\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 363}

\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=363}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0363}
\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=0.6281, line join=round, line cap=round]
\draw[line width=0.5pt] (3.919,11.49) -- (4.171,11.44) -- (4.424,11.396) -- (4.679,11.358) -- (4.934,11.326) -- (5.191,11.3) -- (5.447,11.28) -- (5.705,11.266) -- (5.962,11.258) -- (6.22,11.257) -- (6.477,11.261) -- (6.734,11.271) -- (6.991,11.287) -- (7.248,11.31) -- (7.503,11.338);
\draw[line width=0.5pt] (3.919,11.49) -- (3.994,11.237) -- (4.063,10.982) -- (4.126,10.726) -- (4.183,10.468) -- (4.234,10.208) -- (4.279,9.948) -- (4.318,9.686) -- (4.351,9.424) -- (4.377,9.161) -- (4.398,8.897) -- (4.412,8.633) -- (4.42,8.369) -- (4.421,8.105) -- (4.417,7.841);
\draw[line width=0.5pt] (1.664,11.482) -- (1.824,11.458) -- (1.984,11.438) -- (2.145,11.421) -- (2.307,11.409) -- (2.468,11.4) -- (2.63,11.395) -- (2.792,11.393) -- (2.953,11.396) -- (3.115,11.402) -- (3.277,11.412) -- (3.438,11.426) -- (3.599,11.444) -- (3.759,11.465) -- (3.919,11.49);
\draw[line width=0.5pt] (1.664,11.482) -- (1.681,11.241) -- (1.692,11.0) -- (1.698,10.758) -- (1.697,10.516) -- (1.691,10.275) -- (1.68,10.033) -- (1.662,9.792) -- (1.639,9.551) -- (1.611,9.311) -- (1.577,9.071) -- (1.537,8.833) -- (1.491,8.596) -- (1.44,8.359) -- (1.384,8.124);
\draw[line width=0.5pt] (7.503,11.338) -- (7.691,11.304) -- (7.88,11.274) -- (8.07,11.248) -- (8.26,11.228) -- (8.451,11.211) -- (8.642,11.199) -- (8.833,11.192) -- (9.024,11.189) -- (9.216,11.19) -- (9.407,11.196) -- (9.598,11.207) -- (9.789,11.222) -- (9.979,11.241) -- (10.169,11.265);
\draw[line width=0.5pt] (7.503,11.338) -- (7.564,11.146) -- (7.619,10.952) -- (7.671,10.757) -- (7.717,10.56) -- (7.759,10.363) -- (7.797,10.164) -- (7.829,9.965) -- (7.857,9.765) -- (7.881,9.565) -- (7.899,9.364) -- (7.913,9.162) -- (7.922,8.961) -- (7.927,8.759) -- (7.926,8.557);
\draw[line width=0.5pt] (10.169,11.265) -- (10.178,11.004) -- (10.194,10.743) -- (10.215,10.482) -- (10.243,10.222) -- (10.277,9.963) -- (10.317,9.704) -- (10.363,9.447) -- (10.416,9.19) -- (10.474,8.935) -- (10.538,8.682) -- (10.608,8.43) -- (10.684,8.179) -- (10.766,7.931) -- (10.853,7.685);
\draw[line width=0.5pt] (7.926,8.557) -- (8.143,8.528) -- (8.359,8.493) -- (8.575,8.453) -- (8.789,8.407) -- (9.003,8.357) -- (9.215,8.302) -- (9.426,8.242) -- (9.635,8.177) -- (9.843,8.107) -- (10.049,8.032) -- (10.253,7.952) -- (10.455,7.868) -- (10.655,7.779) -- (10.853,7.685);
\draw[line width=0.5pt] (7.926,8.557) -- (7.845,8.309) -- (7.77,8.059) -- (7.7,7.808) -- (7.637,7.554) -- (7.58,7.3) -- (7.528,7.043) -- (7.483,6.786) -- (7.444,6.528) -- (7.41,6.269) -- (7.383,6.009) -- (7.362,5.749) -- (7.347,5.488) -- (7.338,5.227) -- (7.336,4.966);
\draw[line width=0.5pt] (1.384,8.124) -- (1.602,8.138) -- (1.82,8.145) -- (2.039,8.148) -- (2.257,8.146) -- (2.476,8.138) -- (2.694,8.126) -- (2.912,8.108) -- (3.129,8.085) -- (3.346,8.057) -- (3.562,8.024) -- (3.778,7.986) -- (3.992,7.942) -- (4.205,7.894) -- (4.417,7.841);
\draw[line width=0.5pt] (1.384,8.124) -- (1.352,7.91) -- (1.326,7.694) -- (1.305,7.478) -- (1.289,7.262) -- (1.278,7.045) -- (1.273,6.828) -- (1.272,6.61) -- (1.277,6.393) -- (1.286,6.176) -- (1.301,5.96) -- (1.32,5.743) -- (1.345,5.528) -- (1.375,5.313) -- (1.41,5.099);
\draw[line width=0.5pt] (4.417,7.841) -- (4.658,7.93) -- (4.901,8.014) -- (5.146,8.091) -- (5.392,8.163) -- (5.641,8.229) -- (5.891,8.29) -- (6.142,8.344) -- (6.394,8.392) -- (6.648,8.435) -- (6.902,8.472) -- (7.158,8.502) -- (7.413,8.527) -- (7.67,8.545) -- (7.926,8.557);
\draw[line width=0.5pt] (4.417,7.841) -- (4.387,7.652) -- (4.361,7.461) -- (4.34,7.271) -- (4.323,7.08) -- (4.311,6.888) -- (4.303,6.696) -- (4.3,6.504) -- (4.301,6.312) -- (4.307,6.12) -- (4.317,5.929) -- (4.332,5.737) -- (4.351,5.546) -- (4.375,5.356) -- (4.403,5.166);
\draw[line width=0.5pt] (4.403,5.166) -- (4.613,5.184) -- (4.824,5.197) -- (5.035,5.205) -- (5.245,5.208) -- (5.456,5.206) -- (5.667,5.199) -- (5.878,5.187) -- (6.088,5.17) -- (6.298,5.149) -- (6.507,5.122) -- (6.716,5.09) -- (6.924,5.054) -- (7.13,5.012) -- (7.336,4.966);
\draw[line width=0.5pt] (1.41,5.099) -- (1.623,5.071) -- (1.837,5.048) -- (2.051,5.03) -- (2.265,5.017) -- (2.48,5.009) -- (2.695,5.007) -- (2.91,5.009) -- (3.124,5.016) -- (3.339,5.029) -- (3.553,5.046) -- (3.767,5.069) -- (3.98,5.096) -- (4.192,5.129) -- (4.403,5.166);
\draw[line width=0.5pt] (1.41,5.099) -- (1.465,4.845) -- (1.513,4.591) -- (1.556,4.335) -- (1.593,4.078) -- (1.623,3.82) -- (1.648,3.562) -- (1.666,3.303) -- (1.678,3.044) -- (1.685,2.785) -- (1.685,2.525) -- (1.679,2.266) -- (1.667,2.007) -- (1.649,1.748) -- (1.624,1.49);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.079,11.49) -- (4.057,11.57) -- (3.999,11.629) -- (3.919,11.65) -- (3.839,11.629) -- (3.78,11.57) -- (3.759,11.49) -- (3.78,11.41) -- (3.839,11.352) -- (3.919,11.33) -- (3.999,11.352) -- (4.057,11.41) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.824,11.482) -- (1.803,11.562) -- (1.744,11.621) -- (1.664,11.642) -- (1.584,11.621) -- (1.526,11.562) -- (1.504,11.482) -- (1.526,11.402) -- (1.584,11.344) -- (1.664,11.322) -- (1.744,11.344) -- (1.803,11.402) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.663,11.338) -- (7.642,11.418) -- (7.583,11.477) -- (7.503,11.498) -- (7.423,11.477) -- (7.365,11.418) -- (7.343,11.338) -- (7.365,11.258) -- (7.423,11.199) -- (7.503,11.178) -- (7.583,11.199) -- (7.642,11.258) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.329,11.265) -- (10.307,11.345) -- (10.249,11.404) -- (10.169,11.425) -- (10.089,11.404) -- (10.03,11.345) -- (10.009,11.265) -- (10.03,11.185) -- (10.089,11.127) -- (10.169,11.105) -- (10.249,11.127) -- (10.307,11.185) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.086,8.557) -- (8.065,8.637) -- (8.006,8.696) -- (7.926,8.717) -- (7.846,8.696) -- (7.788,8.637) -- (7.766,8.557) -- (7.788,8.477) -- (7.846,8.419) -- (7.926,8.397) -- (8.006,8.419) -- (8.065,8.477) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.544,8.124) -- (1.522,8.204) -- (1.464,8.263) -- (1.384,8.284) -- (1.304,8.263) -- (1.245,8.204) -- (1.224,8.124) -- (1.245,8.044) -- (1.304,7.986) -- (1.384,7.964) -- (1.464,7.986) -- (1.522,8.044) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.577,7.841) -- (4.555,7.921) -- (4.497,7.979) -- (4.417,8.001) -- (4.337,7.979) -- (4.278,7.921) -- (4.257,7.841) -- (4.278,7.761) -- (4.337,7.702) -- (4.417,7.681) -- (4.497,7.702) -- (4.555,7.761) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.013,7.685) -- (10.992,7.765) -- (10.933,7.824) -- (10.853,7.845) -- (10.773,7.824) -- (10.715,7.765) -- (10.693,7.685) -- (10.715,7.605) -- (10.773,7.546) -- (10.853,7.525) -- (10.933,7.546) -- (10.992,7.605) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.563,5.166) -- (4.542,5.246) -- (4.483,5.305) -- (4.403,5.326) -- (4.323,5.305) -- (4.265,5.246) -- (4.243,5.166) -- (4.265,5.086) -- (4.323,5.028) -- (4.403,5.006) -- (4.483,5.028) -- (4.542,5.086) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.57,5.099) -- (1.549,5.179) -- (1.49,5.237) -- (1.41,5.259) -- (1.33,5.237) -- (1.272,5.179) -- (1.25,5.099) -- (1.272,5.019) -- (1.33,4.96) -- (1.41,4.939) -- (1.49,4.96) -- (1.549,5.019) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.496,4.966) -- (7.475,5.046) -- (7.416,5.105) -- (7.336,5.126) -- (7.256,5.105) -- (7.197,5.046) -- (7.176,4.966) -- (7.197,4.886) -- (7.256,4.827) -- (7.336,4.806) -- (7.416,4.827) -- (7.475,4.886) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.784,1.49) -- (1.763,1.57) -- (1.704,1.629) -- (1.624,1.65) -- (1.544,1.629) -- (1.486,1.57) -- (1.464,1.49) -- (1.486,1.41) -- (1.544,1.352) -- (1.624,1.33) -- (1.704,1.352) -- (1.763,1.41) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.721,11.266) -- (5.843,11.285) -- (5.953,11.341) -- (6.04,11.428) -- (6.096,11.538) -- (6.115,11.66) -- (6.096,11.782) -- (6.04,11.891) -- (5.953,11.979) -- (5.843,12.034) -- (5.721,12.054) -- (5.721,12.054) -- (5.6,12.034) -- (5.49,11.979) -- (5.403,11.891) -- (5.347,11.782) -- (5.327,11.66) -- (5.347,11.538) -- (5.403,11.428) -- (5.49,11.341) -- (5.6,11.285) -- (5.721,11.266) -- cycle;
\node at (5.721,11.66) {\scalebox{1.592}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.709,9.346) -- (4.83,9.365) -- (4.94,9.421) -- (5.027,9.508) -- (5.083,9.618) -- (5.102,9.739) -- (5.083,9.861) -- (5.027,9.971) -- (4.94,10.058) -- (4.83,10.114) -- (4.709,10.133) -- (4.709,10.133) -- (4.587,10.114) -- (4.477,10.058) -- (4.39,9.971) -- (4.334,9.861) -- (4.315,9.739) -- (4.334,9.618) -- (4.39,9.508) -- (4.477,9.421) -- (4.587,9.365) -- (4.709,9.346) -- cycle;
\node at (4.709,9.739) {\scalebox{1.592}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.79,11.393) -- (2.912,11.413) -- (3.022,11.469) -- (3.109,11.556) -- (3.165,11.665) -- (3.184,11.787) -- (3.165,11.909) -- (3.109,12.019) -- (3.022,12.106) -- (2.912,12.162) -- (2.79,12.181) -- (2.79,12.181) -- (2.669,12.162) -- (2.559,12.106) -- (2.472,12.019) -- (2.416,11.909) -- (2.396,11.787) -- (2.416,11.665) -- (2.472,11.556) -- (2.559,11.469) -- (2.669,11.413) -- (2.79,11.393) -- cycle;
\node at (2.79,11.787) {\scalebox{1.592}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.055,9.365) -- (2.177,9.384) -- (2.286,9.44) -- (2.374,9.527) -- (2.43,9.637) -- (2.449,9.759) -- (2.43,9.881) -- (2.374,9.991) -- (2.286,10.078) -- (2.177,10.134) -- (2.055,10.153) -- (2.055,10.153) -- (1.933,10.134) -- (1.823,10.078) -- (1.736,9.991) -- (1.68,9.881) -- (1.661,9.759) -- (1.68,9.637) -- (1.736,9.527) -- (1.823,9.44) -- (1.933,9.384) -- (2.055,9.365) -- cycle;
\node at (2.055,9.759) {\scalebox{1.592}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.844,11.192) -- (8.965,11.211) -- (9.075,11.267) -- (9.162,11.354) -- (9.218,11.464) -- (9.238,11.585) -- (9.218,11.707) -- (9.162,11.817) -- (9.075,11.904) -- (8.965,11.96) -- (8.844,11.979) -- (8.844,11.979) -- (8.722,11.96) -- (8.612,11.904) -- (8.525,11.817) -- (8.469,11.707) -- (8.45,11.585) -- (8.469,11.464) -- (8.525,11.354) -- (8.612,11.267) -- (8.722,11.211) -- (8.844,11.192) -- cycle;
\node at (8.844,11.585) {\scalebox{1.592}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.44,9.512) -- (7.562,9.531) -- (7.671,9.587) -- (7.759,9.674) -- (7.815,9.784) -- (7.834,9.906) -- (7.815,10.028) -- (7.759,10.137) -- (7.671,10.225) -- (7.562,10.281) -- (7.44,10.3) -- (7.44,10.3) -- (7.318,10.281) -- (7.208,10.225) -- (7.121,10.137) -- (7.065,10.028) -- (7.046,9.906) -- (7.065,9.784) -- (7.121,9.674) -- (7.208,9.587) -- (7.318,9.531) -- (7.44,9.512) -- cycle;
\node at (7.44,9.906) {\scalebox{1.592}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.75,9.127) -- (10.872,9.146) -- (10.982,9.202) -- (11.069,9.289) -- (11.125,9.399) -- (11.144,9.521) -- (11.125,9.643) -- (11.069,9.752) -- (10.982,9.84) -- (10.872,9.895) -- (10.75,9.915) -- (10.75,9.915) -- (10.629,9.895) -- (10.519,9.84) -- (10.432,9.752) -- (10.376,9.643) -- (10.356,9.521) -- (10.376,9.399) -- (10.432,9.289) -- (10.519,9.202) -- (10.629,9.146) -- (10.75,9.127) -- cycle;
\node at (10.75,9.521) {\scalebox{1.592}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.313,7.471) -- (9.435,7.49) -- (9.545,7.546) -- (9.632,7.633) -- (9.688,7.743) -- (9.707,7.864) -- (9.688,7.986) -- (9.632,8.096) -- (9.545,8.183) -- (9.435,8.239) -- (9.313,8.258) -- (9.313,8.258) -- (9.191,8.239) -- (9.082,8.183) -- (8.995,8.096) -- (8.939,7.986) -- (8.919,7.864) -- (8.939,7.743) -- (8.995,7.633) -- (9.082,7.546) -- (9.191,7.49) -- (9.313,7.471) -- cycle;
\node at (9.313,7.864) {\scalebox{1.592}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.872,6.328) -- (7.993,6.348) -- (8.103,6.404) -- (8.19,6.491) -- (8.246,6.6) -- (8.265,6.722) -- (8.246,6.844) -- (8.19,6.954) -- (8.103,7.041) -- (7.993,7.097) -- (7.872,7.116) -- (7.872,7.116) -- (7.75,7.097) -- (7.64,7.041) -- (7.553,6.954) -- (7.497,6.844) -- (7.478,6.722) -- (7.497,6.6) -- (7.553,6.491) -- (7.64,6.404) -- (7.75,6.348) -- (7.872,6.328) -- cycle;
\node at (7.872,6.722) {\scalebox{1.592}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.134,8.106) -- (3.256,8.125) -- (3.366,8.181) -- (3.453,8.268) -- (3.509,8.378) -- (3.528,8.5) -- (3.509,8.622) -- (3.453,8.732) -- (3.366,8.819) -- (3.256,8.875) -- (3.134,8.894) -- (2.763,8.894) -- (2.642,8.875) -- (2.532,8.819) -- (2.445,8.732) -- (2.389,8.622) -- (2.369,8.5) -- (2.389,8.378) -- (2.445,8.268) -- (2.532,8.181) -- (2.642,8.125) -- (2.763,8.106) -- cycle;
\node at (2.949,8.5) {\scalebox{1.592}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.851,6.22) -- (1.973,6.239) -- (2.083,6.295) -- (2.17,6.382) -- (2.226,6.492) -- (2.245,6.614) -- (2.226,6.736) -- (2.17,6.845) -- (2.083,6.932) -- (1.973,6.988) -- (1.851,7.008) -- (1.481,7.008) -- (1.359,6.988) -- (1.249,6.932) -- (1.162,6.845) -- (1.106,6.736) -- (1.087,6.614) -- (1.106,6.492) -- (1.162,6.382) -- (1.249,6.295) -- (1.359,6.239) -- (1.481,6.22) -- cycle;
\node at (1.666,6.614) {\scalebox{1.592}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.248,8.336) -- (6.37,8.355) -- (6.48,8.411) -- (6.567,8.498) -- (6.623,8.608) -- (6.642,8.73) -- (6.623,8.852) -- (6.567,8.961) -- (6.48,9.049) -- (6.37,9.105) -- (6.248,9.124) -- (5.878,9.124) -- (5.756,9.105) -- (5.646,9.049) -- (5.559,8.961) -- (5.503,8.852) -- (5.484,8.73) -- (5.503,8.608) -- (5.559,8.498) -- (5.646,8.411) -- (5.756,8.355) -- (5.878,8.336) -- cycle;
\node at (6.063,8.73) {\scalebox{1.592}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.879,6.108) -- (5.001,6.128) -- (5.111,6.184) -- (5.198,6.271) -- (5.254,6.38) -- (5.273,6.502) -- (5.254,6.624) -- (5.198,6.734) -- (5.111,6.821) -- (5.001,6.877) -- (4.879,6.896) -- (4.508,6.896) -- (4.387,6.877) -- (4.277,6.821) -- (4.19,6.734) -- (4.134,6.624) -- (4.114,6.502) -- (4.134,6.38) -- (4.19,6.271) -- (4.277,6.184) -- (4.387,6.128) -- (4.508,6.108) -- cycle;
\node at (4.694,6.502) {\scalebox{1.592}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.529,5.146) -- (6.65,5.165) -- (6.76,5.221) -- (6.847,5.308) -- (6.903,5.418) -- (6.923,5.54) -- (6.903,5.662) -- (6.847,5.771) -- (6.76,5.859) -- (6.65,5.914) -- (6.529,5.934) -- (6.158,5.934) -- (6.036,5.914) -- (5.926,5.859) -- (5.839,5.771) -- (5.783,5.662) -- (5.764,5.54) -- (5.783,5.418) -- (5.839,5.308) -- (5.926,5.221) -- (6.036,5.165) -- (6.158,5.146) -- cycle;
\node at (6.343,5.54) {\scalebox{1.592}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.086,5.009) -- (3.208,5.028) -- (3.318,5.084) -- (3.405,5.171) -- (3.461,5.281) -- (3.48,5.403) -- (3.461,5.524) -- (3.405,5.634) -- (3.318,5.721) -- (3.208,5.777) -- (3.086,5.797) -- (2.715,5.797) -- (2.594,5.777) -- (2.484,5.721) -- (2.397,5.634) -- (2.341,5.524) -- (2.321,5.403) -- (2.341,5.281) -- (2.397,5.171) -- (2.484,5.084) -- (2.594,5.028) -- (2.715,5.009) -- cycle;
\node at (2.901,5.403) {\scalebox{1.592}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.245,2.933) -- (2.366,2.952) -- (2.476,3.008) -- (2.563,3.095) -- (2.619,3.205) -- (2.639,3.327) -- (2.619,3.448) -- (2.563,3.558) -- (2.476,3.645) -- (2.366,3.701) -- (2.245,3.721) -- (1.874,3.721) -- (1.752,3.701) -- (1.642,3.645) -- (1.555,3.558) -- (1.499,3.448) -- (1.48,3.327) -- (1.499,3.205) -- (1.555,3.095) -- (1.642,3.008) -- (1.752,2.952) -- (1.874,2.933) -- cycle;
\node at (2.059,3.327) {\scalebox{1.592}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (10.553,7.385) -- (11.153,7.385) -- (11.153,7.985) -- (10.553,7.985) -- cycle;
\node at (10.853,7.685) {\scalebox{1.592}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.835,8.591) -- (9.435,8.591) -- (9.435,9.191) -- (8.835,9.191) -- cycle;
\node at (9.135,8.891) {\scalebox{1.592}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.999,10.171) -- (8.599,10.171) -- (8.599,10.771) -- (7.999,10.771) -- cycle;
\node at (8.299,10.471) {\scalebox{1.592}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.297,4.364) -- (5.897,4.364) -- (5.897,4.964) -- (5.297,4.964) -- cycle;
\node at (5.597,4.664) {\scalebox{1.592}{\small d}};
\node at (3.709,12.074) {\scalebox{1.592}{\small A}};
\node at (1.229,11.924) {\scalebox{1.592}{\small B}};
\node at (7.845,11.855) {\scalebox{1.592}{\small C}};
\node at (10.68,11.615) {\scalebox{1.592}{\small D}};
\node at (8.527,8.709) {\scalebox{1.592}{\small E}};
\node at (0.765,8.165) {\scalebox{1.592}{\small F}};
\node at (3.797,7.814) {\scalebox{1.592}{\small G}};
\node at (11.473,7.664) {\scalebox{1.592}{\small H}};
\node at (4.225,4.573) {\scalebox{1.592}{\small I}};
\node at (0.909,4.733) {\scalebox{1.592}{\small J}};
\node at (7.701,4.465) {\scalebox{1.592}{\small K}};
\node at (1.32,0.95) {\scalebox{1.592}{\small L}};
\draw[line width=0.5pt] (12.153,10.54) -- (12.153,11.64);
\draw[line width=0.5pt] (11.922,11.255) -- (12.153,11.64) -- (12.384,11.255);
\draw[line width=0.5pt] (11.851,11.09) -- (12.456,11.09);
\node at (12.153,12.025) {\scalebox{1.592}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour H, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ rue Pasteur & Ⓐ Carrefour de la Mairie & Ⓙ Carrefour Saint-Roch \\ \hline
① rue Verte & ⑩ avenue de la Gare & Ⓑ Place du Marché & Ⓚ Placette des Ormes \\ \hline
② chemin du Lavoir & ⑪ allée des Tilleuls & Ⓒ Rond-point de la Poste & Ⓛ Coin du Lavoir \\ \hline
③ quai aux Fleurs & ⑫ rue des Lilas & Ⓓ Place Sainte-Croix & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ rue du Puits & ⑬ rue de l'Église & Ⓔ Rond-point de la Gare & \textbf{a} l'école \\ \hline
⑤ chemin de Ronde & ⑭ rue du Moulin & Ⓕ Carrefour de l'Hôpital & \textbf{b} le cimetière \\ \hline
⑥ rue des Écoles & ⑮ sentier des Prés & Ⓖ Rond-point des Vaches & \textbf{c} le gymnase \\ \hline
⑦ rue Basse & ⑯ impasse des Roses & Ⓗ Place de l'Église & \textbf{d} la boulangerie \\ \hline
⑧ passage du Marché & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour des Tilleuls &  \\ \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=0.6434, line join=round, line cap=round]
\draw[line width=0.5pt] (3.919,11.49) -- (4.171,11.44) -- (4.424,11.396) -- (4.679,11.358) -- (4.934,11.326) -- (5.191,11.3) -- (5.447,11.28) -- (5.705,11.266) -- (5.962,11.258) -- (6.22,11.257) -- (6.477,11.261) -- (6.734,11.271) -- (6.991,11.287) -- (7.248,11.31) -- (7.503,11.338);
\draw[line width=0.5pt] (3.919,11.49) -- (3.994,11.237) -- (4.063,10.982) -- (4.126,10.726) -- (4.183,10.468) -- (4.234,10.208) -- (4.279,9.948) -- (4.318,9.686) -- (4.351,9.424) -- (4.377,9.161) -- (4.398,8.897) -- (4.412,8.633) -- (4.42,8.369) -- (4.421,8.105) -- (4.417,7.841);
\draw[line width=0.5pt] (1.664,11.482) -- (1.824,11.458) -- (1.984,11.438) -- (2.145,11.421) -- (2.307,11.409) -- (2.468,11.4) -- (2.63,11.395) -- (2.792,11.393) -- (2.953,11.396) -- (3.115,11.402) -- (3.277,11.412) -- (3.438,11.426) -- (3.599,11.444) -- (3.759,11.465) -- (3.919,11.49);
\draw[line width=0.5pt] (1.664,11.482) -- (1.681,11.241) -- (1.692,11.0) -- (1.698,10.758) -- (1.697,10.516) -- (1.691,10.275) -- (1.68,10.033) -- (1.662,9.792) -- (1.639,9.551) -- (1.611,9.311) -- (1.577,9.071) -- (1.537,8.833) -- (1.491,8.596) -- (1.44,8.359) -- (1.384,8.124);
\draw[line width=0.5pt] (7.503,11.338) -- (7.691,11.304) -- (7.88,11.274) -- (8.07,11.248) -- (8.26,11.228) -- (8.451,11.211) -- (8.642,11.199) -- (8.833,11.192) -- (9.024,11.189) -- (9.216,11.19) -- (9.407,11.196) -- (9.598,11.207) -- (9.789,11.222) -- (9.979,11.241) -- (10.169,11.265);
\draw[line width=0.5pt] (7.503,11.338) -- (7.564,11.146) -- (7.619,10.952) -- (7.671,10.757) -- (7.717,10.56) -- (7.759,10.363) -- (7.797,10.164) -- (7.829,9.965) -- (7.857,9.765) -- (7.881,9.565) -- (7.899,9.364) -- (7.913,9.162) -- (7.922,8.961) -- (7.927,8.759) -- (7.926,8.557);
\draw[line width=0.5pt] (10.169,11.265) -- (10.178,11.004) -- (10.194,10.743) -- (10.215,10.482) -- (10.243,10.222) -- (10.277,9.963) -- (10.317,9.704) -- (10.363,9.447) -- (10.416,9.19) -- (10.474,8.935) -- (10.538,8.682) -- (10.608,8.43) -- (10.684,8.179) -- (10.766,7.931) -- (10.853,7.685);
\draw[line width=0.5pt] (7.926,8.557) -- (8.143,8.528) -- (8.359,8.493) -- (8.575,8.453) -- (8.789,8.407) -- (9.003,8.357) -- (9.215,8.302) -- (9.426,8.242) -- (9.635,8.177) -- (9.843,8.107) -- (10.049,8.032) -- (10.253,7.952) -- (10.455,7.868) -- (10.655,7.779) -- (10.853,7.685);
\draw[line width=0.5pt] (7.926,8.557) -- (7.845,8.309) -- (7.77,8.059) -- (7.7,7.808) -- (7.637,7.554) -- (7.58,7.3) -- (7.528,7.043) -- (7.483,6.786) -- (7.444,6.528) -- (7.41,6.269) -- (7.383,6.009) -- (7.362,5.749) -- (7.347,5.488) -- (7.338,5.227) -- (7.336,4.966);
\draw[line width=0.5pt] (1.384,8.124) -- (1.602,8.138) -- (1.82,8.145) -- (2.039,8.148) -- (2.257,8.146) -- (2.476,8.138) -- (2.694,8.126) -- (2.912,8.108) -- (3.129,8.085) -- (3.346,8.057) -- (3.562,8.024) -- (3.778,7.986) -- (3.992,7.942) -- (4.205,7.894) -- (4.417,7.841);
\draw[line width=0.5pt] (1.384,8.124) -- (1.352,7.91) -- (1.326,7.694) -- (1.305,7.478) -- (1.289,7.262) -- (1.278,7.045) -- (1.273,6.828) -- (1.272,6.61) -- (1.277,6.393) -- (1.286,6.176) -- (1.301,5.96) -- (1.32,5.743) -- (1.345,5.528) -- (1.375,5.313) -- (1.41,5.099);
\draw[line width=0.5pt] (4.417,7.841) -- (4.658,7.93) -- (4.901,8.014) -- (5.146,8.091) -- (5.392,8.163) -- (5.641,8.229) -- (5.891,8.29) -- (6.142,8.344) -- (6.394,8.392) -- (6.648,8.435) -- (6.902,8.472) -- (7.158,8.502) -- (7.413,8.527) -- (7.67,8.545) -- (7.926,8.557);
\draw[line width=0.5pt] (4.417,7.841) -- (4.387,7.652) -- (4.361,7.461) -- (4.34,7.271) -- (4.323,7.08) -- (4.311,6.888) -- (4.303,6.696) -- (4.3,6.504) -- (4.301,6.312) -- (4.307,6.12) -- (4.317,5.929) -- (4.332,5.737) -- (4.351,5.546) -- (4.375,5.356) -- (4.403,5.166);
\draw[line width=0.5pt] (4.403,5.166) -- (4.613,5.184) -- (4.824,5.197) -- (5.035,5.205) -- (5.245,5.208) -- (5.456,5.206) -- (5.667,5.199) -- (5.878,5.187) -- (6.088,5.17) -- (6.298,5.149) -- (6.507,5.122) -- (6.716,5.09) -- (6.924,5.054) -- (7.13,5.012) -- (7.336,4.966);
\draw[line width=0.5pt] (1.41,5.099) -- (1.623,5.071) -- (1.837,5.048) -- (2.051,5.03) -- (2.265,5.017) -- (2.48,5.009) -- (2.695,5.007) -- (2.91,5.009) -- (3.124,5.016) -- (3.339,5.029) -- (3.553,5.046) -- (3.767,5.069) -- (3.98,5.096) -- (4.192,5.129) -- (4.403,5.166);
\draw[line width=0.5pt] (1.41,5.099) -- (1.465,4.845) -- (1.513,4.591) -- (1.556,4.335) -- (1.593,4.078) -- (1.623,3.82) -- (1.648,3.562) -- (1.666,3.303) -- (1.678,3.044) -- (1.685,2.785) -- (1.685,2.525) -- (1.679,2.266) -- (1.667,2.007) -- (1.649,1.748) -- (1.624,1.49);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.292,11.49) -- (4.263,11.633) -- (4.182,11.754) -- (4.061,11.835) -- (3.919,11.864) -- (3.776,11.835) -- (3.655,11.754) -- (3.574,11.633) -- (3.545,11.49) -- (3.574,11.348) -- (3.655,11.226) -- (3.776,11.146) -- (3.919,11.117) -- (4.061,11.146) -- (4.182,11.226) -- (4.263,11.348) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.037,11.482) -- (2.009,11.625) -- (1.928,11.746) -- (1.807,11.827) -- (1.664,11.855) -- (1.521,11.827) -- (1.4,11.746) -- (1.319,11.625) -- (1.291,11.482) -- (1.319,11.339) -- (1.4,11.218) -- (1.521,11.137) -- (1.664,11.109) -- (1.807,11.137) -- (1.928,11.218) -- (2.009,11.339) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.876,11.338) -- (7.848,11.481) -- (7.767,11.602) -- (7.646,11.683) -- (7.503,11.711) -- (7.36,11.683) -- (7.239,11.602) -- (7.158,11.481) -- (7.13,11.338) -- (7.158,11.195) -- (7.239,11.074) -- (7.36,10.993) -- (7.503,10.965) -- (7.646,10.993) -- (7.767,11.074) -- (7.848,11.195) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (10.542,11.265) -- (10.513,11.408) -- (10.433,11.529) -- (10.312,11.61) -- (10.169,11.638) -- (10.026,11.61) -- (9.905,11.529) -- (9.824,11.408) -- (9.795,11.265) -- (9.824,11.122) -- (9.905,11.001) -- (10.026,10.92) -- (10.169,10.892) -- (10.312,10.92) -- (10.433,11.001) -- (10.513,11.122) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.299,8.557) -- (8.271,8.7) -- (8.19,8.821) -- (8.069,8.902) -- (7.926,8.931) -- (7.783,8.902) -- (7.662,8.821) -- (7.581,8.7) -- (7.553,8.557) -- (7.581,8.415) -- (7.662,8.294) -- (7.783,8.213) -- (7.926,8.184) -- (8.069,8.213) -- (8.19,8.294) -- (8.271,8.415) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.757,8.124) -- (1.729,8.267) -- (1.648,8.388) -- (1.527,8.469) -- (1.384,8.498) -- (1.241,8.469) -- (1.12,8.388) -- (1.039,8.267) -- (1.01,8.124) -- (1.039,7.982) -- (1.12,7.861) -- (1.241,7.78) -- (1.384,7.751) -- (1.527,7.78) -- (1.648,7.861) -- (1.729,7.982) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.79,7.841) -- (4.762,7.984) -- (4.681,8.105) -- (4.56,8.186) -- (4.417,8.214) -- (4.274,8.186) -- (4.153,8.105) -- (4.072,7.984) -- (4.044,7.841) -- (4.072,7.698) -- (4.153,7.577) -- (4.274,7.496) -- (4.417,7.468) -- (4.56,7.496) -- (4.681,7.577) -- (4.762,7.698) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.227,7.685) -- (11.198,7.828) -- (11.117,7.949) -- (10.996,8.03) -- (10.853,8.058) -- (10.711,8.03) -- (10.589,7.949) -- (10.509,7.828) -- (10.48,7.685) -- (10.509,7.542) -- (10.589,7.421) -- (10.711,7.34) -- (10.853,7.312) -- (10.996,7.34) -- (11.117,7.421) -- (11.198,7.542) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.777,5.166) -- (4.748,5.309) -- (4.667,5.43) -- (4.546,5.511) -- (4.403,5.54) -- (4.261,5.511) -- (4.14,5.43) -- (4.059,5.309) -- (4.03,5.166) -- (4.059,5.024) -- (4.14,4.902) -- (4.261,4.822) -- (4.403,4.793) -- (4.546,4.822) -- (4.667,4.902) -- (4.748,5.024) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.783,5.099) -- (1.755,5.241) -- (1.674,5.362) -- (1.553,5.443) -- (1.41,5.472) -- (1.267,5.443) -- (1.146,5.362) -- (1.065,5.241) -- (1.037,5.099) -- (1.065,4.956) -- (1.146,4.835) -- (1.267,4.754) -- (1.41,4.725) -- (1.553,4.754) -- (1.674,4.835) -- (1.755,4.956) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.709,4.966) -- (7.681,5.109) -- (7.6,5.23) -- (7.479,5.311) -- (7.336,5.339) -- (7.193,5.311) -- (7.072,5.23) -- (6.991,5.109) -- (6.963,4.966) -- (6.991,4.823) -- (7.072,4.702) -- (7.193,4.621) -- (7.336,4.593) -- (7.479,4.621) -- (7.6,4.702) -- (7.681,4.823) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.998,1.49) -- (1.969,1.633) -- (1.888,1.754) -- (1.767,1.835) -- (1.624,1.864) -- (1.482,1.835) -- (1.361,1.754) -- (1.28,1.633) -- (1.251,1.49) -- (1.28,1.348) -- (1.361,1.226) -- (1.482,1.146) -- (1.624,1.117) -- (1.767,1.146) -- (1.888,1.226) -- (1.969,1.348) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.721,11.266) -- (5.84,11.285) -- (5.947,11.339) -- (6.032,11.424) -- (6.087,11.532) -- (6.105,11.65) -- (6.087,11.769) -- (6.032,11.876) -- (5.947,11.962) -- (5.84,12.016) -- (5.721,12.035) -- (5.721,12.035) -- (5.602,12.016) -- (5.495,11.962) -- (5.41,11.876) -- (5.355,11.769) -- (5.336,11.65) -- (5.355,11.532) -- (5.41,11.424) -- (5.495,11.339) -- (5.602,11.285) -- (5.721,11.266) -- cycle;
\node at (5.721,11.65) {\scalebox{1.554}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.699,9.354) -- (4.818,9.372) -- (4.925,9.427) -- (5.01,9.512) -- (5.065,9.619) -- (5.084,9.738) -- (5.065,9.857) -- (5.01,9.964) -- (4.925,10.049) -- (4.818,10.104) -- (4.699,10.123) -- (4.699,10.123) -- (4.58,10.104) -- (4.473,10.049) -- (4.388,9.964) -- (4.334,9.857) -- (4.315,9.738) -- (4.334,9.619) -- (4.388,9.512) -- (4.473,9.427) -- (4.58,9.372) -- (4.699,9.354) -- cycle;
\node at (4.699,9.738) {\scalebox{1.554}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.79,11.393) -- (2.909,11.412) -- (3.016,11.467) -- (3.101,11.552) -- (3.156,11.659) -- (3.175,11.778) -- (3.156,11.897) -- (3.101,12.004) -- (3.016,12.089) -- (2.909,12.144) -- (2.79,12.162) -- (2.79,12.162) -- (2.671,12.144) -- (2.564,12.089) -- (2.479,12.004) -- (2.425,11.897) -- (2.406,11.778) -- (2.425,11.659) -- (2.479,11.552) -- (2.564,11.467) -- (2.671,11.412) -- (2.79,11.393) -- cycle;
\node at (2.79,11.778) {\scalebox{1.554}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.046,9.375) -- (2.164,9.394) -- (2.272,9.449) -- (2.357,9.534) -- (2.411,9.641) -- (2.43,9.76) -- (2.411,9.879) -- (2.357,9.986) -- (2.272,10.071) -- (2.164,10.125) -- (2.046,10.144) -- (2.046,10.144) -- (1.927,10.125) -- (1.82,10.071) -- (1.735,9.986) -- (1.68,9.879) -- (1.661,9.76) -- (1.68,9.641) -- (1.735,9.534) -- (1.82,9.449) -- (1.927,9.394) -- (2.046,9.375) -- cycle;
\node at (2.046,9.76) {\scalebox{1.554}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.843,11.192) -- (8.962,11.21) -- (9.069,11.265) -- (9.155,11.35) -- (9.209,11.457) -- (9.228,11.576) -- (9.209,11.695) -- (9.155,11.802) -- (9.069,11.887) -- (8.962,11.942) -- (8.843,11.961) -- (8.843,11.961) -- (8.725,11.942) -- (8.617,11.887) -- (8.532,11.802) -- (8.478,11.695) -- (8.459,11.576) -- (8.478,11.457) -- (8.532,11.35) -- (8.617,11.265) -- (8.725,11.21) -- (8.843,11.192) -- cycle;
\node at (8.843,11.576) {\scalebox{1.554}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.449,9.523) -- (7.568,9.542) -- (7.675,9.596) -- (7.76,9.681) -- (7.815,9.789) -- (7.834,9.907) -- (7.815,10.026) -- (7.76,10.133) -- (7.675,10.218) -- (7.568,10.273) -- (7.449,10.292) -- (7.449,10.292) -- (7.33,10.273) -- (7.223,10.218) -- (7.138,10.133) -- (7.083,10.026) -- (7.065,9.907) -- (7.083,9.789) -- (7.138,9.681) -- (7.223,9.596) -- (7.33,9.542) -- (7.449,9.523) -- cycle;
\node at (7.449,9.907) {\scalebox{1.554}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.741,9.135) -- (10.86,9.153) -- (10.967,9.208) -- (11.052,9.293) -- (11.107,9.4) -- (11.126,9.519) -- (11.107,9.638) -- (11.052,9.745) -- (10.967,9.83) -- (10.86,9.885) -- (10.741,9.904) -- (10.741,9.904) -- (10.622,9.885) -- (10.515,9.83) -- (10.43,9.745) -- (10.375,9.638) -- (10.356,9.519) -- (10.375,9.4) -- (10.43,9.293) -- (10.515,9.208) -- (10.622,9.153) -- (10.741,9.135) -- cycle;
\node at (10.741,9.519) {\scalebox{1.554}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.316,7.489) -- (9.435,7.508) -- (9.542,7.562) -- (9.627,7.647) -- (9.682,7.755) -- (9.7,7.873) -- (9.682,7.992) -- (9.627,8.099) -- (9.542,8.185) -- (9.435,8.239) -- (9.316,8.258) -- (9.316,8.258) -- (9.197,8.239) -- (9.09,8.185) -- (9.005,8.099) -- (8.95,7.992) -- (8.931,7.873) -- (8.95,7.755) -- (9.005,7.647) -- (9.09,7.562) -- (9.197,7.508) -- (9.316,7.489) -- cycle;
\node at (9.316,7.873) {\scalebox{1.554}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.862,6.339) -- (7.981,6.358) -- (8.088,6.413) -- (8.173,6.498) -- (8.228,6.605) -- (8.247,6.724) -- (8.228,6.843) -- (8.173,6.95) -- (8.088,7.035) -- (7.981,7.089) -- (7.862,7.108) -- (7.862,7.108) -- (7.743,7.089) -- (7.636,7.035) -- (7.551,6.95) -- (7.497,6.843) -- (7.478,6.724) -- (7.497,6.605) -- (7.551,6.498) -- (7.636,6.413) -- (7.743,6.358) -- (7.862,6.339) -- cycle;
\node at (7.862,6.724) {\scalebox{1.554}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.129,8.106) -- (3.248,8.125) -- (3.355,8.18) -- (3.44,8.265) -- (3.494,8.372) -- (3.513,8.491) -- (3.494,8.609) -- (3.44,8.717) -- (3.355,8.802) -- (3.248,8.856) -- (3.129,8.875) -- (2.767,8.875) -- (2.648,8.856) -- (2.541,8.802) -- (2.456,8.717) -- (2.401,8.609) -- (2.382,8.491) -- (2.401,8.372) -- (2.456,8.265) -- (2.541,8.18) -- (2.648,8.125) -- (2.767,8.106) -- cycle;
\node at (2.948,8.491) {\scalebox{1.554}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.838,6.229) -- (1.956,6.248) -- (2.064,6.303) -- (2.149,6.388) -- (2.203,6.495) -- (2.222,6.614) -- (2.203,6.733) -- (2.149,6.84) -- (2.064,6.925) -- (1.956,6.979) -- (1.838,6.998) -- (1.476,6.998) -- (1.357,6.979) -- (1.25,6.925) -- (1.165,6.84) -- (1.11,6.733) -- (1.091,6.614) -- (1.11,6.495) -- (1.165,6.388) -- (1.25,6.303) -- (1.357,6.248) -- (1.476,6.229) -- cycle;
\node at (1.657,6.614) {\scalebox{1.554}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.246,8.336) -- (6.365,8.355) -- (6.472,8.41) -- (6.557,8.495) -- (6.612,8.602) -- (6.63,8.721) -- (6.612,8.84) -- (6.557,8.947) -- (6.472,9.032) -- (6.365,9.086) -- (6.246,9.105) -- (5.884,9.105) -- (5.765,9.086) -- (5.658,9.032) -- (5.573,8.947) -- (5.518,8.84) -- (5.499,8.721) -- (5.518,8.602) -- (5.573,8.495) -- (5.658,8.41) -- (5.765,8.355) -- (5.884,8.336) -- cycle;
\node at (6.065,8.721) {\scalebox{1.554}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.865,6.118) -- (4.984,6.137) -- (5.091,6.191) -- (5.176,6.276) -- (5.231,6.383) -- (5.25,6.502) -- (5.231,6.621) -- (5.176,6.728) -- (5.091,6.813) -- (4.984,6.868) -- (4.865,6.887) -- (4.503,6.887) -- (4.385,6.868) -- (4.277,6.813) -- (4.192,6.728) -- (4.138,6.621) -- (4.119,6.502) -- (4.138,6.383) -- (4.192,6.276) -- (4.277,6.191) -- (4.385,6.137) -- (4.503,6.118) -- cycle;
\node at (4.684,6.502) {\scalebox{1.554}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.523,5.146) -- (6.642,5.165) -- (6.749,5.219) -- (6.834,5.304) -- (6.889,5.412) -- (6.908,5.531) -- (6.889,5.649) -- (6.834,5.757) -- (6.749,5.842) -- (6.642,5.896) -- (6.523,5.915) -- (6.161,5.915) -- (6.043,5.896) -- (5.935,5.842) -- (5.85,5.757) -- (5.796,5.649) -- (5.777,5.531) -- (5.796,5.412) -- (5.85,5.304) -- (5.935,5.219) -- (6.043,5.165) -- (6.161,5.146) -- cycle;
\node at (6.342,5.531) {\scalebox{1.554}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.082,5.009) -- (3.201,5.028) -- (3.308,5.082) -- (3.393,5.167) -- (3.448,5.275) -- (3.466,5.393) -- (3.448,5.512) -- (3.393,5.619) -- (3.308,5.704) -- (3.201,5.759) -- (3.082,5.778) -- (2.72,5.778) -- (2.601,5.759) -- (2.494,5.704) -- (2.409,5.619) -- (2.354,5.512) -- (2.335,5.393) -- (2.354,5.275) -- (2.409,5.167) -- (2.494,5.082) -- (2.601,5.028) -- (2.72,5.009) -- cycle;
\node at (2.901,5.393) {\scalebox{1.554}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.231,2.942) -- (2.35,2.96) -- (2.457,3.015) -- (2.542,3.1) -- (2.597,3.207) -- (2.615,3.326) -- (2.597,3.445) -- (2.542,3.552) -- (2.457,3.637) -- (2.35,3.692) -- (2.231,3.711) -- (1.869,3.711) -- (1.75,3.692) -- (1.643,3.637) -- (1.558,3.552) -- (1.503,3.445) -- (1.484,3.326) -- (1.503,3.207) -- (1.558,3.1) -- (1.643,3.015) -- (1.75,2.96) -- (1.869,2.942) -- cycle;
\node at (2.05,3.326) {\scalebox{1.554}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (10.553,7.385) -- (11.153,7.385) -- (11.153,7.985) -- (10.553,7.985) -- cycle;
\node at (10.853,7.685) {\scalebox{1.554}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.835,8.591) -- (9.435,8.591) -- (9.435,9.191) -- (8.835,9.191) -- cycle;
\node at (9.135,8.891) {\scalebox{1.554}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.999,10.171) -- (8.599,10.171) -- (8.599,10.771) -- (7.999,10.771) -- cycle;
\node at (8.299,10.471) {\scalebox{1.554}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.297,4.364) -- (5.897,4.364) -- (5.897,4.964) -- (5.297,4.964) -- cycle;
\node at (5.597,4.664) {\scalebox{1.554}{\small d}};
\node at (3.709,12.074) {\scalebox{1.554}{\small A}};
\node at (1.229,11.924) {\scalebox{1.554}{\small B}};
\node at (7.845,11.855) {\scalebox{1.554}{\small C}};
\node at (10.68,11.615) {\scalebox{1.554}{\small D}};
\node at (8.527,8.709) {\scalebox{1.554}{\small E}};
\node at (0.765,8.165) {\scalebox{1.554}{\small F}};
\node at (3.797,7.814) {\scalebox{1.554}{\small G}};
\node at (11.473,7.664) {\scalebox{1.554}{\small H}};
\node at (4.225,4.573) {\scalebox{1.554}{\small I}};
\node at (0.909,4.733) {\scalebox{1.554}{\small J}};
\node at (7.701,4.465) {\scalebox{1.554}{\small K}};
\node at (1.32,0.95) {\scalebox{1.554}{\small L}};
\draw[line width=0.5pt] (12.153,10.54) -- (12.153,11.64);
\draw[line width=0.5pt] (11.922,11.255) -- (12.153,11.64) -- (12.384,11.255);
\draw[line width=0.5pt] (11.851,11.09) -- (12.456,11.09);
\node at (12.153,12.025) {\scalebox{1.554}{\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, E, F, I, L & 6 \\ \hline
clairs & B, C, G, H, J, K & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{6} clairs. Il y en a \textbf{autant} de chaque côté.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 12 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Si l'on retire le carrefour \textbf{J}, 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{H} :}\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 Basse ⑦} vers le nord jusqu'à Place Sainte-Croix \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin de Ronde ⑤}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Verte ①}, vers l'ouest, jusqu'à Carrefour de la Mairie \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{quai aux Fleurs ③}, vers l'ouest, jusqu'à Place du Marché \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ④}, vers le sud, jusqu'à Carrefour de l'Hôpital \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{avenue de la Gare ⑩}, vers l'est, jusqu'à Rond-point des Vaches \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Lilas ⑫}, vers l'est, jusqu'à Rond-point de la Gare \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Pasteur ⑨}, vers le sud, jusqu'à Placette des Ormes \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Moulin ⑭}, vers l'ouest, jusqu'à Carrefour des Tilleuls \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{sentier des Prés ⑮}, vers l'ouest, jusqu'à Carrefour Saint-Roch \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse des Roses ⑯}, vers le sud, jusqu'à Coin du Lavoir \textbf{(L)}.\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}
