\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 80}

\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=80}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0080}
\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=0.6119, line join=round, line cap=round]
\draw[line width=0.5pt] (10.763,10.851) -- (10.755,10.598) -- (10.754,10.346) -- (10.758,10.093) -- (10.769,9.84) -- (10.785,9.588) -- (10.808,9.335) -- (10.836,9.084) -- (10.87,8.833) -- (10.91,8.583) -- (10.956,8.335) -- (11.008,8.087) -- (11.066,7.841) -- (11.129,7.596) -- (11.198,7.353);
\draw[line width=0.5pt] (5.179,10.646) -- (5.372,10.674) -- (5.566,10.697) -- (5.761,10.716) -- (5.956,10.73) -- (6.152,10.74) -- (6.347,10.745) -- (6.543,10.745) -- (6.738,10.741) -- (6.934,10.732) -- (7.129,10.719) -- (7.324,10.701) -- (7.518,10.678) -- (7.712,10.651) -- (7.904,10.619);
\draw[line width=0.5pt] (5.179,10.646) -- (5.146,10.446) -- (5.118,10.246) -- (5.094,10.045) -- (5.075,9.843) -- (5.061,9.641) -- (5.052,9.439) -- (5.047,9.236) -- (5.048,9.034) -- (5.053,8.831) -- (5.062,8.629) -- (5.077,8.427) -- (5.096,8.225) -- (5.12,8.024) -- (5.149,7.824);
\draw[line width=0.5pt] (7.904,10.619) -- (8.104,10.667) -- (8.306,10.71) -- (8.508,10.748) -- (8.711,10.782) -- (8.915,10.81) -- (9.119,10.834) -- (9.324,10.853) -- (9.529,10.867) -- (9.735,10.877) -- (9.941,10.881) -- (10.146,10.881) -- (10.352,10.876) -- (10.557,10.866) -- (10.763,10.851);
\draw[line width=0.5pt] (7.904,10.619) -- (7.848,10.39) -- (7.797,10.159) -- (7.752,9.927) -- (7.712,9.693) -- (7.678,9.459) -- (7.649,9.224) -- (7.626,8.989) -- (7.608,8.753) -- (7.596,8.517) -- (7.589,8.28) -- (7.588,8.043) -- (7.593,7.807) -- (7.603,7.571) -- (7.618,7.335);
\draw[line width=0.5pt] (5.149,7.824) -- (5.329,7.816) -- (5.509,7.804) -- (5.689,7.788) -- (5.869,7.768) -- (6.048,7.743) -- (6.226,7.714) -- (6.404,7.681) -- (6.581,7.644) -- (6.757,7.603) -- (6.931,7.557) -- (7.105,7.508) -- (7.278,7.454) -- (7.449,7.397) -- (7.618,7.335);
\draw[line width=0.5pt] (5.149,7.824) -- (5.09,7.551) -- (5.038,7.277) -- (4.992,7.002) -- (4.953,6.726) -- (4.921,6.448) -- (4.894,6.17) -- (4.875,5.892) -- (4.862,5.613) -- (4.855,5.334) -- (4.855,5.055) -- (4.862,4.776) -- (4.875,4.497) -- (4.895,4.219) -- (4.921,3.941);
\draw[line width=0.5pt] (7.618,7.335) -- (7.872,7.375) -- (8.126,7.41) -- (8.382,7.438) -- (8.638,7.461) -- (8.894,7.477) -- (9.151,7.487) -- (9.408,7.492) -- (9.665,7.49) -- (9.921,7.482) -- (10.178,7.468) -- (10.434,7.448) -- (10.69,7.423) -- (10.944,7.391) -- (11.198,7.353);
\draw[line width=0.5pt] (7.618,7.335) -- (7.634,7.126) -- (7.655,6.917) -- (7.68,6.709) -- (7.711,6.502) -- (7.746,6.295) -- (7.786,6.089) -- (7.831,5.884) -- (7.881,5.68) -- (7.935,5.477) -- (7.995,5.276) -- (8.059,5.076) -- (8.127,4.878) -- (8.201,4.682) -- (8.279,4.487);
\draw[line width=0.5pt] (1.892,7.03) -- (2.114,7.122) -- (2.339,7.208) -- (2.565,7.29) -- (2.793,7.366) -- (3.023,7.436) -- (3.255,7.501) -- (3.488,7.561) -- (3.722,7.615) -- (3.958,7.664) -- (4.195,7.707) -- (4.432,7.745) -- (4.67,7.777) -- (4.909,7.803) -- (5.149,7.824);
\draw[line width=0.5pt] (1.892,7.03) -- (1.833,6.848) -- (1.778,6.665) -- (1.727,6.48) -- (1.681,6.295) -- (1.639,6.108) -- (1.601,5.921) -- (1.568,5.732) -- (1.539,5.543) -- (1.515,5.353) -- (1.495,5.163) -- (1.48,4.972) -- (1.469,4.782) -- (1.462,4.59) -- (1.461,4.399);
\draw[line width=0.5pt] (1.461,4.399) -- (1.711,4.405) -- (1.961,4.404) -- (2.212,4.398) -- (2.462,4.385) -- (2.712,4.367) -- (2.961,4.343) -- (3.21,4.313) -- (3.458,4.277) -- (3.705,4.236) -- (3.951,4.188) -- (4.196,4.135) -- (4.439,4.076) -- (4.681,4.011) -- (4.921,3.941);
\draw[line width=0.5pt] (1.461,4.399) -- (1.41,4.193) -- (1.364,3.986) -- (1.323,3.777) -- (1.286,3.567) -- (1.255,3.357) -- (1.229,3.146) -- (1.208,2.934) -- (1.191,2.722) -- (1.18,2.51) -- (1.174,2.297) -- (1.172,2.085) -- (1.176,1.872) -- (1.185,1.66) -- (1.198,1.448);
\draw[line width=0.5pt] (4.921,3.941) -- (5.153,4.016) -- (5.387,4.086) -- (5.622,4.151) -- (5.859,4.21) -- (6.097,4.263) -- (6.337,4.311) -- (6.577,4.353) -- (6.819,4.389) -- (7.061,4.42) -- (7.304,4.445) -- (7.547,4.464) -- (7.791,4.478) -- (8.035,4.485) -- (8.279,4.487);
\draw[line width=0.5pt] (4.921,3.941) -- (4.88,3.774) -- (4.842,3.606) -- (4.809,3.438) -- (4.779,3.268) -- (4.754,3.098) -- (4.732,2.927) -- (4.715,2.756) -- (4.702,2.585) -- (4.692,2.413) -- (4.687,2.241) -- (4.685,2.069) -- (4.688,1.897) -- (4.695,1.725) -- (4.706,1.554);
\draw[line width=0.5pt] (1.198,1.448) -- (1.446,1.494) -- (1.694,1.534) -- (1.944,1.568) -- (2.194,1.596) -- (2.445,1.618) -- (2.696,1.635) -- (2.948,1.645) -- (3.199,1.65) -- (3.451,1.649) -- (3.703,1.642) -- (3.955,1.628) -- (4.206,1.609) -- (4.456,1.585) -- (4.706,1.554);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.923,10.851) -- (10.901,10.931) -- (10.843,10.99) -- (10.763,11.011) -- (10.683,10.99) -- (10.624,10.931) -- (10.603,10.851) -- (10.624,10.771) -- (10.683,10.713) -- (10.763,10.691) -- (10.843,10.713) -- (10.901,10.771) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.339,10.646) -- (5.317,10.726) -- (5.259,10.785) -- (5.179,10.806) -- (5.099,10.785) -- (5.04,10.726) -- (5.019,10.646) -- (5.04,10.566) -- (5.099,10.507) -- (5.179,10.486) -- (5.259,10.507) -- (5.317,10.566) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.064,10.619) -- (8.043,10.699) -- (7.984,10.758) -- (7.904,10.779) -- (7.824,10.758) -- (7.766,10.699) -- (7.744,10.619) -- (7.766,10.539) -- (7.824,10.481) -- (7.904,10.459) -- (7.984,10.481) -- (8.043,10.539) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.309,7.824) -- (5.287,7.904) -- (5.229,7.963) -- (5.149,7.984) -- (5.069,7.963) -- (5.01,7.904) -- (4.989,7.824) -- (5.01,7.744) -- (5.069,7.685) -- (5.149,7.664) -- (5.229,7.685) -- (5.287,7.744) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.358,7.353) -- (11.337,7.433) -- (11.278,7.491) -- (11.198,7.513) -- (11.118,7.491) -- (11.06,7.433) -- (11.038,7.353) -- (11.06,7.273) -- (11.118,7.214) -- (11.198,7.193) -- (11.278,7.214) -- (11.337,7.273) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.778,7.335) -- (7.757,7.415) -- (7.698,7.473) -- (7.618,7.495) -- (7.538,7.473) -- (7.48,7.415) -- (7.458,7.335) -- (7.48,7.255) -- (7.538,7.196) -- (7.618,7.175) -- (7.698,7.196) -- (7.757,7.255) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.052,7.03) -- (2.031,7.11) -- (1.972,7.168) -- (1.892,7.19) -- (1.812,7.168) -- (1.754,7.11) -- (1.732,7.03) -- (1.754,6.95) -- (1.812,6.891) -- (1.892,6.87) -- (1.972,6.891) -- (2.031,6.95) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.439,4.487) -- (8.417,4.567) -- (8.359,4.626) -- (8.279,4.647) -- (8.199,4.626) -- (8.14,4.567) -- (8.119,4.487) -- (8.14,4.407) -- (8.199,4.349) -- (8.279,4.327) -- (8.359,4.349) -- (8.417,4.407) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.621,4.399) -- (1.599,4.479) -- (1.541,4.538) -- (1.461,4.559) -- (1.381,4.538) -- (1.322,4.479) -- (1.301,4.399) -- (1.322,4.319) -- (1.381,4.261) -- (1.461,4.239) -- (1.541,4.261) -- (1.599,4.319) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.081,3.941) -- (5.06,4.021) -- (5.001,4.08) -- (4.921,4.101) -- (4.841,4.08) -- (4.783,4.021) -- (4.761,3.941) -- (4.783,3.861) -- (4.841,3.802) -- (4.921,3.781) -- (5.001,3.802) -- (5.06,3.861) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.866,1.554) -- (4.844,1.634) -- (4.786,1.692) -- (4.706,1.714) -- (4.626,1.692) -- (4.567,1.634) -- (4.546,1.554) -- (4.567,1.474) -- (4.626,1.415) -- (4.706,1.394) -- (4.786,1.415) -- (4.844,1.474) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.358,1.448) -- (1.337,1.528) -- (1.278,1.586) -- (1.198,1.608) -- (1.118,1.586) -- (1.06,1.528) -- (1.038,1.448) -- (1.06,1.368) -- (1.118,1.309) -- (1.198,1.288) -- (1.278,1.309) -- (1.337,1.368) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.227,8.739) -- (11.349,8.758) -- (11.458,8.814) -- (11.546,8.901) -- (11.602,9.011) -- (11.621,9.133) -- (11.602,9.254) -- (11.546,9.364) -- (11.458,9.451) -- (11.349,9.507) -- (11.227,9.527) -- (11.227,9.527) -- (11.105,9.507) -- (10.995,9.451) -- (10.908,9.364) -- (10.852,9.254) -- (10.833,9.133) -- (10.852,9.011) -- (10.908,8.901) -- (10.995,8.814) -- (11.105,8.758) -- (11.227,8.739) -- cycle;
\node at (11.227,9.133) {\scalebox{1.634}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.547,10.745) -- (6.668,10.764) -- (6.778,10.82) -- (6.865,10.907) -- (6.921,11.017) -- (6.941,11.139) -- (6.921,11.261) -- (6.865,11.37) -- (6.778,11.458) -- (6.668,11.514) -- (6.547,11.533) -- (6.547,11.533) -- (6.425,11.514) -- (6.315,11.458) -- (6.228,11.37) -- (6.172,11.261) -- (6.153,11.139) -- (6.172,11.017) -- (6.228,10.907) -- (6.315,10.82) -- (6.425,10.764) -- (6.547,10.745) -- cycle;
\node at (6.547,11.139) {\scalebox{1.634}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.441,8.838) -- (5.563,8.857) -- (5.673,8.913) -- (5.76,9.0) -- (5.816,9.11) -- (5.835,9.232) -- (5.816,9.354) -- (5.76,9.464) -- (5.673,9.551) -- (5.563,9.607) -- (5.441,9.626) -- (5.441,9.626) -- (5.32,9.607) -- (5.21,9.551) -- (5.123,9.464) -- (5.067,9.354) -- (5.047,9.232) -- (5.067,9.11) -- (5.123,9.0) -- (5.21,8.913) -- (5.32,8.857) -- (5.441,8.838) -- cycle;
\node at (5.441,9.232) {\scalebox{1.634}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.292,10.852) -- (9.414,10.871) -- (9.524,10.927) -- (9.611,11.014) -- (9.667,11.124) -- (9.686,11.246) -- (9.667,11.367) -- (9.611,11.477) -- (9.524,11.564) -- (9.414,11.62) -- (9.292,11.64) -- (9.292,11.64) -- (9.17,11.62) -- (9.061,11.564) -- (8.973,11.477) -- (8.917,11.367) -- (8.898,11.246) -- (8.917,11.124) -- (8.973,11.014) -- (9.061,10.927) -- (9.17,10.871) -- (9.292,10.852) -- cycle;
\node at (9.292,11.246) {\scalebox{1.634}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.018,8.561) -- (8.14,8.58) -- (8.25,8.636) -- (8.337,8.723) -- (8.393,8.833) -- (8.412,8.955) -- (8.393,9.076) -- (8.337,9.186) -- (8.25,9.273) -- (8.14,9.329) -- (8.018,9.349) -- (8.018,9.349) -- (7.897,9.329) -- (7.787,9.273) -- (7.7,9.186) -- (7.644,9.076) -- (7.624,8.955) -- (7.644,8.833) -- (7.7,8.723) -- (7.787,8.636) -- (7.897,8.58) -- (8.018,8.561) -- cycle;
\node at (8.018,8.955) {\scalebox{1.634}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.48,7.674) -- (6.602,7.693) -- (6.712,7.749) -- (6.799,7.836) -- (6.855,7.946) -- (6.874,8.068) -- (6.855,8.189) -- (6.799,8.299) -- (6.712,8.386) -- (6.602,8.442) -- (6.48,8.462) -- (6.48,8.462) -- (6.359,8.442) -- (6.249,8.386) -- (6.162,8.299) -- (6.106,8.189) -- (6.086,8.068) -- (6.106,7.946) -- (6.162,7.836) -- (6.249,7.749) -- (6.359,7.693) -- (6.48,7.674) -- cycle;
\node at (6.48,8.068) {\scalebox{1.634}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.268,5.475) -- (5.39,5.494) -- (5.5,5.55) -- (5.587,5.637) -- (5.643,5.747) -- (5.662,5.869) -- (5.643,5.991) -- (5.587,6.1) -- (5.5,6.187) -- (5.39,6.243) -- (5.268,6.263) -- (5.268,6.263) -- (5.146,6.243) -- (5.037,6.187) -- (4.949,6.1) -- (4.893,5.991) -- (4.874,5.869) -- (4.893,5.747) -- (4.949,5.637) -- (5.037,5.55) -- (5.146,5.494) -- (5.268,5.475) -- cycle;
\node at (5.268,5.869) {\scalebox{1.634}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.406,7.492) -- (9.527,7.511) -- (9.637,7.567) -- (9.724,7.654) -- (9.78,7.764) -- (9.8,7.885) -- (9.78,8.007) -- (9.724,8.117) -- (9.637,8.204) -- (9.527,8.26) -- (9.406,8.279) -- (9.406,8.279) -- (9.284,8.26) -- (9.174,8.204) -- (9.087,8.117) -- (9.031,8.007) -- (9.012,7.885) -- (9.031,7.764) -- (9.087,7.654) -- (9.174,7.567) -- (9.284,7.511) -- (9.406,7.492) -- cycle;
\node at (9.406,7.885) {\scalebox{1.634}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.215,5.579) -- (8.337,5.598) -- (8.446,5.654) -- (8.533,5.741) -- (8.589,5.851) -- (8.609,5.973) -- (8.589,6.095) -- (8.533,6.204) -- (8.446,6.291) -- (8.337,6.347) -- (8.215,6.367) -- (8.215,6.367) -- (8.093,6.347) -- (7.983,6.291) -- (7.896,6.204) -- (7.84,6.095) -- (7.821,5.973) -- (7.84,5.851) -- (7.896,5.741) -- (7.983,5.654) -- (8.093,5.598) -- (8.215,5.579) -- cycle;
\node at (8.215,5.973) {\scalebox{1.634}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.767,6.785) -- (3.888,6.804) -- (3.998,6.86) -- (4.085,6.947) -- (4.141,7.057) -- (4.16,7.178) -- (4.141,7.3) -- (4.085,7.41) -- (3.998,7.497) -- (3.888,7.553) -- (3.767,7.572) -- (3.396,7.572) -- (3.274,7.553) -- (3.164,7.497) -- (3.077,7.41) -- (3.021,7.3) -- (3.002,7.178) -- (3.021,7.057) -- (3.077,6.947) -- (3.164,6.86) -- (3.274,6.804) -- (3.396,6.785) -- cycle;
\node at (3.581,7.178) {\scalebox{1.634}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.365,5.402) -- (1.486,5.421) -- (1.596,5.477) -- (1.683,5.565) -- (1.739,5.674) -- (1.759,5.796) -- (1.739,5.918) -- (1.683,6.028) -- (1.596,6.115) -- (1.486,6.171) -- (1.365,6.19) -- (0.994,6.19) -- (0.872,6.171) -- (0.762,6.115) -- (0.675,6.028) -- (0.619,5.918) -- (0.6,5.796) -- (0.619,5.674) -- (0.675,5.565) -- (0.762,5.477) -- (0.872,5.421) -- (0.994,5.402) -- cycle;
\node at (1.179,5.796) {\scalebox{1.634}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.343,3.529) -- (3.465,3.548) -- (3.575,3.604) -- (3.662,3.691) -- (3.718,3.801) -- (3.737,3.922) -- (3.718,4.044) -- (3.662,4.154) -- (3.575,4.241) -- (3.465,4.297) -- (3.343,4.316) -- (2.973,4.316) -- (2.851,4.297) -- (2.741,4.241) -- (2.654,4.154) -- (2.598,4.044) -- (2.579,3.922) -- (2.598,3.801) -- (2.654,3.691) -- (2.741,3.604) -- (2.851,3.548) -- (2.973,3.529) -- cycle;
\node at (3.158,3.922) {\scalebox{1.634}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.785,2.506) -- (1.907,2.525) -- (2.017,2.581) -- (2.104,2.668) -- (2.16,2.778) -- (2.179,2.9) -- (2.16,3.021) -- (2.104,3.131) -- (2.017,3.218) -- (1.907,3.274) -- (1.785,3.293) -- (1.415,3.293) -- (1.293,3.274) -- (1.183,3.218) -- (1.096,3.131) -- (1.04,3.021) -- (1.021,2.9) -- (1.04,2.778) -- (1.096,2.668) -- (1.183,2.581) -- (1.293,2.525) -- (1.415,2.506) -- cycle;
\node at (1.6,2.9) {\scalebox{1.634}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.699,4.348) -- (6.821,4.367) -- (6.931,4.423) -- (7.018,4.51) -- (7.074,4.62) -- (7.093,4.741) -- (7.074,4.863) -- (7.018,4.973) -- (6.931,5.06) -- (6.821,5.116) -- (6.699,5.135) -- (6.329,5.135) -- (6.207,5.116) -- (6.097,5.06) -- (6.01,4.973) -- (5.954,4.863) -- (5.935,4.741) -- (5.954,4.62) -- (6.01,4.51) -- (6.097,4.423) -- (6.207,4.367) -- (6.329,4.348) -- cycle;
\node at (6.514,4.741) {\scalebox{1.634}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.618,1.987) -- (5.74,2.006) -- (5.85,2.062) -- (5.937,2.149) -- (5.993,2.259) -- (6.012,2.381) -- (5.993,2.503) -- (5.937,2.612) -- (5.85,2.7) -- (5.74,2.756) -- (5.618,2.775) -- (5.248,2.775) -- (5.126,2.756) -- (5.016,2.7) -- (4.929,2.612) -- (4.873,2.503) -- (4.854,2.381) -- (4.873,2.259) -- (4.929,2.149) -- (5.016,2.062) -- (5.126,2.006) -- (5.248,1.987) -- cycle;
\node at (5.433,2.381) {\scalebox{1.634}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.643,1.649) -- (3.765,1.668) -- (3.875,1.724) -- (3.962,1.811) -- (4.018,1.921) -- (4.037,2.043) -- (4.018,2.164) -- (3.962,2.274) -- (3.875,2.361) -- (3.765,2.417) -- (3.643,2.437) -- (3.273,2.437) -- (3.151,2.417) -- (3.041,2.361) -- (2.954,2.274) -- (2.898,2.164) -- (2.879,2.043) -- (2.898,1.921) -- (2.954,1.811) -- (3.041,1.724) -- (3.151,1.668) -- (3.273,1.649) -- cycle;
\node at (3.458,2.043) {\scalebox{1.634}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.979,4.187) -- (8.579,4.187) -- (8.579,4.787) -- (7.979,4.787) -- cycle;
\node at (8.279,4.487) {\scalebox{1.634}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.966,4.37) -- (2.566,4.37) -- (2.566,4.97) -- (1.966,4.97) -- cycle;
\node at (2.266,4.67) {\scalebox{1.634}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.077,7.911) -- (2.677,7.911) -- (2.677,8.511) -- (2.077,8.511) -- cycle;
\node at (2.377,8.211) {\scalebox{1.634}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.757,1.438) -- (2.357,1.438) -- (2.357,2.038) -- (1.757,2.038) -- cycle;
\node at (2.057,1.738) {\scalebox{1.634}{\small d}};
\node at (11.224,11.265) {\scalebox{1.634}{\small A}};
\node at (5.08,11.258) {\scalebox{1.634}{\small B}};
\node at (8.178,11.176) {\scalebox{1.634}{\small C}};
\node at (4.864,8.375) {\scalebox{1.634}{\small D}};
\node at (11.81,7.455) {\scalebox{1.634}{\small E}};
\node at (8.173,7.611) {\scalebox{1.634}{\small F}};
\node at (1.279,7.118) {\scalebox{1.634}{\small G}};
\node at (8.76,4.096) {\scalebox{1.634}{\small H}};
\node at (0.899,4.136) {\scalebox{1.634}{\small I}};
\node at (4.705,3.36) {\scalebox{1.634}{\small J}};
\node at (4.564,0.95) {\scalebox{1.634}{\small K}};
\node at (0.776,0.994) {\scalebox{1.634}{\small L}};
\draw[line width=0.5pt] (12.498,9.901) -- (12.498,11.001);
\draw[line width=0.5pt] (12.267,10.616) -- (12.498,11.001) -- (12.729,10.616);
\draw[line width=0.5pt] (12.196,10.451) -- (12.801,10.451);
\node at (12.498,11.386) {\scalebox{1.634}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour H, 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 avenue de la Gare & Ⓐ Place des Tanneurs & Ⓙ Carrefour de la Mairie \\ \hline
① rue du Pont & ⑩ allée des Peupliers & Ⓑ Fontaine Notre-Dame & Ⓚ Placette des Ormes \\ \hline
② rue des Écoles & ⑪ rue du Moulin & Ⓒ Carrefour de l'Hôpital & Ⓛ Place aux Herbes \\ \hline
③ rue du Four & ⑫ rue de la Poste & Ⓓ Carrefour du Moulin & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ rue Haute & ⑬ chemin de Ronde & Ⓔ Place du Marché & \textbf{a} l'école \\ \hline
⑤ quai aux Fleurs & ⑭ rue du Puits & Ⓕ Rond-point du Stade & \textbf{b} le cimetière \\ \hline
⑥ venelle du Chat & ⑮ rue de la Mairie & Ⓖ Place de la Fontaine & \textbf{c} la boulangerie \\ \hline
⑦ rue Verte & ⑯ chemin des Vignes & Ⓗ Place de l'Église & \textbf{d} l'église \\ \hline
⑧ sentier des Prés & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Rond-point de la Gare &  \\ \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=0.6273, line join=round, line cap=round]
\draw[line width=0.5pt] (10.751,10.851) -- (10.744,10.598) -- (10.743,10.346) -- (10.747,10.093) -- (10.758,9.84) -- (10.774,9.588) -- (10.797,9.335) -- (10.825,9.084) -- (10.859,8.833) -- (10.899,8.583) -- (10.945,8.335) -- (10.997,8.087) -- (11.055,7.841) -- (11.118,7.596) -- (11.187,7.353);
\draw[line width=0.5pt] (5.168,10.646) -- (5.361,10.674) -- (5.555,10.697) -- (5.75,10.716) -- (5.945,10.73) -- (6.14,10.74) -- (6.336,10.745) -- (6.532,10.745) -- (6.727,10.741) -- (6.923,10.732) -- (7.118,10.719) -- (7.313,10.701) -- (7.507,10.678) -- (7.701,10.651) -- (7.893,10.619);
\draw[line width=0.5pt] (5.168,10.646) -- (5.135,10.446) -- (5.107,10.246) -- (5.083,10.045) -- (5.064,9.843) -- (5.05,9.641) -- (5.041,9.439) -- (5.036,9.236) -- (5.037,9.034) -- (5.042,8.831) -- (5.051,8.629) -- (5.066,8.427) -- (5.085,8.225) -- (5.109,8.024) -- (5.138,7.824);
\draw[line width=0.5pt] (7.893,10.619) -- (8.093,10.667) -- (8.294,10.71) -- (8.497,10.748) -- (8.7,10.782) -- (8.903,10.81) -- (9.108,10.834) -- (9.313,10.853) -- (9.518,10.867) -- (9.724,10.877) -- (9.93,10.881) -- (10.135,10.881) -- (10.341,10.876) -- (10.546,10.866) -- (10.751,10.851);
\draw[line width=0.5pt] (7.893,10.619) -- (7.837,10.39) -- (7.786,10.159) -- (7.741,9.927) -- (7.701,9.693) -- (7.667,9.459) -- (7.638,9.224) -- (7.615,8.989) -- (7.597,8.753) -- (7.585,8.517) -- (7.578,8.28) -- (7.577,8.043) -- (7.582,7.807) -- (7.592,7.571) -- (7.607,7.335);
\draw[line width=0.5pt] (5.138,7.824) -- (5.318,7.816) -- (5.498,7.804) -- (5.678,7.788) -- (5.858,7.768) -- (6.037,7.743) -- (6.215,7.714) -- (6.393,7.681) -- (6.57,7.644) -- (6.746,7.603) -- (6.92,7.557) -- (7.094,7.508) -- (7.267,7.454) -- (7.438,7.397) -- (7.607,7.335);
\draw[line width=0.5pt] (5.138,7.824) -- (5.079,7.551) -- (5.027,7.277) -- (4.981,7.002) -- (4.942,6.726) -- (4.909,6.448) -- (4.883,6.17) -- (4.864,5.892) -- (4.851,5.613) -- (4.844,5.334) -- (4.844,5.055) -- (4.851,4.776) -- (4.864,4.497) -- (4.884,4.219) -- (4.91,3.941);
\draw[line width=0.5pt] (7.607,7.335) -- (7.861,7.375) -- (8.115,7.41) -- (8.371,7.438) -- (8.626,7.461) -- (8.883,7.477) -- (9.14,7.487) -- (9.397,7.492) -- (9.654,7.49) -- (9.91,7.482) -- (10.167,7.468) -- (10.423,7.448) -- (10.679,7.423) -- (10.933,7.391) -- (11.187,7.353);
\draw[line width=0.5pt] (7.607,7.335) -- (7.623,7.126) -- (7.644,6.917) -- (7.669,6.709) -- (7.7,6.502) -- (7.735,6.295) -- (7.775,6.089) -- (7.82,5.884) -- (7.87,5.68) -- (7.924,5.477) -- (7.984,5.276) -- (8.048,5.076) -- (8.116,4.878) -- (8.19,4.682) -- (8.268,4.487);
\draw[line width=0.5pt] (1.881,7.03) -- (2.103,7.122) -- (2.328,7.208) -- (2.554,7.29) -- (2.782,7.366) -- (3.012,7.436) -- (3.244,7.501) -- (3.477,7.561) -- (3.711,7.615) -- (3.947,7.664) -- (4.183,7.707) -- (4.421,7.745) -- (4.659,7.777) -- (4.898,7.803) -- (5.138,7.824);
\draw[line width=0.5pt] (1.881,7.03) -- (1.822,6.848) -- (1.767,6.665) -- (1.716,6.48) -- (1.67,6.295) -- (1.628,6.108) -- (1.59,5.921) -- (1.557,5.732) -- (1.528,5.543) -- (1.504,5.353) -- (1.484,5.163) -- (1.469,4.972) -- (1.458,4.782) -- (1.451,4.59) -- (1.45,4.399);
\draw[line width=0.5pt] (1.45,4.399) -- (1.7,4.405) -- (1.95,4.404) -- (2.2,4.398) -- (2.451,4.385) -- (2.701,4.367) -- (2.95,4.343) -- (3.199,4.313) -- (3.447,4.277) -- (3.694,4.236) -- (3.94,4.188) -- (4.185,4.135) -- (4.428,4.076) -- (4.67,4.011) -- (4.91,3.941);
\draw[line width=0.5pt] (1.45,4.399) -- (1.399,4.193) -- (1.353,3.986) -- (1.312,3.777) -- (1.275,3.567) -- (1.244,3.357) -- (1.218,3.146) -- (1.197,2.934) -- (1.18,2.722) -- (1.169,2.51) -- (1.163,2.297) -- (1.161,2.085) -- (1.165,1.872) -- (1.173,1.66) -- (1.187,1.448);
\draw[line width=0.5pt] (4.91,3.941) -- (5.142,4.016) -- (5.376,4.086) -- (5.611,4.151) -- (5.848,4.21) -- (6.086,4.263) -- (6.326,4.311) -- (6.566,4.353) -- (6.808,4.389) -- (7.05,4.42) -- (7.293,4.445) -- (7.536,4.464) -- (7.78,4.478) -- (8.024,4.485) -- (8.268,4.487);
\draw[line width=0.5pt] (4.91,3.941) -- (4.869,3.774) -- (4.831,3.606) -- (4.798,3.438) -- (4.768,3.268) -- (4.743,3.098) -- (4.721,2.927) -- (4.704,2.756) -- (4.69,2.585) -- (4.681,2.413) -- (4.676,2.241) -- (4.674,2.069) -- (4.677,1.897) -- (4.684,1.725) -- (4.695,1.554);
\draw[line width=0.5pt] (1.187,1.448) -- (1.435,1.494) -- (1.683,1.534) -- (1.932,1.568) -- (2.183,1.596) -- (2.434,1.618) -- (2.685,1.635) -- (2.937,1.645) -- (3.188,1.65) -- (3.44,1.649) -- (3.692,1.642) -- (3.944,1.628) -- (4.195,1.609) -- (4.445,1.585) -- (4.695,1.554);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.125,10.851) -- (11.096,10.994) -- (11.015,11.115) -- (10.894,11.196) -- (10.751,11.224) -- (10.609,11.196) -- (10.488,11.115) -- (10.407,10.994) -- (10.378,10.851) -- (10.407,10.708) -- (10.488,10.587) -- (10.609,10.506) -- (10.751,10.478) -- (10.894,10.506) -- (11.015,10.587) -- (11.096,10.708) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.541,10.646) -- (5.513,10.789) -- (5.432,10.91) -- (5.311,10.991) -- (5.168,11.019) -- (5.025,10.991) -- (4.904,10.91) -- (4.823,10.789) -- (4.795,10.646) -- (4.823,10.503) -- (4.904,10.382) -- (5.025,10.301) -- (5.168,10.273) -- (5.311,10.301) -- (5.432,10.382) -- (5.513,10.503) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.267,10.619) -- (8.238,10.762) -- (8.157,10.883) -- (8.036,10.964) -- (7.893,10.993) -- (7.751,10.964) -- (7.629,10.883) -- (7.549,10.762) -- (7.52,10.619) -- (7.549,10.476) -- (7.629,10.355) -- (7.751,10.275) -- (7.893,10.246) -- (8.036,10.275) -- (8.157,10.355) -- (8.238,10.476) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.511,7.824) -- (5.483,7.967) -- (5.402,8.088) -- (5.281,8.169) -- (5.138,8.197) -- (4.995,8.169) -- (4.874,8.088) -- (4.793,7.967) -- (4.765,7.824) -- (4.793,7.681) -- (4.874,7.56) -- (4.995,7.479) -- (5.138,7.451) -- (5.281,7.479) -- (5.402,7.56) -- (5.483,7.681) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.56,7.353) -- (11.532,7.496) -- (11.451,7.617) -- (11.33,7.698) -- (11.187,7.726) -- (11.044,7.698) -- (10.923,7.617) -- (10.842,7.496) -- (10.814,7.353) -- (10.842,7.21) -- (10.923,7.089) -- (11.044,7.008) -- (11.187,6.98) -- (11.33,7.008) -- (11.451,7.089) -- (11.532,7.21) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.981,7.335) -- (7.952,7.478) -- (7.871,7.599) -- (7.75,7.68) -- (7.607,7.708) -- (7.465,7.68) -- (7.343,7.599) -- (7.263,7.478) -- (7.234,7.335) -- (7.263,7.192) -- (7.343,7.071) -- (7.465,6.99) -- (7.607,6.962) -- (7.75,6.99) -- (7.871,7.071) -- (7.952,7.192) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.254,7.03) -- (2.226,7.172) -- (2.145,7.293) -- (2.024,7.374) -- (1.881,7.403) -- (1.738,7.374) -- (1.617,7.293) -- (1.536,7.172) -- (1.508,7.03) -- (1.536,6.887) -- (1.617,6.766) -- (1.738,6.685) -- (1.881,6.656) -- (2.024,6.685) -- (2.145,6.766) -- (2.226,6.887) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.641,4.487) -- (8.612,4.63) -- (8.531,4.751) -- (8.41,4.832) -- (8.268,4.861) -- (8.125,4.832) -- (8.004,4.751) -- (7.923,4.63) -- (7.894,4.487) -- (7.923,4.344) -- (8.004,4.223) -- (8.125,4.142) -- (8.268,4.114) -- (8.41,4.142) -- (8.531,4.223) -- (8.612,4.344) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.823,4.399) -- (1.794,4.542) -- (1.713,4.663) -- (1.592,4.744) -- (1.45,4.773) -- (1.307,4.744) -- (1.186,4.663) -- (1.105,4.542) -- (1.076,4.399) -- (1.105,4.257) -- (1.186,4.135) -- (1.307,4.055) -- (1.45,4.026) -- (1.592,4.055) -- (1.713,4.135) -- (1.794,4.257) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.283,3.941) -- (5.255,4.084) -- (5.174,4.205) -- (5.053,4.286) -- (4.91,4.314) -- (4.767,4.286) -- (4.646,4.205) -- (4.565,4.084) -- (4.537,3.941) -- (4.565,3.798) -- (4.646,3.677) -- (4.767,3.596) -- (4.91,3.568) -- (5.053,3.596) -- (5.174,3.677) -- (5.255,3.798) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.068,1.554) -- (5.04,1.696) -- (4.959,1.818) -- (4.838,1.898) -- (4.695,1.927) -- (4.552,1.898) -- (4.431,1.818) -- (4.35,1.696) -- (4.321,1.554) -- (4.35,1.411) -- (4.431,1.29) -- (4.552,1.209) -- (4.695,1.18) -- (4.838,1.209) -- (4.959,1.29) -- (5.04,1.411) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.56,1.448) -- (1.532,1.591) -- (1.451,1.712) -- (1.33,1.793) -- (1.187,1.821) -- (1.044,1.793) -- (0.923,1.712) -- (0.842,1.591) -- (0.814,1.448) -- (0.842,1.305) -- (0.923,1.184) -- (1.044,1.103) -- (1.187,1.075) -- (1.33,1.103) -- (1.451,1.184) -- (1.532,1.305) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.207,8.747) -- (11.325,8.766) -- (11.433,8.82) -- (11.518,8.906) -- (11.572,9.013) -- (11.591,9.132) -- (11.572,9.25) -- (11.518,9.358) -- (11.433,9.443) -- (11.325,9.497) -- (11.207,9.516) -- (11.207,9.516) -- (11.088,9.497) -- (10.981,9.443) -- (10.895,9.358) -- (10.841,9.25) -- (10.822,9.132) -- (10.841,9.013) -- (10.895,8.906) -- (10.981,8.82) -- (11.088,8.766) -- (11.207,8.747) -- cycle;
\node at (11.207,9.132) {\scalebox{1.594}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.535,10.745) -- (6.654,10.764) -- (6.761,10.819) -- (6.847,10.904) -- (6.901,11.011) -- (6.92,11.13) -- (6.901,11.248) -- (6.847,11.356) -- (6.761,11.441) -- (6.654,11.495) -- (6.535,11.514) -- (6.535,11.514) -- (6.417,11.495) -- (6.309,11.441) -- (6.224,11.356) -- (6.17,11.248) -- (6.151,11.13) -- (6.17,11.011) -- (6.224,10.904) -- (6.309,10.819) -- (6.417,10.764) -- (6.535,10.745) -- cycle;
\node at (6.535,11.13) {\scalebox{1.594}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.421,8.848) -- (5.54,8.866) -- (5.647,8.921) -- (5.732,9.006) -- (5.787,9.113) -- (5.805,9.232) -- (5.787,9.351) -- (5.732,9.458) -- (5.647,9.543) -- (5.54,9.598) -- (5.421,9.617) -- (5.421,9.617) -- (5.302,9.598) -- (5.195,9.543) -- (5.11,9.458) -- (5.055,9.351) -- (5.036,9.232) -- (5.055,9.113) -- (5.11,9.006) -- (5.195,8.921) -- (5.302,8.866) -- (5.421,8.848) -- cycle;
\node at (5.421,9.232) {\scalebox{1.594}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.282,10.852) -- (9.401,10.871) -- (9.508,10.925) -- (9.593,11.01) -- (9.647,11.118) -- (9.666,11.236) -- (9.647,11.355) -- (9.593,11.462) -- (9.508,11.547) -- (9.401,11.602) -- (9.282,11.621) -- (9.282,11.621) -- (9.163,11.602) -- (9.056,11.547) -- (8.971,11.462) -- (8.916,11.355) -- (8.897,11.236) -- (8.916,11.118) -- (8.971,11.01) -- (9.056,10.925) -- (9.163,10.871) -- (9.282,10.852) -- cycle;
\node at (9.282,11.236) {\scalebox{1.594}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.998,8.571) -- (8.117,8.59) -- (8.224,8.644) -- (8.309,8.729) -- (8.364,8.837) -- (8.382,8.956) -- (8.364,9.074) -- (8.309,9.182) -- (8.224,9.267) -- (8.117,9.321) -- (7.998,9.34) -- (7.998,9.34) -- (7.879,9.321) -- (7.772,9.267) -- (7.687,9.182) -- (7.632,9.074) -- (7.613,8.956) -- (7.632,8.837) -- (7.687,8.729) -- (7.772,8.644) -- (7.879,8.59) -- (7.998,8.571) -- cycle;
\node at (7.998,8.956) {\scalebox{1.594}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.467,7.674) -- (6.586,7.693) -- (6.693,7.747) -- (6.779,7.832) -- (6.833,7.94) -- (6.852,8.058) -- (6.833,8.177) -- (6.779,8.284) -- (6.693,8.37) -- (6.586,8.424) -- (6.467,8.443) -- (6.467,8.443) -- (6.349,8.424) -- (6.241,8.37) -- (6.156,8.284) -- (6.102,8.177) -- (6.083,8.058) -- (6.102,7.94) -- (6.156,7.832) -- (6.241,7.747) -- (6.349,7.693) -- (6.467,7.674) -- cycle;
\node at (6.467,8.058) {\scalebox{1.594}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.248,5.485) -- (5.366,5.504) -- (5.474,5.558) -- (5.559,5.643) -- (5.613,5.751) -- (5.632,5.869) -- (5.613,5.988) -- (5.559,6.095) -- (5.474,6.18) -- (5.366,6.235) -- (5.248,6.254) -- (5.248,6.254) -- (5.129,6.235) -- (5.022,6.18) -- (4.937,6.095) -- (4.882,5.988) -- (4.863,5.869) -- (4.882,5.751) -- (4.937,5.643) -- (5.022,5.558) -- (5.129,5.504) -- (5.248,5.485) -- cycle;
\node at (5.248,5.869) {\scalebox{1.594}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.395,7.492) -- (9.513,7.51) -- (9.621,7.565) -- (9.706,7.65) -- (9.76,7.757) -- (9.779,7.876) -- (9.76,7.995) -- (9.706,8.102) -- (9.621,8.187) -- (9.513,8.242) -- (9.395,8.261) -- (9.395,8.261) -- (9.276,8.242) -- (9.169,8.187) -- (9.084,8.102) -- (9.029,7.995) -- (9.01,7.876) -- (9.029,7.757) -- (9.084,7.65) -- (9.169,7.565) -- (9.276,7.51) -- (9.395,7.492) -- cycle;
\node at (9.395,7.876) {\scalebox{1.594}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.195,5.586) -- (8.313,5.605) -- (8.421,5.66) -- (8.506,5.745) -- (8.56,5.852) -- (8.579,5.971) -- (8.56,6.089) -- (8.506,6.197) -- (8.421,6.282) -- (8.313,6.336) -- (8.195,6.355) -- (8.195,6.355) -- (8.076,6.336) -- (7.969,6.282) -- (7.884,6.197) -- (7.829,6.089) -- (7.81,5.971) -- (7.829,5.852) -- (7.884,5.745) -- (7.969,5.66) -- (8.076,5.605) -- (8.195,5.586) -- cycle;
\node at (8.195,5.971) {\scalebox{1.594}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.567,7.55) -- (3.685,7.569) -- (3.793,7.624) -- (3.878,7.709) -- (3.932,7.816) -- (3.951,7.935) -- (3.932,8.053) -- (3.878,8.161) -- (3.793,8.246) -- (3.685,8.3) -- (3.567,8.319) -- (3.205,8.319) -- (3.086,8.3) -- (2.979,8.246) -- (2.894,8.161) -- (2.839,8.053) -- (2.82,7.935) -- (2.839,7.816) -- (2.894,7.709) -- (2.979,7.624) -- (3.086,7.569) -- (3.205,7.55) -- cycle;
\node at (3.386,7.935) {\scalebox{1.594}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.117,5.285) -- (2.236,5.304) -- (2.343,5.359) -- (2.428,5.444) -- (2.483,5.551) -- (2.502,5.67) -- (2.483,5.789) -- (2.428,5.896) -- (2.343,5.981) -- (2.236,6.036) -- (2.117,6.055) -- (1.755,6.055) -- (1.637,6.036) -- (1.529,5.981) -- (1.444,5.896) -- (1.39,5.789) -- (1.371,5.67) -- (1.39,5.551) -- (1.444,5.444) -- (1.529,5.359) -- (1.637,5.304) -- (1.755,5.285) -- cycle;
\node at (1.936,5.67) {\scalebox{1.594}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.329,3.547) -- (3.448,3.566) -- (3.555,3.621) -- (3.64,3.706) -- (3.695,3.813) -- (3.714,3.932) -- (3.695,4.051) -- (3.64,4.158) -- (3.555,4.243) -- (3.448,4.297) -- (3.329,4.316) -- (2.967,4.316) -- (2.848,4.297) -- (2.741,4.243) -- (2.656,4.158) -- (2.602,4.051) -- (2.583,3.932) -- (2.602,3.813) -- (2.656,3.706) -- (2.741,3.621) -- (2.848,3.566) -- (2.967,3.547) -- cycle;
\node at (3.148,3.932) {\scalebox{1.594}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.761,2.516) -- (1.879,2.535) -- (1.987,2.589) -- (2.072,2.674) -- (2.126,2.782) -- (2.145,2.9) -- (2.126,3.019) -- (2.072,3.126) -- (1.987,3.211) -- (1.879,3.266) -- (1.761,3.285) -- (1.399,3.285) -- (1.28,3.266) -- (1.173,3.211) -- (1.088,3.126) -- (1.033,3.019) -- (1.014,2.9) -- (1.033,2.782) -- (1.088,2.674) -- (1.173,2.589) -- (1.28,2.535) -- (1.399,2.516) -- cycle;
\node at (1.58,2.9) {\scalebox{1.594}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.685,4.348) -- (6.804,4.366) -- (6.911,4.421) -- (6.997,4.506) -- (7.051,4.613) -- (7.07,4.732) -- (7.051,4.851) -- (6.997,4.958) -- (6.911,5.043) -- (6.804,5.098) -- (6.685,5.117) -- (6.324,5.117) -- (6.205,5.098) -- (6.098,5.043) -- (6.012,4.958) -- (5.958,4.851) -- (5.939,4.732) -- (5.958,4.613) -- (6.012,4.506) -- (6.098,4.421) -- (6.205,4.366) -- (6.324,4.348) -- cycle;
\node at (6.505,4.732) {\scalebox{1.594}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.32,2.052) -- (4.438,2.071) -- (4.546,2.125) -- (4.631,2.21) -- (4.685,2.317) -- (4.704,2.436) -- (4.685,2.555) -- (4.631,2.662) -- (4.546,2.747) -- (4.438,2.802) -- (4.32,2.821) -- (3.958,2.821) -- (3.839,2.802) -- (3.732,2.747) -- (3.647,2.662) -- (3.592,2.555) -- (3.573,2.436) -- (3.592,2.317) -- (3.647,2.21) -- (3.732,2.125) -- (3.839,2.071) -- (3.958,2.052) -- cycle;
\node at (4.139,2.436) {\scalebox{1.594}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.615,0.88) -- (3.734,0.899) -- (3.841,0.953) -- (3.926,1.038) -- (3.981,1.145) -- (3.999,1.264) -- (3.981,1.383) -- (3.926,1.49) -- (3.841,1.575) -- (3.734,1.63) -- (3.615,1.649) -- (3.253,1.649) -- (3.134,1.63) -- (3.027,1.575) -- (2.942,1.49) -- (2.887,1.383) -- (2.868,1.264) -- (2.887,1.145) -- (2.942,1.038) -- (3.027,0.953) -- (3.134,0.899) -- (3.253,0.88) -- cycle;
\node at (3.434,1.264) {\scalebox{1.594}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.968,4.187) -- (8.568,4.187) -- (8.568,4.787) -- (7.968,4.787) -- cycle;
\node at (8.268,4.487) {\scalebox{1.594}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.955,4.37) -- (2.555,4.37) -- (2.555,4.97) -- (1.955,4.97) -- cycle;
\node at (2.255,4.67) {\scalebox{1.594}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.066,7.911) -- (2.666,7.911) -- (2.666,8.511) -- (2.066,8.511) -- cycle;
\node at (2.366,8.211) {\scalebox{1.594}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.746,1.438) -- (2.346,1.438) -- (2.346,2.038) -- (1.746,2.038) -- cycle;
\node at (2.046,1.738) {\scalebox{1.594}{\small d}};
\node at (11.213,11.265) {\scalebox{1.594}{\small A}};
\node at (5.069,11.258) {\scalebox{1.594}{\small B}};
\node at (8.167,11.176) {\scalebox{1.594}{\small C}};
\node at (4.853,8.375) {\scalebox{1.594}{\small D}};
\node at (11.799,7.455) {\scalebox{1.594}{\small E}};
\node at (8.162,7.611) {\scalebox{1.594}{\small F}};
\node at (1.268,7.118) {\scalebox{1.594}{\small G}};
\node at (8.749,4.096) {\scalebox{1.594}{\small H}};
\node at (0.888,4.136) {\scalebox{1.594}{\small I}};
\node at (4.694,3.36) {\scalebox{1.594}{\small J}};
\node at (4.553,0.95) {\scalebox{1.594}{\small K}};
\node at (0.765,0.994) {\scalebox{1.594}{\small L}};
\draw[line width=0.5pt] (12.487,9.901) -- (12.487,11.001);
\draw[line width=0.5pt] (12.256,10.616) -- (12.487,11.001) -- (12.718,10.616);
\draw[line width=0.5pt] (12.185,10.451) -- (12.79,10.451);
\node at (12.487,11.386) {\scalebox{1.594}{\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, B, F, G, J, L & 6 \\ \hline
clairs & C, D, E, H, I, K & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{6} 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 12 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 l'\textbf{avenue de la Gare ⑨} vers le nord jusqu'à Rond-point du Stade \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ⑧}, vers l'est, jusqu'à Place du Marché \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Pont ①}, vers le nord, jusqu'à Place des Tanneurs \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Haute ④}, vers l'ouest, jusqu'à Carrefour de l'Hôpital \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Écoles ②}, vers l'ouest, jusqu'à Fontaine Notre-Dame \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Four ③}, vers le sud, jusqu'à Carrefour du Moulin \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Peupliers ⑩}, vers l'ouest, jusqu'à Place de la Fontaine \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Moulin ⑪}, vers le sud, jusqu'à Rond-point de la Gare \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin de Ronde ⑬}, vers le sud, jusqu'à Place aux Herbes \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin des Vignes ⑯}, vers l'est, jusqu'à Placette des Ormes \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de la Mairie ⑮}, vers le nord, jusqu'à Carrefour de la Mairie \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Puits ⑭}, vers l'est, jusqu'à Place de l'Église \textbf{(H)}.\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}
