\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 145}

\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=145}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0145}
\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=0.6052, line join=round, line cap=round]
\draw[line width=0.5pt] (4.249,8.753) -- (4.517,8.69) -- (4.785,8.633) -- (5.055,8.583) -- (5.327,8.538) -- (5.599,8.501) -- (5.872,8.469) -- (6.146,8.445) -- (6.42,8.426) -- (6.695,8.414) -- (6.97,8.409) -- (7.244,8.409) -- (7.519,8.417) -- (7.793,8.431) -- (8.067,8.451);
\draw[line width=0.5pt] (4.249,8.753) -- (4.268,8.453) -- (4.28,8.152) -- (4.284,7.85) -- (4.282,7.549) -- (4.272,7.248) -- (4.255,6.947) -- (4.231,6.647) -- (4.2,6.347) -- (4.162,6.048) -- (4.118,5.75) -- (4.066,5.453) -- (4.007,5.157) -- (3.941,4.864) -- (3.868,4.572);
\draw[line width=0.5pt] (8.067,8.451) -- (8.007,8.203) -- (7.953,7.953) -- (7.905,7.702) -- (7.863,7.45) -- (7.826,7.197) -- (7.796,6.943) -- (7.772,6.688) -- (7.753,6.433) -- (7.741,6.178) -- (7.734,5.922) -- (7.734,5.666) -- (7.739,5.411) -- (7.751,5.155) -- (7.769,4.901);
\draw[line width=0.5pt] (11.346,5.302) -- (11.347,5.062) -- (11.342,4.822) -- (11.331,4.582) -- (11.315,4.343) -- (11.293,4.103) -- (11.265,3.865) -- (11.232,3.627) -- (11.193,3.39) -- (11.148,3.154) -- (11.099,2.919) -- (11.043,2.685) -- (10.982,2.453) -- (10.916,2.223) -- (10.844,1.994);
\draw[line width=0.5pt] (1.374,5.288) -- (1.543,5.21) -- (1.714,5.136) -- (1.886,5.066) -- (2.06,5.0) -- (2.236,4.938) -- (2.413,4.88) -- (2.592,4.827) -- (2.771,4.777) -- (2.952,4.732) -- (3.134,4.692) -- (3.316,4.655) -- (3.5,4.623) -- (3.684,4.595) -- (3.868,4.572);
\draw[line width=0.5pt] (1.374,5.288) -- (1.412,5.027) -- (1.444,4.766) -- (1.47,4.503) -- (1.49,4.241) -- (1.503,3.977) -- (1.511,3.714) -- (1.512,3.45) -- (1.507,3.186) -- (1.495,2.923) -- (1.478,2.66) -- (1.454,2.397) -- (1.424,2.135) -- (1.388,1.874) -- (1.346,1.614);
\draw[line width=0.5pt] (7.769,4.901) -- (8.027,4.89) -- (8.285,4.885) -- (8.543,4.887) -- (8.801,4.894) -- (9.059,4.908) -- (9.317,4.928) -- (9.574,4.954) -- (9.831,4.985) -- (10.086,5.023) -- (10.341,5.067) -- (10.594,5.117) -- (10.846,5.173) -- (11.097,5.234) -- (11.346,5.302);
\draw[line width=0.5pt] (7.769,4.901) -- (7.824,4.661) -- (7.874,4.419) -- (7.918,4.177) -- (7.957,3.933) -- (7.989,3.689) -- (8.016,3.444) -- (8.037,3.198) -- (8.053,2.952) -- (8.062,2.705) -- (8.066,2.459) -- (8.064,2.212) -- (8.056,1.966) -- (8.042,1.72) -- (8.023,1.474);
\draw[line width=0.5pt] (3.868,4.572) -- (4.148,4.552) -- (4.429,4.539) -- (4.709,4.533) -- (4.99,4.534) -- (5.271,4.541) -- (5.552,4.555) -- (5.832,4.575) -- (6.112,4.602) -- (6.391,4.635) -- (6.669,4.675) -- (6.946,4.722) -- (7.222,4.775) -- (7.496,4.835) -- (7.769,4.901);
\draw[line width=0.5pt] (3.868,4.572) -- (3.928,4.356) -- (3.983,4.139) -- (4.033,3.92) -- (4.077,3.701) -- (4.116,3.48) -- (4.151,3.259) -- (4.18,3.036) -- (4.203,2.814) -- (4.222,2.59) -- (4.235,2.367) -- (4.243,2.143) -- (4.246,1.919) -- (4.243,1.695) -- (4.235,1.471);
\draw[line width=0.5pt] (1.346,1.614) -- (1.549,1.572) -- (1.754,1.536) -- (1.959,1.503) -- (2.164,1.476) -- (2.371,1.454) -- (2.578,1.436) -- (2.785,1.423) -- (2.992,1.416) -- (3.2,1.413) -- (3.408,1.415) -- (3.615,1.421) -- (3.822,1.433) -- (4.029,1.45) -- (4.235,1.471);
\draw[line width=0.5pt] (8.023,1.474) -- (8.217,1.542) -- (8.413,1.605) -- (8.61,1.663) -- (8.809,1.717) -- (9.009,1.766) -- (9.21,1.811) -- (9.412,1.85) -- (9.615,1.885) -- (9.819,1.915) -- (10.023,1.941) -- (10.228,1.961) -- (10.433,1.977) -- (10.639,1.988) -- (10.844,1.994);
\draw[line width=0.5pt] (4.235,1.471) -- (4.504,1.43) -- (4.773,1.395) -- (5.043,1.367) -- (5.314,1.344) -- (5.586,1.329) -- (5.857,1.319) -- (6.129,1.316) -- (6.401,1.32) -- (6.673,1.33) -- (6.944,1.346) -- (7.215,1.368) -- (7.485,1.397) -- (7.755,1.433) -- (8.023,1.474);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.409,8.753) -- (4.388,8.833) -- (4.329,8.892) -- (4.249,8.913) -- (4.169,8.892) -- (4.111,8.833) -- (4.089,8.753) -- (4.111,8.673) -- (4.169,8.614) -- (4.249,8.593) -- (4.329,8.614) -- (4.388,8.673) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.227,8.451) -- (8.206,8.531) -- (8.147,8.59) -- (8.067,8.611) -- (7.987,8.59) -- (7.929,8.531) -- (7.907,8.451) -- (7.929,8.371) -- (7.987,8.313) -- (8.067,8.291) -- (8.147,8.313) -- (8.206,8.371) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.506,5.302) -- (11.485,5.382) -- (11.426,5.44) -- (11.346,5.462) -- (11.266,5.44) -- (11.208,5.382) -- (11.186,5.302) -- (11.208,5.222) -- (11.266,5.163) -- (11.346,5.142) -- (11.426,5.163) -- (11.485,5.222) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.534,5.288) -- (1.513,5.368) -- (1.454,5.426) -- (1.374,5.448) -- (1.294,5.426) -- (1.235,5.368) -- (1.214,5.288) -- (1.235,5.208) -- (1.294,5.149) -- (1.374,5.128) -- (1.454,5.149) -- (1.513,5.208) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.929,4.901) -- (7.907,4.981) -- (7.849,5.039) -- (7.769,5.061) -- (7.689,5.039) -- (7.63,4.981) -- (7.609,4.901) -- (7.63,4.821) -- (7.689,4.762) -- (7.769,4.741) -- (7.849,4.762) -- (7.907,4.821) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.028,4.572) -- (4.007,4.652) -- (3.948,4.71) -- (3.868,4.732) -- (3.788,4.71) -- (3.73,4.652) -- (3.708,4.572) -- (3.73,4.492) -- (3.788,4.433) -- (3.868,4.412) -- (3.948,4.433) -- (4.007,4.492) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.004,1.994) -- (10.983,2.074) -- (10.924,2.132) -- (10.844,2.154) -- (10.764,2.132) -- (10.706,2.074) -- (10.684,1.994) -- (10.706,1.914) -- (10.764,1.855) -- (10.844,1.834) -- (10.924,1.855) -- (10.983,1.914) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.506,1.614) -- (1.485,1.694) -- (1.426,1.753) -- (1.346,1.774) -- (1.266,1.753) -- (1.208,1.694) -- (1.186,1.614) -- (1.208,1.534) -- (1.266,1.476) -- (1.346,1.454) -- (1.426,1.476) -- (1.485,1.534) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.183,1.474) -- (8.161,1.554) -- (8.103,1.613) -- (8.023,1.634) -- (7.943,1.613) -- (7.884,1.554) -- (7.863,1.474) -- (7.884,1.394) -- (7.943,1.336) -- (8.023,1.314) -- (8.103,1.336) -- (8.161,1.394) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.395,1.471) -- (4.374,1.551) -- (4.315,1.61) -- (4.235,1.631) -- (4.155,1.61) -- (4.097,1.551) -- (4.075,1.471) -- (4.097,1.391) -- (4.155,1.333) -- (4.235,1.311) -- (4.315,1.333) -- (4.374,1.391) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.177,8.443) -- (6.299,8.463) -- (6.408,8.519) -- (6.496,8.606) -- (6.551,8.716) -- (6.571,8.837) -- (6.551,8.959) -- (6.496,9.069) -- (6.408,9.156) -- (6.299,9.212) -- (6.177,9.231) -- (6.177,9.231) -- (6.055,9.212) -- (5.945,9.156) -- (5.858,9.069) -- (5.802,8.959) -- (5.783,8.837) -- (5.802,8.716) -- (5.858,8.606) -- (5.945,8.519) -- (6.055,8.463) -- (6.177,8.443) -- cycle;
\node at (6.177,8.837) {\scalebox{1.652}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.623,6.217) -- (4.745,6.236) -- (4.855,6.292) -- (4.942,6.379) -- (4.998,6.489) -- (5.017,6.611) -- (4.998,6.732) -- (4.942,6.842) -- (4.855,6.929) -- (4.745,6.985) -- (4.623,7.005) -- (4.623,7.005) -- (4.502,6.985) -- (4.392,6.929) -- (4.305,6.842) -- (4.249,6.732) -- (4.23,6.611) -- (4.249,6.489) -- (4.305,6.379) -- (4.392,6.292) -- (4.502,6.236) -- (4.623,6.217) -- cycle;
\node at (4.623,6.611) {\scalebox{1.652}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.164,6.261) -- (8.286,6.281) -- (8.396,6.336) -- (8.483,6.424) -- (8.539,6.533) -- (8.558,6.655) -- (8.539,6.777) -- (8.483,6.887) -- (8.396,6.974) -- (8.286,7.03) -- (8.164,7.049) -- (8.164,7.049) -- (8.042,7.03) -- (7.933,6.974) -- (7.845,6.887) -- (7.789,6.777) -- (7.77,6.655) -- (7.789,6.533) -- (7.845,6.424) -- (7.933,6.336) -- (8.042,6.281) -- (8.164,6.261) -- cycle;
\node at (8.164,6.655) {\scalebox{1.652}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.621,3.174) -- (11.743,3.193) -- (11.853,3.249) -- (11.94,3.336) -- (11.996,3.446) -- (12.015,3.568) -- (11.996,3.69) -- (11.94,3.799) -- (11.853,3.887) -- (11.743,3.943) -- (11.621,3.962) -- (11.621,3.962) -- (11.499,3.943) -- (11.39,3.887) -- (11.302,3.799) -- (11.247,3.69) -- (11.227,3.568) -- (11.247,3.446) -- (11.302,3.336) -- (11.39,3.249) -- (11.499,3.193) -- (11.621,3.174) -- cycle;
\node at (11.621,3.568) {\scalebox{1.652}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.295,4.111) -- (2.417,4.13) -- (2.527,4.186) -- (2.614,4.273) -- (2.67,4.383) -- (2.689,4.504) -- (2.67,4.626) -- (2.614,4.736) -- (2.527,4.823) -- (2.417,4.879) -- (2.295,4.898) -- (2.295,4.898) -- (2.174,4.879) -- (2.064,4.823) -- (1.977,4.736) -- (1.921,4.626) -- (1.902,4.504) -- (1.921,4.383) -- (1.977,4.273) -- (2.064,4.186) -- (2.174,4.13) -- (2.295,4.111) -- cycle;
\node at (2.295,4.504) {\scalebox{1.652}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.906,3.053) -- (2.027,3.072) -- (2.137,3.128) -- (2.224,3.215) -- (2.28,3.325) -- (2.299,3.447) -- (2.28,3.569) -- (2.224,3.678) -- (2.137,3.766) -- (2.027,3.822) -- (1.906,3.841) -- (1.906,3.841) -- (1.784,3.822) -- (1.674,3.766) -- (1.587,3.678) -- (1.531,3.569) -- (1.512,3.447) -- (1.531,3.325) -- (1.587,3.215) -- (1.674,3.128) -- (1.784,3.072) -- (1.906,3.053) -- cycle;
\node at (1.906,3.447) {\scalebox{1.652}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.53,4.951) -- (9.652,4.97) -- (9.762,5.026) -- (9.849,5.114) -- (9.905,5.223) -- (9.924,5.345) -- (9.905,5.467) -- (9.849,5.577) -- (9.762,5.664) -- (9.652,5.72) -- (9.53,5.739) -- (9.53,5.739) -- (9.408,5.72) -- (9.299,5.664) -- (9.211,5.577) -- (9.156,5.467) -- (9.136,5.345) -- (9.156,5.223) -- (9.211,5.114) -- (9.299,5.026) -- (9.408,4.97) -- (9.53,4.951) -- cycle;
\node at (9.53,5.345) {\scalebox{1.652}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.43,2.833) -- (8.552,2.852) -- (8.662,2.908) -- (8.749,2.996) -- (8.805,3.105) -- (8.824,3.227) -- (8.805,3.349) -- (8.749,3.459) -- (8.662,3.546) -- (8.552,3.602) -- (8.43,3.621) -- (8.43,3.621) -- (8.308,3.602) -- (8.198,3.546) -- (8.111,3.459) -- (8.055,3.349) -- (8.036,3.227) -- (8.055,3.105) -- (8.111,2.996) -- (8.198,2.908) -- (8.308,2.852) -- (8.43,2.833) -- cycle;
\node at (8.43,3.227) {\scalebox{1.652}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.799,4.574) -- (5.921,4.593) -- (6.03,4.649) -- (6.118,4.736) -- (6.174,4.846) -- (6.193,4.968) -- (6.174,5.089) -- (6.118,5.199) -- (6.03,5.286) -- (5.921,5.342) -- (5.799,5.362) -- (5.799,5.362) -- (5.677,5.342) -- (5.567,5.286) -- (5.48,5.199) -- (5.424,5.089) -- (5.405,4.968) -- (5.424,4.846) -- (5.48,4.736) -- (5.567,4.649) -- (5.677,4.593) -- (5.799,4.574) -- cycle;
\node at (5.799,4.968) {\scalebox{1.652}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.756,2.689) -- (4.878,2.708) -- (4.988,2.764) -- (5.075,2.851) -- (5.131,2.961) -- (5.15,3.083) -- (5.131,3.204) -- (5.075,3.314) -- (4.988,3.401) -- (4.878,3.457) -- (4.756,3.477) -- (4.385,3.477) -- (4.264,3.457) -- (4.154,3.401) -- (4.067,3.314) -- (4.011,3.204) -- (3.992,3.083) -- (4.011,2.961) -- (4.067,2.851) -- (4.154,2.764) -- (4.264,2.708) -- (4.385,2.689) -- cycle;
\node at (4.571,3.083) {\scalebox{1.652}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.99,1.423) -- (3.111,1.442) -- (3.221,1.498) -- (3.308,1.585) -- (3.364,1.695) -- (3.384,1.817) -- (3.364,1.939) -- (3.308,2.048) -- (3.221,2.136) -- (3.111,2.192) -- (2.99,2.211) -- (2.619,2.211) -- (2.497,2.192) -- (2.387,2.136) -- (2.3,2.048) -- (2.244,1.939) -- (2.225,1.817) -- (2.244,1.695) -- (2.3,1.585) -- (2.387,1.498) -- (2.497,1.442) -- (2.619,1.423) -- cycle;
\node at (2.804,1.817) {\scalebox{1.652}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.526,1.844) -- (9.648,1.863) -- (9.758,1.919) -- (9.845,2.006) -- (9.901,2.116) -- (9.92,2.238) -- (9.901,2.359) -- (9.845,2.469) -- (9.758,2.556) -- (9.648,2.612) -- (9.526,2.632) -- (9.155,2.632) -- (9.034,2.612) -- (8.924,2.556) -- (8.837,2.469) -- (8.781,2.359) -- (8.762,2.238) -- (8.781,2.116) -- (8.837,2.006) -- (8.924,1.919) -- (9.034,1.863) -- (9.155,1.844) -- cycle;
\node at (9.341,2.238) {\scalebox{1.652}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.314,1.316) -- (6.436,1.336) -- (6.546,1.392) -- (6.633,1.479) -- (6.689,1.589) -- (6.708,1.71) -- (6.689,1.832) -- (6.633,1.942) -- (6.546,2.029) -- (6.436,2.085) -- (6.314,2.104) -- (5.944,2.104) -- (5.822,2.085) -- (5.712,2.029) -- (5.625,1.942) -- (5.569,1.832) -- (5.55,1.71) -- (5.569,1.589) -- (5.625,1.479) -- (5.712,1.392) -- (5.822,1.336) -- (5.944,1.316) -- cycle;
\node at (6.129,1.71) {\scalebox{1.652}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (3.949,8.453) -- (4.549,8.453) -- (4.549,9.053) -- (3.949,9.053) -- cycle;
\node at (4.249,8.753) {\scalebox{1.652}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.199,1.105) -- (1.799,1.105) -- (1.799,1.705) -- (1.199,1.705) -- cycle;
\node at (1.499,1.405) {\scalebox{1.652}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.584,5.313) -- (2.184,5.313) -- (2.184,5.913) -- (1.584,5.913) -- cycle;
\node at (1.884,5.613) {\scalebox{1.652}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.318,5.746) -- (10.918,5.746) -- (10.918,6.346) -- (10.318,6.346) -- cycle;
\node at (10.618,6.046) {\scalebox{1.652}{\small d}};
\node at (4.006,9.323) {\scalebox{1.652}{\small A}};
\node at (8.336,9.01) {\scalebox{1.652}{\small B}};
\node at (11.957,5.409) {\scalebox{1.652}{\small C}};
\node at (0.765,5.404) {\scalebox{1.652}{\small D}};
\node at (8.36,5.086) {\scalebox{1.652}{\small E}};
\node at (3.25,4.624) {\scalebox{1.652}{\small F}};
\node at (11.398,1.714) {\scalebox{1.652}{\small G}};
\node at (0.81,1.303) {\scalebox{1.652}{\small H}};
\node at (8.363,0.956) {\scalebox{1.652}{\small I}};
\node at (3.899,0.95) {\scalebox{1.652}{\small J}};
\draw[line width=0.5pt] (12.646,7.803) -- (12.646,8.903);
\draw[line width=0.5pt] (12.415,8.518) -- (12.646,8.903) -- (12.877,8.518);
\draw[line width=0.5pt] (12.344,8.353) -- (12.949,8.353);
\node at (12.646,9.288) {\scalebox{1.652}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour A, 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 & ⑧ quai aux Fleurs & Ⓑ Place aux Herbes & Ⓙ Rond-point du Pont \\ \hline
① rue du Four & ⑨ rue du Pont & Ⓒ Coin du Lavoir & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
② rue Verte & ⑩ chemin de Ronde & Ⓓ Place Sainte-Croix & \textbf{a} l'école \\ \hline
③ rue de la Mairie & ⑪ rue du Stade & Ⓔ Carrefour des Tilleuls & \textbf{b} le stade \\ \hline
④ rue de la Poste & ⑫ sentier des Prés & Ⓕ Carrefour de la Mairie & \textbf{c} la gendarmerie \\ \hline
⑤ rue Neuve & ⑬ allée des Peupliers & Ⓖ Place de l'Église & \textbf{d} la boulangerie \\ \hline
⑥ avenue de la Gare & \cellcolor{black!32}\bfseries Les carrefours & Ⓗ Placette des Ormes &  \\ \hline
⑦ boulevard du Parc & Ⓐ Place des Tanneurs & Ⓘ Carrefour du Lavoir &  \\ \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=0.62, line join=round, line cap=round]
\draw[line width=0.5pt] (4.249,8.753) -- (4.517,8.69) -- (4.785,8.633) -- (5.055,8.583) -- (5.327,8.538) -- (5.599,8.501) -- (5.872,8.469) -- (6.146,8.445) -- (6.42,8.426) -- (6.695,8.414) -- (6.97,8.409) -- (7.244,8.409) -- (7.519,8.417) -- (7.793,8.431) -- (8.067,8.451);
\draw[line width=0.5pt] (4.249,8.753) -- (4.268,8.453) -- (4.28,8.152) -- (4.284,7.85) -- (4.282,7.549) -- (4.272,7.248) -- (4.255,6.947) -- (4.231,6.647) -- (4.2,6.347) -- (4.162,6.048) -- (4.118,5.75) -- (4.066,5.453) -- (4.007,5.157) -- (3.941,4.864) -- (3.868,4.572);
\draw[line width=0.5pt] (8.067,8.451) -- (8.007,8.203) -- (7.953,7.953) -- (7.905,7.702) -- (7.863,7.45) -- (7.826,7.197) -- (7.796,6.943) -- (7.772,6.688) -- (7.753,6.433) -- (7.741,6.178) -- (7.734,5.922) -- (7.734,5.666) -- (7.739,5.411) -- (7.751,5.155) -- (7.769,4.901);
\draw[line width=0.5pt] (11.346,5.302) -- (11.347,5.062) -- (11.342,4.822) -- (11.331,4.582) -- (11.315,4.343) -- (11.293,4.103) -- (11.265,3.865) -- (11.232,3.627) -- (11.193,3.39) -- (11.148,3.154) -- (11.099,2.919) -- (11.043,2.685) -- (10.982,2.453) -- (10.916,2.223) -- (10.844,1.994);
\draw[line width=0.5pt] (1.374,5.288) -- (1.543,5.21) -- (1.714,5.136) -- (1.886,5.066) -- (2.06,5.0) -- (2.236,4.938) -- (2.413,4.88) -- (2.592,4.827) -- (2.771,4.777) -- (2.952,4.732) -- (3.134,4.692) -- (3.316,4.655) -- (3.5,4.623) -- (3.684,4.595) -- (3.868,4.572);
\draw[line width=0.5pt] (1.374,5.288) -- (1.412,5.027) -- (1.444,4.766) -- (1.47,4.503) -- (1.49,4.241) -- (1.503,3.977) -- (1.511,3.714) -- (1.512,3.45) -- (1.507,3.186) -- (1.495,2.923) -- (1.478,2.66) -- (1.454,2.397) -- (1.424,2.135) -- (1.388,1.874) -- (1.346,1.614);
\draw[line width=0.5pt] (7.769,4.901) -- (8.027,4.89) -- (8.285,4.885) -- (8.543,4.887) -- (8.801,4.894) -- (9.059,4.908) -- (9.317,4.928) -- (9.574,4.954) -- (9.831,4.985) -- (10.086,5.023) -- (10.341,5.067) -- (10.594,5.117) -- (10.846,5.173) -- (11.097,5.234) -- (11.346,5.302);
\draw[line width=0.5pt] (7.769,4.901) -- (7.824,4.661) -- (7.874,4.419) -- (7.918,4.177) -- (7.957,3.933) -- (7.989,3.689) -- (8.016,3.444) -- (8.037,3.198) -- (8.053,2.952) -- (8.062,2.705) -- (8.066,2.459) -- (8.064,2.212) -- (8.056,1.966) -- (8.042,1.72) -- (8.023,1.474);
\draw[line width=0.5pt] (3.868,4.572) -- (4.148,4.552) -- (4.429,4.539) -- (4.709,4.533) -- (4.99,4.534) -- (5.271,4.541) -- (5.552,4.555) -- (5.832,4.575) -- (6.112,4.602) -- (6.391,4.635) -- (6.669,4.675) -- (6.946,4.722) -- (7.222,4.775) -- (7.496,4.835) -- (7.769,4.901);
\draw[line width=0.5pt] (3.868,4.572) -- (3.928,4.356) -- (3.983,4.139) -- (4.033,3.92) -- (4.077,3.701) -- (4.116,3.48) -- (4.151,3.259) -- (4.18,3.036) -- (4.203,2.814) -- (4.222,2.59) -- (4.235,2.367) -- (4.243,2.143) -- (4.246,1.919) -- (4.243,1.695) -- (4.235,1.471);
\draw[line width=0.5pt] (1.346,1.614) -- (1.549,1.572) -- (1.754,1.536) -- (1.959,1.503) -- (2.164,1.476) -- (2.371,1.454) -- (2.578,1.436) -- (2.785,1.423) -- (2.992,1.416) -- (3.2,1.413) -- (3.408,1.415) -- (3.615,1.421) -- (3.822,1.433) -- (4.029,1.45) -- (4.235,1.471);
\draw[line width=0.5pt] (8.023,1.474) -- (8.217,1.542) -- (8.413,1.605) -- (8.61,1.663) -- (8.809,1.717) -- (9.009,1.766) -- (9.21,1.811) -- (9.412,1.85) -- (9.615,1.885) -- (9.819,1.915) -- (10.023,1.941) -- (10.228,1.961) -- (10.433,1.977) -- (10.639,1.988) -- (10.844,1.994);
\draw[line width=0.5pt] (4.235,1.471) -- (4.504,1.43) -- (4.773,1.395) -- (5.043,1.367) -- (5.314,1.344) -- (5.586,1.329) -- (5.857,1.319) -- (6.129,1.316) -- (6.401,1.32) -- (6.673,1.33) -- (6.944,1.346) -- (7.215,1.368) -- (7.485,1.397) -- (7.755,1.433) -- (8.023,1.474);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.622,8.753) -- (4.594,8.896) -- (4.513,9.017) -- (4.392,9.098) -- (4.249,9.126) -- (4.106,9.098) -- (3.985,9.017) -- (3.904,8.896) -- (3.876,8.753) -- (3.904,8.61) -- (3.985,8.489) -- (4.106,8.408) -- (4.249,8.38) -- (4.392,8.408) -- (4.513,8.489) -- (4.594,8.61) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.44,8.451) -- (8.412,8.594) -- (8.331,8.715) -- (8.21,8.796) -- (8.067,8.824) -- (7.924,8.796) -- (7.803,8.715) -- (7.722,8.594) -- (7.694,8.451) -- (7.722,8.308) -- (7.803,8.187) -- (7.924,8.106) -- (8.067,8.078) -- (8.21,8.106) -- (8.331,8.187) -- (8.412,8.308) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.719,5.302) -- (11.691,5.445) -- (11.61,5.566) -- (11.489,5.647) -- (11.346,5.675) -- (11.203,5.647) -- (11.082,5.566) -- (11.001,5.445) -- (10.973,5.302) -- (11.001,5.159) -- (11.082,5.038) -- (11.203,4.957) -- (11.346,4.929) -- (11.489,4.957) -- (11.61,5.038) -- (11.691,5.159) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.747,5.288) -- (1.719,5.431) -- (1.638,5.552) -- (1.517,5.633) -- (1.374,5.661) -- (1.231,5.633) -- (1.11,5.552) -- (1.029,5.431) -- (1.001,5.288) -- (1.029,5.145) -- (1.11,5.024) -- (1.231,4.943) -- (1.374,4.915) -- (1.517,4.943) -- (1.638,5.024) -- (1.719,5.145) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.142,4.901) -- (8.114,5.043) -- (8.033,5.164) -- (7.912,5.245) -- (7.769,5.274) -- (7.626,5.245) -- (7.505,5.164) -- (7.424,5.043) -- (7.396,4.901) -- (7.424,4.758) -- (7.505,4.637) -- (7.626,4.556) -- (7.769,4.527) -- (7.912,4.556) -- (8.033,4.637) -- (8.114,4.758) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.241,4.572) -- (4.213,4.714) -- (4.132,4.835) -- (4.011,4.916) -- (3.868,4.945) -- (3.725,4.916) -- (3.604,4.835) -- (3.523,4.714) -- (3.495,4.572) -- (3.523,4.429) -- (3.604,4.308) -- (3.725,4.227) -- (3.868,4.198) -- (4.011,4.227) -- (4.132,4.308) -- (4.213,4.429) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.217,1.994) -- (11.189,2.136) -- (11.108,2.258) -- (10.987,2.338) -- (10.844,2.367) -- (10.701,2.338) -- (10.58,2.258) -- (10.499,2.136) -- (10.471,1.994) -- (10.499,1.851) -- (10.58,1.73) -- (10.701,1.649) -- (10.844,1.62) -- (10.987,1.649) -- (11.108,1.73) -- (11.189,1.851) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.719,1.614) -- (1.691,1.757) -- (1.61,1.878) -- (1.489,1.959) -- (1.346,1.987) -- (1.203,1.959) -- (1.082,1.878) -- (1.001,1.757) -- (0.973,1.614) -- (1.001,1.471) -- (1.082,1.35) -- (1.203,1.269) -- (1.346,1.241) -- (1.489,1.269) -- (1.61,1.35) -- (1.691,1.471) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.396,1.474) -- (8.368,1.617) -- (8.287,1.738) -- (8.166,1.819) -- (8.023,1.848) -- (7.88,1.819) -- (7.759,1.738) -- (7.678,1.617) -- (7.65,1.474) -- (7.678,1.331) -- (7.759,1.21) -- (7.88,1.13) -- (8.023,1.101) -- (8.166,1.13) -- (8.287,1.21) -- (8.368,1.331) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.608,1.471) -- (4.58,1.614) -- (4.499,1.735) -- (4.378,1.816) -- (4.235,1.844) -- (4.092,1.816) -- (3.971,1.735) -- (3.89,1.614) -- (3.862,1.471) -- (3.89,1.328) -- (3.971,1.207) -- (4.092,1.126) -- (4.235,1.098) -- (4.378,1.126) -- (4.499,1.207) -- (4.58,1.328) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.176,8.443) -- (6.295,8.462) -- (6.402,8.517) -- (6.487,8.602) -- (6.542,8.709) -- (6.561,8.828) -- (6.542,8.947) -- (6.487,9.054) -- (6.402,9.139) -- (6.295,9.194) -- (6.176,9.212) -- (6.176,9.212) -- (6.057,9.194) -- (5.95,9.139) -- (5.865,9.054) -- (5.81,8.947) -- (5.792,8.828) -- (5.81,8.709) -- (5.865,8.602) -- (5.95,8.517) -- (6.057,8.462) -- (6.176,8.443) -- cycle;
\node at (6.176,8.828) {\scalebox{1.613}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.614,6.227) -- (4.733,6.246) -- (4.84,6.301) -- (4.925,6.386) -- (4.98,6.493) -- (4.999,6.612) -- (4.98,6.73) -- (4.925,6.838) -- (4.84,6.923) -- (4.733,6.977) -- (4.614,6.996) -- (4.614,6.996) -- (4.495,6.977) -- (4.388,6.923) -- (4.303,6.838) -- (4.248,6.73) -- (4.23,6.612) -- (4.248,6.493) -- (4.303,6.386) -- (4.388,6.301) -- (4.495,6.246) -- (4.614,6.227) -- cycle;
\node at (4.614,6.612) {\scalebox{1.613}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.155,6.271) -- (8.274,6.29) -- (8.381,6.345) -- (8.466,6.43) -- (8.52,6.537) -- (8.539,6.656) -- (8.52,6.775) -- (8.466,6.882) -- (8.381,6.967) -- (8.274,7.022) -- (8.155,7.04) -- (8.155,7.04) -- (8.036,7.022) -- (7.929,6.967) -- (7.844,6.882) -- (7.789,6.775) -- (7.77,6.656) -- (7.789,6.537) -- (7.844,6.43) -- (7.929,6.345) -- (8.036,6.29) -- (8.155,6.271) -- cycle;
\node at (8.155,6.656) {\scalebox{1.613}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.612,3.185) -- (11.731,3.204) -- (11.838,3.258) -- (11.923,3.343) -- (11.978,3.451) -- (11.996,3.569) -- (11.978,3.688) -- (11.923,3.795) -- (11.838,3.88) -- (11.731,3.935) -- (11.612,3.954) -- (11.612,3.954) -- (11.493,3.935) -- (11.386,3.88) -- (11.301,3.795) -- (11.246,3.688) -- (11.227,3.569) -- (11.246,3.451) -- (11.301,3.343) -- (11.386,3.258) -- (11.493,3.204) -- (11.612,3.185) -- cycle;
\node at (11.612,3.569) {\scalebox{1.613}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.298,4.129) -- (2.417,4.148) -- (2.524,4.202) -- (2.609,4.287) -- (2.664,4.395) -- (2.683,4.513) -- (2.664,4.632) -- (2.609,4.739) -- (2.524,4.824) -- (2.417,4.879) -- (2.298,4.898) -- (2.298,4.898) -- (2.179,4.879) -- (2.072,4.824) -- (1.987,4.739) -- (1.933,4.632) -- (1.914,4.513) -- (1.933,4.395) -- (1.987,4.287) -- (2.072,4.202) -- (2.179,4.148) -- (2.298,4.129) -- cycle;
\node at (2.298,4.513) {\scalebox{1.613}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.896,3.062) -- (2.015,3.081) -- (2.122,3.136) -- (2.207,3.221) -- (2.262,3.328) -- (2.281,3.447) -- (2.262,3.566) -- (2.207,3.673) -- (2.122,3.758) -- (2.015,3.813) -- (1.896,3.832) -- (1.896,3.832) -- (1.777,3.813) -- (1.67,3.758) -- (1.585,3.673) -- (1.53,3.566) -- (1.512,3.447) -- (1.53,3.328) -- (1.585,3.221) -- (1.67,3.136) -- (1.777,3.081) -- (1.896,3.062) -- cycle;
\node at (1.896,3.447) {\scalebox{1.613}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.531,4.951) -- (9.65,4.97) -- (9.757,5.025) -- (9.842,5.11) -- (9.897,5.217) -- (9.916,5.336) -- (9.897,5.455) -- (9.842,5.562) -- (9.757,5.647) -- (9.65,5.701) -- (9.531,5.72) -- (9.531,5.72) -- (9.412,5.701) -- (9.305,5.647) -- (9.22,5.562) -- (9.166,5.455) -- (9.147,5.336) -- (9.166,5.217) -- (9.22,5.11) -- (9.305,5.025) -- (9.412,4.97) -- (9.531,4.951) -- cycle;
\node at (9.531,5.336) {\scalebox{1.613}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.421,2.842) -- (8.539,2.861) -- (8.647,2.915) -- (8.732,3.0) -- (8.786,3.108) -- (8.805,3.226) -- (8.786,3.345) -- (8.732,3.452) -- (8.647,3.537) -- (8.539,3.592) -- (8.421,3.611) -- (8.421,3.611) -- (8.302,3.592) -- (8.195,3.537) -- (8.11,3.452) -- (8.055,3.345) -- (8.036,3.226) -- (8.055,3.108) -- (8.11,3.0) -- (8.195,2.915) -- (8.302,2.861) -- (8.421,2.842) -- cycle;
\node at (8.421,3.226) {\scalebox{1.613}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.8,4.574) -- (5.919,4.593) -- (6.026,4.647) -- (6.111,4.732) -- (6.165,4.839) -- (6.184,4.958) -- (6.165,5.077) -- (6.111,5.184) -- (6.026,5.269) -- (5.919,5.324) -- (5.8,5.343) -- (5.8,5.343) -- (5.681,5.324) -- (5.574,5.269) -- (5.489,5.184) -- (5.434,5.077) -- (5.415,4.958) -- (5.434,4.839) -- (5.489,4.732) -- (5.574,4.647) -- (5.681,4.593) -- (5.8,4.574) -- cycle;
\node at (5.8,4.958) {\scalebox{1.613}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.742,2.697) -- (4.861,2.716) -- (4.968,2.771) -- (5.054,2.856) -- (5.108,2.963) -- (5.127,3.082) -- (5.108,3.2) -- (5.054,3.308) -- (4.968,3.393) -- (4.861,3.447) -- (4.742,3.466) -- (4.381,3.466) -- (4.262,3.447) -- (4.155,3.393) -- (4.069,3.308) -- (4.015,3.2) -- (3.996,3.082) -- (4.015,2.963) -- (4.069,2.856) -- (4.155,2.771) -- (4.262,2.716) -- (4.381,2.697) -- cycle;
\node at (4.561,3.082) {\scalebox{1.613}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.985,1.423) -- (3.104,1.442) -- (3.211,1.496) -- (3.296,1.582) -- (3.351,1.689) -- (3.369,1.808) -- (3.351,1.926) -- (3.296,2.034) -- (3.211,2.119) -- (3.104,2.173) -- (2.985,2.192) -- (2.623,2.192) -- (2.504,2.173) -- (2.397,2.119) -- (2.312,2.034) -- (2.257,1.926) -- (2.238,1.808) -- (2.257,1.689) -- (2.312,1.582) -- (2.397,1.496) -- (2.504,1.442) -- (2.623,1.423) -- cycle;
\node at (2.804,1.808) {\scalebox{1.613}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.523,1.844) -- (9.642,1.863) -- (9.749,1.917) -- (9.835,2.003) -- (9.889,2.11) -- (9.908,2.229) -- (9.889,2.347) -- (9.835,2.455) -- (9.749,2.54) -- (9.642,2.594) -- (9.523,2.613) -- (9.162,2.613) -- (9.043,2.594) -- (8.936,2.54) -- (8.85,2.455) -- (8.796,2.347) -- (8.777,2.229) -- (8.796,2.11) -- (8.85,2.003) -- (8.936,1.917) -- (9.043,1.863) -- (9.162,1.844) -- cycle;
\node at (9.343,2.229) {\scalebox{1.613}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.31,1.316) -- (6.429,1.335) -- (6.536,1.39) -- (6.621,1.475) -- (6.676,1.582) -- (6.694,1.701) -- (6.676,1.82) -- (6.621,1.927) -- (6.536,2.012) -- (6.429,2.067) -- (6.31,2.085) -- (5.948,2.085) -- (5.829,2.067) -- (5.722,2.012) -- (5.637,1.927) -- (5.582,1.82) -- (5.563,1.701) -- (5.582,1.582) -- (5.637,1.475) -- (5.722,1.39) -- (5.829,1.335) -- (5.948,1.316) -- cycle;
\node at (6.129,1.701) {\scalebox{1.613}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (3.949,8.453) -- (4.549,8.453) -- (4.549,9.053) -- (3.949,9.053) -- cycle;
\node at (4.249,8.753) {\scalebox{1.613}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.199,1.105) -- (1.799,1.105) -- (1.799,1.705) -- (1.199,1.705) -- cycle;
\node at (1.499,1.405) {\scalebox{1.613}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.584,5.313) -- (2.184,5.313) -- (2.184,5.913) -- (1.584,5.913) -- cycle;
\node at (1.884,5.613) {\scalebox{1.613}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.318,5.746) -- (10.918,5.746) -- (10.918,6.346) -- (10.318,6.346) -- cycle;
\node at (10.618,6.046) {\scalebox{1.613}{\small d}};
\node at (4.006,9.323) {\scalebox{1.613}{\small A}};
\node at (8.336,9.01) {\scalebox{1.613}{\small B}};
\node at (11.957,5.409) {\scalebox{1.613}{\small C}};
\node at (0.765,5.404) {\scalebox{1.613}{\small D}};
\node at (8.36,5.086) {\scalebox{1.613}{\small E}};
\node at (3.25,4.624) {\scalebox{1.613}{\small F}};
\node at (11.398,1.714) {\scalebox{1.613}{\small G}};
\node at (0.81,1.303) {\scalebox{1.613}{\small H}};
\node at (8.363,0.956) {\scalebox{1.613}{\small I}};
\node at (3.899,0.95) {\scalebox{1.613}{\small J}};
\draw[line width=0.5pt] (12.646,7.803) -- (12.646,8.903);
\draw[line width=0.5pt] (12.415,8.518) -- (12.646,8.903) -- (12.877,8.518);
\draw[line width=0.5pt] (12.344,8.353) -- (12.949,8.353);
\node at (12.646,9.288) {\scalebox{1.613}{\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, G, J & 5 \\ \hline
clairs & B, C, F, H, I & 5 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{5} 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 10 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui.} Un circuit existe :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{rue du Four ①} vers l'est jusqu'à Place aux Herbes \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Mairie ③}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{boulevard du Parc ⑦}, vers l'est, jusqu'à Coin du Lavoir \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Poste ④}, vers le sud, jusqu'à Place de l'Église \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ⑫}, vers l'ouest, jusqu'à Carrefour du Lavoir \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Peupliers ⑬}, vers l'ouest, jusqu'à Rond-point du Pont \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Stade ⑪}, vers l'ouest, jusqu'à Placette des Ormes \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{avenue de la Gare ⑥}, vers le nord, jusqu'à Place Sainte-Croix \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Neuve ⑤}, vers l'est, jusqu'à Carrefour de la Mairie \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Verte ②}, vers le nord, jusqu'à Place des Tanneurs \textbf{(A)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Le car est passé une fois par chaque carrefour et se retrouve à l'école.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, et c'est déjà fait : le trajet ci-dessus dessert tous les carrefours. Il suffit au car de s'arrêter avant le dernier trajet de retour.}\par\addvspace{0.4em}
\end{document}
