\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 352}

\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=352}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0352}
\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=0.6035, line join=round, line cap=round]
\draw[line width=0.5pt] (4.396,7.124) -- (4.318,6.931) -- (4.245,6.736) -- (4.176,6.539) -- (4.112,6.34) -- (4.053,6.14) -- (3.999,5.939) -- (3.949,5.736) -- (3.904,5.532) -- (3.864,5.328) -- (3.829,5.122) -- (3.798,4.916) -- (3.773,4.709) -- (3.752,4.501) -- (3.736,4.293);
\draw[line width=0.5pt] (1.604,6.777) -- (1.798,6.833) -- (1.993,6.883) -- (2.19,6.929) -- (2.388,6.97) -- (2.586,7.007) -- (2.785,7.039) -- (2.986,7.066) -- (3.186,7.089) -- (3.387,7.107) -- (3.589,7.12) -- (3.791,7.128) -- (3.992,7.132) -- (4.194,7.13) -- (4.396,7.124);
\draw[line width=0.5pt] (1.604,6.777) -- (1.561,6.603) -- (1.522,6.427) -- (1.488,6.25) -- (1.457,6.072) -- (1.431,5.894) -- (1.409,5.715) -- (1.391,5.536) -- (1.378,5.356) -- (1.368,5.176) -- (1.363,4.996) -- (1.362,4.816) -- (1.365,4.636) -- (1.373,4.456) -- (1.385,4.276);
\draw[line width=0.5pt] (11.366,6.747) -- (11.397,6.555) -- (11.423,6.362) -- (11.445,6.169) -- (11.463,5.975) -- (11.475,5.781) -- (11.484,5.586) -- (11.487,5.392) -- (11.486,5.197) -- (11.481,5.002) -- (11.471,4.808) -- (11.456,4.614) -- (11.437,4.42) -- (11.413,4.227) -- (11.385,4.034);
\draw[line width=0.5pt] (9.078,6.519) -- (9.243,6.51) -- (9.408,6.506) -- (9.573,6.504) -- (9.738,6.507) -- (9.903,6.514) -- (10.067,6.525) -- (10.232,6.539) -- (10.396,6.557) -- (10.559,6.579) -- (10.722,6.605) -- (10.884,6.635) -- (11.046,6.669) -- (11.206,6.706) -- (11.366,6.747);
\draw[line width=0.5pt] (9.078,6.519) -- (9.104,6.349) -- (9.126,6.178) -- (9.144,6.007) -- (9.158,5.835) -- (9.167,5.663) -- (9.173,5.491) -- (9.174,5.319) -- (9.172,5.146) -- (9.166,4.974) -- (9.155,4.802) -- (9.14,4.631) -- (9.122,4.459) -- (9.099,4.289) -- (9.072,4.119);
\draw[line width=0.5pt] (3.736,4.293) -- (3.922,4.254) -- (4.108,4.219) -- (4.295,4.188) -- (4.483,4.161) -- (4.672,4.139) -- (4.861,4.122) -- (5.05,4.109) -- (5.24,4.1) -- (5.43,4.096) -- (5.62,4.096) -- (5.809,4.101) -- (5.999,4.11) -- (6.188,4.124) -- (6.377,4.142);
\draw[line width=0.5pt] (3.736,4.293) -- (3.794,4.107) -- (3.848,3.919) -- (3.898,3.729) -- (3.943,3.539) -- (3.983,3.347) -- (4.019,3.155) -- (4.05,2.961) -- (4.077,2.768) -- (4.099,2.573) -- (4.117,2.378) -- (4.13,2.183) -- (4.139,1.987) -- (4.143,1.792) -- (4.142,1.596);
\draw[line width=0.5pt] (1.385,4.276) -- (1.552,4.252) -- (1.719,4.231) -- (1.887,4.215) -- (2.055,4.202) -- (2.224,4.193) -- (2.392,4.189) -- (2.561,4.188) -- (2.73,4.191) -- (2.899,4.198) -- (3.067,4.209) -- (3.235,4.224) -- (3.403,4.243) -- (3.57,4.266) -- (3.736,4.293);
\draw[line width=0.5pt] (1.385,4.276) -- (1.436,4.085) -- (1.482,3.893) -- (1.524,3.7) -- (1.561,3.505) -- (1.594,3.31) -- (1.622,3.114) -- (1.645,2.918) -- (1.664,2.721) -- (1.678,2.523) -- (1.688,2.326) -- (1.693,2.128) -- (1.693,1.93) -- (1.689,1.732) -- (1.679,1.535);
\draw[line width=0.5pt] (6.377,4.142) -- (6.568,4.111) -- (6.759,4.084) -- (6.951,4.062) -- (7.144,4.045) -- (7.337,4.032) -- (7.53,4.023) -- (7.724,4.019) -- (7.917,4.02) -- (8.111,4.025) -- (8.304,4.035) -- (8.497,4.049) -- (8.689,4.068) -- (8.881,4.091) -- (9.072,4.119);
\draw[line width=0.5pt] (6.377,4.142) -- (6.433,3.962) -- (6.485,3.781) -- (6.533,3.599) -- (6.577,3.415) -- (6.616,3.231) -- (6.651,3.045) -- (6.681,2.859) -- (6.708,2.672) -- (6.729,2.485) -- (6.747,2.297) -- (6.76,2.109) -- (6.768,1.921) -- (6.772,1.732) -- (6.772,1.544);
\draw[line width=0.5pt] (9.072,4.119) -- (9.235,4.087) -- (9.399,4.06) -- (9.563,4.036) -- (9.728,4.017) -- (9.894,4.001) -- (10.059,3.989) -- (10.225,3.981) -- (10.391,3.977) -- (10.557,3.977) -- (10.723,3.981) -- (10.889,3.988) -- (11.055,4.0) -- (11.22,4.015) -- (11.385,4.034);
\draw[line width=0.5pt] (4.142,1.596) -- (4.329,1.621) -- (4.516,1.642) -- (4.704,1.658) -- (4.893,1.67) -- (5.081,1.677) -- (5.27,1.68) -- (5.459,1.678) -- (5.648,1.672) -- (5.836,1.662) -- (6.024,1.647) -- (6.212,1.628) -- (6.399,1.604) -- (6.586,1.576) -- (6.772,1.544);
\draw[line width=0.5pt] (1.679,1.535) -- (1.855,1.512) -- (2.03,1.494) -- (2.206,1.48) -- (2.383,1.469) -- (2.56,1.463) -- (2.736,1.462) -- (2.913,1.464) -- (3.09,1.47) -- (3.266,1.481) -- (3.443,1.496) -- (3.618,1.515) -- (3.794,1.538) -- (3.968,1.565) -- (4.142,1.596);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.556,7.124) -- (4.534,7.204) -- (4.476,7.263) -- (4.396,7.284) -- (4.316,7.263) -- (4.257,7.204) -- (4.236,7.124) -- (4.257,7.044) -- (4.316,6.986) -- (4.396,6.964) -- (4.476,6.986) -- (4.534,7.044) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.764,6.777) -- (1.743,6.857) -- (1.684,6.916) -- (1.604,6.937) -- (1.524,6.916) -- (1.465,6.857) -- (1.444,6.777) -- (1.465,6.697) -- (1.524,6.639) -- (1.604,6.617) -- (1.684,6.639) -- (1.743,6.697) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.526,6.747) -- (11.505,6.827) -- (11.446,6.886) -- (11.366,6.907) -- (11.286,6.886) -- (11.227,6.827) -- (11.206,6.747) -- (11.227,6.667) -- (11.286,6.609) -- (11.366,6.587) -- (11.446,6.609) -- (11.505,6.667) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.238,6.519) -- (9.217,6.599) -- (9.158,6.658) -- (9.078,6.679) -- (8.998,6.658) -- (8.94,6.599) -- (8.918,6.519) -- (8.94,6.439) -- (8.998,6.381) -- (9.078,6.359) -- (9.158,6.381) -- (9.217,6.439) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.896,4.293) -- (3.875,4.373) -- (3.816,4.432) -- (3.736,4.453) -- (3.656,4.432) -- (3.597,4.373) -- (3.576,4.293) -- (3.597,4.213) -- (3.656,4.155) -- (3.736,4.133) -- (3.816,4.155) -- (3.875,4.213) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.545,4.276) -- (1.523,4.356) -- (1.465,4.415) -- (1.385,4.436) -- (1.305,4.415) -- (1.246,4.356) -- (1.225,4.276) -- (1.246,4.196) -- (1.305,4.138) -- (1.385,4.116) -- (1.465,4.138) -- (1.523,4.196) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.537,4.142) -- (6.515,4.222) -- (6.457,4.281) -- (6.377,4.302) -- (6.297,4.281) -- (6.238,4.222) -- (6.217,4.142) -- (6.238,4.062) -- (6.297,4.004) -- (6.377,3.982) -- (6.457,4.004) -- (6.515,4.062) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.232,4.119) -- (9.211,4.199) -- (9.152,4.257) -- (9.072,4.279) -- (8.992,4.257) -- (8.934,4.199) -- (8.912,4.119) -- (8.934,4.039) -- (8.992,3.98) -- (9.072,3.959) -- (9.152,3.98) -- (9.211,4.039) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.545,4.034) -- (11.523,4.114) -- (11.465,4.173) -- (11.385,4.194) -- (11.305,4.173) -- (11.246,4.114) -- (11.225,4.034) -- (11.246,3.954) -- (11.305,3.896) -- (11.385,3.874) -- (11.465,3.896) -- (11.523,3.954) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.302,1.596) -- (4.281,1.676) -- (4.222,1.735) -- (4.142,1.756) -- (4.062,1.735) -- (4.003,1.676) -- (3.982,1.596) -- (4.003,1.516) -- (4.062,1.458) -- (4.142,1.436) -- (4.222,1.458) -- (4.281,1.516) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.932,1.544) -- (6.91,1.624) -- (6.852,1.682) -- (6.772,1.704) -- (6.692,1.682) -- (6.633,1.624) -- (6.612,1.544) -- (6.633,1.464) -- (6.692,1.405) -- (6.772,1.384) -- (6.852,1.405) -- (6.91,1.464) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.839,1.535) -- (1.818,1.615) -- (1.759,1.673) -- (1.679,1.695) -- (1.599,1.673) -- (1.541,1.615) -- (1.519,1.535) -- (1.541,1.455) -- (1.599,1.396) -- (1.679,1.375) -- (1.759,1.396) -- (1.818,1.455) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.333,5.253) -- (4.454,5.272) -- (4.564,5.328) -- (4.651,5.415) -- (4.707,5.525) -- (4.727,5.647) -- (4.707,5.768) -- (4.651,5.878) -- (4.564,5.965) -- (4.454,6.021) -- (4.333,6.041) -- (4.333,6.041) -- (4.211,6.021) -- (4.101,5.965) -- (4.014,5.878) -- (3.958,5.768) -- (3.939,5.647) -- (3.958,5.525) -- (4.014,5.415) -- (4.101,5.328) -- (4.211,5.272) -- (4.333,5.253) -- cycle;
\node at (4.333,5.647) {\scalebox{1.657}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.937,7.063) -- (3.059,7.082) -- (3.168,7.138) -- (3.256,7.225) -- (3.312,7.335) -- (3.331,7.457) -- (3.312,7.579) -- (3.256,7.689) -- (3.168,7.776) -- (3.059,7.832) -- (2.937,7.851) -- (2.937,7.851) -- (2.815,7.832) -- (2.705,7.776) -- (2.618,7.689) -- (2.562,7.579) -- (2.543,7.457) -- (2.562,7.335) -- (2.618,7.225) -- (2.705,7.138) -- (2.815,7.082) -- (2.937,7.063) -- cycle;
\node at (2.937,7.457) {\scalebox{1.657}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.784,5.108) -- (1.905,5.127) -- (2.015,5.183) -- (2.102,5.27) -- (2.158,5.38) -- (2.178,5.501) -- (2.158,5.623) -- (2.102,5.733) -- (2.015,5.82) -- (1.905,5.876) -- (1.784,5.895) -- (1.784,5.895) -- (1.662,5.876) -- (1.552,5.82) -- (1.465,5.733) -- (1.409,5.623) -- (1.39,5.501) -- (1.409,5.38) -- (1.465,5.27) -- (1.552,5.183) -- (1.662,5.127) -- (1.784,5.108) -- cycle;
\node at (1.784,5.501) {\scalebox{1.657}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.881,5.0) -- (12.003,5.02) -- (12.113,5.076) -- (12.2,5.163) -- (12.256,5.273) -- (12.275,5.394) -- (12.256,5.516) -- (12.2,5.626) -- (12.113,5.713) -- (12.003,5.769) -- (11.881,5.788) -- (11.881,5.788) -- (11.759,5.769) -- (11.65,5.713) -- (11.563,5.626) -- (11.507,5.516) -- (11.487,5.394) -- (11.507,5.273) -- (11.563,5.163) -- (11.65,5.076) -- (11.759,5.02) -- (11.881,5.0) -- cycle;
\node at (11.881,5.394) {\scalebox{1.657}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.271,5.753) -- (10.392,5.772) -- (10.502,5.828) -- (10.589,5.915) -- (10.645,6.025) -- (10.665,6.147) -- (10.645,6.269) -- (10.589,6.379) -- (10.502,6.466) -- (10.392,6.522) -- (10.271,6.541) -- (10.271,6.541) -- (10.149,6.522) -- (10.039,6.466) -- (9.952,6.379) -- (9.896,6.269) -- (9.877,6.147) -- (9.896,6.025) -- (9.952,5.915) -- (10.039,5.828) -- (10.149,5.772) -- (10.271,5.753) -- cycle;
\node at (10.271,6.147) {\scalebox{1.657}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.568,4.924) -- (9.69,4.943) -- (9.8,4.999) -- (9.887,5.086) -- (9.943,5.196) -- (9.962,5.318) -- (9.943,5.439) -- (9.887,5.549) -- (9.8,5.636) -- (9.69,5.692) -- (9.568,5.712) -- (9.568,5.712) -- (9.447,5.692) -- (9.337,5.636) -- (9.25,5.549) -- (9.194,5.439) -- (9.174,5.318) -- (9.194,5.196) -- (9.25,5.086) -- (9.337,4.999) -- (9.447,4.943) -- (9.568,4.924) -- cycle;
\node at (9.568,5.318) {\scalebox{1.657}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.073,4.108) -- (5.194,4.127) -- (5.304,4.183) -- (5.391,4.271) -- (5.447,4.38) -- (5.467,4.502) -- (5.447,4.624) -- (5.391,4.734) -- (5.304,4.821) -- (5.194,4.877) -- (5.073,4.896) -- (5.073,4.896) -- (4.951,4.877) -- (4.841,4.821) -- (4.754,4.734) -- (4.698,4.624) -- (4.679,4.502) -- (4.698,4.38) -- (4.754,4.271) -- (4.841,4.183) -- (4.951,4.127) -- (5.073,4.108) -- cycle;
\node at (5.073,4.502) {\scalebox{1.657}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.44,2.626) -- (4.562,2.645) -- (4.671,2.701) -- (4.758,2.789) -- (4.814,2.898) -- (4.834,3.02) -- (4.814,3.142) -- (4.758,3.252) -- (4.671,3.339) -- (4.562,3.395) -- (4.44,3.414) -- (4.44,3.414) -- (4.318,3.395) -- (4.208,3.339) -- (4.121,3.252) -- (4.065,3.142) -- (4.046,3.02) -- (4.065,2.898) -- (4.121,2.789) -- (4.208,2.701) -- (4.318,2.645) -- (4.44,2.626) -- cycle;
\node at (4.44,3.02) {\scalebox{1.657}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.383,3.239) -- (2.505,3.258) -- (2.615,3.314) -- (2.702,3.401) -- (2.758,3.511) -- (2.777,3.632) -- (2.758,3.754) -- (2.702,3.864) -- (2.615,3.951) -- (2.505,4.007) -- (2.383,4.026) -- (2.383,4.026) -- (2.262,4.007) -- (2.152,3.951) -- (2.065,3.864) -- (2.009,3.754) -- (1.989,3.632) -- (2.009,3.511) -- (2.065,3.401) -- (2.152,3.314) -- (2.262,3.258) -- (2.383,3.239) -- cycle;
\node at (2.383,3.632) {\scalebox{1.657}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.439,2.482) -- (1.561,2.501) -- (1.67,2.557) -- (1.758,2.644) -- (1.814,2.754) -- (1.833,2.876) -- (1.814,2.997) -- (1.758,3.107) -- (1.67,3.194) -- (1.561,3.25) -- (1.439,3.269) -- (1.068,3.269) -- (0.946,3.25) -- (0.837,3.194) -- (0.749,3.107) -- (0.694,2.997) -- (0.674,2.876) -- (0.694,2.754) -- (0.749,2.644) -- (0.837,2.557) -- (0.946,2.501) -- (1.068,2.482) -- cycle;
\node at (1.254,2.876) {\scalebox{1.657}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.914,4.181) -- (8.036,4.201) -- (8.145,4.257) -- (8.233,4.344) -- (8.289,4.454) -- (8.308,4.575) -- (8.289,4.697) -- (8.233,4.807) -- (8.145,4.894) -- (8.036,4.95) -- (7.914,4.969) -- (7.543,4.969) -- (7.421,4.95) -- (7.312,4.894) -- (7.225,4.807) -- (7.169,4.697) -- (7.149,4.575) -- (7.169,4.454) -- (7.225,4.344) -- (7.312,4.257) -- (7.421,4.201) -- (7.543,4.181) -- cycle;
\node at (7.729,4.575) {\scalebox{1.657}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.477,2.406) -- (6.599,2.426) -- (6.709,2.481) -- (6.796,2.569) -- (6.852,2.678) -- (6.871,2.8) -- (6.852,2.922) -- (6.796,3.032) -- (6.709,3.119) -- (6.599,3.175) -- (6.477,3.194) -- (6.107,3.194) -- (5.985,3.175) -- (5.875,3.119) -- (5.788,3.032) -- (5.732,2.922) -- (5.713,2.8) -- (5.732,2.678) -- (5.788,2.569) -- (5.875,2.481) -- (5.985,2.426) -- (6.107,2.406) -- cycle;
\node at (6.292,2.8) {\scalebox{1.657}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.735,4.324) -- (10.857,4.344) -- (10.966,4.4) -- (11.054,4.487) -- (11.109,4.597) -- (11.129,4.718) -- (11.109,4.84) -- (11.054,4.95) -- (10.966,5.037) -- (10.857,5.093) -- (10.735,5.112) -- (10.364,5.112) -- (10.242,5.093) -- (10.133,5.037) -- (10.045,4.95) -- (9.989,4.84) -- (9.97,4.718) -- (9.989,4.597) -- (10.045,4.487) -- (10.133,4.4) -- (10.242,4.344) -- (10.364,4.324) -- cycle;
\node at (10.549,4.718) {\scalebox{1.657}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.636,0.891) -- (5.758,0.91) -- (5.868,0.966) -- (5.955,1.053) -- (6.011,1.163) -- (6.03,1.285) -- (6.011,1.406) -- (5.955,1.516) -- (5.868,1.603) -- (5.758,1.659) -- (5.636,1.679) -- (5.266,1.679) -- (5.144,1.659) -- (5.034,1.603) -- (4.947,1.516) -- (4.891,1.406) -- (4.872,1.285) -- (4.891,1.163) -- (4.947,1.053) -- (5.034,0.966) -- (5.144,0.91) -- (5.266,0.891) -- cycle;
\node at (5.451,1.285) {\scalebox{1.657}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.089,1.464) -- (3.211,1.483) -- (3.32,1.539) -- (3.407,1.626) -- (3.463,1.736) -- (3.483,1.858) -- (3.463,1.979) -- (3.407,2.089) -- (3.32,2.176) -- (3.211,2.232) -- (3.089,2.252) -- (2.718,2.252) -- (2.596,2.232) -- (2.487,2.176) -- (2.399,2.089) -- (2.343,1.979) -- (2.324,1.858) -- (2.343,1.736) -- (2.399,1.626) -- (2.487,1.539) -- (2.596,1.483) -- (2.718,1.464) -- cycle;
\node at (2.903,1.858) {\scalebox{1.657}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.304,6.477) -- (1.904,6.477) -- (1.904,7.077) -- (1.304,7.077) -- cycle;
\node at (1.604,6.777) {\scalebox{1.657}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.375,1.076) -- (1.975,1.076) -- (1.975,1.676) -- (1.375,1.676) -- cycle;
\node at (1.675,1.376) {\scalebox{1.657}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.693,2.326) -- (8.293,2.326) -- (8.293,2.926) -- (7.693,2.926) -- cycle;
\node at (7.993,2.626) {\scalebox{1.657}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (0.642,5.895) -- (1.242,5.895) -- (1.242,6.495) -- (0.642,6.495) -- cycle;
\node at (0.942,6.195) {\scalebox{1.657}{\small d}};
\node at (4.094,7.666) {\scalebox{1.657}{\small A}};
\node at (1.062,7.077) {\scalebox{1.657}{\small B}};
\node at (11.935,6.993) {\scalebox{1.657}{\small C}};
\node at (9.593,6.865) {\scalebox{1.657}{\small D}};
\node at (3.117,4.265) {\scalebox{1.657}{\small E}};
\node at (0.765,4.26) {\scalebox{1.657}{\small F}};
\node at (6.922,3.847) {\scalebox{1.657}{\small G}};
\node at (9.69,4.065) {\scalebox{1.657}{\small H}};
\node at (12.003,3.994) {\scalebox{1.657}{\small I}};
\node at (3.81,1.073) {\scalebox{1.657}{\small J}};
\node at (6.95,0.95) {\scalebox{1.657}{\small K}};
\node at (1.165,1.188) {\scalebox{1.657}{\small L}};
\draw[line width=0.5pt] (12.685,6.174) -- (12.685,7.274);
\draw[line width=0.5pt] (12.454,6.889) -- (12.685,7.274) -- (12.916,6.889);
\draw[line width=0.5pt] (12.382,6.724) -- (12.987,6.724);
\node at (12.685,7.659) {\scalebox{1.657}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour B, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ rue Haute & Ⓑ Place des Tanneurs & Ⓚ Fontaine Notre-Dame \\ \hline
① chemin Creux & ⑩ sentier des Prés & Ⓒ Place de l'Église & Ⓛ Place du Marché \\ \hline
② rue du Stade & ⑪ quai aux Fleurs & Ⓓ Place du Vieux-Puits & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ rue du Puits & ⑫ rue Basse & Ⓔ Carrefour des Tilleuls & \textbf{a} l'école \\ \hline
④ passage du Marché & ⑬ rue de la Poste & Ⓕ Carrefour du Moulin & \textbf{b} la salle des fêtes \\ \hline
⑤ chemin du Lavoir & ⑭ rue Verte & Ⓖ Carrefour de l'Hôpital & \textbf{c} le cimetière \\ \hline
⑥ allée des Tilleuls & ⑮ rue du Pont & Ⓗ Rond-point de la Poste & \textbf{d} la boulangerie \\ \hline
⑦ rue Pasteur & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Place de la Fontaine &  \\ \hline
⑧ rue du Moulin & Ⓐ Place Sainte-Croix & Ⓙ Rond-point de la Gare &  \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\begin{enumerate}
  \item Colorier les carrefours du plan avec \textbf{deux couleurs}, de sorte que deux carrefours \textbf{reliés par une rue} ne soient jamais de la même couleur. Combien y en a-t-il de chaque couleur ?
  \item Le car roule d'un carrefour au suivant. Que peut-on dire de la \textbf{couleur} des arrêts successifs ? Qu'en déduire sur les couleurs des arrêts d'une tournée qui les dessert tous ?
  \item \textbf{Le car peut-il desservir tous les carrefours, une fois chacun, et revenir à l'école ?} Si oui, décrire un trajet ; sinon, démontrer que c'est impossible.
  \item \textbf{Et s'il n'est pas obligé de revenir à l'école ?} Même consigne.
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage — Correction}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6182, line join=round, line cap=round]
\draw[line width=0.5pt] (4.396,7.124) -- (4.318,6.931) -- (4.245,6.736) -- (4.176,6.539) -- (4.112,6.34) -- (4.053,6.14) -- (3.999,5.939) -- (3.949,5.736) -- (3.904,5.532) -- (3.864,5.328) -- (3.829,5.122) -- (3.798,4.916) -- (3.773,4.709) -- (3.752,4.501) -- (3.736,4.293);
\draw[line width=0.5pt] (1.604,6.777) -- (1.798,6.833) -- (1.993,6.883) -- (2.19,6.929) -- (2.388,6.97) -- (2.586,7.007) -- (2.785,7.039) -- (2.986,7.066) -- (3.186,7.089) -- (3.387,7.107) -- (3.589,7.12) -- (3.791,7.128) -- (3.992,7.132) -- (4.194,7.13) -- (4.396,7.124);
\draw[line width=0.5pt] (1.604,6.777) -- (1.561,6.603) -- (1.522,6.427) -- (1.488,6.25) -- (1.457,6.072) -- (1.431,5.894) -- (1.409,5.715) -- (1.391,5.536) -- (1.378,5.356) -- (1.368,5.176) -- (1.363,4.996) -- (1.362,4.816) -- (1.365,4.636) -- (1.373,4.456) -- (1.385,4.276);
\draw[line width=0.5pt] (11.366,6.747) -- (11.397,6.555) -- (11.423,6.362) -- (11.445,6.169) -- (11.463,5.975) -- (11.475,5.781) -- (11.484,5.586) -- (11.487,5.392) -- (11.486,5.197) -- (11.481,5.002) -- (11.471,4.808) -- (11.456,4.614) -- (11.437,4.42) -- (11.413,4.227) -- (11.385,4.034);
\draw[line width=0.5pt] (9.078,6.519) -- (9.243,6.51) -- (9.408,6.506) -- (9.573,6.504) -- (9.738,6.507) -- (9.903,6.514) -- (10.067,6.525) -- (10.232,6.539) -- (10.396,6.557) -- (10.559,6.579) -- (10.722,6.605) -- (10.884,6.635) -- (11.046,6.669) -- (11.206,6.706) -- (11.366,6.747);
\draw[line width=0.5pt] (9.078,6.519) -- (9.104,6.349) -- (9.126,6.178) -- (9.144,6.007) -- (9.158,5.835) -- (9.167,5.663) -- (9.173,5.491) -- (9.174,5.319) -- (9.172,5.146) -- (9.166,4.974) -- (9.155,4.802) -- (9.14,4.631) -- (9.122,4.459) -- (9.099,4.289) -- (9.072,4.119);
\draw[line width=0.5pt] (3.736,4.293) -- (3.922,4.254) -- (4.108,4.219) -- (4.295,4.188) -- (4.483,4.161) -- (4.672,4.139) -- (4.861,4.122) -- (5.05,4.109) -- (5.24,4.1) -- (5.43,4.096) -- (5.62,4.096) -- (5.809,4.101) -- (5.999,4.11) -- (6.188,4.124) -- (6.377,4.142);
\draw[line width=0.5pt] (3.736,4.293) -- (3.794,4.107) -- (3.848,3.919) -- (3.898,3.729) -- (3.943,3.539) -- (3.983,3.347) -- (4.019,3.155) -- (4.05,2.961) -- (4.077,2.768) -- (4.099,2.573) -- (4.117,2.378) -- (4.13,2.183) -- (4.139,1.987) -- (4.143,1.792) -- (4.142,1.596);
\draw[line width=0.5pt] (1.385,4.276) -- (1.552,4.252) -- (1.719,4.231) -- (1.887,4.215) -- (2.055,4.202) -- (2.224,4.193) -- (2.392,4.189) -- (2.561,4.188) -- (2.73,4.191) -- (2.899,4.198) -- (3.067,4.209) -- (3.235,4.224) -- (3.403,4.243) -- (3.57,4.266) -- (3.736,4.293);
\draw[line width=0.5pt] (1.385,4.276) -- (1.436,4.085) -- (1.482,3.893) -- (1.524,3.7) -- (1.561,3.505) -- (1.594,3.31) -- (1.622,3.114) -- (1.645,2.918) -- (1.664,2.721) -- (1.678,2.523) -- (1.688,2.326) -- (1.693,2.128) -- (1.693,1.93) -- (1.689,1.732) -- (1.679,1.535);
\draw[line width=0.5pt] (6.377,4.142) -- (6.568,4.111) -- (6.759,4.084) -- (6.951,4.062) -- (7.144,4.045) -- (7.337,4.032) -- (7.53,4.023) -- (7.724,4.019) -- (7.917,4.02) -- (8.111,4.025) -- (8.304,4.035) -- (8.497,4.049) -- (8.689,4.068) -- (8.881,4.091) -- (9.072,4.119);
\draw[line width=0.5pt] (6.377,4.142) -- (6.433,3.962) -- (6.485,3.781) -- (6.533,3.599) -- (6.577,3.415) -- (6.616,3.231) -- (6.651,3.045) -- (6.681,2.859) -- (6.708,2.672) -- (6.729,2.485) -- (6.747,2.297) -- (6.76,2.109) -- (6.768,1.921) -- (6.772,1.732) -- (6.772,1.544);
\draw[line width=0.5pt] (9.072,4.119) -- (9.235,4.087) -- (9.399,4.06) -- (9.563,4.036) -- (9.728,4.017) -- (9.894,4.001) -- (10.059,3.989) -- (10.225,3.981) -- (10.391,3.977) -- (10.557,3.977) -- (10.723,3.981) -- (10.889,3.988) -- (11.055,4.0) -- (11.22,4.015) -- (11.385,4.034);
\draw[line width=0.5pt] (4.142,1.596) -- (4.329,1.621) -- (4.516,1.642) -- (4.704,1.658) -- (4.893,1.67) -- (5.081,1.677) -- (5.27,1.68) -- (5.459,1.678) -- (5.648,1.672) -- (5.836,1.662) -- (6.024,1.647) -- (6.212,1.628) -- (6.399,1.604) -- (6.586,1.576) -- (6.772,1.544);
\draw[line width=0.5pt] (1.679,1.535) -- (1.855,1.512) -- (2.03,1.494) -- (2.206,1.48) -- (2.383,1.469) -- (2.56,1.463) -- (2.736,1.462) -- (2.913,1.464) -- (3.09,1.47) -- (3.266,1.481) -- (3.443,1.496) -- (3.618,1.515) -- (3.794,1.538) -- (3.968,1.565) -- (4.142,1.596);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.769,7.124) -- (4.74,7.267) -- (4.66,7.388) -- (4.539,7.469) -- (4.396,7.498) -- (4.253,7.469) -- (4.132,7.388) -- (4.051,7.267) -- (4.022,7.124) -- (4.051,6.982) -- (4.132,6.861) -- (4.253,6.78) -- (4.396,6.751) -- (4.539,6.78) -- (4.66,6.861) -- (4.74,6.982) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.977,6.777) -- (1.949,6.92) -- (1.868,7.041) -- (1.747,7.122) -- (1.604,7.151) -- (1.461,7.122) -- (1.34,7.041) -- (1.259,6.92) -- (1.231,6.777) -- (1.259,6.635) -- (1.34,6.513) -- (1.461,6.433) -- (1.604,6.404) -- (1.747,6.433) -- (1.868,6.513) -- (1.949,6.635) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.739,6.747) -- (11.711,6.89) -- (11.63,7.011) -- (11.509,7.092) -- (11.366,7.12) -- (11.223,7.092) -- (11.102,7.011) -- (11.021,6.89) -- (10.993,6.747) -- (11.021,6.604) -- (11.102,6.483) -- (11.223,6.402) -- (11.366,6.374) -- (11.509,6.402) -- (11.63,6.483) -- (11.711,6.604) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.452,6.519) -- (9.423,6.662) -- (9.342,6.783) -- (9.221,6.864) -- (9.078,6.893) -- (8.936,6.864) -- (8.815,6.783) -- (8.734,6.662) -- (8.705,6.519) -- (8.734,6.377) -- (8.815,6.255) -- (8.936,6.175) -- (9.078,6.146) -- (9.221,6.175) -- (9.342,6.255) -- (9.423,6.377) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.109,4.293) -- (4.081,4.436) -- (4.0,4.557) -- (3.879,4.638) -- (3.736,4.667) -- (3.593,4.638) -- (3.472,4.557) -- (3.391,4.436) -- (3.363,4.293) -- (3.391,4.151) -- (3.472,4.029) -- (3.593,3.949) -- (3.736,3.92) -- (3.879,3.949) -- (4.0,4.029) -- (4.081,4.151) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.758,4.276) -- (1.73,4.419) -- (1.649,4.54) -- (1.528,4.621) -- (1.385,4.649) -- (1.242,4.621) -- (1.121,4.54) -- (1.04,4.419) -- (1.012,4.276) -- (1.04,4.133) -- (1.121,4.012) -- (1.242,3.931) -- (1.385,3.903) -- (1.528,3.931) -- (1.649,4.012) -- (1.73,4.133) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.75,4.142) -- (6.722,4.285) -- (6.641,4.406) -- (6.52,4.487) -- (6.377,4.515) -- (6.234,4.487) -- (6.113,4.406) -- (6.032,4.285) -- (6.004,4.142) -- (6.032,3.999) -- (6.113,3.878) -- (6.234,3.797) -- (6.377,3.769) -- (6.52,3.797) -- (6.641,3.878) -- (6.722,3.999) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.446,4.119) -- (9.417,4.261) -- (9.336,4.383) -- (9.215,4.463) -- (9.072,4.492) -- (8.93,4.463) -- (8.809,4.383) -- (8.728,4.261) -- (8.699,4.119) -- (8.728,3.976) -- (8.809,3.855) -- (8.93,3.774) -- (9.072,3.745) -- (9.215,3.774) -- (9.336,3.855) -- (9.417,3.976) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.758,4.034) -- (11.73,4.177) -- (11.649,4.298) -- (11.528,4.379) -- (11.385,4.408) -- (11.242,4.379) -- (11.121,4.298) -- (11.04,4.177) -- (11.012,4.034) -- (11.04,3.892) -- (11.121,3.77) -- (11.242,3.69) -- (11.385,3.661) -- (11.528,3.69) -- (11.649,3.77) -- (11.73,3.892) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.515,1.596) -- (4.487,1.739) -- (4.406,1.86) -- (4.285,1.941) -- (4.142,1.969) -- (3.999,1.941) -- (3.878,1.86) -- (3.797,1.739) -- (3.769,1.596) -- (3.797,1.453) -- (3.878,1.332) -- (3.999,1.251) -- (4.142,1.223) -- (4.285,1.251) -- (4.406,1.332) -- (4.487,1.453) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.145,1.544) -- (7.116,1.687) -- (7.035,1.808) -- (6.914,1.889) -- (6.772,1.917) -- (6.629,1.889) -- (6.508,1.808) -- (6.427,1.687) -- (6.398,1.544) -- (6.427,1.401) -- (6.508,1.28) -- (6.629,1.199) -- (6.772,1.171) -- (6.914,1.199) -- (7.035,1.28) -- (7.116,1.401) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.053,1.535) -- (2.024,1.678) -- (1.943,1.799) -- (1.822,1.88) -- (1.679,1.908) -- (1.537,1.88) -- (1.415,1.799) -- (1.335,1.678) -- (1.306,1.535) -- (1.335,1.392) -- (1.415,1.271) -- (1.537,1.19) -- (1.679,1.162) -- (1.822,1.19) -- (1.943,1.271) -- (2.024,1.392) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.324,5.264) -- (4.442,5.283) -- (4.55,5.338) -- (4.635,5.423) -- (4.689,5.53) -- (4.708,5.649) -- (4.689,5.768) -- (4.635,5.875) -- (4.55,5.96) -- (4.442,6.015) -- (4.324,6.033) -- (4.324,6.033) -- (4.205,6.015) -- (4.098,5.96) -- (4.012,5.875) -- (3.958,5.768) -- (3.939,5.649) -- (3.958,5.53) -- (4.012,5.423) -- (4.098,5.338) -- (4.205,5.283) -- (4.324,5.264) -- cycle;
\node at (4.324,5.649) {\scalebox{1.618}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.938,7.063) -- (3.057,7.082) -- (3.164,7.137) -- (3.249,7.222) -- (3.304,7.329) -- (3.323,7.448) -- (3.304,7.567) -- (3.249,7.674) -- (3.164,7.759) -- (3.057,7.813) -- (2.938,7.832) -- (2.938,7.832) -- (2.819,7.813) -- (2.712,7.759) -- (2.627,7.674) -- (2.572,7.567) -- (2.554,7.448) -- (2.572,7.329) -- (2.627,7.222) -- (2.712,7.137) -- (2.819,7.082) -- (2.938,7.063) -- cycle;
\node at (2.938,7.448) {\scalebox{1.618}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.774,5.118) -- (1.893,5.137) -- (2.0,5.191) -- (2.085,5.276) -- (2.14,5.383) -- (2.159,5.502) -- (2.14,5.621) -- (2.085,5.728) -- (2.0,5.813) -- (1.893,5.868) -- (1.774,5.887) -- (1.774,5.887) -- (1.655,5.868) -- (1.548,5.813) -- (1.463,5.728) -- (1.409,5.621) -- (1.39,5.502) -- (1.409,5.383) -- (1.463,5.276) -- (1.548,5.191) -- (1.655,5.137) -- (1.774,5.118) -- cycle;
\node at (1.774,5.502) {\scalebox{1.618}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.872,5.01) -- (11.991,5.029) -- (12.098,5.083) -- (12.183,5.168) -- (12.238,5.275) -- (12.256,5.394) -- (12.238,5.513) -- (12.183,5.62) -- (12.098,5.705) -- (11.991,5.76) -- (11.872,5.779) -- (11.872,5.779) -- (11.753,5.76) -- (11.646,5.705) -- (11.561,5.62) -- (11.506,5.513) -- (11.487,5.394) -- (11.506,5.275) -- (11.561,5.168) -- (11.646,5.083) -- (11.753,5.029) -- (11.872,5.01) -- cycle;
\node at (11.872,5.394) {\scalebox{1.618}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.27,5.772) -- (10.389,5.791) -- (10.496,5.845) -- (10.581,5.93) -- (10.635,6.037) -- (10.654,6.156) -- (10.635,6.275) -- (10.581,6.382) -- (10.496,6.467) -- (10.389,6.522) -- (10.27,6.541) -- (10.27,6.541) -- (10.151,6.522) -- (10.044,6.467) -- (9.959,6.382) -- (9.904,6.275) -- (9.885,6.156) -- (9.904,6.037) -- (9.959,5.93) -- (10.044,5.845) -- (10.151,5.791) -- (10.27,5.772) -- cycle;
\node at (10.27,6.156) {\scalebox{1.618}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.559,4.933) -- (9.678,4.952) -- (9.785,5.007) -- (9.87,5.092) -- (9.925,5.199) -- (9.944,5.318) -- (9.925,5.437) -- (9.87,5.544) -- (9.785,5.629) -- (9.678,5.684) -- (9.559,5.702) -- (9.559,5.702) -- (9.44,5.684) -- (9.333,5.629) -- (9.248,5.544) -- (9.193,5.437) -- (9.174,5.318) -- (9.193,5.199) -- (9.248,5.092) -- (9.333,5.007) -- (9.44,4.952) -- (9.559,4.933) -- cycle;
\node at (9.559,5.318) {\scalebox{1.618}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.072,4.108) -- (5.191,4.127) -- (5.298,4.182) -- (5.383,4.267) -- (5.438,4.374) -- (5.457,4.493) -- (5.438,4.612) -- (5.383,4.719) -- (5.298,4.804) -- (5.191,4.858) -- (5.072,4.877) -- (5.072,4.877) -- (4.953,4.858) -- (4.846,4.804) -- (4.761,4.719) -- (4.707,4.612) -- (4.688,4.493) -- (4.707,4.374) -- (4.761,4.267) -- (4.846,4.182) -- (4.953,4.127) -- (5.072,4.108) -- cycle;
\node at (5.072,4.493) {\scalebox{1.618}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.431,2.634) -- (4.549,2.653) -- (4.657,2.708) -- (4.742,2.793) -- (4.796,2.9) -- (4.815,3.019) -- (4.796,3.138) -- (4.742,3.245) -- (4.657,3.33) -- (4.549,3.384) -- (4.431,3.403) -- (4.431,3.403) -- (4.312,3.384) -- (4.205,3.33) -- (4.119,3.245) -- (4.065,3.138) -- (4.046,3.019) -- (4.065,2.9) -- (4.119,2.793) -- (4.205,2.708) -- (4.312,2.653) -- (4.431,2.634) -- cycle;
\node at (4.431,3.019) {\scalebox{1.618}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.386,3.42) -- (2.505,3.438) -- (2.612,3.493) -- (2.697,3.578) -- (2.752,3.685) -- (2.771,3.804) -- (2.752,3.923) -- (2.697,4.03) -- (2.612,4.115) -- (2.505,4.17) -- (2.386,4.189) -- (2.386,4.189) -- (2.267,4.17) -- (2.16,4.115) -- (2.075,4.03) -- (2.02,3.923) -- (2.002,3.804) -- (2.02,3.685) -- (2.075,3.578) -- (2.16,3.493) -- (2.267,3.438) -- (2.386,3.42) -- cycle;
\node at (2.386,3.804) {\scalebox{1.618}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.444,2.492) -- (1.563,2.511) -- (1.67,2.565) -- (1.755,2.651) -- (1.81,2.758) -- (1.828,2.877) -- (1.81,2.995) -- (1.755,3.103) -- (1.67,3.188) -- (1.563,3.242) -- (1.444,3.261) -- (1.082,3.261) -- (0.963,3.242) -- (0.856,3.188) -- (0.771,3.103) -- (0.716,2.995) -- (0.697,2.877) -- (0.716,2.758) -- (0.771,2.651) -- (0.856,2.565) -- (0.963,2.511) -- (1.082,2.492) -- cycle;
\node at (1.263,2.877) {\scalebox{1.618}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.909,4.178) -- (8.028,4.196) -- (8.135,4.251) -- (8.22,4.336) -- (8.275,4.443) -- (8.294,4.562) -- (8.275,4.681) -- (8.22,4.788) -- (8.135,4.873) -- (8.028,4.928) -- (7.909,4.947) -- (7.547,4.947) -- (7.429,4.928) -- (7.321,4.873) -- (7.236,4.788) -- (7.182,4.681) -- (7.163,4.562) -- (7.182,4.443) -- (7.236,4.336) -- (7.321,4.251) -- (7.429,4.196) -- (7.547,4.178) -- cycle;
\node at (7.728,4.562) {\scalebox{1.618}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.482,2.417) -- (6.601,2.436) -- (6.708,2.49) -- (6.793,2.576) -- (6.848,2.683) -- (6.867,2.802) -- (6.848,2.92) -- (6.793,3.028) -- (6.708,3.113) -- (6.601,3.167) -- (6.482,3.186) -- (6.12,3.186) -- (6.001,3.167) -- (5.894,3.113) -- (5.809,3.028) -- (5.755,2.92) -- (5.736,2.802) -- (5.755,2.683) -- (5.809,2.576) -- (5.894,2.49) -- (6.001,2.436) -- (6.12,2.417) -- cycle;
\node at (6.301,2.802) {\scalebox{1.618}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.731,4.316) -- (10.849,4.335) -- (10.957,4.39) -- (11.042,4.475) -- (11.096,4.582) -- (11.115,4.701) -- (11.096,4.819) -- (11.042,4.927) -- (10.957,5.012) -- (10.849,5.066) -- (10.731,5.085) -- (10.369,5.085) -- (10.25,5.066) -- (10.143,5.012) -- (10.058,4.927) -- (10.003,4.819) -- (9.984,4.701) -- (10.003,4.582) -- (10.058,4.475) -- (10.143,4.39) -- (10.25,4.335) -- (10.369,4.316) -- cycle;
\node at (10.55,4.701) {\scalebox{1.618}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.632,0.909) -- (5.751,0.928) -- (5.858,0.983) -- (5.943,1.068) -- (5.998,1.175) -- (6.017,1.294) -- (5.998,1.413) -- (5.943,1.52) -- (5.858,1.605) -- (5.751,1.66) -- (5.632,1.679) -- (5.27,1.679) -- (5.152,1.66) -- (5.044,1.605) -- (4.959,1.52) -- (4.905,1.413) -- (4.886,1.294) -- (4.905,1.175) -- (4.959,1.068) -- (5.044,0.983) -- (5.152,0.928) -- (5.27,0.909) -- cycle;
\node at (5.451,1.294) {\scalebox{1.618}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.085,1.464) -- (3.203,1.483) -- (3.311,1.537) -- (3.396,1.622) -- (3.45,1.729) -- (3.469,1.848) -- (3.45,1.967) -- (3.396,2.074) -- (3.311,2.159) -- (3.203,2.214) -- (3.085,2.233) -- (2.723,2.233) -- (2.604,2.214) -- (2.497,2.159) -- (2.412,2.074) -- (2.357,1.967) -- (2.338,1.848) -- (2.357,1.729) -- (2.412,1.622) -- (2.497,1.537) -- (2.604,1.483) -- (2.723,1.464) -- cycle;
\node at (2.904,1.848) {\scalebox{1.618}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.304,6.477) -- (1.904,6.477) -- (1.904,7.077) -- (1.304,7.077) -- cycle;
\node at (1.604,6.777) {\scalebox{1.618}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.375,1.076) -- (1.975,1.076) -- (1.975,1.676) -- (1.375,1.676) -- cycle;
\node at (1.675,1.376) {\scalebox{1.618}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.693,2.326) -- (8.293,2.326) -- (8.293,2.926) -- (7.693,2.926) -- cycle;
\node at (7.993,2.626) {\scalebox{1.618}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (0.642,5.895) -- (1.242,5.895) -- (1.242,6.495) -- (0.642,6.495) -- cycle;
\node at (0.942,6.195) {\scalebox{1.618}{\small d}};
\node at (4.094,7.666) {\scalebox{1.618}{\small A}};
\node at (1.062,7.077) {\scalebox{1.618}{\small B}};
\node at (11.935,6.993) {\scalebox{1.618}{\small C}};
\node at (9.593,6.865) {\scalebox{1.618}{\small D}};
\node at (3.117,4.265) {\scalebox{1.618}{\small E}};
\node at (0.765,4.26) {\scalebox{1.618}{\small F}};
\node at (6.922,3.847) {\scalebox{1.618}{\small G}};
\node at (9.69,4.065) {\scalebox{1.618}{\small H}};
\node at (12.003,3.994) {\scalebox{1.618}{\small I}};
\node at (3.81,1.073) {\scalebox{1.618}{\small J}};
\node at (6.95,0.95) {\scalebox{1.618}{\small K}};
\node at (1.165,1.188) {\scalebox{1.618}{\small L}};
\draw[line width=0.5pt] (12.685,6.174) -- (12.685,7.274);
\draw[line width=0.5pt] (12.454,6.889) -- (12.685,7.274) -- (12.916,6.889);
\draw[line width=0.5pt] (12.382,6.724) -- (12.987,6.724);
\node at (12.685,7.659) {\scalebox{1.618}{\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, F, G, I, J & 6 \\ \hline
clairs & B, C, E, H, K, 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{Non.} Si l'on retire le carrefour \textbf{G}, le quartier tombe en 2 morceaux : toute route de l'un à l'autre passe par lui. Un circuit qui reviendrait à l'école devrait donc s'y arrêter au moins \textbf{deux fois} — c'est interdit.}\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{B} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{rue du Stade ②} vers l'est jusqu'à Place Sainte-Croix \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin Creux ①}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Haute ⑨}, vers l'ouest, jusqu'à Carrefour du Moulin \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{sentier des Prés ⑩}, vers le sud, jusqu'à Place du Marché \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Pont ⑮}, vers l'est, jusqu'à Rond-point de la Gare \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Verte ⑭}, vers l'est, jusqu'à Fontaine Notre-Dame \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ⑫}, vers le nord, jusqu'à Carrefour de l'Hôpital \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{quai aux Fleurs ⑪}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{allée des Tilleuls ⑥}, vers le nord, jusqu'à Place du Vieux-Puits \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin du Lavoir ⑤}, vers l'est, jusqu'à Place de l'Église \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{passage du Marché ④}, vers le sud, jusqu'à Place de la Fontaine \textbf{(I)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 12 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}
