\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 37}

\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=37}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0037}
\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=0.6332, line join=round, line cap=round]
\draw[line width=0.5pt] (7.686,11.471) -- (7.634,11.196) -- (7.588,10.919) -- (7.549,10.642) -- (7.517,10.363) -- (7.491,10.084) -- (7.472,9.804) -- (7.459,9.524) -- (7.453,9.243) -- (7.453,8.963) -- (7.46,8.682) -- (7.473,8.402) -- (7.494,8.123) -- (7.52,7.844) -- (7.554,7.565);
\draw[line width=0.5pt] (4.151,11.169) -- (4.398,11.229) -- (4.647,11.284) -- (4.897,11.332) -- (5.148,11.374) -- (5.4,11.411) -- (5.652,11.442) -- (5.906,11.466) -- (6.16,11.485) -- (6.414,11.498) -- (6.669,11.504) -- (6.924,11.505) -- (7.178,11.5) -- (7.432,11.489) -- (7.686,11.471);
\draw[line width=0.5pt] (4.151,11.169) -- (4.153,10.924) -- (4.161,10.678) -- (4.175,10.433) -- (4.195,10.188) -- (4.22,9.943) -- (4.251,9.699) -- (4.288,9.456) -- (4.331,9.214) -- (4.379,8.973) -- (4.433,8.733) -- (4.492,8.494) -- (4.557,8.257) -- (4.628,8.022) -- (4.704,7.789);
\draw[line width=0.5pt] (1.704,11.054) -- (1.731,10.83) -- (1.752,10.605) -- (1.767,10.379) -- (1.778,10.154) -- (1.783,9.927) -- (1.783,9.701) -- (1.777,9.475) -- (1.766,9.249) -- (1.75,9.023) -- (1.729,8.798) -- (1.702,8.574) -- (1.67,8.35) -- (1.633,8.127) -- (1.59,7.905);
\draw[line width=0.5pt] (1.59,7.905) -- (1.81,7.863) -- (2.03,7.826) -- (2.251,7.794) -- (2.473,7.767) -- (2.696,7.746) -- (2.919,7.729) -- (3.143,7.718) -- (3.366,7.713) -- (3.59,7.712) -- (3.813,7.717) -- (4.037,7.727) -- (4.26,7.742) -- (4.482,7.763) -- (4.704,7.789);
\draw[line width=0.5pt] (1.59,7.905) -- (1.607,7.666) -- (1.618,7.426) -- (1.623,7.186) -- (1.623,6.947) -- (1.616,6.707) -- (1.605,6.467) -- (1.587,6.228) -- (1.564,5.989) -- (1.536,5.75) -- (1.502,5.513) -- (1.462,5.276) -- (1.417,5.041) -- (1.366,4.806) -- (1.31,4.573);
\draw[line width=0.5pt] (4.704,7.789) -- (4.904,7.742) -- (5.104,7.699) -- (5.306,7.662) -- (5.508,7.629) -- (5.712,7.601) -- (5.915,7.578) -- (6.12,7.56) -- (6.325,7.546) -- (6.53,7.537) -- (6.735,7.533) -- (6.94,7.534) -- (7.145,7.54) -- (7.349,7.55) -- (7.554,7.565);
\draw[line width=0.5pt] (4.704,7.789) -- (4.733,7.596) -- (4.757,7.403) -- (4.777,7.209) -- (4.792,7.014) -- (4.803,6.819) -- (4.809,6.624) -- (4.81,6.429) -- (4.807,6.234) -- (4.799,6.039) -- (4.787,5.845) -- (4.77,5.65) -- (4.748,5.457) -- (4.722,5.263) -- (4.691,5.071);
\draw[line width=0.5pt] (10.432,7.624) -- (10.416,7.419) -- (10.405,7.213) -- (10.398,7.007) -- (10.397,6.801) -- (10.4,6.595) -- (10.408,6.389) -- (10.421,6.183) -- (10.438,5.978) -- (10.461,5.773) -- (10.488,5.569) -- (10.521,5.365) -- (10.557,5.162) -- (10.599,4.961) -- (10.646,4.76);
\draw[line width=0.5pt] (7.554,7.565) -- (7.757,7.601) -- (7.961,7.632) -- (8.166,7.658) -- (8.372,7.679) -- (8.578,7.695) -- (8.784,7.707) -- (8.991,7.714) -- (9.197,7.715) -- (9.404,7.712) -- (9.61,7.704) -- (9.817,7.692) -- (10.022,7.674) -- (10.228,7.652) -- (10.432,7.624);
\draw[line width=0.5pt] (7.501,5.224) -- (7.719,5.156) -- (7.938,5.094) -- (8.159,5.037) -- (8.381,4.986) -- (8.605,4.939) -- (8.829,4.898) -- (9.054,4.862) -- (9.28,4.831) -- (9.507,4.806) -- (9.734,4.786) -- (9.962,4.771) -- (10.19,4.762) -- (10.418,4.758) -- (10.646,4.76);
\draw[line width=0.5pt] (4.691,5.071) -- (4.889,5.112) -- (5.087,5.149) -- (5.287,5.182) -- (5.487,5.209) -- (5.687,5.232) -- (5.889,5.25) -- (6.09,5.263) -- (6.292,5.272) -- (6.494,5.276) -- (6.696,5.275) -- (6.898,5.269) -- (7.099,5.259) -- (7.301,5.243) -- (7.501,5.224);
\draw[line width=0.5pt] (4.691,5.071) -- (4.666,4.843) -- (4.647,4.615) -- (4.632,4.386) -- (4.623,4.157) -- (4.62,3.927) -- (4.622,3.698) -- (4.629,3.468) -- (4.641,3.239) -- (4.659,3.011) -- (4.683,2.782) -- (4.712,2.555) -- (4.746,2.328) -- (4.785,2.102) -- (4.83,1.877);
\draw[line width=0.5pt] (1.31,4.573) -- (1.544,4.646) -- (1.78,4.712) -- (2.017,4.773) -- (2.256,4.829) -- (2.496,4.879) -- (2.738,4.923) -- (2.98,4.962) -- (3.223,4.995) -- (3.467,5.022) -- (3.711,5.043) -- (3.956,5.059) -- (4.201,5.069) -- (4.446,5.073) -- (4.691,5.071);
\draw[line width=0.5pt] (1.31,4.573) -- (1.317,4.347) -- (1.33,4.121) -- (1.348,3.895) -- (1.372,3.67) -- (1.401,3.445) -- (1.435,3.221) -- (1.474,2.998) -- (1.519,2.776) -- (1.569,2.555) -- (1.624,2.335) -- (1.684,2.117) -- (1.749,1.9) -- (1.82,1.685) -- (1.895,1.471);
\draw[line width=0.5pt] (1.895,1.471) -- (2.099,1.532) -- (2.304,1.588) -- (2.51,1.639) -- (2.717,1.686) -- (2.926,1.727) -- (3.135,1.764) -- (3.346,1.795) -- (3.557,1.822) -- (3.768,1.844) -- (3.98,1.861) -- (4.192,1.872) -- (4.405,1.879) -- (4.617,1.881) -- (4.83,1.877);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.846,11.471) -- (7.825,11.551) -- (7.766,11.61) -- (7.686,11.631) -- (7.606,11.61) -- (7.548,11.551) -- (7.526,11.471) -- (7.548,11.391) -- (7.606,11.333) -- (7.686,11.311) -- (7.766,11.333) -- (7.825,11.391) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.311,11.169) -- (4.289,11.249) -- (4.231,11.308) -- (4.151,11.329) -- (4.071,11.308) -- (4.012,11.249) -- (3.991,11.169) -- (4.012,11.089) -- (4.071,11.031) -- (4.151,11.009) -- (4.231,11.031) -- (4.289,11.089) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.864,11.054) -- (1.843,11.134) -- (1.784,11.193) -- (1.704,11.214) -- (1.624,11.193) -- (1.566,11.134) -- (1.544,11.054) -- (1.566,10.974) -- (1.624,10.916) -- (1.704,10.894) -- (1.784,10.916) -- (1.843,10.974) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.75,7.905) -- (1.729,7.985) -- (1.67,8.044) -- (1.59,8.065) -- (1.51,8.044) -- (1.452,7.985) -- (1.43,7.905) -- (1.452,7.825) -- (1.51,7.766) -- (1.59,7.745) -- (1.67,7.766) -- (1.729,7.825) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.864,7.789) -- (4.843,7.869) -- (4.784,7.927) -- (4.704,7.949) -- (4.624,7.927) -- (4.566,7.869) -- (4.544,7.789) -- (4.566,7.709) -- (4.624,7.65) -- (4.704,7.629) -- (4.784,7.65) -- (4.843,7.709) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.592,7.624) -- (10.571,7.704) -- (10.512,7.763) -- (10.432,7.784) -- (10.352,7.763) -- (10.294,7.704) -- (10.272,7.624) -- (10.294,7.544) -- (10.352,7.486) -- (10.432,7.464) -- (10.512,7.486) -- (10.571,7.544) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.714,7.565) -- (7.692,7.645) -- (7.634,7.704) -- (7.554,7.725) -- (7.474,7.704) -- (7.415,7.645) -- (7.394,7.565) -- (7.415,7.485) -- (7.474,7.427) -- (7.554,7.405) -- (7.634,7.427) -- (7.692,7.485) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.661,5.224) -- (7.64,5.304) -- (7.581,5.362) -- (7.501,5.384) -- (7.421,5.362) -- (7.363,5.304) -- (7.341,5.224) -- (7.363,5.144) -- (7.421,5.085) -- (7.501,5.064) -- (7.581,5.085) -- (7.64,5.144) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.851,5.071) -- (4.83,5.151) -- (4.771,5.209) -- (4.691,5.231) -- (4.611,5.209) -- (4.553,5.151) -- (4.531,5.071) -- (4.553,4.991) -- (4.611,4.932) -- (4.691,4.911) -- (4.771,4.932) -- (4.83,4.991) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.806,4.76) -- (10.784,4.84) -- (10.726,4.899) -- (10.646,4.92) -- (10.566,4.899) -- (10.507,4.84) -- (10.486,4.76) -- (10.507,4.68) -- (10.566,4.621) -- (10.646,4.6) -- (10.726,4.621) -- (10.784,4.68) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.47,4.573) -- (1.448,4.653) -- (1.39,4.712) -- (1.31,4.733) -- (1.23,4.712) -- (1.171,4.653) -- (1.15,4.573) -- (1.171,4.493) -- (1.23,4.435) -- (1.31,4.413) -- (1.39,4.435) -- (1.448,4.493) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.99,1.877) -- (4.968,1.957) -- (4.91,2.016) -- (4.83,2.037) -- (4.75,2.016) -- (4.691,1.957) -- (4.67,1.877) -- (4.691,1.797) -- (4.75,1.739) -- (4.83,1.717) -- (4.91,1.739) -- (4.968,1.797) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.055,1.471) -- (2.034,1.551) -- (1.975,1.61) -- (1.895,1.631) -- (1.815,1.61) -- (1.757,1.551) -- (1.735,1.471) -- (1.757,1.391) -- (1.815,1.333) -- (1.895,1.311) -- (1.975,1.333) -- (2.034,1.391) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.853,9.117) -- (7.974,9.136) -- (8.084,9.192) -- (8.171,9.279) -- (8.227,9.389) -- (8.246,9.51) -- (8.227,9.632) -- (8.171,9.742) -- (8.084,9.829) -- (7.974,9.885) -- (7.853,9.904) -- (7.853,9.904) -- (7.731,9.885) -- (7.621,9.829) -- (7.534,9.742) -- (7.478,9.632) -- (7.459,9.51) -- (7.478,9.389) -- (7.534,9.279) -- (7.621,9.192) -- (7.731,9.136) -- (7.853,9.117) -- cycle;
\node at (7.853,9.51) {\scalebox{1.579}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.872,11.465) -- (5.994,11.484) -- (6.104,11.54) -- (6.191,11.627) -- (6.247,11.737) -- (6.266,11.859) -- (6.247,11.98) -- (6.191,12.09) -- (6.104,12.177) -- (5.994,12.233) -- (5.872,12.253) -- (5.872,12.253) -- (5.751,12.233) -- (5.641,12.177) -- (5.554,12.09) -- (5.498,11.98) -- (5.479,11.859) -- (5.498,11.737) -- (5.554,11.627) -- (5.641,11.54) -- (5.751,11.484) -- (5.872,11.465) -- cycle;
\node at (5.872,11.859) {\scalebox{1.579}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.677,9.126) -- (4.799,9.145) -- (4.908,9.201) -- (4.995,9.288) -- (5.051,9.398) -- (5.071,9.52) -- (5.051,9.642) -- (4.995,9.751) -- (4.908,9.839) -- (4.799,9.894) -- (4.677,9.914) -- (4.677,9.914) -- (4.555,9.894) -- (4.445,9.839) -- (4.358,9.751) -- (4.302,9.642) -- (4.283,9.52) -- (4.302,9.398) -- (4.358,9.288) -- (4.445,9.201) -- (4.555,9.145) -- (4.677,9.126) -- cycle;
\node at (4.677,9.52) {\scalebox{1.579}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.171,9.067) -- (2.293,9.086) -- (2.402,9.142) -- (2.49,9.229) -- (2.545,9.339) -- (2.565,9.461) -- (2.545,9.583) -- (2.49,9.692) -- (2.402,9.779) -- (2.293,9.835) -- (2.171,9.855) -- (2.171,9.855) -- (2.049,9.835) -- (1.939,9.779) -- (1.852,9.692) -- (1.796,9.583) -- (1.777,9.461) -- (1.796,9.339) -- (1.852,9.229) -- (1.939,9.142) -- (2.049,9.086) -- (2.171,9.067) -- cycle;
\node at (2.171,9.461) {\scalebox{1.579}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.157,7.718) -- (3.279,7.737) -- (3.389,7.793) -- (3.476,7.88) -- (3.532,7.99) -- (3.551,8.112) -- (3.532,8.234) -- (3.476,8.344) -- (3.389,8.431) -- (3.279,8.487) -- (3.157,8.506) -- (3.157,8.506) -- (3.036,8.487) -- (2.926,8.431) -- (2.839,8.344) -- (2.783,8.234) -- (2.763,8.112) -- (2.783,7.99) -- (2.839,7.88) -- (2.926,7.793) -- (3.036,7.737) -- (3.157,7.718) -- cycle;
\node at (3.157,8.112) {\scalebox{1.579}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.98,5.801) -- (2.102,5.82) -- (2.211,5.876) -- (2.299,5.963) -- (2.355,6.073) -- (2.374,6.195) -- (2.355,6.316) -- (2.299,6.426) -- (2.211,6.513) -- (2.102,6.569) -- (1.98,6.588) -- (1.98,6.588) -- (1.858,6.569) -- (1.748,6.513) -- (1.661,6.426) -- (1.605,6.316) -- (1.586,6.195) -- (1.605,6.073) -- (1.661,5.963) -- (1.748,5.876) -- (1.858,5.82) -- (1.98,5.801) -- cycle;
\node at (1.98,6.195) {\scalebox{1.579}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.151,7.558) -- (6.272,7.578) -- (6.382,7.634) -- (6.469,7.721) -- (6.525,7.831) -- (6.544,7.952) -- (6.525,8.074) -- (6.469,8.184) -- (6.382,8.271) -- (6.272,8.327) -- (6.151,8.346) -- (6.151,8.346) -- (6.029,8.327) -- (5.919,8.271) -- (5.832,8.184) -- (5.776,8.074) -- (5.757,7.952) -- (5.776,7.831) -- (5.832,7.721) -- (5.919,7.634) -- (6.029,7.578) -- (6.151,7.558) -- cycle;
\node at (6.151,7.952) {\scalebox{1.579}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.204,6.034) -- (5.326,6.053) -- (5.435,6.109) -- (5.523,6.196) -- (5.579,6.306) -- (5.598,6.427) -- (5.579,6.549) -- (5.523,6.659) -- (5.435,6.746) -- (5.326,6.802) -- (5.204,6.821) -- (5.204,6.821) -- (5.082,6.802) -- (4.972,6.746) -- (4.885,6.659) -- (4.829,6.549) -- (4.81,6.427) -- (4.829,6.306) -- (4.885,6.196) -- (4.972,6.109) -- (5.082,6.053) -- (5.204,6.034) -- cycle;
\node at (5.204,6.427) {\scalebox{1.579}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.814,5.819) -- (10.935,5.838) -- (11.045,5.894) -- (11.132,5.981) -- (11.188,6.091) -- (11.207,6.213) -- (11.188,6.334) -- (11.132,6.444) -- (11.045,6.531) -- (10.935,6.587) -- (10.814,6.606) -- (10.814,6.606) -- (10.692,6.587) -- (10.582,6.531) -- (10.495,6.444) -- (10.439,6.334) -- (10.42,6.213) -- (10.439,6.091) -- (10.495,5.981) -- (10.582,5.894) -- (10.692,5.838) -- (10.814,5.819) -- cycle;
\node at (10.814,6.213) {\scalebox{1.579}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.6,7.712) -- (9.721,7.731) -- (9.831,7.787) -- (9.918,7.875) -- (9.974,7.984) -- (9.994,8.106) -- (9.974,8.228) -- (9.918,8.338) -- (9.831,8.425) -- (9.721,8.481) -- (9.6,8.5) -- (9.229,8.5) -- (9.107,8.481) -- (8.997,8.425) -- (8.91,8.338) -- (8.854,8.228) -- (8.835,8.106) -- (8.854,7.984) -- (8.91,7.875) -- (8.997,7.787) -- (9.107,7.731) -- (9.229,7.712) -- cycle;
\node at (9.414,8.106) {\scalebox{1.579}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.297,4.858) -- (9.419,4.877) -- (9.529,4.933) -- (9.616,5.02) -- (9.672,5.13) -- (9.691,5.252) -- (9.672,5.373) -- (9.616,5.483) -- (9.529,5.57) -- (9.419,5.626) -- (9.297,5.646) -- (8.926,5.646) -- (8.805,5.626) -- (8.695,5.57) -- (8.608,5.483) -- (8.552,5.373) -- (8.533,5.252) -- (8.552,5.13) -- (8.608,5.02) -- (8.695,4.933) -- (8.805,4.877) -- (8.926,4.858) -- cycle;
\node at (9.112,5.252) {\scalebox{1.579}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.254,5.263) -- (6.376,5.282) -- (6.486,5.338) -- (6.573,5.425) -- (6.629,5.535) -- (6.648,5.656) -- (6.629,5.778) -- (6.573,5.888) -- (6.486,5.975) -- (6.376,6.031) -- (6.254,6.05) -- (5.883,6.05) -- (5.762,6.031) -- (5.652,5.975) -- (5.565,5.888) -- (5.509,5.778) -- (5.49,5.656) -- (5.509,5.535) -- (5.565,5.425) -- (5.652,5.338) -- (5.762,5.282) -- (5.883,5.263) -- cycle;
\node at (6.069,5.656) {\scalebox{1.579}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.208,3.092) -- (5.329,3.111) -- (5.439,3.167) -- (5.526,3.254) -- (5.582,3.364) -- (5.602,3.486) -- (5.582,3.607) -- (5.526,3.717) -- (5.439,3.804) -- (5.329,3.86) -- (5.208,3.879) -- (4.837,3.879) -- (4.715,3.86) -- (4.605,3.804) -- (4.518,3.717) -- (4.462,3.607) -- (4.443,3.486) -- (4.462,3.364) -- (4.518,3.254) -- (4.605,3.167) -- (4.715,3.111) -- (4.837,3.092) -- cycle;
\node at (5.022,3.486) {\scalebox{1.579}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.108,4.957) -- (3.23,4.977) -- (3.34,5.033) -- (3.427,5.12) -- (3.483,5.23) -- (3.502,5.351) -- (3.483,5.473) -- (3.427,5.583) -- (3.34,5.67) -- (3.23,5.726) -- (3.108,5.745) -- (2.737,5.745) -- (2.616,5.726) -- (2.506,5.67) -- (2.419,5.583) -- (2.363,5.473) -- (2.343,5.351) -- (2.363,5.23) -- (2.419,5.12) -- (2.506,5.033) -- (2.616,4.977) -- (2.737,4.957) -- cycle;
\node at (2.923,5.351) {\scalebox{1.579}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.047,2.677) -- (2.169,2.697) -- (2.278,2.753) -- (2.366,2.84) -- (2.421,2.95) -- (2.441,3.071) -- (2.421,3.193) -- (2.366,3.303) -- (2.278,3.39) -- (2.169,3.446) -- (2.047,3.465) -- (1.676,3.465) -- (1.554,3.446) -- (1.445,3.39) -- (1.357,3.303) -- (1.302,3.193) -- (1.282,3.071) -- (1.302,2.95) -- (1.357,2.84) -- (1.445,2.753) -- (1.554,2.697) -- (1.676,2.677) -- cycle;
\node at (1.862,3.071) {\scalebox{1.579}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.477,1.792) -- (3.599,1.811) -- (3.709,1.867) -- (3.796,1.954) -- (3.852,2.064) -- (3.871,2.186) -- (3.852,2.307) -- (3.796,2.417) -- (3.709,2.504) -- (3.599,2.56) -- (3.477,2.58) -- (3.106,2.58) -- (2.985,2.56) -- (2.875,2.504) -- (2.788,2.417) -- (2.732,2.307) -- (2.712,2.186) -- (2.732,2.064) -- (2.788,1.954) -- (2.875,1.867) -- (2.985,1.811) -- (3.106,1.792) -- cycle;
\node at (3.292,2.186) {\scalebox{1.579}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.01,4.273) -- (1.61,4.273) -- (1.61,4.873) -- (1.01,4.873) -- cycle;
\node at (1.31,4.573) {\scalebox{1.579}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.66,10.038) -- (2.26,10.038) -- (2.26,10.638) -- (1.66,10.638) -- cycle;
\node at (1.96,10.338) {\scalebox{1.579}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (11.749,6.286) -- (12.349,6.286) -- (12.349,6.886) -- (11.749,6.886) -- cycle;
\node at (12.049,6.586) {\scalebox{1.579}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.766,6.369) -- (1.366,6.369) -- (1.366,6.969) -- (0.766,6.969) -- cycle;
\node at (1.066,6.669) {\scalebox{1.579}{\small d}};
\node at (7.967,12.024) {\scalebox{1.579}{\small A}};
\node at (3.997,11.77) {\scalebox{1.579}{\small B}};
\node at (1.309,11.532) {\scalebox{1.579}{\small C}};
\node at (0.999,8.092) {\scalebox{1.579}{\small D}};
\node at (4.405,8.332) {\scalebox{1.579}{\small E}};
\node at (11.043,7.73) {\scalebox{1.579}{\small F}};
\node at (8.136,7.779) {\scalebox{1.579}{\small G}};
\node at (8.014,4.874) {\scalebox{1.579}{\small H}};
\node at (4.483,4.487) {\scalebox{1.579}{\small I}};
\node at (11.227,4.546) {\scalebox{1.579}{\small J}};
\node at (0.765,4.277) {\scalebox{1.579}{\small K}};
\node at (4.772,1.26) {\scalebox{1.579}{\small L}};
\node at (1.559,0.95) {\scalebox{1.579}{\small M}};
\draw[line width=0.5pt] (11.946,10.521) -- (11.946,11.621);
\draw[line width=0.5pt] (11.715,11.236) -- (11.946,11.621) -- (12.177,11.236);
\draw[line width=0.5pt] (11.643,11.071) -- (12.248,11.071);
\node at (11.946,12.006) {\scalebox{1.579}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour K, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ rue du Stade & Ⓐ Place de l'Église & Ⓙ Place aux Herbes \\ \hline
① allée des Peupliers & ⑩ rue des Lilas & Ⓑ Place de la Fontaine & Ⓚ Rond-point du Stade \\ \hline
② chemin de Ronde & ⑪ rue du Pont & Ⓒ Place Sainte-Croix & Ⓛ Place du Marché \\ \hline
③ boulevard du Parc & ⑫ chemin du Lavoir & Ⓓ Carrefour du Moulin & Ⓜ Coin du Lavoir \\ \hline
④ impasse du Vieux-Four & ⑬ rue Verte & Ⓔ Carrefour de l'Hôpital & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ quai aux Fleurs & ⑭ rue de la Poste & Ⓕ Placette des Ormes & \textbf{a} l'école \\ \hline
⑥ rue Neuve & ⑮ chemin des Vignes & Ⓖ Rond-point du Pont & \textbf{b} la salle des fêtes \\ \hline
⑦ allée des Tilleuls & ⑯ rue Pasteur & Ⓗ Place des Tanneurs & \textbf{c} le gymnase \\ \hline
⑧ rue du Four & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour du Lavoir & \textbf{d} l'église \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\begin{enumerate}
  \item Colorier les carrefours du plan avec \textbf{deux couleurs}, de sorte que deux carrefours \textbf{reliés par une rue} ne soient jamais de la même couleur. Combien y en a-t-il de chaque couleur ?
  \item Le car roule d'un carrefour au suivant. Que peut-on dire de la \textbf{couleur} des arrêts successifs ? Qu'en déduire sur les couleurs des arrêts d'une tournée qui les dessert tous ?
  \item \textbf{Le car peut-il desservir tous les carrefours, une fois chacun, et revenir à l'école ?} Si oui, décrire un trajet ; sinon, démontrer que c'est impossible.
  \item \textbf{Et s'il n'est pas obligé de revenir à l'école ?} Même consigne.
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage — Correction}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6487, line join=round, line cap=round]
\draw[line width=0.5pt] (7.686,11.471) -- (7.634,11.196) -- (7.588,10.919) -- (7.549,10.642) -- (7.517,10.363) -- (7.491,10.084) -- (7.472,9.804) -- (7.459,9.524) -- (7.453,9.243) -- (7.453,8.963) -- (7.46,8.682) -- (7.473,8.402) -- (7.494,8.123) -- (7.52,7.844) -- (7.554,7.565);
\draw[line width=0.5pt] (4.151,11.169) -- (4.398,11.229) -- (4.647,11.284) -- (4.897,11.332) -- (5.148,11.374) -- (5.4,11.411) -- (5.652,11.442) -- (5.906,11.466) -- (6.16,11.485) -- (6.414,11.498) -- (6.669,11.504) -- (6.924,11.505) -- (7.178,11.5) -- (7.432,11.489) -- (7.686,11.471);
\draw[line width=0.5pt] (4.151,11.169) -- (4.153,10.924) -- (4.161,10.678) -- (4.175,10.433) -- (4.195,10.188) -- (4.22,9.943) -- (4.251,9.699) -- (4.288,9.456) -- (4.331,9.214) -- (4.379,8.973) -- (4.433,8.733) -- (4.492,8.494) -- (4.557,8.257) -- (4.628,8.022) -- (4.704,7.789);
\draw[line width=0.5pt] (1.704,11.054) -- (1.731,10.83) -- (1.752,10.605) -- (1.767,10.379) -- (1.778,10.154) -- (1.783,9.927) -- (1.783,9.701) -- (1.777,9.475) -- (1.766,9.249) -- (1.75,9.023) -- (1.729,8.798) -- (1.702,8.574) -- (1.67,8.35) -- (1.633,8.127) -- (1.59,7.905);
\draw[line width=0.5pt] (1.59,7.905) -- (1.81,7.863) -- (2.03,7.826) -- (2.251,7.794) -- (2.473,7.767) -- (2.696,7.746) -- (2.919,7.729) -- (3.143,7.718) -- (3.366,7.713) -- (3.59,7.712) -- (3.813,7.717) -- (4.037,7.727) -- (4.26,7.742) -- (4.482,7.763) -- (4.704,7.789);
\draw[line width=0.5pt] (1.59,7.905) -- (1.607,7.666) -- (1.618,7.426) -- (1.623,7.186) -- (1.623,6.947) -- (1.616,6.707) -- (1.605,6.467) -- (1.587,6.228) -- (1.564,5.989) -- (1.536,5.75) -- (1.502,5.513) -- (1.462,5.276) -- (1.417,5.041) -- (1.366,4.806) -- (1.31,4.573);
\draw[line width=0.5pt] (4.704,7.789) -- (4.904,7.742) -- (5.104,7.699) -- (5.306,7.662) -- (5.508,7.629) -- (5.712,7.601) -- (5.915,7.578) -- (6.12,7.56) -- (6.325,7.546) -- (6.53,7.537) -- (6.735,7.533) -- (6.94,7.534) -- (7.145,7.54) -- (7.349,7.55) -- (7.554,7.565);
\draw[line width=0.5pt] (4.704,7.789) -- (4.733,7.596) -- (4.757,7.403) -- (4.777,7.209) -- (4.792,7.014) -- (4.803,6.819) -- (4.809,6.624) -- (4.81,6.429) -- (4.807,6.234) -- (4.799,6.039) -- (4.787,5.845) -- (4.77,5.65) -- (4.748,5.457) -- (4.722,5.263) -- (4.691,5.071);
\draw[line width=0.5pt] (10.432,7.624) -- (10.416,7.419) -- (10.405,7.213) -- (10.398,7.007) -- (10.397,6.801) -- (10.4,6.595) -- (10.408,6.389) -- (10.421,6.183) -- (10.438,5.978) -- (10.461,5.773) -- (10.488,5.569) -- (10.521,5.365) -- (10.557,5.162) -- (10.599,4.961) -- (10.646,4.76);
\draw[line width=0.5pt] (7.554,7.565) -- (7.757,7.601) -- (7.961,7.632) -- (8.166,7.658) -- (8.372,7.679) -- (8.578,7.695) -- (8.784,7.707) -- (8.991,7.714) -- (9.197,7.715) -- (9.404,7.712) -- (9.61,7.704) -- (9.817,7.692) -- (10.022,7.674) -- (10.228,7.652) -- (10.432,7.624);
\draw[line width=0.5pt] (7.501,5.224) -- (7.719,5.156) -- (7.938,5.094) -- (8.159,5.037) -- (8.381,4.986) -- (8.605,4.939) -- (8.829,4.898) -- (9.054,4.862) -- (9.28,4.831) -- (9.507,4.806) -- (9.734,4.786) -- (9.962,4.771) -- (10.19,4.762) -- (10.418,4.758) -- (10.646,4.76);
\draw[line width=0.5pt] (4.691,5.071) -- (4.889,5.112) -- (5.087,5.149) -- (5.287,5.182) -- (5.487,5.209) -- (5.687,5.232) -- (5.889,5.25) -- (6.09,5.263) -- (6.292,5.272) -- (6.494,5.276) -- (6.696,5.275) -- (6.898,5.269) -- (7.099,5.259) -- (7.301,5.243) -- (7.501,5.224);
\draw[line width=0.5pt] (4.691,5.071) -- (4.666,4.843) -- (4.647,4.615) -- (4.632,4.386) -- (4.623,4.157) -- (4.62,3.927) -- (4.622,3.698) -- (4.629,3.468) -- (4.641,3.239) -- (4.659,3.011) -- (4.683,2.782) -- (4.712,2.555) -- (4.746,2.328) -- (4.785,2.102) -- (4.83,1.877);
\draw[line width=0.5pt] (1.31,4.573) -- (1.544,4.646) -- (1.78,4.712) -- (2.017,4.773) -- (2.256,4.829) -- (2.496,4.879) -- (2.738,4.923) -- (2.98,4.962) -- (3.223,4.995) -- (3.467,5.022) -- (3.711,5.043) -- (3.956,5.059) -- (4.201,5.069) -- (4.446,5.073) -- (4.691,5.071);
\draw[line width=0.5pt] (1.31,4.573) -- (1.317,4.347) -- (1.33,4.121) -- (1.348,3.895) -- (1.372,3.67) -- (1.401,3.445) -- (1.435,3.221) -- (1.474,2.998) -- (1.519,2.776) -- (1.569,2.555) -- (1.624,2.335) -- (1.684,2.117) -- (1.749,1.9) -- (1.82,1.685) -- (1.895,1.471);
\draw[line width=0.5pt] (1.895,1.471) -- (2.099,1.532) -- (2.304,1.588) -- (2.51,1.639) -- (2.717,1.686) -- (2.926,1.727) -- (3.135,1.764) -- (3.346,1.795) -- (3.557,1.822) -- (3.768,1.844) -- (3.98,1.861) -- (4.192,1.872) -- (4.405,1.879) -- (4.617,1.881) -- (4.83,1.877);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.059,11.471) -- (8.031,11.614) -- (7.95,11.735) -- (7.829,11.816) -- (7.686,11.845) -- (7.543,11.816) -- (7.422,11.735) -- (7.341,11.614) -- (7.313,11.471) -- (7.341,11.328) -- (7.422,11.207) -- (7.543,11.126) -- (7.686,11.098) -- (7.829,11.126) -- (7.95,11.207) -- (8.031,11.328) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.524,11.169) -- (4.495,11.312) -- (4.415,11.433) -- (4.294,11.514) -- (4.151,11.543) -- (4.008,11.514) -- (3.887,11.433) -- (3.806,11.312) -- (3.777,11.169) -- (3.806,11.027) -- (3.887,10.906) -- (4.008,10.825) -- (4.151,10.796) -- (4.294,10.825) -- (4.415,10.906) -- (4.495,11.027) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.078,11.054) -- (2.049,11.197) -- (1.968,11.318) -- (1.847,11.399) -- (1.704,11.428) -- (1.561,11.399) -- (1.44,11.318) -- (1.359,11.197) -- (1.331,11.054) -- (1.359,10.912) -- (1.44,10.791) -- (1.561,10.71) -- (1.704,10.681) -- (1.847,10.71) -- (1.968,10.791) -- (2.049,10.912) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.964,7.905) -- (1.935,8.048) -- (1.854,8.169) -- (1.733,8.25) -- (1.59,8.278) -- (1.447,8.25) -- (1.326,8.169) -- (1.245,8.048) -- (1.217,7.905) -- (1.245,7.762) -- (1.326,7.641) -- (1.447,7.56) -- (1.59,7.532) -- (1.733,7.56) -- (1.854,7.641) -- (1.935,7.762) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.078,7.789) -- (5.049,7.932) -- (4.968,8.053) -- (4.847,8.133) -- (4.704,8.162) -- (4.562,8.133) -- (4.44,8.053) -- (4.36,7.932) -- (4.331,7.789) -- (4.36,7.646) -- (4.44,7.525) -- (4.562,7.444) -- (4.704,7.415) -- (4.847,7.444) -- (4.968,7.525) -- (5.049,7.646) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (10.805,7.624) -- (10.777,7.767) -- (10.696,7.888) -- (10.575,7.969) -- (10.432,7.997) -- (10.289,7.969) -- (10.168,7.888) -- (10.087,7.767) -- (10.059,7.624) -- (10.087,7.481) -- (10.168,7.36) -- (10.289,7.279) -- (10.432,7.251) -- (10.575,7.279) -- (10.696,7.36) -- (10.777,7.481) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.927,7.565) -- (7.899,7.708) -- (7.818,7.829) -- (7.697,7.91) -- (7.554,7.939) -- (7.411,7.91) -- (7.29,7.829) -- (7.209,7.708) -- (7.181,7.565) -- (7.209,7.423) -- (7.29,7.302) -- (7.411,7.221) -- (7.554,7.192) -- (7.697,7.221) -- (7.818,7.302) -- (7.899,7.423) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.875,5.224) -- (7.846,5.366) -- (7.765,5.487) -- (7.644,5.568) -- (7.501,5.597) -- (7.359,5.568) -- (7.238,5.487) -- (7.157,5.366) -- (7.128,5.224) -- (7.157,5.081) -- (7.238,4.96) -- (7.359,4.879) -- (7.501,4.85) -- (7.644,4.879) -- (7.765,4.96) -- (7.846,5.081) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.065,5.071) -- (5.036,5.214) -- (4.955,5.335) -- (4.834,5.416) -- (4.691,5.444) -- (4.549,5.416) -- (4.428,5.335) -- (4.347,5.214) -- (4.318,5.071) -- (4.347,4.928) -- (4.428,4.807) -- (4.549,4.726) -- (4.691,4.698) -- (4.834,4.726) -- (4.955,4.807) -- (5.036,4.928) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.019,4.76) -- (10.99,4.903) -- (10.91,5.024) -- (10.788,5.105) -- (10.646,5.133) -- (10.503,5.105) -- (10.382,5.024) -- (10.301,4.903) -- (10.272,4.76) -- (10.301,4.617) -- (10.382,4.496) -- (10.503,4.415) -- (10.646,4.387) -- (10.788,4.415) -- (10.91,4.496) -- (10.99,4.617) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.683,4.573) -- (1.654,4.716) -- (1.574,4.837) -- (1.452,4.918) -- (1.31,4.947) -- (1.167,4.918) -- (1.046,4.837) -- (0.965,4.716) -- (0.936,4.573) -- (0.965,4.431) -- (1.046,4.309) -- (1.167,4.229) -- (1.31,4.2) -- (1.452,4.229) -- (1.574,4.309) -- (1.654,4.431) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.203,1.877) -- (5.175,2.02) -- (5.094,2.141) -- (4.973,2.222) -- (4.83,2.251) -- (4.687,2.222) -- (4.566,2.141) -- (4.485,2.02) -- (4.457,1.877) -- (4.485,1.735) -- (4.566,1.614) -- (4.687,1.533) -- (4.83,1.504) -- (4.973,1.533) -- (5.094,1.614) -- (5.175,1.735) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.268,1.471) -- (2.24,1.614) -- (2.159,1.735) -- (2.038,1.816) -- (1.895,1.845) -- (1.752,1.816) -- (1.631,1.735) -- (1.55,1.614) -- (1.522,1.471) -- (1.55,1.328) -- (1.631,1.207) -- (1.752,1.126) -- (1.895,1.098) -- (2.038,1.126) -- (2.159,1.207) -- (2.24,1.328) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.843,9.126) -- (7.962,9.145) -- (8.069,9.2) -- (8.154,9.285) -- (8.209,9.392) -- (8.228,9.511) -- (8.209,9.63) -- (8.154,9.737) -- (8.069,9.822) -- (7.962,9.877) -- (7.843,9.895) -- (7.843,9.895) -- (7.724,9.877) -- (7.617,9.822) -- (7.532,9.737) -- (7.477,9.63) -- (7.459,9.511) -- (7.477,9.392) -- (7.532,9.285) -- (7.617,9.2) -- (7.724,9.145) -- (7.843,9.126) -- cycle;
\node at (7.843,9.511) {\scalebox{1.542}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.873,11.465) -- (5.992,11.484) -- (6.099,11.538) -- (6.184,11.623) -- (6.239,11.73) -- (6.258,11.849) -- (6.239,11.968) -- (6.184,12.075) -- (6.099,12.16) -- (5.992,12.215) -- (5.873,12.234) -- (5.873,12.234) -- (5.754,12.215) -- (5.647,12.16) -- (5.562,12.075) -- (5.508,11.968) -- (5.489,11.849) -- (5.508,11.73) -- (5.562,11.623) -- (5.647,11.538) -- (5.754,11.484) -- (5.873,11.465) -- cycle;
\node at (5.873,11.849) {\scalebox{1.542}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.668,9.134) -- (4.786,9.153) -- (4.894,9.207) -- (4.979,9.292) -- (5.033,9.4) -- (5.052,9.518) -- (5.033,9.637) -- (4.979,9.744) -- (4.894,9.829) -- (4.786,9.884) -- (4.668,9.903) -- (4.668,9.903) -- (4.549,9.884) -- (4.442,9.829) -- (4.356,9.744) -- (4.302,9.637) -- (4.283,9.518) -- (4.302,9.4) -- (4.356,9.292) -- (4.442,9.207) -- (4.549,9.153) -- (4.668,9.134) -- cycle;
\node at (4.668,9.518) {\scalebox{1.542}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.161,9.077) -- (2.28,9.095) -- (2.388,9.15) -- (2.473,9.235) -- (2.527,9.342) -- (2.546,9.461) -- (2.527,9.58) -- (2.473,9.687) -- (2.388,9.772) -- (2.28,9.827) -- (2.161,9.846) -- (2.161,9.846) -- (2.043,9.827) -- (1.935,9.772) -- (1.85,9.687) -- (1.796,9.58) -- (1.777,9.461) -- (1.796,9.342) -- (1.85,9.235) -- (1.935,9.15) -- (2.043,9.095) -- (2.161,9.077) -- cycle;
\node at (2.161,9.461) {\scalebox{1.542}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.157,7.718) -- (3.276,7.737) -- (3.383,7.792) -- (3.468,7.877) -- (3.523,7.984) -- (3.541,8.103) -- (3.523,8.221) -- (3.468,8.329) -- (3.383,8.414) -- (3.276,8.468) -- (3.157,8.487) -- (3.157,8.487) -- (3.038,8.468) -- (2.931,8.414) -- (2.846,8.329) -- (2.791,8.221) -- (2.772,8.103) -- (2.791,7.984) -- (2.846,7.877) -- (2.931,7.792) -- (3.038,7.737) -- (3.157,7.718) -- cycle;
\node at (3.157,8.103) {\scalebox{1.542}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.971,5.811) -- (2.089,5.83) -- (2.197,5.884) -- (2.282,5.969) -- (2.336,6.077) -- (2.355,6.195) -- (2.336,6.314) -- (2.282,6.421) -- (2.197,6.506) -- (2.089,6.561) -- (1.971,6.58) -- (1.971,6.58) -- (1.852,6.561) -- (1.745,6.506) -- (1.659,6.421) -- (1.605,6.314) -- (1.586,6.195) -- (1.605,6.077) -- (1.659,5.969) -- (1.745,5.884) -- (1.852,5.83) -- (1.971,5.811) -- cycle;
\node at (1.971,6.195) {\scalebox{1.542}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.15,7.558) -- (6.269,7.577) -- (6.376,7.632) -- (6.461,7.717) -- (6.516,7.824) -- (6.534,7.943) -- (6.516,8.062) -- (6.461,8.169) -- (6.376,8.254) -- (6.269,8.309) -- (6.15,8.327) -- (6.15,8.327) -- (6.031,8.309) -- (5.924,8.254) -- (5.839,8.169) -- (5.784,8.062) -- (5.765,7.943) -- (5.784,7.824) -- (5.839,7.717) -- (5.924,7.632) -- (6.031,7.577) -- (6.15,7.558) -- cycle;
\node at (6.15,7.943) {\scalebox{1.542}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.195,6.043) -- (5.313,6.062) -- (5.421,6.116) -- (5.506,6.201) -- (5.56,6.309) -- (5.579,6.427) -- (5.56,6.546) -- (5.506,6.653) -- (5.421,6.739) -- (5.313,6.793) -- (5.195,6.812) -- (5.195,6.812) -- (5.076,6.793) -- (4.969,6.739) -- (4.883,6.653) -- (4.829,6.546) -- (4.81,6.427) -- (4.829,6.309) -- (4.883,6.201) -- (4.969,6.116) -- (5.076,6.062) -- (5.195,6.043) -- cycle;
\node at (5.195,6.427) {\scalebox{1.542}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.804,5.827) -- (10.923,5.846) -- (11.03,5.901) -- (11.115,5.986) -- (11.17,6.093) -- (11.189,6.212) -- (11.17,6.331) -- (11.115,6.438) -- (11.03,6.523) -- (10.923,6.578) -- (10.804,6.596) -- (10.804,6.596) -- (10.685,6.578) -- (10.578,6.523) -- (10.493,6.438) -- (10.439,6.331) -- (10.42,6.212) -- (10.439,6.093) -- (10.493,5.986) -- (10.578,5.901) -- (10.685,5.846) -- (10.804,5.827) -- cycle;
\node at (10.804,6.212) {\scalebox{1.542}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.595,7.712) -- (9.714,7.731) -- (9.821,7.786) -- (9.906,7.871) -- (9.961,7.978) -- (9.98,8.097) -- (9.961,8.216) -- (9.906,8.323) -- (9.821,8.408) -- (9.714,8.462) -- (9.595,8.481) -- (9.233,8.481) -- (9.114,8.462) -- (9.007,8.408) -- (8.922,8.323) -- (8.867,8.216) -- (8.849,8.097) -- (8.867,7.978) -- (8.922,7.871) -- (9.007,7.786) -- (9.114,7.731) -- (9.233,7.712) -- cycle;
\node at (9.414,8.097) {\scalebox{1.542}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.291,4.858) -- (9.41,4.877) -- (9.517,4.931) -- (9.603,5.016) -- (9.657,5.124) -- (9.676,5.242) -- (9.657,5.361) -- (9.603,5.468) -- (9.517,5.554) -- (9.41,5.608) -- (9.291,5.627) -- (8.93,5.627) -- (8.811,5.608) -- (8.704,5.554) -- (8.618,5.468) -- (8.564,5.361) -- (8.545,5.242) -- (8.564,5.124) -- (8.618,5.016) -- (8.704,4.931) -- (8.811,4.877) -- (8.93,4.858) -- cycle;
\node at (9.11,5.242) {\scalebox{1.542}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.25,5.263) -- (6.369,5.281) -- (6.476,5.336) -- (6.561,5.421) -- (6.616,5.528) -- (6.635,5.647) -- (6.616,5.766) -- (6.561,5.873) -- (6.476,5.958) -- (6.369,6.013) -- (6.25,6.032) -- (5.888,6.032) -- (5.77,6.013) -- (5.662,5.958) -- (5.577,5.873) -- (5.523,5.766) -- (5.504,5.647) -- (5.523,5.528) -- (5.577,5.421) -- (5.662,5.336) -- (5.77,5.281) -- (5.888,5.263) -- cycle;
\node at (6.069,5.647) {\scalebox{1.542}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.194,3.101) -- (5.313,3.119) -- (5.42,3.174) -- (5.505,3.259) -- (5.56,3.366) -- (5.579,3.485) -- (5.56,3.604) -- (5.505,3.711) -- (5.42,3.796) -- (5.313,3.851) -- (5.194,3.87) -- (4.832,3.87) -- (4.713,3.851) -- (4.606,3.796) -- (4.521,3.711) -- (4.466,3.604) -- (4.448,3.485) -- (4.466,3.366) -- (4.521,3.259) -- (4.606,3.174) -- (4.713,3.119) -- (4.832,3.101) -- cycle;
\node at (5.013,3.485) {\scalebox{1.542}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.105,4.958) -- (3.224,4.976) -- (3.331,5.031) -- (3.416,5.116) -- (3.471,5.223) -- (3.49,5.342) -- (3.471,5.461) -- (3.416,5.568) -- (3.331,5.653) -- (3.224,5.708) -- (3.105,5.727) -- (2.743,5.727) -- (2.624,5.708) -- (2.517,5.653) -- (2.432,5.568) -- (2.377,5.461) -- (2.359,5.342) -- (2.377,5.223) -- (2.432,5.116) -- (2.517,5.031) -- (2.624,4.976) -- (2.743,4.958) -- cycle;
\node at (2.924,5.342) {\scalebox{1.542}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.033,2.685) -- (2.152,2.704) -- (2.259,2.758) -- (2.344,2.843) -- (2.399,2.951) -- (2.418,3.07) -- (2.399,3.188) -- (2.344,3.296) -- (2.259,3.381) -- (2.152,3.435) -- (2.033,3.454) -- (1.671,3.454) -- (1.553,3.435) -- (1.445,3.381) -- (1.36,3.296) -- (1.306,3.188) -- (1.287,3.07) -- (1.306,2.951) -- (1.36,2.843) -- (1.445,2.758) -- (1.553,2.704) -- (1.671,2.685) -- cycle;
\node at (1.852,3.07) {\scalebox{1.542}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.474,1.792) -- (3.593,1.811) -- (3.7,1.865) -- (3.785,1.95) -- (3.84,2.058) -- (3.858,2.176) -- (3.84,2.295) -- (3.785,2.402) -- (3.7,2.487) -- (3.593,2.542) -- (3.474,2.561) -- (3.112,2.561) -- (2.993,2.542) -- (2.886,2.487) -- (2.801,2.402) -- (2.746,2.295) -- (2.727,2.176) -- (2.746,2.058) -- (2.801,1.95) -- (2.886,1.865) -- (2.993,1.811) -- (3.112,1.792) -- cycle;
\node at (3.293,2.176) {\scalebox{1.542}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.01,4.273) -- (1.61,4.273) -- (1.61,4.873) -- (1.01,4.873) -- cycle;
\node at (1.31,4.573) {\scalebox{1.542}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.66,10.038) -- (2.26,10.038) -- (2.26,10.638) -- (1.66,10.638) -- cycle;
\node at (1.96,10.338) {\scalebox{1.542}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (11.749,6.286) -- (12.349,6.286) -- (12.349,6.886) -- (11.749,6.886) -- cycle;
\node at (12.049,6.586) {\scalebox{1.542}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.766,6.369) -- (1.366,6.369) -- (1.366,6.969) -- (0.766,6.969) -- cycle;
\node at (1.066,6.669) {\scalebox{1.542}{\small d}};
\node at (7.967,12.024) {\scalebox{1.542}{\small A}};
\node at (3.997,11.77) {\scalebox{1.542}{\small B}};
\node at (1.309,11.532) {\scalebox{1.542}{\small C}};
\node at (0.999,8.092) {\scalebox{1.542}{\small D}};
\node at (4.405,8.332) {\scalebox{1.542}{\small E}};
\node at (11.043,7.73) {\scalebox{1.542}{\small F}};
\node at (8.136,7.779) {\scalebox{1.542}{\small G}};
\node at (8.014,4.874) {\scalebox{1.542}{\small H}};
\node at (4.483,4.487) {\scalebox{1.542}{\small I}};
\node at (11.227,4.546) {\scalebox{1.542}{\small J}};
\node at (0.765,4.277) {\scalebox{1.542}{\small K}};
\node at (4.772,1.26) {\scalebox{1.542}{\small L}};
\node at (1.559,0.95) {\scalebox{1.542}{\small M}};
\draw[line width=0.5pt] (11.946,10.521) -- (11.946,11.621);
\draw[line width=0.5pt] (11.715,11.236) -- (11.946,11.621) -- (12.177,11.236);
\draw[line width=0.5pt] (11.643,11.071) -- (12.248,11.071);
\node at (11.946,12.006) {\scalebox{1.542}{\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, F, H, K, L & 7 \\ \hline
clairs & B, D, G, I, J, M & 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{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{K} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend le \textbf{chemin des Vignes ⑮} vers le sud jusqu'à Coin du Lavoir \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Pasteur ⑯}, vers l'est, jusqu'à Place du Marché \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Verte ⑬}, vers le nord, jusqu'à Carrefour du Lavoir \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin du Lavoir ⑫}, vers l'est, jusqu'à Place des Tanneurs \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Pont ⑪}, vers l'est, jusqu'à Place aux Herbes \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Stade ⑨}, vers le nord, jusqu'à Placette des Ormes \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ⑩}, vers l'ouest, jusqu'à Rond-point du Pont \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Peupliers ①}, vers le nord, jusqu'à Place de l'Église \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin de Ronde ②}, vers l'ouest, jusqu'à Place de la Fontaine \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{boulevard du Parc ③}, vers le sud, jusqu'à Carrefour de l'Hôpital \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{quai aux Fleurs ⑤}, vers l'ouest, jusqu'à Carrefour du Moulin \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{impasse du Vieux-Four ④}, vers le nord, jusqu'à Place Sainte-Croix \textbf{(C)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 13 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
