\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 220}

\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=220}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0220}
\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=0.6056, line join=round, line cap=round]
\draw[line width=0.5pt] (8.887,7.2) -- (9.018,7.141) -- (9.151,7.084) -- (9.285,7.031) -- (9.42,6.981) -- (9.556,6.934) -- (9.693,6.891) -- (9.832,6.85) -- (9.971,6.813) -- (10.111,6.779) -- (10.251,6.749) -- (10.393,6.722) -- (10.535,6.698) -- (10.677,6.677) -- (10.82,6.66);
\draw[line width=0.5pt] (8.887,7.2) -- (8.889,6.999) -- (8.886,6.799) -- (8.879,6.598) -- (8.867,6.398) -- (8.85,6.198) -- (8.828,5.998) -- (8.802,5.799) -- (8.77,5.6) -- (8.735,5.403) -- (8.694,5.206) -- (8.649,5.01) -- (8.6,4.816) -- (8.546,4.622) -- (8.487,4.43);
\draw[line width=0.5pt] (6.008,7.061) -- (6.214,7.039) -- (6.42,7.022) -- (6.626,7.011) -- (6.833,7.004) -- (7.04,7.001) -- (7.247,7.004) -- (7.454,7.012) -- (7.66,7.024) -- (7.866,7.041) -- (8.072,7.063) -- (8.277,7.09) -- (8.481,7.122) -- (8.685,7.159) -- (8.887,7.2);
\draw[line width=0.5pt] (6.008,7.061) -- (6.038,6.901) -- (6.063,6.74) -- (6.085,6.578) -- (6.103,6.416) -- (6.118,6.253) -- (6.128,6.09) -- (6.135,5.927) -- (6.138,5.764) -- (6.136,5.601) -- (6.132,5.438) -- (6.123,5.275) -- (6.11,5.113) -- (6.094,4.95) -- (6.074,4.789);
\draw[line width=0.5pt] (1.337,6.966) -- (1.52,6.935) -- (1.704,6.908) -- (1.889,6.886) -- (2.074,6.868) -- (2.259,6.854) -- (2.445,6.845) -- (2.631,6.84) -- (2.817,6.839) -- (3.002,6.843) -- (3.188,6.851) -- (3.374,6.864) -- (3.559,6.881) -- (3.743,6.902) -- (3.927,6.927);
\draw[line width=0.5pt] (1.337,6.966) -- (1.384,6.784) -- (1.426,6.6) -- (1.463,6.416) -- (1.497,6.23) -- (1.526,6.044) -- (1.55,5.857) -- (1.571,5.67) -- (1.586,5.482) -- (1.598,5.294) -- (1.605,5.105) -- (1.607,4.917) -- (1.605,4.729) -- (1.599,4.54) -- (1.588,4.352);
\draw[line width=0.5pt] (3.927,6.927) -- (4.076,6.914) -- (4.225,6.904) -- (4.375,6.898) -- (4.524,6.895) -- (4.674,6.896) -- (4.824,6.901) -- (4.973,6.908) -- (5.122,6.92) -- (5.271,6.934) -- (5.42,6.953) -- (5.568,6.975) -- (5.715,7.0) -- (5.862,7.029) -- (6.008,7.061);
\draw[line width=0.5pt] (3.927,6.927) -- (3.977,6.782) -- (4.022,6.636) -- (4.065,6.489) -- (4.104,6.34) -- (4.139,6.191) -- (4.171,6.041) -- (4.2,5.89) -- (4.225,5.739) -- (4.246,5.587) -- (4.264,5.435) -- (4.278,5.282) -- (4.288,5.129) -- (4.295,4.976) -- (4.298,4.823);
\draw[line width=0.5pt] (10.82,6.66) -- (10.836,6.516) -- (10.854,6.373) -- (10.876,6.23) -- (10.902,6.087) -- (10.931,5.945) -- (10.963,5.804) -- (10.998,5.664) -- (11.037,5.524) -- (11.079,5.386) -- (11.124,5.248) -- (11.173,5.112) -- (11.224,4.977) -- (11.279,4.843) -- (11.337,4.71);
\draw[line width=0.5pt] (4.298,4.823) -- (4.425,4.84) -- (4.551,4.854) -- (4.678,4.865) -- (4.805,4.873) -- (4.933,4.878) -- (5.06,4.88) -- (5.187,4.879) -- (5.315,4.875) -- (5.442,4.868) -- (5.569,4.858) -- (5.696,4.845) -- (5.822,4.829) -- (5.948,4.81) -- (6.074,4.789);
\draw[line width=0.5pt] (4.298,4.823) -- (4.246,4.631) -- (4.199,4.437) -- (4.155,4.243) -- (4.117,4.047) -- (4.083,3.851) -- (4.054,3.654) -- (4.029,3.456) -- (4.009,3.257) -- (3.994,3.059) -- (3.983,2.86) -- (3.977,2.661) -- (3.976,2.461) -- (3.979,2.262) -- (3.988,2.063);
\draw[line width=0.5pt] (6.074,4.789) -- (6.249,4.79) -- (6.424,4.787) -- (6.599,4.779) -- (6.773,4.768) -- (6.948,4.752) -- (7.122,4.733) -- (7.295,4.709) -- (7.468,4.681) -- (7.64,4.65) -- (7.812,4.614) -- (7.982,4.574) -- (8.152,4.53) -- (8.32,4.482) -- (8.487,4.43);
\draw[line width=0.5pt] (8.487,4.43) -- (8.692,4.419) -- (8.897,4.413) -- (9.103,4.411) -- (9.308,4.414) -- (9.514,4.422) -- (9.719,4.435) -- (9.924,4.453) -- (10.128,4.475) -- (10.332,4.502) -- (10.535,4.534) -- (10.737,4.571) -- (10.938,4.613) -- (11.138,4.659) -- (11.337,4.71);
\draw[line width=0.5pt] (1.588,4.352) -- (1.775,4.415) -- (1.964,4.474) -- (2.153,4.528) -- (2.344,4.578) -- (2.537,4.623) -- (2.73,4.663) -- (2.924,4.699) -- (3.119,4.731) -- (3.314,4.758) -- (3.511,4.78) -- (3.707,4.798) -- (3.904,4.811) -- (4.101,4.819) -- (4.298,4.823);
\draw[line width=0.5pt] (1.588,4.352) -- (1.567,4.168) -- (1.55,3.983) -- (1.538,3.797) -- (1.529,3.611) -- (1.526,3.425) -- (1.526,3.239) -- (1.531,3.053) -- (1.541,2.867) -- (1.554,2.681) -- (1.573,2.496) -- (1.595,2.311) -- (1.622,2.127) -- (1.653,1.944) -- (1.688,1.761);
\draw[line width=0.5pt] (1.688,1.761) -- (1.848,1.808) -- (2.009,1.851) -- (2.17,1.89) -- (2.333,1.925) -- (2.497,1.956) -- (2.661,1.984) -- (2.826,2.007) -- (2.991,2.027) -- (3.157,2.043) -- (3.323,2.055) -- (3.489,2.063) -- (3.655,2.067) -- (3.821,2.067) -- (3.988,2.063);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.047,7.2) -- (9.026,7.28) -- (8.967,7.339) -- (8.887,7.36) -- (8.807,7.339) -- (8.749,7.28) -- (8.727,7.2) -- (8.749,7.12) -- (8.807,7.061) -- (8.887,7.04) -- (8.967,7.061) -- (9.026,7.12) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.168,7.061) -- (6.147,7.141) -- (6.088,7.199) -- (6.008,7.221) -- (5.928,7.199) -- (5.87,7.141) -- (5.848,7.061) -- (5.87,6.981) -- (5.928,6.922) -- (6.008,6.901) -- (6.088,6.922) -- (6.147,6.981) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.497,6.966) -- (1.476,7.046) -- (1.417,7.105) -- (1.337,7.126) -- (1.257,7.105) -- (1.199,7.046) -- (1.177,6.966) -- (1.199,6.886) -- (1.257,6.828) -- (1.337,6.806) -- (1.417,6.828) -- (1.476,6.886) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.087,6.927) -- (4.066,7.007) -- (4.007,7.066) -- (3.927,7.087) -- (3.847,7.066) -- (3.789,7.007) -- (3.767,6.927) -- (3.789,6.847) -- (3.847,6.789) -- (3.927,6.767) -- (4.007,6.789) -- (4.066,6.847) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.98,6.66) -- (10.959,6.74) -- (10.9,6.799) -- (10.82,6.82) -- (10.74,6.799) -- (10.682,6.74) -- (10.66,6.66) -- (10.682,6.58) -- (10.74,6.522) -- (10.82,6.5) -- (10.9,6.522) -- (10.959,6.58) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.458,4.823) -- (4.437,4.903) -- (4.378,4.961) -- (4.298,4.983) -- (4.218,4.961) -- (4.16,4.903) -- (4.138,4.823) -- (4.16,4.743) -- (4.218,4.684) -- (4.298,4.663) -- (4.378,4.684) -- (4.437,4.743) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.234,4.789) -- (6.212,4.869) -- (6.154,4.927) -- (6.074,4.949) -- (5.994,4.927) -- (5.935,4.869) -- (5.914,4.789) -- (5.935,4.709) -- (5.994,4.65) -- (6.074,4.629) -- (6.154,4.65) -- (6.212,4.709) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.497,4.71) -- (11.476,4.79) -- (11.417,4.849) -- (11.337,4.87) -- (11.257,4.849) -- (11.199,4.79) -- (11.177,4.71) -- (11.199,4.63) -- (11.257,4.572) -- (11.337,4.55) -- (11.417,4.572) -- (11.476,4.63) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.647,4.43) -- (8.626,4.51) -- (8.567,4.569) -- (8.487,4.59) -- (8.407,4.569) -- (8.349,4.51) -- (8.327,4.43) -- (8.349,4.35) -- (8.407,4.292) -- (8.487,4.27) -- (8.567,4.292) -- (8.626,4.35) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.748,4.352) -- (1.727,4.432) -- (1.668,4.491) -- (1.588,4.512) -- (1.508,4.491) -- (1.45,4.432) -- (1.428,4.352) -- (1.45,4.272) -- (1.508,4.214) -- (1.588,4.192) -- (1.668,4.214) -- (1.727,4.272) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.148,2.063) -- (4.126,2.143) -- (4.068,2.202) -- (3.988,2.223) -- (3.908,2.202) -- (3.849,2.143) -- (3.828,2.063) -- (3.849,1.983) -- (3.908,1.925) -- (3.988,1.903) -- (4.068,1.925) -- (4.126,1.983) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.848,1.761) -- (1.827,1.841) -- (1.768,1.9) -- (1.688,1.921) -- (1.608,1.9) -- (1.55,1.841) -- (1.528,1.761) -- (1.55,1.681) -- (1.608,1.623) -- (1.688,1.601) -- (1.768,1.623) -- (1.827,1.681) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.726,6.077) -- (9.847,6.096) -- (9.957,6.152) -- (10.044,6.239) -- (10.1,6.349) -- (10.119,6.471) -- (10.1,6.593) -- (10.044,6.702) -- (9.957,6.79) -- (9.847,6.846) -- (9.726,6.865) -- (9.726,6.865) -- (9.604,6.846) -- (9.494,6.79) -- (9.407,6.702) -- (9.351,6.593) -- (9.332,6.471) -- (9.351,6.349) -- (9.407,6.239) -- (9.494,6.152) -- (9.604,6.096) -- (9.726,6.077) -- cycle;
\node at (9.726,6.471) {\scalebox{1.651}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.191,5.348) -- (9.313,5.368) -- (9.423,5.424) -- (9.51,5.511) -- (9.566,5.621) -- (9.585,5.742) -- (9.566,5.864) -- (9.51,5.974) -- (9.423,6.061) -- (9.313,6.117) -- (9.191,6.136) -- (9.191,6.136) -- (9.07,6.117) -- (8.96,6.061) -- (8.873,5.974) -- (8.817,5.864) -- (8.797,5.742) -- (8.817,5.621) -- (8.873,5.511) -- (8.96,5.424) -- (9.07,5.368) -- (9.191,5.348) -- cycle;
\node at (9.191,5.742) {\scalebox{1.651}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.435,7.011) -- (7.556,7.03) -- (7.666,7.086) -- (7.753,7.174) -- (7.809,7.283) -- (7.828,7.405) -- (7.809,7.527) -- (7.753,7.637) -- (7.666,7.724) -- (7.556,7.78) -- (7.435,7.799) -- (7.435,7.799) -- (7.313,7.78) -- (7.203,7.724) -- (7.116,7.637) -- (7.06,7.527) -- (7.041,7.405) -- (7.06,7.283) -- (7.116,7.174) -- (7.203,7.086) -- (7.313,7.03) -- (7.435,7.011) -- cycle;
\node at (7.435,7.405) {\scalebox{1.651}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.741,5.522) -- (5.863,5.541) -- (5.973,5.597) -- (6.06,5.685) -- (6.116,5.794) -- (6.135,5.916) -- (6.116,6.038) -- (6.06,6.148) -- (5.973,6.235) -- (5.863,6.291) -- (5.741,6.31) -- (5.741,6.31) -- (5.619,6.291) -- (5.509,6.235) -- (5.422,6.148) -- (5.366,6.038) -- (5.347,5.916) -- (5.366,5.794) -- (5.422,5.685) -- (5.509,5.597) -- (5.619,5.541) -- (5.741,5.522) -- cycle;
\node at (5.741,5.916) {\scalebox{1.651}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.625,6.052) -- (2.746,6.071) -- (2.856,6.127) -- (2.943,6.215) -- (2.999,6.324) -- (3.019,6.446) -- (2.999,6.568) -- (2.943,6.678) -- (2.856,6.765) -- (2.746,6.821) -- (2.625,6.84) -- (2.625,6.84) -- (2.503,6.821) -- (2.393,6.765) -- (2.306,6.678) -- (2.25,6.568) -- (2.231,6.446) -- (2.25,6.324) -- (2.306,6.215) -- (2.393,6.127) -- (2.503,6.071) -- (2.625,6.052) -- cycle;
\node at (2.625,6.446) {\scalebox{1.651}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.963,5.313) -- (2.084,5.333) -- (2.194,5.389) -- (2.281,5.476) -- (2.337,5.586) -- (2.357,5.707) -- (2.337,5.829) -- (2.281,5.939) -- (2.194,6.026) -- (2.084,6.082) -- (1.963,6.101) -- (1.963,6.101) -- (1.841,6.082) -- (1.731,6.026) -- (1.644,5.939) -- (1.588,5.829) -- (1.569,5.707) -- (1.588,5.586) -- (1.644,5.476) -- (1.731,5.389) -- (1.841,5.333) -- (1.963,5.313) -- cycle;
\node at (1.963,5.707) {\scalebox{1.651}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.998,6.121) -- (5.12,6.141) -- (5.23,6.197) -- (5.317,6.284) -- (5.373,6.394) -- (5.392,6.515) -- (5.373,6.637) -- (5.317,6.747) -- (5.23,6.834) -- (5.12,6.89) -- (4.998,6.909) -- (4.998,6.909) -- (4.877,6.89) -- (4.767,6.834) -- (4.68,6.747) -- (4.624,6.637) -- (4.604,6.515) -- (4.624,6.394) -- (4.68,6.284) -- (4.767,6.197) -- (4.877,6.141) -- (4.998,6.121) -- cycle;
\node at (4.998,6.515) {\scalebox{1.651}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.812,5.428) -- (3.933,5.447) -- (4.043,5.503) -- (4.13,5.59) -- (4.186,5.7) -- (4.206,5.822) -- (4.186,5.944) -- (4.13,6.053) -- (4.043,6.141) -- (3.933,6.197) -- (3.812,6.216) -- (3.812,6.216) -- (3.69,6.197) -- (3.58,6.141) -- (3.493,6.053) -- (3.437,5.944) -- (3.418,5.822) -- (3.437,5.7) -- (3.493,5.59) -- (3.58,5.503) -- (3.69,5.447) -- (3.812,5.428) -- cycle;
\node at (3.812,5.822) {\scalebox{1.651}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.379,5.371) -- (11.501,5.39) -- (11.611,5.446) -- (11.698,5.533) -- (11.754,5.643) -- (11.773,5.765) -- (11.754,5.886) -- (11.698,5.996) -- (11.611,6.083) -- (11.501,6.139) -- (11.379,6.159) -- (11.379,6.159) -- (11.257,6.139) -- (11.148,6.083) -- (11.06,5.996) -- (11.004,5.886) -- (10.985,5.765) -- (11.004,5.643) -- (11.06,5.533) -- (11.148,5.446) -- (11.257,5.39) -- (11.379,5.371) -- cycle;
\node at (11.379,5.765) {\scalebox{1.651}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.359,3.744) -- (5.48,3.763) -- (5.59,3.819) -- (5.677,3.906) -- (5.733,4.016) -- (5.753,4.138) -- (5.733,4.259) -- (5.677,4.369) -- (5.59,4.456) -- (5.48,4.512) -- (5.359,4.531) -- (4.988,4.531) -- (4.866,4.512) -- (4.756,4.456) -- (4.669,4.369) -- (4.613,4.259) -- (4.594,4.138) -- (4.613,4.016) -- (4.669,3.906) -- (4.756,3.819) -- (4.866,3.763) -- (4.988,3.744) -- cycle;
\node at (5.173,4.138) {\scalebox{1.651}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.823,3.106) -- (3.945,3.125) -- (4.055,3.181) -- (4.142,3.268) -- (4.198,3.378) -- (4.217,3.5) -- (4.198,3.622) -- (4.142,3.731) -- (4.055,3.819) -- (3.945,3.875) -- (3.823,3.894) -- (3.452,3.894) -- (3.331,3.875) -- (3.221,3.819) -- (3.134,3.731) -- (3.078,3.622) -- (3.059,3.5) -- (3.078,3.378) -- (3.134,3.268) -- (3.221,3.181) -- (3.331,3.125) -- (3.452,3.106) -- cycle;
\node at (3.638,3.5) {\scalebox{1.651}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.683,3.528) -- (7.804,3.547) -- (7.914,3.603) -- (8.001,3.69) -- (8.057,3.8) -- (8.076,3.922) -- (8.057,4.044) -- (8.001,4.154) -- (7.914,4.241) -- (7.804,4.297) -- (7.683,4.316) -- (7.312,4.316) -- (7.19,4.297) -- (7.08,4.241) -- (6.993,4.154) -- (6.937,4.044) -- (6.918,3.922) -- (6.937,3.8) -- (6.993,3.69) -- (7.08,3.603) -- (7.19,3.547) -- (7.312,3.528) -- cycle;
\node at (7.497,3.922) {\scalebox{1.651}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.46,4.498) -- (10.582,4.518) -- (10.692,4.574) -- (10.779,4.661) -- (10.835,4.77) -- (10.854,4.892) -- (10.835,5.014) -- (10.779,5.124) -- (10.692,5.211) -- (10.582,5.267) -- (10.46,5.286) -- (10.09,5.286) -- (9.968,5.267) -- (9.858,5.211) -- (9.771,5.124) -- (9.715,5.014) -- (9.696,4.892) -- (9.715,4.77) -- (9.771,4.661) -- (9.858,4.574) -- (9.968,4.518) -- (10.09,4.498) -- cycle;
\node at (10.275,4.892) {\scalebox{1.651}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.055,4.621) -- (3.176,4.64) -- (3.286,4.696) -- (3.373,4.783) -- (3.429,4.893) -- (3.449,5.015) -- (3.429,5.136) -- (3.373,5.246) -- (3.286,5.333) -- (3.176,5.389) -- (3.055,5.409) -- (2.684,5.409) -- (2.562,5.389) -- (2.452,5.333) -- (2.365,5.246) -- (2.309,5.136) -- (2.29,5.015) -- (2.309,4.893) -- (2.365,4.783) -- (2.452,4.696) -- (2.562,4.64) -- (2.684,4.621) -- cycle;
\node at (2.869,5.015) {\scalebox{1.651}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.11,2.674) -- (2.232,2.693) -- (2.342,2.749) -- (2.429,2.836) -- (2.485,2.946) -- (2.504,3.068) -- (2.485,3.19) -- (2.429,3.299) -- (2.342,3.387) -- (2.232,3.443) -- (2.11,3.462) -- (1.74,3.462) -- (1.618,3.443) -- (1.508,3.387) -- (1.421,3.299) -- (1.365,3.19) -- (1.346,3.068) -- (1.365,2.946) -- (1.421,2.836) -- (1.508,2.749) -- (1.618,2.693) -- (1.74,2.674) -- cycle;
\node at (1.925,3.068) {\scalebox{1.651}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.708,0.6) -- (2.83,0.619) -- (2.94,0.675) -- (3.027,0.762) -- (3.083,0.872) -- (3.102,0.994) -- (3.083,1.116) -- (3.027,1.225) -- (2.94,1.313) -- (2.83,1.369) -- (2.708,1.388) -- (2.337,1.388) -- (2.216,1.369) -- (2.106,1.313) -- (2.019,1.225) -- (1.963,1.116) -- (1.944,0.994) -- (1.963,0.872) -- (2.019,0.762) -- (2.106,0.675) -- (2.216,0.619) -- (2.337,0.6) -- cycle;
\node at (2.523,0.994) {\scalebox{1.651}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (11.037,4.41) -- (11.637,4.41) -- (11.637,5.01) -- (11.037,5.01) -- cycle;
\node at (11.337,4.71) {\scalebox{1.651}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (6.366,5.995) -- (6.966,5.995) -- (6.966,6.595) -- (6.366,6.595) -- cycle;
\node at (6.666,6.295) {\scalebox{1.651}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.703,5.0) -- (7.303,5.0) -- (7.303,5.6) -- (6.703,5.6) -- cycle;
\node at (7.003,5.3) {\scalebox{1.651}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (4.138,7.647) -- (4.738,7.647) -- (4.738,8.247) -- (4.138,8.247) -- cycle;
\node at (4.438,7.947) {\scalebox{1.651}{\small d}};
\node at (9.408,7.536) {\scalebox{1.651}{\small A}};
\node at (6.106,7.673) {\scalebox{1.651}{\small B}};
\node at (0.765,7.205) {\scalebox{1.651}{\small C}};
\node at (3.49,7.367) {\scalebox{1.651}{\small D}};
\node at (11.415,6.836) {\scalebox{1.651}{\small E}};
\node at (3.694,4.684) {\scalebox{1.651}{\small F}};
\node at (6.52,4.359) {\scalebox{1.651}{\small G}};
\node at (11.955,4.662) {\scalebox{1.651}{\small H}};
\node at (9.088,4.276) {\scalebox{1.651}{\small I}};
\node at (0.979,4.235) {\scalebox{1.651}{\small J}};
\node at (3.686,1.522) {\scalebox{1.651}{\small K}};
\node at (1.214,1.362) {\scalebox{1.651}{\small L}};
\draw[line width=0.5pt] (12.637,6.25) -- (12.637,7.35);
\draw[line width=0.5pt] (12.406,6.965) -- (12.637,7.35) -- (12.868,6.965);
\draw[line width=0.5pt] (12.335,6.8) -- (12.94,6.8);
\node at (12.637,7.735) {\scalebox{1.651}{\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 chemin de Ronde & Ⓐ Carrefour de l'Hôpital & Ⓙ Rond-point des Vaches \\ \hline
① quai aux Fleurs & ⑩ rue Pasteur & Ⓑ Carrefour de la Mairie & Ⓚ Placette des Ormes \\ \hline
② rue des Lilas & ⑪ rue de la Mairie & Ⓒ Place de la Fontaine & Ⓛ Coin du Lavoir \\ \hline
③ rue des Écoles & ⑫ venelle du Chat & Ⓓ Carrefour Saint-Roch & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ passage du Marché & ⑬ rue du Puits & Ⓔ Place de l'Église & \textbf{a} l'école \\ \hline
⑤ rue Haute & ⑭ rue du Four & Ⓕ Carrefour des Tilleuls & \textbf{b} le stade \\ \hline
⑥ rue Neuve & ⑮ rue des Jardins & Ⓖ Carrefour du Lavoir & \textbf{c} le marché couvert \\ \hline
⑦ rue de l'Église & ⑯ allée des Peupliers & Ⓗ Place aux Herbes & \textbf{d} le cimetière \\ \hline
⑧ rue du Stade & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Rond-point de la Poste &  \\ \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=0.6196, line join=round, line cap=round]
\draw[line width=0.5pt] (8.906,6.788) -- (9.037,6.729) -- (9.169,6.673) -- (9.303,6.619) -- (9.438,6.569) -- (9.574,6.523) -- (9.712,6.479) -- (9.85,6.439) -- (9.989,6.401) -- (10.129,6.368) -- (10.27,6.337) -- (10.411,6.31) -- (10.553,6.286) -- (10.696,6.265) -- (10.839,6.248);
\draw[line width=0.5pt] (8.906,6.788) -- (8.908,6.588) -- (8.905,6.387) -- (8.897,6.186) -- (8.885,5.986) -- (8.868,5.786) -- (8.846,5.586) -- (8.82,5.387) -- (8.789,5.188) -- (8.753,4.991) -- (8.713,4.794) -- (8.668,4.598) -- (8.618,4.404) -- (8.564,4.21) -- (8.505,4.019);
\draw[line width=0.5pt] (6.027,6.649) -- (6.232,6.627) -- (6.438,6.611) -- (6.645,6.599) -- (6.851,6.592) -- (7.058,6.59) -- (7.265,6.592) -- (7.472,6.6) -- (7.678,6.612) -- (7.885,6.63) -- (8.09,6.652) -- (8.295,6.679) -- (8.5,6.71) -- (8.703,6.747) -- (8.906,6.788);
\draw[line width=0.5pt] (6.027,6.649) -- (6.056,6.489) -- (6.082,6.328) -- (6.104,6.166) -- (6.122,6.004) -- (6.136,5.842) -- (6.146,5.679) -- (6.153,5.516) -- (6.156,5.353) -- (6.155,5.189) -- (6.15,5.026) -- (6.141,4.863) -- (6.129,4.701) -- (6.112,4.539) -- (6.092,4.377);
\draw[line width=0.5pt] (1.356,6.554) -- (1.539,6.523) -- (1.723,6.497) -- (1.907,6.474) -- (2.092,6.456) -- (2.277,6.442) -- (2.463,6.433) -- (2.649,6.428) -- (2.835,6.428) -- (3.021,6.431) -- (3.206,6.44) -- (3.392,6.452) -- (3.577,6.469) -- (3.762,6.49) -- (3.946,6.516);
\draw[line width=0.5pt] (1.356,6.554) -- (1.402,6.372) -- (1.444,6.188) -- (1.482,6.004) -- (1.515,5.818) -- (1.544,5.632) -- (1.569,5.445) -- (1.589,5.258) -- (1.605,5.07) -- (1.616,4.882) -- (1.623,4.694) -- (1.626,4.505) -- (1.624,4.317) -- (1.617,4.129) -- (1.607,3.941);
\draw[line width=0.5pt] (3.946,6.516) -- (4.094,6.502) -- (4.244,6.493) -- (4.393,6.486) -- (4.543,6.484) -- (4.692,6.484) -- (4.842,6.489) -- (4.992,6.497) -- (5.141,6.508) -- (5.29,6.523) -- (5.438,6.541) -- (5.586,6.563) -- (5.734,6.588) -- (5.881,6.617) -- (6.027,6.649);
\draw[line width=0.5pt] (3.946,6.516) -- (3.995,6.371) -- (4.041,6.224) -- (4.083,6.077) -- (4.122,5.929) -- (4.158,5.779) -- (4.19,5.629) -- (4.218,5.479) -- (4.243,5.327) -- (4.264,5.175) -- (4.282,5.023) -- (4.296,4.87) -- (4.307,4.717) -- (4.314,4.564) -- (4.317,4.411);
\draw[line width=0.5pt] (10.839,6.248) -- (10.854,6.105) -- (10.873,5.961) -- (10.895,5.818) -- (10.92,5.676) -- (10.949,5.534) -- (10.981,5.392) -- (11.017,5.252) -- (11.055,5.112) -- (11.097,4.974) -- (11.143,4.836) -- (11.191,4.7) -- (11.243,4.565) -- (11.298,4.431) -- (11.356,4.298);
\draw[line width=0.5pt] (4.317,4.411) -- (4.443,4.428) -- (4.57,4.442) -- (4.696,4.453) -- (4.824,4.461) -- (4.951,4.466) -- (5.078,4.468) -- (5.206,4.467) -- (5.333,4.463) -- (5.46,4.456) -- (5.588,4.446) -- (5.714,4.433) -- (5.841,4.418) -- (5.967,4.399) -- (6.092,4.377);
\draw[line width=0.5pt] (4.317,4.411) -- (4.265,4.219) -- (4.217,4.025) -- (4.174,3.831) -- (4.135,3.635) -- (4.101,3.439) -- (4.072,3.242) -- (4.048,3.044) -- (4.028,2.846) -- (4.012,2.647) -- (4.002,2.448) -- (3.996,2.249) -- (3.994,2.05) -- (3.998,1.85) -- (4.006,1.652);
\draw[line width=0.5pt] (6.092,4.377) -- (6.267,4.378) -- (6.442,4.375) -- (6.617,4.367) -- (6.792,4.356) -- (6.966,4.341) -- (7.14,4.321) -- (7.314,4.297) -- (7.486,4.27) -- (7.659,4.238) -- (7.83,4.202) -- (8.001,4.162) -- (8.17,4.118) -- (8.338,4.07) -- (8.505,4.019);
\draw[line width=0.5pt] (8.505,4.019) -- (8.711,4.007) -- (8.916,4.001) -- (9.121,3.999) -- (9.327,4.002) -- (9.532,4.01) -- (9.737,4.023) -- (9.942,4.041) -- (10.146,4.063) -- (10.35,4.091) -- (10.553,4.123) -- (10.755,4.159) -- (10.957,4.201) -- (11.157,4.247) -- (11.356,4.298);
\draw[line width=0.5pt] (1.607,3.941) -- (1.793,4.004) -- (1.982,4.062) -- (2.172,4.116) -- (2.363,4.166) -- (2.555,4.211) -- (2.748,4.252) -- (2.942,4.288) -- (3.137,4.319) -- (3.333,4.346) -- (3.529,4.368) -- (3.726,4.386) -- (3.922,4.399) -- (4.12,4.407) -- (4.317,4.411);
\draw[line width=0.5pt] (1.607,3.941) -- (1.585,3.756) -- (1.568,3.571) -- (1.556,3.385) -- (1.548,3.199) -- (1.544,3.013) -- (1.545,2.827) -- (1.55,2.641) -- (1.559,2.455) -- (1.573,2.269) -- (1.591,2.084) -- (1.613,1.9) -- (1.64,1.715) -- (1.671,1.532) -- (1.707,1.35);
\draw[line width=0.5pt] (1.707,1.35) -- (1.866,1.396) -- (2.027,1.439) -- (2.189,1.478) -- (2.351,1.513) -- (2.515,1.544) -- (2.679,1.572) -- (2.844,1.595) -- (3.009,1.615) -- (3.175,1.631) -- (3.341,1.643) -- (3.507,1.651) -- (3.673,1.655) -- (3.84,1.655) -- (4.006,1.652);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.279,6.788) -- (9.25,6.931) -- (9.17,7.052) -- (9.049,7.133) -- (8.906,7.161) -- (8.763,7.133) -- (8.642,7.052) -- (8.561,6.931) -- (8.532,6.788) -- (8.561,6.645) -- (8.642,6.524) -- (8.763,6.443) -- (8.906,6.415) -- (9.049,6.443) -- (9.17,6.524) -- (9.25,6.645) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.4,6.649) -- (6.371,6.792) -- (6.29,6.913) -- (6.169,6.994) -- (6.027,7.022) -- (5.884,6.994) -- (5.763,6.913) -- (5.682,6.792) -- (5.653,6.649) -- (5.682,6.506) -- (5.763,6.385) -- (5.884,6.304) -- (6.027,6.276) -- (6.169,6.304) -- (6.29,6.385) -- (6.371,6.506) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.729,6.554) -- (1.7,6.697) -- (1.619,6.818) -- (1.498,6.899) -- (1.356,6.928) -- (1.213,6.899) -- (1.092,6.818) -- (1.011,6.697) -- (0.982,6.554) -- (1.011,6.412) -- (1.092,6.29) -- (1.213,6.21) -- (1.356,6.181) -- (1.498,6.21) -- (1.619,6.29) -- (1.7,6.412) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.319,6.516) -- (4.29,6.659) -- (4.209,6.78) -- (4.088,6.861) -- (3.946,6.889) -- (3.803,6.861) -- (3.682,6.78) -- (3.601,6.659) -- (3.572,6.516) -- (3.601,6.373) -- (3.682,6.252) -- (3.803,6.171) -- (3.946,6.143) -- (4.088,6.171) -- (4.209,6.252) -- (4.29,6.373) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.212,6.248) -- (11.183,6.391) -- (11.102,6.512) -- (10.981,6.593) -- (10.839,6.622) -- (10.696,6.593) -- (10.575,6.512) -- (10.494,6.391) -- (10.465,6.248) -- (10.494,6.106) -- (10.575,5.984) -- (10.696,5.904) -- (10.839,5.875) -- (10.981,5.904) -- (11.102,5.984) -- (11.183,6.106) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.69,4.411) -- (4.662,4.554) -- (4.581,4.675) -- (4.46,4.756) -- (4.317,4.784) -- (4.174,4.756) -- (4.053,4.675) -- (3.972,4.554) -- (3.944,4.411) -- (3.972,4.268) -- (4.053,4.147) -- (4.174,4.066) -- (4.317,4.038) -- (4.46,4.066) -- (4.581,4.147) -- (4.662,4.268) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.465,4.377) -- (6.437,4.52) -- (6.356,4.641) -- (6.235,4.722) -- (6.092,4.75) -- (5.949,4.722) -- (5.828,4.641) -- (5.747,4.52) -- (5.719,4.377) -- (5.747,4.234) -- (5.828,4.113) -- (5.949,4.032) -- (6.092,4.004) -- (6.235,4.032) -- (6.356,4.113) -- (6.437,4.234) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.729,4.298) -- (11.7,4.441) -- (11.619,4.562) -- (11.498,4.643) -- (11.356,4.672) -- (11.213,4.643) -- (11.092,4.562) -- (11.011,4.441) -- (10.982,4.298) -- (11.011,4.156) -- (11.092,4.034) -- (11.213,3.954) -- (11.356,3.925) -- (11.498,3.954) -- (11.619,4.034) -- (11.7,4.156) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.879,4.019) -- (8.85,4.161) -- (8.769,4.283) -- (8.648,4.363) -- (8.505,4.392) -- (8.363,4.363) -- (8.242,4.283) -- (8.161,4.161) -- (8.132,4.019) -- (8.161,3.876) -- (8.242,3.755) -- (8.363,3.674) -- (8.505,3.645) -- (8.648,3.674) -- (8.769,3.755) -- (8.85,3.876) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.98,3.941) -- (1.951,4.084) -- (1.87,4.205) -- (1.749,4.286) -- (1.607,4.314) -- (1.464,4.286) -- (1.343,4.205) -- (1.262,4.084) -- (1.233,3.941) -- (1.262,3.798) -- (1.343,3.677) -- (1.464,3.596) -- (1.607,3.568) -- (1.749,3.596) -- (1.87,3.677) -- (1.951,3.798) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.379,1.652) -- (4.351,1.794) -- (4.27,1.915) -- (4.149,1.996) -- (4.006,2.025) -- (3.863,1.996) -- (3.742,1.915) -- (3.661,1.794) -- (3.633,1.652) -- (3.661,1.509) -- (3.742,1.388) -- (3.863,1.307) -- (4.006,1.278) -- (4.149,1.307) -- (4.27,1.388) -- (4.351,1.509) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.08,1.35) -- (2.051,1.492) -- (1.971,1.613) -- (1.849,1.694) -- (1.707,1.723) -- (1.564,1.694) -- (1.443,1.613) -- (1.362,1.492) -- (1.333,1.35) -- (1.362,1.207) -- (1.443,1.086) -- (1.564,1.005) -- (1.707,0.976) -- (1.849,1.005) -- (1.971,1.086) -- (2.051,1.207) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.746,5.684) -- (9.865,5.703) -- (9.972,5.757) -- (10.057,5.842) -- (10.112,5.949) -- (10.131,6.068) -- (10.112,6.187) -- (10.057,6.294) -- (9.972,6.379) -- (9.865,6.434) -- (9.746,6.453) -- (9.746,6.453) -- (9.628,6.434) -- (9.52,6.379) -- (9.435,6.294) -- (9.381,6.187) -- (9.362,6.068) -- (9.381,5.949) -- (9.435,5.842) -- (9.52,5.757) -- (9.628,5.703) -- (9.746,5.684) -- cycle;
\node at (9.746,6.068) {\scalebox{1.614}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.2,4.947) -- (9.319,4.966) -- (9.426,5.021) -- (9.511,5.106) -- (9.566,5.213) -- (9.585,5.332) -- (9.566,5.451) -- (9.511,5.558) -- (9.426,5.643) -- (9.319,5.698) -- (9.2,5.716) -- (9.2,5.716) -- (9.082,5.698) -- (8.974,5.643) -- (8.889,5.558) -- (8.835,5.451) -- (8.816,5.332) -- (8.835,5.213) -- (8.889,5.106) -- (8.974,5.021) -- (9.082,4.966) -- (9.2,4.947) -- cycle;
\node at (9.2,5.332) {\scalebox{1.614}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.453,6.599) -- (7.572,6.618) -- (7.679,6.673) -- (7.764,6.758) -- (7.819,6.865) -- (7.838,6.984) -- (7.819,7.103) -- (7.764,7.21) -- (7.679,7.295) -- (7.572,7.35) -- (7.453,7.368) -- (7.453,7.368) -- (7.334,7.35) -- (7.227,7.295) -- (7.142,7.21) -- (7.088,7.103) -- (7.069,6.984) -- (7.088,6.865) -- (7.142,6.758) -- (7.227,6.673) -- (7.334,6.618) -- (7.453,6.599) -- cycle;
\node at (7.453,6.984) {\scalebox{1.614}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.769,5.12) -- (5.888,5.139) -- (5.995,5.194) -- (6.08,5.279) -- (6.134,5.386) -- (6.153,5.505) -- (6.134,5.623) -- (6.08,5.731) -- (5.995,5.816) -- (5.888,5.87) -- (5.769,5.889) -- (5.769,5.889) -- (5.65,5.87) -- (5.543,5.816) -- (5.458,5.731) -- (5.403,5.623) -- (5.384,5.505) -- (5.403,5.386) -- (5.458,5.279) -- (5.543,5.194) -- (5.65,5.139) -- (5.769,5.12) -- cycle;
\node at (5.769,5.505) {\scalebox{1.614}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.655,6.428) -- (2.773,6.447) -- (2.881,6.502) -- (2.966,6.587) -- (3.02,6.694) -- (3.039,6.813) -- (3.02,6.932) -- (2.966,7.039) -- (2.881,7.124) -- (2.773,7.178) -- (2.655,7.197) -- (2.655,7.197) -- (2.536,7.178) -- (2.429,7.124) -- (2.344,7.039) -- (2.289,6.932) -- (2.27,6.813) -- (2.289,6.694) -- (2.344,6.587) -- (2.429,6.502) -- (2.536,6.447) -- (2.655,6.428) -- cycle;
\node at (2.655,6.813) {\scalebox{1.614}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.972,4.91) -- (2.09,4.929) -- (2.198,4.984) -- (2.283,5.069) -- (2.337,5.176) -- (2.356,5.295) -- (2.337,5.413) -- (2.283,5.521) -- (2.198,5.606) -- (2.09,5.66) -- (1.972,5.679) -- (1.972,5.679) -- (1.853,5.66) -- (1.746,5.606) -- (1.661,5.521) -- (1.606,5.413) -- (1.587,5.295) -- (1.606,5.176) -- (1.661,5.069) -- (1.746,4.984) -- (1.853,4.929) -- (1.972,4.91) -- cycle;
\node at (1.972,5.295) {\scalebox{1.614}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.016,5.728) -- (5.135,5.747) -- (5.242,5.802) -- (5.327,5.887) -- (5.382,5.994) -- (5.401,6.113) -- (5.382,6.232) -- (5.327,6.339) -- (5.242,6.424) -- (5.135,6.479) -- (5.016,6.497) -- (5.016,6.497) -- (4.897,6.479) -- (4.79,6.424) -- (4.705,6.339) -- (4.65,6.232) -- (4.632,6.113) -- (4.65,5.994) -- (4.705,5.887) -- (4.79,5.802) -- (4.897,5.747) -- (5.016,5.728) -- cycle;
\node at (5.016,6.113) {\scalebox{1.614}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.839,5.027) -- (3.958,5.046) -- (4.065,5.101) -- (4.15,5.186) -- (4.205,5.293) -- (4.224,5.412) -- (4.205,5.531) -- (4.15,5.638) -- (4.065,5.723) -- (3.958,5.778) -- (3.839,5.796) -- (3.839,5.796) -- (3.72,5.778) -- (3.613,5.723) -- (3.528,5.638) -- (3.474,5.531) -- (3.455,5.412) -- (3.474,5.293) -- (3.528,5.186) -- (3.613,5.101) -- (3.72,5.046) -- (3.839,5.027) -- cycle;
\node at (3.839,5.412) {\scalebox{1.614}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.388,4.966) -- (11.507,4.985) -- (11.614,5.039) -- (11.699,5.125) -- (11.754,5.232) -- (11.773,5.351) -- (11.754,5.469) -- (11.699,5.577) -- (11.614,5.662) -- (11.507,5.716) -- (11.388,5.735) -- (11.388,5.735) -- (11.269,5.716) -- (11.162,5.662) -- (11.077,5.577) -- (11.023,5.469) -- (11.004,5.351) -- (11.023,5.232) -- (11.077,5.125) -- (11.162,5.039) -- (11.269,4.985) -- (11.388,4.966) -- cycle;
\node at (11.388,5.351) {\scalebox{1.614}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.373,3.359) -- (5.492,3.378) -- (5.599,3.432) -- (5.684,3.517) -- (5.739,3.625) -- (5.757,3.743) -- (5.739,3.862) -- (5.684,3.969) -- (5.599,4.055) -- (5.492,4.109) -- (5.373,4.128) -- (5.011,4.128) -- (4.892,4.109) -- (4.785,4.055) -- (4.7,3.969) -- (4.645,3.862) -- (4.626,3.743) -- (4.645,3.625) -- (4.7,3.517) -- (4.785,3.432) -- (4.892,3.378) -- (5.011,3.359) -- cycle;
\node at (5.192,3.743) {\scalebox{1.614}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.846,2.703) -- (3.965,2.721) -- (4.072,2.776) -- (4.157,2.861) -- (4.212,2.968) -- (4.231,3.087) -- (4.212,3.206) -- (4.157,3.313) -- (4.072,3.398) -- (3.965,3.453) -- (3.846,3.472) -- (3.485,3.472) -- (3.366,3.453) -- (3.258,3.398) -- (3.173,3.313) -- (3.119,3.206) -- (3.1,3.087) -- (3.119,2.968) -- (3.173,2.861) -- (3.258,2.776) -- (3.366,2.721) -- (3.485,2.703) -- cycle;
\node at (3.665,3.087) {\scalebox{1.614}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.7,3.143) -- (7.819,3.162) -- (7.926,3.216) -- (8.011,3.302) -- (8.066,3.409) -- (8.084,3.528) -- (8.066,3.646) -- (8.011,3.754) -- (7.926,3.839) -- (7.819,3.893) -- (7.7,3.912) -- (7.338,3.912) -- (7.219,3.893) -- (7.112,3.839) -- (7.027,3.754) -- (6.972,3.646) -- (6.953,3.528) -- (6.972,3.409) -- (7.027,3.302) -- (7.112,3.216) -- (7.219,3.162) -- (7.338,3.143) -- cycle;
\node at (7.519,3.528) {\scalebox{1.614}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.476,4.087) -- (10.594,4.105) -- (10.702,4.16) -- (10.787,4.245) -- (10.841,4.352) -- (10.86,4.471) -- (10.841,4.59) -- (10.787,4.697) -- (10.702,4.782) -- (10.594,4.837) -- (10.476,4.856) -- (10.114,4.856) -- (9.995,4.837) -- (9.888,4.782) -- (9.803,4.697) -- (9.748,4.59) -- (9.729,4.471) -- (9.748,4.352) -- (9.803,4.245) -- (9.888,4.16) -- (9.995,4.105) -- (10.114,4.087) -- cycle;
\node at (10.295,4.471) {\scalebox{1.614}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.596,2.837) -- (2.715,2.856) -- (2.822,2.91) -- (2.907,2.995) -- (2.962,3.103) -- (2.981,3.221) -- (2.962,3.34) -- (2.907,3.448) -- (2.822,3.533) -- (2.715,3.587) -- (2.596,3.606) -- (2.234,3.606) -- (2.115,3.587) -- (2.008,3.533) -- (1.923,3.448) -- (1.869,3.34) -- (1.85,3.221) -- (1.869,3.103) -- (1.923,2.995) -- (2.008,2.91) -- (2.115,2.856) -- (2.234,2.837) -- cycle;
\node at (2.415,3.221) {\scalebox{1.614}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.346,2.242) -- (1.465,2.26) -- (1.572,2.315) -- (1.658,2.4) -- (1.712,2.507) -- (1.731,2.626) -- (1.712,2.745) -- (1.658,2.852) -- (1.572,2.937) -- (1.465,2.992) -- (1.346,3.011) -- (0.985,3.011) -- (0.866,2.992) -- (0.759,2.937) -- (0.673,2.852) -- (0.619,2.745) -- (0.6,2.626) -- (0.619,2.507) -- (0.673,2.4) -- (0.759,2.315) -- (0.866,2.26) -- (0.985,2.242) -- cycle;
\node at (1.165,2.626) {\scalebox{1.614}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.975,1.592) -- (3.094,1.611) -- (3.201,1.666) -- (3.286,1.751) -- (3.34,1.858) -- (3.359,1.977) -- (3.34,2.095) -- (3.286,2.203) -- (3.201,2.288) -- (3.094,2.342) -- (2.975,2.361) -- (2.613,2.361) -- (2.494,2.342) -- (2.387,2.288) -- (2.302,2.203) -- (2.247,2.095) -- (2.228,1.977) -- (2.247,1.858) -- (2.302,1.751) -- (2.387,1.666) -- (2.494,1.611) -- (2.613,1.592) -- cycle;
\node at (2.794,1.977) {\scalebox{1.614}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (11.056,3.998) -- (11.656,3.998) -- (11.656,4.598) -- (11.056,4.598) -- cycle;
\node at (11.356,4.298) {\scalebox{1.614}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (6.384,5.583) -- (6.984,5.583) -- (6.984,6.183) -- (6.384,6.183) -- cycle;
\node at (6.684,5.883) {\scalebox{1.614}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.721,4.588) -- (7.321,4.588) -- (7.321,5.188) -- (6.721,5.188) -- cycle;
\node at (7.021,4.888) {\scalebox{1.614}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (4.157,7.236) -- (4.757,7.236) -- (4.757,7.836) -- (4.157,7.836) -- cycle;
\node at (4.457,7.536) {\scalebox{1.614}{\small d}};
\node at (9.427,7.124) {\scalebox{1.614}{\small A}};
\node at (6.124,7.261) {\scalebox{1.614}{\small B}};
\node at (0.783,6.793) {\scalebox{1.614}{\small C}};
\node at (3.508,6.955) {\scalebox{1.614}{\small D}};
\node at (11.433,6.424) {\scalebox{1.614}{\small E}};
\node at (3.713,4.272) {\scalebox{1.614}{\small F}};
\node at (6.539,3.947) {\scalebox{1.614}{\small G}};
\node at (11.974,4.251) {\scalebox{1.614}{\small H}};
\node at (9.106,3.864) {\scalebox{1.614}{\small I}};
\node at (0.998,3.823) {\scalebox{1.614}{\small J}};
\node at (3.704,1.11) {\scalebox{1.614}{\small K}};
\node at (1.233,0.95) {\scalebox{1.614}{\small L}};
\draw[line width=0.5pt] (12.656,5.838) -- (12.656,6.938);
\draw[line width=0.5pt] (12.425,6.553) -- (12.656,6.938) -- (12.887,6.553);
\draw[line width=0.5pt] (12.353,6.388) -- (12.958,6.388);
\node at (12.656,7.323) {\scalebox{1.614}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, D, G, H, J, K & 6 \\ \hline
clairs & B, C, E, F, I, L & 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 le \textbf{chemin de Ronde ⑨} vers le nord jusqu'à Place de l'Église \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{quai aux Fleurs ①}, vers l'ouest, jusqu'à Carrefour de l'Hôpital \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Écoles ③}, vers l'ouest, jusqu'à Carrefour de la Mairie \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de l'Église ⑦}, vers l'ouest, jusqu'à Carrefour Saint-Roch \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Haute ⑤}, vers l'ouest, jusqu'à Place de la Fontaine \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Neuve ⑥}, vers le sud, jusqu'à Rond-point des Vaches \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Jardins ⑮}, vers le sud, jusqu'à Coin du Lavoir \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{allée des Peupliers 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 ⑪}, vers le nord, jusqu'à Carrefour des Tilleuls \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Pasteur ⑩}, vers l'est, jusqu'à Carrefour du Lavoir \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{venelle du Chat ⑫}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Puits ⑬}, vers l'est, jusqu'à Place aux Herbes \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}
