\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 44}

\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=44}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0044}
\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=0.6073, line join=round, line cap=round]
\draw[line width=0.5pt] (4.875,11.562) -- (5.082,11.518) -- (5.291,11.479) -- (5.501,11.444) -- (5.711,11.415) -- (5.922,11.39) -- (6.134,11.37) -- (6.346,11.356) -- (6.559,11.346) -- (6.771,11.341) -- (6.984,11.342) -- (7.196,11.347) -- (7.408,11.357) -- (7.62,11.373) -- (7.832,11.393);
\draw[line width=0.5pt] (4.875,11.562) -- (4.898,11.31) -- (4.915,11.056) -- (4.926,10.802) -- (4.931,10.548) -- (4.93,10.294) -- (4.923,10.04) -- (4.91,9.786) -- (4.891,9.533) -- (4.866,9.28) -- (4.836,9.027) -- (4.799,8.776) -- (4.757,8.525) -- (4.708,8.276) -- (4.654,8.028);
\draw[line width=0.5pt] (7.832,11.393) -- (8.074,11.336) -- (8.319,11.284) -- (8.564,11.239) -- (8.81,11.199) -- (9.058,11.165) -- (9.306,11.136) -- (9.554,11.114) -- (9.803,11.097) -- (10.053,11.086) -- (10.303,11.082) -- (10.552,11.083) -- (10.802,11.089) -- (11.051,11.102) -- (11.299,11.121);
\draw[line width=0.5pt] (7.832,11.393) -- (7.89,11.19) -- (7.944,10.986) -- (7.993,10.78) -- (8.037,10.573) -- (8.077,10.366) -- (8.111,10.157) -- (8.14,9.948) -- (8.165,9.738) -- (8.185,9.527) -- (8.199,9.316) -- (8.209,9.105) -- (8.214,8.894) -- (8.213,8.683) -- (8.208,8.471);
\draw[line width=0.5pt] (11.299,11.121) -- (11.265,10.939) -- (11.235,10.756) -- (11.209,10.572) -- (11.187,10.387) -- (11.17,10.203) -- (11.157,10.017) -- (11.149,9.832) -- (11.145,9.646) -- (11.145,9.461) -- (11.149,9.275) -- (11.158,9.09) -- (11.172,8.905) -- (11.189,8.72) -- (11.211,8.536);
\draw[line width=0.5pt] (11.211,8.536) -- (11.241,8.282) -- (11.264,8.028) -- (11.281,7.773) -- (11.293,7.517) -- (11.298,7.262) -- (11.297,7.006) -- (11.29,6.751) -- (11.277,6.495) -- (11.259,6.24) -- (11.234,5.986) -- (11.203,5.732) -- (11.167,5.48) -- (11.124,5.228) -- (11.075,4.977);
\draw[line width=0.5pt] (8.208,8.471) -- (8.422,8.443) -- (8.636,8.42) -- (8.85,8.402) -- (9.066,8.389) -- (9.281,8.381) -- (9.497,8.378) -- (9.712,8.38) -- (9.928,8.387) -- (10.143,8.399) -- (10.358,8.416) -- (10.572,8.439) -- (10.786,8.466) -- (10.999,8.498) -- (11.211,8.536);
\draw[line width=0.5pt] (8.208,8.471) -- (8.242,8.205) -- (8.271,7.939) -- (8.292,7.671) -- (8.308,7.403) -- (8.317,7.135) -- (8.32,6.867) -- (8.316,6.598) -- (8.307,6.33) -- (8.291,6.062) -- (8.268,5.794) -- (8.24,5.527) -- (8.205,5.261) -- (8.164,4.996) -- (8.116,4.732);
\draw[line width=0.5pt] (4.654,8.028) -- (4.91,8.02) -- (5.167,8.019) -- (5.424,8.024) -- (5.681,8.035) -- (5.938,8.051) -- (6.194,8.074) -- (6.449,8.103) -- (6.704,8.138) -- (6.958,8.179) -- (7.211,8.225) -- (7.462,8.278) -- (7.712,8.337) -- (7.961,8.401) -- (8.208,8.471);
\draw[line width=0.5pt] (1.365,8.024) -- (1.598,8.061) -- (1.832,8.091) -- (2.067,8.116) -- (2.302,8.136) -- (2.537,8.15) -- (2.773,8.159) -- (3.009,8.162) -- (3.245,8.159) -- (3.481,8.151) -- (3.717,8.138) -- (3.952,8.118) -- (4.187,8.094) -- (4.421,8.064) -- (4.654,8.028);
\draw[line width=0.5pt] (11.075,4.977) -- (11.005,4.756) -- (10.94,4.533) -- (10.88,4.308) -- (10.826,4.082) -- (10.777,3.855) -- (10.733,3.627) -- (10.695,3.397) -- (10.662,3.167) -- (10.634,2.936) -- (10.612,2.705) -- (10.595,2.473) -- (10.584,2.241) -- (10.578,2.009) -- (10.578,1.777);
\draw[line width=0.5pt] (8.116,4.732) -- (8.323,4.782) -- (8.532,4.827) -- (8.741,4.867) -- (8.951,4.902) -- (9.162,4.932) -- (9.374,4.957) -- (9.586,4.977) -- (9.798,4.992) -- (10.011,5.002) -- (10.224,5.007) -- (10.437,5.007) -- (10.65,5.002) -- (10.863,4.992) -- (11.075,4.977);
\draw[line width=0.5pt] (8.116,4.732) -- (8.072,4.509) -- (8.033,4.285) -- (7.999,4.06) -- (7.971,3.834) -- (7.947,3.608) -- (7.929,3.381) -- (7.917,3.153) -- (7.91,2.925) -- (7.908,2.698) -- (7.911,2.47) -- (7.92,2.242) -- (7.935,2.015) -- (7.954,1.789) -- (7.979,1.562);
\draw[line width=0.5pt] (4.985,2.455) -- (5.207,2.424) -- (5.428,2.389) -- (5.648,2.348) -- (5.868,2.302) -- (6.086,2.25) -- (6.303,2.194) -- (6.519,2.132) -- (6.733,2.066) -- (6.945,1.994) -- (7.156,1.917) -- (7.365,1.836) -- (7.572,1.75) -- (7.777,1.658) -- (7.979,1.562);
\draw[line width=0.5pt] (1.389,2.015) -- (1.649,2.007) -- (1.909,2.005) -- (2.168,2.009) -- (2.428,2.019) -- (2.688,2.035) -- (2.947,2.058) -- (3.205,2.087) -- (3.463,2.121) -- (3.72,2.162) -- (3.975,2.208) -- (4.23,2.261) -- (4.483,2.32) -- (4.735,2.384) -- (4.985,2.455);
\draw[line width=0.5pt] (7.979,1.562) -- (8.161,1.606) -- (8.344,1.645) -- (8.528,1.68) -- (8.712,1.711) -- (8.898,1.737) -- (9.083,1.759) -- (9.27,1.777) -- (9.457,1.79) -- (9.643,1.799) -- (9.831,1.803) -- (10.018,1.803) -- (10.205,1.799) -- (10.392,1.79) -- (10.578,1.777);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.035,11.562) -- (5.013,11.642) -- (4.955,11.701) -- (4.875,11.722) -- (4.795,11.701) -- (4.736,11.642) -- (4.715,11.562) -- (4.736,11.482) -- (4.795,11.424) -- (4.875,11.402) -- (4.955,11.424) -- (5.013,11.482) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.992,11.393) -- (7.97,11.473) -- (7.912,11.531) -- (7.832,11.553) -- (7.752,11.531) -- (7.693,11.473) -- (7.672,11.393) -- (7.693,11.313) -- (7.752,11.254) -- (7.832,11.233) -- (7.912,11.254) -- (7.97,11.313) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.459,11.121) -- (11.438,11.201) -- (11.379,11.259) -- (11.299,11.281) -- (11.219,11.259) -- (11.161,11.201) -- (11.139,11.121) -- (11.161,11.041) -- (11.219,10.982) -- (11.299,10.961) -- (11.379,10.982) -- (11.438,11.041) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.371,8.536) -- (11.35,8.616) -- (11.291,8.674) -- (11.211,8.696) -- (11.131,8.674) -- (11.073,8.616) -- (11.051,8.536) -- (11.073,8.456) -- (11.131,8.397) -- (11.211,8.376) -- (11.291,8.397) -- (11.35,8.456) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.368,8.471) -- (8.347,8.551) -- (8.288,8.61) -- (8.208,8.631) -- (8.128,8.61) -- (8.069,8.551) -- (8.048,8.471) -- (8.069,8.391) -- (8.128,8.333) -- (8.208,8.311) -- (8.288,8.333) -- (8.347,8.391) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.814,8.028) -- (4.792,8.108) -- (4.734,8.166) -- (4.654,8.188) -- (4.574,8.166) -- (4.515,8.108) -- (4.494,8.028) -- (4.515,7.948) -- (4.574,7.889) -- (4.654,7.868) -- (4.734,7.889) -- (4.792,7.948) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.525,8.024) -- (1.504,8.104) -- (1.445,8.163) -- (1.365,8.184) -- (1.285,8.163) -- (1.227,8.104) -- (1.205,8.024) -- (1.227,7.944) -- (1.285,7.886) -- (1.365,7.864) -- (1.445,7.886) -- (1.504,7.944) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.235,4.977) -- (11.214,5.057) -- (11.155,5.116) -- (11.075,5.137) -- (10.995,5.116) -- (10.937,5.057) -- (10.915,4.977) -- (10.937,4.897) -- (10.995,4.838) -- (11.075,4.817) -- (11.155,4.838) -- (11.214,4.897) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.276,4.732) -- (8.255,4.812) -- (8.196,4.871) -- (8.116,4.892) -- (8.036,4.871) -- (7.978,4.812) -- (7.956,4.732) -- (7.978,4.652) -- (8.036,4.594) -- (8.116,4.572) -- (8.196,4.594) -- (8.255,4.652) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.145,2.455) -- (5.123,2.535) -- (5.065,2.593) -- (4.985,2.615) -- (4.905,2.593) -- (4.846,2.535) -- (4.825,2.455) -- (4.846,2.375) -- (4.905,2.316) -- (4.985,2.295) -- (5.065,2.316) -- (5.123,2.375) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.549,2.015) -- (1.528,2.095) -- (1.469,2.153) -- (1.389,2.175) -- (1.309,2.153) -- (1.251,2.095) -- (1.229,2.015) -- (1.251,1.935) -- (1.309,1.876) -- (1.389,1.855) -- (1.469,1.876) -- (1.528,1.935) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.738,1.777) -- (10.717,1.857) -- (10.658,1.915) -- (10.578,1.937) -- (10.498,1.915) -- (10.44,1.857) -- (10.418,1.777) -- (10.44,1.697) -- (10.498,1.638) -- (10.578,1.617) -- (10.658,1.638) -- (10.717,1.697) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.139,1.562) -- (8.118,1.642) -- (8.059,1.701) -- (7.979,1.722) -- (7.899,1.701) -- (7.841,1.642) -- (7.819,1.562) -- (7.841,1.482) -- (7.899,1.424) -- (7.979,1.402) -- (8.059,1.424) -- (8.118,1.482) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.369,11.355) -- (6.491,11.374) -- (6.6,11.43) -- (6.687,11.517) -- (6.743,11.627) -- (6.763,11.749) -- (6.743,11.871) -- (6.687,11.98) -- (6.6,12.068) -- (6.491,12.124) -- (6.369,12.143) -- (6.369,12.143) -- (6.247,12.124) -- (6.137,12.068) -- (6.05,11.98) -- (5.994,11.871) -- (5.975,11.749) -- (5.994,11.627) -- (6.05,11.517) -- (6.137,11.43) -- (6.247,11.374) -- (6.369,11.355) -- cycle;
\node at (6.369,11.749) {\scalebox{1.647}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.303,9.368) -- (5.425,9.387) -- (5.535,9.443) -- (5.622,9.53) -- (5.678,9.64) -- (5.697,9.761) -- (5.678,9.883) -- (5.622,9.993) -- (5.535,10.08) -- (5.425,10.136) -- (5.303,10.155) -- (5.303,10.155) -- (5.181,10.136) -- (5.072,10.08) -- (4.985,9.993) -- (4.929,9.883) -- (4.909,9.761) -- (4.929,9.64) -- (4.985,9.53) -- (5.072,9.443) -- (5.181,9.387) -- (5.303,9.368) -- cycle;
\node at (5.303,9.761) {\scalebox{1.647}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.585,11.113) -- (9.707,11.132) -- (9.817,11.188) -- (9.904,11.275) -- (9.96,11.385) -- (9.979,11.507) -- (9.96,11.628) -- (9.904,11.738) -- (9.817,11.825) -- (9.707,11.881) -- (9.585,11.9) -- (9.585,11.9) -- (9.463,11.881) -- (9.354,11.825) -- (9.266,11.738) -- (9.211,11.628) -- (9.191,11.507) -- (9.211,11.385) -- (9.266,11.275) -- (9.354,11.188) -- (9.463,11.132) -- (9.585,11.113) -- cycle;
\node at (9.585,11.507) {\scalebox{1.647}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.531,9.604) -- (8.653,9.623) -- (8.763,9.679) -- (8.85,9.766) -- (8.906,9.876) -- (8.925,9.998) -- (8.906,10.12) -- (8.85,10.23) -- (8.763,10.317) -- (8.653,10.373) -- (8.531,10.392) -- (8.531,10.392) -- (8.409,10.373) -- (8.3,10.317) -- (8.212,10.23) -- (8.156,10.12) -- (8.137,9.998) -- (8.156,9.876) -- (8.212,9.766) -- (8.3,9.679) -- (8.409,9.623) -- (8.531,9.604) -- cycle;
\node at (8.531,9.998) {\scalebox{1.647}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.542,9.425) -- (11.664,9.444) -- (11.774,9.5) -- (11.861,9.587) -- (11.917,9.697) -- (11.936,9.818) -- (11.917,9.94) -- (11.861,10.05) -- (11.774,10.137) -- (11.664,10.193) -- (11.542,10.212) -- (11.542,10.212) -- (11.421,10.193) -- (11.311,10.137) -- (11.224,10.05) -- (11.168,9.94) -- (11.149,9.818) -- (11.168,9.697) -- (11.224,9.587) -- (11.311,9.5) -- (11.421,9.444) -- (11.542,9.425) -- cycle;
\node at (11.542,9.818) {\scalebox{1.647}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.684,6.342) -- (11.806,6.361) -- (11.915,6.417) -- (12.003,6.504) -- (12.058,6.614) -- (12.078,6.736) -- (12.058,6.857) -- (12.003,6.967) -- (11.915,7.054) -- (11.806,7.11) -- (11.684,7.13) -- (11.684,7.13) -- (11.562,7.11) -- (11.452,7.054) -- (11.365,6.967) -- (11.309,6.857) -- (11.29,6.736) -- (11.309,6.614) -- (11.365,6.504) -- (11.452,6.417) -- (11.562,6.361) -- (11.684,6.342) -- cycle;
\node at (11.684,6.736) {\scalebox{1.647}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.704,8.38) -- (9.826,8.399) -- (9.935,8.455) -- (10.023,8.542) -- (10.079,8.652) -- (10.098,8.773) -- (10.079,8.895) -- (10.023,9.005) -- (9.935,9.092) -- (9.826,9.148) -- (9.704,9.167) -- (9.704,9.167) -- (9.582,9.148) -- (9.472,9.092) -- (9.385,9.005) -- (9.329,8.895) -- (9.31,8.773) -- (9.329,8.652) -- (9.385,8.542) -- (9.472,8.455) -- (9.582,8.399) -- (9.704,8.38) -- cycle;
\node at (9.704,8.773) {\scalebox{1.647}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.71,6.195) -- (8.832,6.214) -- (8.942,6.27) -- (9.029,6.357) -- (9.085,6.467) -- (9.104,6.588) -- (9.085,6.71) -- (9.029,6.82) -- (8.942,6.907) -- (8.832,6.963) -- (8.71,6.982) -- (8.71,6.982) -- (8.588,6.963) -- (8.479,6.907) -- (8.392,6.82) -- (8.336,6.71) -- (8.316,6.588) -- (8.336,6.467) -- (8.392,6.357) -- (8.479,6.27) -- (8.588,6.214) -- (8.71,6.195) -- cycle;
\node at (8.71,6.588) {\scalebox{1.647}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.498,7.318) -- (6.62,7.338) -- (6.729,7.393) -- (6.817,7.481) -- (6.873,7.59) -- (6.892,7.712) -- (6.873,7.834) -- (6.817,7.944) -- (6.729,8.031) -- (6.62,8.087) -- (6.498,8.106) -- (6.498,8.106) -- (6.376,8.087) -- (6.266,8.031) -- (6.179,7.944) -- (6.123,7.834) -- (6.104,7.712) -- (6.123,7.59) -- (6.179,7.481) -- (6.266,7.393) -- (6.376,7.338) -- (6.498,7.318) -- cycle;
\node at (6.498,7.712) {\scalebox{1.647}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.194,8.162) -- (3.316,8.181) -- (3.426,8.237) -- (3.513,8.324) -- (3.569,8.434) -- (3.588,8.556) -- (3.569,8.677) -- (3.513,8.787) -- (3.426,8.874) -- (3.316,8.93) -- (3.194,8.95) -- (2.824,8.95) -- (2.702,8.93) -- (2.592,8.874) -- (2.505,8.787) -- (2.449,8.677) -- (2.43,8.556) -- (2.449,8.434) -- (2.505,8.324) -- (2.592,8.237) -- (2.702,8.181) -- (2.824,8.162) -- cycle;
\node at (3.009,8.556) {\scalebox{1.647}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.269,2.943) -- (11.391,2.962) -- (11.501,3.018) -- (11.588,3.105) -- (11.644,3.215) -- (11.663,3.337) -- (11.644,3.459) -- (11.588,3.568) -- (11.501,3.656) -- (11.391,3.711) -- (11.269,3.731) -- (10.899,3.731) -- (10.777,3.711) -- (10.667,3.656) -- (10.58,3.568) -- (10.524,3.459) -- (10.505,3.337) -- (10.524,3.215) -- (10.58,3.105) -- (10.667,3.018) -- (10.777,2.962) -- (10.899,2.943) -- cycle;
\node at (11.084,3.337) {\scalebox{1.647}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.739,4.975) -- (9.86,4.995) -- (9.97,5.051) -- (10.057,5.138) -- (10.113,5.248) -- (10.133,5.369) -- (10.113,5.491) -- (10.057,5.601) -- (9.97,5.688) -- (9.86,5.744) -- (9.739,5.763) -- (9.368,5.763) -- (9.246,5.744) -- (9.136,5.688) -- (9.049,5.601) -- (8.993,5.491) -- (8.974,5.369) -- (8.993,5.248) -- (9.049,5.138) -- (9.136,5.051) -- (9.246,4.995) -- (9.368,4.975) -- cycle;
\node at (9.553,5.369) {\scalebox{1.647}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.699,2.302) -- (7.821,2.322) -- (7.931,2.378) -- (8.018,2.465) -- (8.074,2.575) -- (8.093,2.696) -- (8.074,2.818) -- (8.018,2.928) -- (7.931,3.015) -- (7.821,3.071) -- (7.699,3.09) -- (7.329,3.09) -- (7.207,3.071) -- (7.097,3.015) -- (7.01,2.928) -- (6.954,2.818) -- (6.935,2.696) -- (6.954,2.575) -- (7.01,2.465) -- (7.097,2.378) -- (7.207,2.322) -- (7.329,2.302) -- cycle;
\node at (7.514,2.696) {\scalebox{1.647}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.174,2.45) -- (6.296,2.47) -- (6.405,2.526) -- (6.493,2.613) -- (6.549,2.723) -- (6.568,2.844) -- (6.549,2.966) -- (6.493,3.076) -- (6.405,3.163) -- (6.296,3.219) -- (6.174,3.238) -- (5.803,3.238) -- (5.681,3.219) -- (5.572,3.163) -- (5.485,3.076) -- (5.429,2.966) -- (5.409,2.844) -- (5.429,2.723) -- (5.485,2.613) -- (5.572,2.526) -- (5.681,2.47) -- (5.803,2.45) -- cycle;
\node at (5.989,2.844) {\scalebox{1.647}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.343,2.084) -- (3.464,2.103) -- (3.574,2.159) -- (3.661,2.246) -- (3.717,2.356) -- (3.736,2.478) -- (3.717,2.599) -- (3.661,2.709) -- (3.574,2.796) -- (3.464,2.852) -- (3.343,2.871) -- (2.972,2.871) -- (2.85,2.852) -- (2.74,2.796) -- (2.653,2.709) -- (2.597,2.599) -- (2.578,2.478) -- (2.597,2.356) -- (2.653,2.246) -- (2.74,2.159) -- (2.85,2.103) -- (2.972,2.084) -- cycle;
\node at (3.157,2.478) {\scalebox{1.647}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.423,1.775) -- (9.545,1.795) -- (9.654,1.851) -- (9.741,1.938) -- (9.797,2.048) -- (9.817,2.169) -- (9.797,2.291) -- (9.741,2.401) -- (9.654,2.488) -- (9.545,2.544) -- (9.423,2.563) -- (9.052,2.563) -- (8.93,2.544) -- (8.821,2.488) -- (8.733,2.401) -- (8.677,2.291) -- (8.658,2.169) -- (8.677,2.048) -- (8.733,1.938) -- (8.821,1.851) -- (8.93,1.795) -- (9.052,1.775) -- cycle;
\node at (9.237,2.169) {\scalebox{1.647}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.816,4.432) -- (8.416,4.432) -- (8.416,5.032) -- (7.816,5.032) -- cycle;
\node at (8.116,4.732) {\scalebox{1.647}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.595,8.3) -- (6.195,8.3) -- (6.195,8.9) -- (5.595,8.9) -- cycle;
\node at (5.895,8.6) {\scalebox{1.647}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.195,3.258) -- (8.795,3.258) -- (8.795,3.858) -- (8.195,3.858) -- cycle;
\node at (8.495,3.558) {\scalebox{1.647}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.639,1.42) -- (6.239,1.42) -- (6.239,2.02) -- (5.639,2.02) -- cycle;
\node at (5.939,1.72) {\scalebox{1.647}{\small d}};
\node at (4.616,12.126) {\scalebox{1.647}{\small A}};
\node at (7.912,12.008) {\scalebox{1.647}{\small B}};
\node at (11.712,11.584) {\scalebox{1.647}{\small C}};
\node at (11.765,8.815) {\scalebox{1.647}{\small D}};
\node at (8.492,9.022) {\scalebox{1.647}{\small E}};
\node at (4.121,8.345) {\scalebox{1.647}{\small F}};
\node at (0.765,8.18) {\scalebox{1.647}{\small G}};
\node at (11.652,4.749) {\scalebox{1.647}{\small H}};
\node at (8.401,4.182) {\scalebox{1.647}{\small I}};
\node at (4.688,1.911) {\scalebox{1.647}{\small J}};
\node at (0.899,1.635) {\scalebox{1.647}{\small K}};
\node at (10.938,1.272) {\scalebox{1.647}{\small L}};
\node at (8.076,0.95) {\scalebox{1.647}{\small M}};
\draw[line width=0.5pt] (12.599,10.612) -- (12.599,11.712);
\draw[line width=0.5pt] (12.368,11.327) -- (12.599,11.712) -- (12.83,11.327);
\draw[line width=0.5pt] (12.297,11.162) -- (12.902,11.162);
\node at (12.599,12.097) {\scalebox{1.647}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour I, 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 de la Fontaine & Ⓙ Place des Tanneurs \\ \hline
① rue du Four & ⑩ impasse des Cerisiers & Ⓑ Carrefour du Lavoir & Ⓚ Coin du Lavoir \\ \hline
② allée des Tilleuls & ⑪ rue de la Poste & Ⓒ Place de l'Église & Ⓛ Place aux Herbes \\ \hline
③ chemin des Vignes & ⑫ rue Basse & Ⓓ Carrefour de la Mairie & Ⓜ Carrefour de l'Hôpital \\ \hline
④ boulevard du Parc & ⑬ rue de la Mairie & Ⓔ Rond-point des Vaches & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ chemin du Lavoir & ⑭ rue Verte & Ⓕ Carrefour des Tilleuls & \textbf{a} l'école \\ \hline
⑥ rue du Puits & ⑮ impasse du Puits & Ⓖ Place du Marché & \textbf{b} le cimetière \\ \hline
⑦ chemin de Ronde & ⑯ rue du Stade & Ⓗ Rond-point de la Poste & \textbf{c} la gendarmerie \\ \hline
⑧ chemin Creux & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour du Moulin & \textbf{d} le stade \\ \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=0.6221, line join=round, line cap=round]
\draw[line width=0.5pt] (4.875,11.562) -- (5.082,11.518) -- (5.291,11.479) -- (5.501,11.444) -- (5.711,11.415) -- (5.922,11.39) -- (6.134,11.37) -- (6.346,11.356) -- (6.559,11.346) -- (6.771,11.341) -- (6.984,11.342) -- (7.196,11.347) -- (7.408,11.357) -- (7.62,11.373) -- (7.832,11.393);
\draw[line width=0.5pt] (4.875,11.562) -- (4.898,11.31) -- (4.915,11.056) -- (4.926,10.802) -- (4.931,10.548) -- (4.93,10.294) -- (4.923,10.04) -- (4.91,9.786) -- (4.891,9.533) -- (4.866,9.28) -- (4.836,9.027) -- (4.799,8.776) -- (4.757,8.525) -- (4.708,8.276) -- (4.654,8.028);
\draw[line width=0.5pt] (7.832,11.393) -- (8.074,11.336) -- (8.319,11.284) -- (8.564,11.239) -- (8.81,11.199) -- (9.058,11.165) -- (9.306,11.136) -- (9.554,11.114) -- (9.803,11.097) -- (10.053,11.086) -- (10.303,11.082) -- (10.552,11.083) -- (10.802,11.089) -- (11.051,11.102) -- (11.299,11.121);
\draw[line width=0.5pt] (7.832,11.393) -- (7.89,11.19) -- (7.944,10.986) -- (7.993,10.78) -- (8.037,10.573) -- (8.077,10.366) -- (8.111,10.157) -- (8.14,9.948) -- (8.165,9.738) -- (8.185,9.527) -- (8.199,9.316) -- (8.209,9.105) -- (8.214,8.894) -- (8.213,8.683) -- (8.208,8.471);
\draw[line width=0.5pt] (11.299,11.121) -- (11.265,10.939) -- (11.235,10.756) -- (11.209,10.572) -- (11.187,10.387) -- (11.17,10.203) -- (11.157,10.017) -- (11.149,9.832) -- (11.145,9.646) -- (11.145,9.461) -- (11.149,9.275) -- (11.158,9.09) -- (11.172,8.905) -- (11.189,8.72) -- (11.211,8.536);
\draw[line width=0.5pt] (11.211,8.536) -- (11.241,8.282) -- (11.264,8.028) -- (11.281,7.773) -- (11.293,7.517) -- (11.298,7.262) -- (11.297,7.006) -- (11.29,6.751) -- (11.277,6.495) -- (11.259,6.24) -- (11.234,5.986) -- (11.203,5.732) -- (11.167,5.48) -- (11.124,5.228) -- (11.075,4.977);
\draw[line width=0.5pt] (8.208,8.471) -- (8.422,8.443) -- (8.636,8.42) -- (8.85,8.402) -- (9.066,8.389) -- (9.281,8.381) -- (9.497,8.378) -- (9.712,8.38) -- (9.928,8.387) -- (10.143,8.399) -- (10.358,8.416) -- (10.572,8.439) -- (10.786,8.466) -- (10.999,8.498) -- (11.211,8.536);
\draw[line width=0.5pt] (8.208,8.471) -- (8.242,8.205) -- (8.271,7.939) -- (8.292,7.671) -- (8.308,7.403) -- (8.317,7.135) -- (8.32,6.867) -- (8.316,6.598) -- (8.307,6.33) -- (8.291,6.062) -- (8.268,5.794) -- (8.24,5.527) -- (8.205,5.261) -- (8.164,4.996) -- (8.116,4.732);
\draw[line width=0.5pt] (4.654,8.028) -- (4.91,8.02) -- (5.167,8.019) -- (5.424,8.024) -- (5.681,8.035) -- (5.938,8.051) -- (6.194,8.074) -- (6.449,8.103) -- (6.704,8.138) -- (6.958,8.179) -- (7.211,8.225) -- (7.462,8.278) -- (7.712,8.337) -- (7.961,8.401) -- (8.208,8.471);
\draw[line width=0.5pt] (1.365,8.024) -- (1.598,8.061) -- (1.832,8.091) -- (2.067,8.116) -- (2.302,8.136) -- (2.537,8.15) -- (2.773,8.159) -- (3.009,8.162) -- (3.245,8.159) -- (3.481,8.151) -- (3.717,8.138) -- (3.952,8.118) -- (4.187,8.094) -- (4.421,8.064) -- (4.654,8.028);
\draw[line width=0.5pt] (11.075,4.977) -- (11.005,4.756) -- (10.94,4.533) -- (10.88,4.308) -- (10.826,4.082) -- (10.777,3.855) -- (10.733,3.627) -- (10.695,3.397) -- (10.662,3.167) -- (10.634,2.936) -- (10.612,2.705) -- (10.595,2.473) -- (10.584,2.241) -- (10.578,2.009) -- (10.578,1.777);
\draw[line width=0.5pt] (8.116,4.732) -- (8.323,4.782) -- (8.532,4.827) -- (8.741,4.867) -- (8.951,4.902) -- (9.162,4.932) -- (9.374,4.957) -- (9.586,4.977) -- (9.798,4.992) -- (10.011,5.002) -- (10.224,5.007) -- (10.437,5.007) -- (10.65,5.002) -- (10.863,4.992) -- (11.075,4.977);
\draw[line width=0.5pt] (8.116,4.732) -- (8.072,4.509) -- (8.033,4.285) -- (7.999,4.06) -- (7.971,3.834) -- (7.947,3.608) -- (7.929,3.381) -- (7.917,3.153) -- (7.91,2.925) -- (7.908,2.698) -- (7.911,2.47) -- (7.92,2.242) -- (7.935,2.015) -- (7.954,1.789) -- (7.979,1.562);
\draw[line width=0.5pt] (4.985,2.455) -- (5.207,2.424) -- (5.428,2.389) -- (5.648,2.348) -- (5.868,2.302) -- (6.086,2.25) -- (6.303,2.194) -- (6.519,2.132) -- (6.733,2.066) -- (6.945,1.994) -- (7.156,1.917) -- (7.365,1.836) -- (7.572,1.75) -- (7.777,1.658) -- (7.979,1.562);
\draw[line width=0.5pt] (1.389,2.015) -- (1.649,2.007) -- (1.909,2.005) -- (2.168,2.009) -- (2.428,2.019) -- (2.688,2.035) -- (2.947,2.058) -- (3.205,2.087) -- (3.463,2.121) -- (3.72,2.162) -- (3.975,2.208) -- (4.23,2.261) -- (4.483,2.32) -- (4.735,2.384) -- (4.985,2.455);
\draw[line width=0.5pt] (7.979,1.562) -- (8.161,1.606) -- (8.344,1.645) -- (8.528,1.68) -- (8.712,1.711) -- (8.898,1.737) -- (9.083,1.759) -- (9.27,1.777) -- (9.457,1.79) -- (9.643,1.799) -- (9.831,1.803) -- (10.018,1.803) -- (10.205,1.799) -- (10.392,1.79) -- (10.578,1.777);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.248,11.562) -- (5.22,11.705) -- (5.139,11.826) -- (5.018,11.907) -- (4.875,11.936) -- (4.732,11.907) -- (4.611,11.826) -- (4.53,11.705) -- (4.502,11.562) -- (4.53,11.42) -- (4.611,11.299) -- (4.732,11.218) -- (4.875,11.189) -- (5.018,11.218) -- (5.139,11.299) -- (5.22,11.42) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.205,11.393) -- (8.177,11.536) -- (8.096,11.657) -- (7.975,11.738) -- (7.832,11.766) -- (7.689,11.738) -- (7.568,11.657) -- (7.487,11.536) -- (7.459,11.393) -- (7.487,11.25) -- (7.568,11.129) -- (7.689,11.048) -- (7.832,11.02) -- (7.975,11.048) -- (8.096,11.129) -- (8.177,11.25) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.673,11.121) -- (11.644,11.264) -- (11.563,11.385) -- (11.442,11.466) -- (11.299,11.494) -- (11.157,11.466) -- (11.036,11.385) -- (10.955,11.264) -- (10.926,11.121) -- (10.955,10.978) -- (11.036,10.857) -- (11.157,10.776) -- (11.299,10.748) -- (11.442,10.776) -- (11.563,10.857) -- (11.644,10.978) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.585,8.536) -- (11.556,8.678) -- (11.475,8.8) -- (11.354,8.88) -- (11.211,8.909) -- (11.069,8.88) -- (10.947,8.8) -- (10.867,8.678) -- (10.838,8.536) -- (10.867,8.393) -- (10.947,8.272) -- (11.069,8.191) -- (11.211,8.162) -- (11.354,8.191) -- (11.475,8.272) -- (11.556,8.393) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.581,8.471) -- (8.553,8.614) -- (8.472,8.735) -- (8.351,8.816) -- (8.208,8.845) -- (8.065,8.816) -- (7.944,8.735) -- (7.863,8.614) -- (7.835,8.471) -- (7.863,8.329) -- (7.944,8.207) -- (8.065,8.127) -- (8.208,8.098) -- (8.351,8.127) -- (8.472,8.207) -- (8.553,8.329) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.027,8.028) -- (4.999,8.171) -- (4.918,8.292) -- (4.797,8.373) -- (4.654,8.401) -- (4.511,8.373) -- (4.39,8.292) -- (4.309,8.171) -- (4.281,8.028) -- (4.309,7.885) -- (4.39,7.764) -- (4.511,7.683) -- (4.654,7.655) -- (4.797,7.683) -- (4.918,7.764) -- (4.999,7.885) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.738,8.024) -- (1.71,8.167) -- (1.629,8.288) -- (1.508,8.369) -- (1.365,8.398) -- (1.222,8.369) -- (1.101,8.288) -- (1.02,8.167) -- (0.992,8.024) -- (1.02,7.881) -- (1.101,7.76) -- (1.222,7.679) -- (1.365,7.651) -- (1.508,7.679) -- (1.629,7.76) -- (1.71,7.881) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.449,4.977) -- (11.42,5.12) -- (11.339,5.241) -- (11.218,5.322) -- (11.075,5.35) -- (10.933,5.322) -- (10.812,5.241) -- (10.731,5.12) -- (10.702,4.977) -- (10.731,4.834) -- (10.812,4.713) -- (10.933,4.632) -- (11.075,4.604) -- (11.218,4.632) -- (11.339,4.713) -- (11.42,4.834) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.49,4.732) -- (8.461,4.875) -- (8.38,4.996) -- (8.259,5.077) -- (8.116,5.106) -- (7.974,5.077) -- (7.852,4.996) -- (7.772,4.875) -- (7.743,4.732) -- (7.772,4.59) -- (7.852,4.469) -- (7.974,4.388) -- (8.116,4.359) -- (8.259,4.388) -- (8.38,4.469) -- (8.461,4.59) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.358,2.455) -- (5.329,2.598) -- (5.249,2.719) -- (5.127,2.8) -- (4.985,2.828) -- (4.842,2.8) -- (4.721,2.719) -- (4.64,2.598) -- (4.611,2.455) -- (4.64,2.312) -- (4.721,2.191) -- (4.842,2.11) -- (4.985,2.082) -- (5.127,2.11) -- (5.249,2.191) -- (5.329,2.312) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.762,2.015) -- (1.734,2.158) -- (1.653,2.279) -- (1.532,2.36) -- (1.389,2.388) -- (1.246,2.36) -- (1.125,2.279) -- (1.044,2.158) -- (1.016,2.015) -- (1.044,1.872) -- (1.125,1.751) -- (1.246,1.67) -- (1.389,1.642) -- (1.532,1.67) -- (1.653,1.751) -- (1.734,1.872) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.951,1.777) -- (10.923,1.919) -- (10.842,2.04) -- (10.721,2.121) -- (10.578,2.15) -- (10.435,2.121) -- (10.314,2.04) -- (10.233,1.919) -- (10.205,1.777) -- (10.233,1.634) -- (10.314,1.513) -- (10.435,1.432) -- (10.578,1.403) -- (10.721,1.432) -- (10.842,1.513) -- (10.923,1.634) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.352,1.562) -- (8.324,1.705) -- (8.243,1.826) -- (8.122,1.907) -- (7.979,1.936) -- (7.836,1.907) -- (7.715,1.826) -- (7.634,1.705) -- (7.606,1.562) -- (7.634,1.42) -- (7.715,1.299) -- (7.836,1.218) -- (7.979,1.189) -- (8.122,1.218) -- (8.243,1.299) -- (8.324,1.42) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.368,11.355) -- (6.487,11.374) -- (6.594,11.428) -- (6.679,11.514) -- (6.734,11.621) -- (6.753,11.74) -- (6.734,11.858) -- (6.679,11.966) -- (6.594,12.051) -- (6.487,12.105) -- (6.368,12.124) -- (6.368,12.124) -- (6.249,12.105) -- (6.142,12.051) -- (6.057,11.966) -- (6.003,11.858) -- (5.984,11.74) -- (6.003,11.621) -- (6.057,11.514) -- (6.142,11.428) -- (6.249,11.374) -- (6.368,11.355) -- cycle;
\node at (6.368,11.74) {\scalebox{1.607}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.294,9.377) -- (5.413,9.396) -- (5.52,9.451) -- (5.605,9.536) -- (5.66,9.643) -- (5.678,9.762) -- (5.66,9.881) -- (5.605,9.988) -- (5.52,10.073) -- (5.413,10.128) -- (5.294,10.147) -- (5.294,10.147) -- (5.175,10.128) -- (5.068,10.073) -- (4.983,9.988) -- (4.928,9.881) -- (4.909,9.762) -- (4.928,9.643) -- (4.983,9.536) -- (5.068,9.451) -- (5.175,9.396) -- (5.294,9.377) -- cycle;
\node at (5.294,9.762) {\scalebox{1.607}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.584,11.113) -- (9.703,11.131) -- (9.81,11.186) -- (9.896,11.271) -- (9.95,11.378) -- (9.969,11.497) -- (9.95,11.616) -- (9.896,11.723) -- (9.81,11.808) -- (9.703,11.863) -- (9.584,11.882) -- (9.584,11.882) -- (9.466,11.863) -- (9.358,11.808) -- (9.273,11.723) -- (9.219,11.616) -- (9.2,11.497) -- (9.219,11.378) -- (9.273,11.271) -- (9.358,11.186) -- (9.466,11.131) -- (9.584,11.113) -- cycle;
\node at (9.584,11.497) {\scalebox{1.607}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.522,9.612) -- (8.641,9.631) -- (8.748,9.686) -- (8.833,9.771) -- (8.887,9.878) -- (8.906,9.997) -- (8.887,10.116) -- (8.833,10.223) -- (8.748,10.308) -- (8.641,10.362) -- (8.522,10.381) -- (8.522,10.381) -- (8.403,10.362) -- (8.296,10.308) -- (8.211,10.223) -- (8.156,10.116) -- (8.137,9.997) -- (8.156,9.878) -- (8.211,9.771) -- (8.296,9.686) -- (8.403,9.631) -- (8.522,9.612) -- cycle;
\node at (8.522,9.997) {\scalebox{1.607}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.533,9.434) -- (11.652,9.453) -- (11.759,9.508) -- (11.844,9.593) -- (11.899,9.7) -- (11.918,9.819) -- (11.899,9.938) -- (11.844,10.045) -- (11.759,10.13) -- (11.652,10.185) -- (11.533,10.203) -- (11.533,10.203) -- (11.414,10.185) -- (11.307,10.13) -- (11.222,10.045) -- (11.167,9.938) -- (11.149,9.819) -- (11.167,9.7) -- (11.222,9.593) -- (11.307,9.508) -- (11.414,9.453) -- (11.533,9.434) -- cycle;
\node at (11.533,9.819) {\scalebox{1.607}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.674,6.352) -- (11.793,6.37) -- (11.9,6.425) -- (11.986,6.51) -- (12.04,6.617) -- (12.059,6.736) -- (12.04,6.855) -- (11.986,6.962) -- (11.9,7.047) -- (11.793,7.102) -- (11.674,7.121) -- (11.674,7.121) -- (11.556,7.102) -- (11.448,7.047) -- (11.363,6.962) -- (11.309,6.855) -- (11.29,6.736) -- (11.309,6.617) -- (11.363,6.51) -- (11.448,6.425) -- (11.556,6.37) -- (11.674,6.352) -- cycle;
\node at (11.674,6.736) {\scalebox{1.607}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.704,8.38) -- (9.823,8.398) -- (9.93,8.453) -- (10.015,8.538) -- (10.07,8.645) -- (10.089,8.764) -- (10.07,8.883) -- (10.015,8.99) -- (9.93,9.075) -- (9.823,9.13) -- (9.704,9.149) -- (9.704,9.149) -- (9.585,9.13) -- (9.478,9.075) -- (9.393,8.99) -- (9.338,8.883) -- (9.32,8.764) -- (9.338,8.645) -- (9.393,8.538) -- (9.478,8.453) -- (9.585,8.398) -- (9.704,8.38) -- cycle;
\node at (9.704,8.764) {\scalebox{1.607}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.701,6.204) -- (8.82,6.223) -- (8.927,6.278) -- (9.012,6.363) -- (9.067,6.47) -- (9.085,6.589) -- (9.067,6.708) -- (9.012,6.815) -- (8.927,6.9) -- (8.82,6.954) -- (8.701,6.973) -- (8.701,6.973) -- (8.582,6.954) -- (8.475,6.9) -- (8.39,6.815) -- (8.335,6.708) -- (8.316,6.589) -- (8.335,6.47) -- (8.39,6.363) -- (8.475,6.278) -- (8.582,6.223) -- (8.701,6.204) -- cycle;
\node at (8.701,6.589) {\scalebox{1.607}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.497,7.337) -- (6.616,7.356) -- (6.723,7.41) -- (6.808,7.495) -- (6.862,7.603) -- (6.881,7.721) -- (6.862,7.84) -- (6.808,7.947) -- (6.723,8.033) -- (6.616,8.087) -- (6.497,8.106) -- (6.497,8.106) -- (6.378,8.087) -- (6.271,8.033) -- (6.186,7.947) -- (6.131,7.84) -- (6.112,7.721) -- (6.131,7.603) -- (6.186,7.495) -- (6.271,7.41) -- (6.378,7.356) -- (6.497,7.337) -- cycle;
\node at (6.497,7.721) {\scalebox{1.607}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.19,8.162) -- (3.309,8.181) -- (3.416,8.235) -- (3.501,8.32) -- (3.556,8.427) -- (3.574,8.546) -- (3.556,8.665) -- (3.501,8.772) -- (3.416,8.857) -- (3.309,8.912) -- (3.19,8.931) -- (2.828,8.931) -- (2.709,8.912) -- (2.602,8.857) -- (2.517,8.772) -- (2.462,8.665) -- (2.443,8.546) -- (2.462,8.427) -- (2.517,8.32) -- (2.602,8.235) -- (2.709,8.181) -- (2.828,8.162) -- cycle;
\node at (3.009,8.546) {\scalebox{1.607}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.256,2.954) -- (11.375,2.973) -- (11.482,3.027) -- (11.567,3.112) -- (11.621,3.219) -- (11.64,3.338) -- (11.621,3.457) -- (11.567,3.564) -- (11.482,3.649) -- (11.375,3.704) -- (11.256,3.723) -- (10.894,3.723) -- (10.775,3.704) -- (10.668,3.649) -- (10.583,3.564) -- (10.528,3.457) -- (10.509,3.338) -- (10.528,3.219) -- (10.583,3.112) -- (10.668,3.027) -- (10.775,2.973) -- (10.894,2.954) -- cycle;
\node at (11.075,3.338) {\scalebox{1.607}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.735,4.975) -- (9.854,4.994) -- (9.961,5.049) -- (10.046,5.134) -- (10.101,5.241) -- (10.12,5.36) -- (10.101,5.479) -- (10.046,5.586) -- (9.961,5.671) -- (9.854,5.726) -- (9.735,5.745) -- (9.373,5.745) -- (9.254,5.726) -- (9.147,5.671) -- (9.062,5.586) -- (9.008,5.479) -- (8.989,5.36) -- (9.008,5.241) -- (9.062,5.134) -- (9.147,5.049) -- (9.254,4.994) -- (9.373,4.975) -- cycle;
\node at (9.554,5.36) {\scalebox{1.607}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.704,2.312) -- (7.823,2.331) -- (7.93,2.385) -- (8.015,2.47) -- (8.07,2.577) -- (8.089,2.696) -- (8.07,2.815) -- (8.015,2.922) -- (7.93,3.007) -- (7.823,3.062) -- (7.704,3.081) -- (7.342,3.081) -- (7.224,3.062) -- (7.116,3.007) -- (7.031,2.922) -- (6.977,2.815) -- (6.958,2.696) -- (6.977,2.577) -- (7.031,2.47) -- (7.116,2.385) -- (7.224,2.331) -- (7.342,2.312) -- cycle;
\node at (7.523,2.696) {\scalebox{1.607}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.167,2.447) -- (6.285,2.466) -- (6.393,2.52) -- (6.478,2.605) -- (6.532,2.713) -- (6.551,2.831) -- (6.532,2.95) -- (6.478,3.057) -- (6.393,3.142) -- (6.285,3.197) -- (6.167,3.216) -- (5.805,3.216) -- (5.686,3.197) -- (5.579,3.142) -- (5.494,3.057) -- (5.439,2.95) -- (5.42,2.831) -- (5.439,2.713) -- (5.494,2.605) -- (5.579,2.52) -- (5.686,2.466) -- (5.805,2.447) -- cycle;
\node at (5.986,2.831) {\scalebox{1.607}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.339,2.084) -- (3.458,2.103) -- (3.565,2.157) -- (3.65,2.242) -- (3.705,2.349) -- (3.724,2.468) -- (3.705,2.587) -- (3.65,2.694) -- (3.565,2.779) -- (3.458,2.834) -- (3.339,2.853) -- (2.977,2.853) -- (2.859,2.834) -- (2.751,2.779) -- (2.666,2.694) -- (2.612,2.587) -- (2.593,2.468) -- (2.612,2.349) -- (2.666,2.242) -- (2.751,2.157) -- (2.859,2.103) -- (2.977,2.084) -- cycle;
\node at (3.158,2.468) {\scalebox{1.607}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.419,1.775) -- (9.538,1.794) -- (9.645,1.849) -- (9.73,1.934) -- (9.785,2.041) -- (9.804,2.16) -- (9.785,2.279) -- (9.73,2.386) -- (9.645,2.471) -- (9.538,2.526) -- (9.419,2.544) -- (9.057,2.544) -- (8.938,2.526) -- (8.831,2.471) -- (8.746,2.386) -- (8.692,2.279) -- (8.673,2.16) -- (8.692,2.041) -- (8.746,1.934) -- (8.831,1.849) -- (8.938,1.794) -- (9.057,1.775) -- cycle;
\node at (9.238,2.16) {\scalebox{1.607}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.816,4.432) -- (8.416,4.432) -- (8.416,5.032) -- (7.816,5.032) -- cycle;
\node at (8.116,4.732) {\scalebox{1.607}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.595,8.3) -- (6.195,8.3) -- (6.195,8.9) -- (5.595,8.9) -- cycle;
\node at (5.895,8.6) {\scalebox{1.607}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.195,3.258) -- (8.795,3.258) -- (8.795,3.858) -- (8.195,3.858) -- cycle;
\node at (8.495,3.558) {\scalebox{1.607}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.639,1.42) -- (6.239,1.42) -- (6.239,2.02) -- (5.639,2.02) -- cycle;
\node at (5.939,1.72) {\scalebox{1.607}{\small d}};
\node at (4.616,12.126) {\scalebox{1.607}{\small A}};
\node at (7.912,12.008) {\scalebox{1.607}{\small B}};
\node at (11.712,11.584) {\scalebox{1.607}{\small C}};
\node at (11.765,8.815) {\scalebox{1.607}{\small D}};
\node at (8.492,9.022) {\scalebox{1.607}{\small E}};
\node at (4.121,8.345) {\scalebox{1.607}{\small F}};
\node at (0.765,8.18) {\scalebox{1.607}{\small G}};
\node at (11.652,4.749) {\scalebox{1.607}{\small H}};
\node at (8.401,4.182) {\scalebox{1.607}{\small I}};
\node at (4.688,1.911) {\scalebox{1.607}{\small J}};
\node at (0.899,1.635) {\scalebox{1.607}{\small K}};
\node at (10.938,1.272) {\scalebox{1.607}{\small L}};
\node at (8.076,0.95) {\scalebox{1.607}{\small M}};
\draw[line width=0.5pt] (12.599,10.612) -- (12.599,11.712);
\draw[line width=0.5pt] (12.368,11.327) -- (12.599,11.712) -- (12.83,11.327);
\draw[line width=0.5pt] (12.297,11.162) -- (12.902,11.162);
\node at (12.599,12.097) {\scalebox{1.607}{\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, E, G, H, K, M & 7 \\ \hline
clairs & B, D, F, I, J, L & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{7} carrefours sombres et \textbf{6} clairs. Il y en a donc \textbf{1 de plus} du côté des sombres.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 13 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 7 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{Non plus.} Avec un carrefour sombre de plus, la tournée doit \textbf{partir et arriver} sur un carrefour sombre. Or l'école est en \textbf{I}, qui est clair.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Une tournée existe pourtant dans ce quartier : il suffirait d'installer l'école sur un carrefour sombre. Ce n'est pas le plan qui est en cause, c'est l'endroit d'où l'on part.}\par\addvspace{0.4em}
\end{document}
