (10.24,2.127) -- (10.121,2.146) -- (9.759,2.146) -- (9.64,2.127) -- (9.533,2.072) -- (9.448,1.987) -- (9.393,1.88) -- (9.374,1.761) -- (9.393,1.642) -- (9.448,1.535) -- (9.533,1.45) -- (9.64,1.395) -- (9.759,1.377) -- cycle;
\node at (9.94,1.761) {\scalebox{1.616}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.302,7.732) -- (7.902,7.732) -- (7.902,8.332) -- (7.302,8.332) -- cycle;
\node at (7.602,8.032) {\scalebox{1.616}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (5.811,8.985) -- (6.411,8.985) -- (6.411,9.585) -- (5.811,9.585) -- cycle;
\node at (6.111,9.285) {\scalebox{1.616}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.376,6.692) -- (9.976,6.692) -- (9.976,7.292) -- (9.376,7.292) -- cycle;
\node at (9.676,6.992) {\scalebox{1.616}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (0.822,3.856) -- (1.422,3.856) -- (1.422,4.456) -- (0.822,4.456) -- cycle;
\node at (1.122,4.156) {\scalebox{1.616}{\small d}};
\node at (4.994,9.026) {\scalebox{1.616}{\small A}};
\node at (7.814,8.615) {\scalebox{1.616}{\small B}};
\node at (8.099,6.085) {\scalebox{1.616}{\small C}};
\node at (11.539,5.793) {\scalebox{1.616}{\small D}};
\node at (3.805,5.948) {\scalebox{1.616}{\small E}};
\node at (0.765,5.62) {\scalebox{1.616}{\small F}};
\node at (1.292,1.315) {\scalebox{1.616}{\small G}};
\node at (4.134,1.071) {\scalebox{1.616}{\small H}};
\node at (11.907,1.22) {\scalebox{1.616}{\small I}};
\node at (8.884,0.95) {\scalebox{1.616}{\small J}};
\draw[line width=0.5pt] (12.673,7.482) -- (12.673,8.582);
\draw[line width=0.5pt] (12.442,8.197) -- (12.673,8.582) -- (12.904,8.197);
\draw[line width=0.5pt] (12.371,8.032) -- (12.976,8.032);
\node at (12.673,8.967) {\scalebox{1.616}{\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, C, F, H, I & 5 \\ \hline
clairs & B, D, E, G, J & 5 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{5} clairs. Il y en a \textbf{autant} de chaque côté.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 10 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{Oui.} Un circuit existe :}\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{sentier des Prés ①} vers l'ouest jusqu'à Place de l'Église \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ②}, vers le sud, jusqu'à Rond-point de la Gare \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Basse ⑨}, vers l'ouest, jusqu'à Coin du Lavoir \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Neuve ⑩}, vers le sud, jusqu'à Place des Tanneurs \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin des Vignes ⑪}, vers l'est, jusqu'à Rond-point du Stade \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de l'Église ⑫}, vers l'est, jusqu'à Carrefour de l'Hôpital \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Peupliers ⑬}, vers l'est, jusqu'à Place Sainte-Croix \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ⑥}, vers le nord, jusqu'à Placette des Ormes \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{allée des Tilleuls ④}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Haute ③}, vers le nord, jusqu'à Fontaine Notre-Dame \textbf{(B)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Le car est passé une fois par chaque carrefour et se retrouve à l'école.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, et c'est déjà fait : le trajet ci-dessus dessert tous les carrefours. Il suffit au car de s'arrêter avant le dernier trajet de retour.}\par\addvspace{0.4em}
\end{document}
