\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 12}

\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=12}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0012}
\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=0.6338, line join=round, line cap=round]
\draw[line width=0.5pt] (7.087,11.558) -- (7.063,11.282) -- (7.047,11.007) -- (7.037,10.731) -- (7.033,10.454) -- (7.036,10.178) -- (7.045,9.901) -- (7.061,9.625) -- (7.083,9.35) -- (7.112,9.075) -- (7.147,8.801) -- (7.188,8.527) -- (7.236,8.255) -- (7.291,7.984) -- (7.352,7.715);
\draw[line width=0.5pt] (1.527,10.853) -- (1.716,10.879) -- (1.905,10.9) -- (2.095,10.916) -- (2.286,10.928) -- (2.476,10.936) -- (2.667,10.939) -- (2.858,10.938) -- (3.049,10.932) -- (3.239,10.921) -- (3.43,10.906) -- (3.619,10.887) -- (3.809,10.863) -- (3.997,10.835) -- (4.185,10.802);
\draw[line width=0.5pt] (1.527,10.853) -- (1.484,10.649) -- (1.446,10.444) -- (1.413,10.238) -- (1.384,10.031) -- (1.361,9.823) -- (1.342,9.615) -- (1.329,9.406) -- (1.32,9.198) -- (1.316,8.989) -- (1.317,8.78) -- (1.323,8.571) -- (1.334,8.362) -- (1.35,8.154) -- (1.37,7.946);
\draw[line width=0.5pt] (4.185,10.802) -- (4.382,10.888) -- (4.582,10.968) -- (4.783,11.044) -- (4.986,11.115) -- (5.191,11.182) -- (5.397,11.243) -- (5.605,11.3) -- (5.814,11.351) -- (6.024,11.398) -- (6.235,11.44) -- (6.447,11.477) -- (6.66,11.509) -- (6.873,11.536) -- (7.087,11.558);
\draw[line width=0.5pt] (4.185,10.802) -- (4.154,10.628) -- (4.128,10.452) -- (4.105,10.276) -- (4.087,10.1) -- (4.073,9.923) -- (4.063,9.745) -- (4.058,9.568) -- (4.056,9.39) -- (4.059,9.213) -- (4.065,9.036) -- (4.076,8.858) -- (4.091,8.682) -- (4.111,8.505) -- (4.134,8.329);
\draw[line width=0.5pt] (10.481,8.423) -- (10.533,8.203) -- (10.58,7.982) -- (10.622,7.759) -- (10.659,7.536) -- (10.69,7.312) -- (10.716,7.087) -- (10.737,6.862) -- (10.753,6.636) -- (10.763,6.41) -- (10.768,6.183) -- (10.768,5.957) -- (10.762,5.731) -- (10.751,5.505) -- (10.734,5.28);
\draw[line width=0.5pt] (4.134,8.329) -- (4.355,8.251) -- (4.579,8.177) -- (4.803,8.109) -- (5.03,8.046) -- (5.258,7.988) -- (5.487,7.936) -- (5.718,7.889) -- (5.949,7.848) -- (6.181,7.812) -- (6.414,7.782) -- (6.648,7.757) -- (6.882,7.737) -- (7.117,7.723) -- (7.352,7.715);
\draw[line width=0.5pt] (4.134,8.329) -- (4.211,8.122) -- (4.284,7.912) -- (4.351,7.701) -- (4.414,7.488) -- (4.471,7.274) -- (4.524,7.059) -- (4.571,6.842) -- (4.613,6.624) -- (4.65,6.406) -- (4.682,6.186) -- (4.709,5.966) -- (4.731,5.745) -- (4.747,5.524) -- (4.758,5.303);
\draw[line width=0.5pt] (1.37,7.946) -- (1.57,7.943) -- (1.77,7.945) -- (1.971,7.951) -- (2.17,7.962) -- (2.37,7.978) -- (2.569,7.999) -- (2.768,8.024) -- (2.966,8.054) -- (3.163,8.088) -- (3.36,8.127) -- (3.555,8.171) -- (3.749,8.219) -- (3.942,8.272) -- (4.134,8.329);
\draw[line width=0.5pt] (7.352,7.715) -- (7.566,7.799) -- (7.782,7.879) -- (7.999,7.953) -- (8.219,8.022) -- (8.44,8.086) -- (8.663,8.144) -- (8.887,8.198) -- (9.112,8.246) -- (9.338,8.289) -- (9.565,8.326) -- (9.793,8.359) -- (10.022,8.385) -- (10.251,8.407) -- (10.481,8.423);
\draw[line width=0.5pt] (7.352,7.715) -- (7.315,7.537) -- (7.283,7.358) -- (7.255,7.178) -- (7.231,6.997) -- (7.212,6.816) -- (7.196,6.634) -- (7.185,6.453) -- (7.179,6.271) -- (7.176,6.088) -- (7.178,5.906) -- (7.184,5.724) -- (7.195,5.543) -- (7.209,5.361) -- (7.228,5.18);
\draw[line width=0.5pt] (4.758,5.303) -- (4.935,5.321) -- (5.111,5.335) -- (5.289,5.345) -- (5.466,5.351) -- (5.643,5.353) -- (5.821,5.35) -- (5.998,5.343) -- (6.176,5.333) -- (6.352,5.318) -- (6.529,5.298) -- (6.705,5.275) -- (6.88,5.248) -- (7.055,5.216) -- (7.228,5.18);
\draw[line width=0.5pt] (4.758,5.303) -- (4.685,5.043) -- (4.618,4.78) -- (4.557,4.517) -- (4.502,4.252) -- (4.454,3.985) -- (4.412,3.718) -- (4.376,3.449) -- (4.346,3.18) -- (4.323,2.91) -- (4.306,2.64) -- (4.296,2.369) -- (4.291,2.099) -- (4.294,1.828) -- (4.302,1.558);
\draw[line width=0.5pt] (7.228,5.18) -- (7.476,5.226) -- (7.724,5.265) -- (7.974,5.299) -- (8.224,5.327) -- (8.474,5.348) -- (8.726,5.364) -- (8.977,5.375) -- (9.229,5.379) -- (9.481,5.377) -- (9.732,5.369) -- (9.984,5.356) -- (10.235,5.336) -- (10.485,5.311) -- (10.734,5.28);
\draw[line width=0.5pt] (1.85,1.635) -- (2.023,1.603) -- (2.197,1.575) -- (2.371,1.551) -- (2.546,1.531) -- (2.722,1.515) -- (2.897,1.503) -- (3.073,1.495) -- (3.249,1.492) -- (3.425,1.492) -- (3.601,1.497) -- (3.777,1.506) -- (3.953,1.519) -- (4.128,1.536) -- (4.302,1.558);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.247,11.558) -- (7.225,11.638) -- (7.167,11.696) -- (7.087,11.718) -- (7.007,11.696) -- (6.948,11.638) -- (6.927,11.558) -- (6.948,11.478) -- (7.007,11.419) -- (7.087,11.398) -- (7.167,11.419) -- (7.225,11.478) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.687,10.853) -- (1.666,10.933) -- (1.607,10.992) -- (1.527,11.013) -- (1.447,10.992) -- (1.388,10.933) -- (1.367,10.853) -- (1.388,10.773) -- (1.447,10.715) -- (1.527,10.693) -- (1.607,10.715) -- (1.666,10.773) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.345,10.802) -- (4.324,10.882) -- (4.265,10.941) -- (4.185,10.962) -- (4.105,10.941) -- (4.047,10.882) -- (4.025,10.802) -- (4.047,10.722) -- (4.105,10.664) -- (4.185,10.642) -- (4.265,10.664) -- (4.324,10.722) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.641,8.423) -- (10.619,8.503) -- (10.561,8.561) -- (10.481,8.583) -- (10.401,8.561) -- (10.342,8.503) -- (10.321,8.423) -- (10.342,8.343) -- (10.401,8.284) -- (10.481,8.263) -- (10.561,8.284) -- (10.619,8.343) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.294,8.329) -- (4.273,8.409) -- (4.214,8.468) -- (4.134,8.489) -- (4.054,8.468) -- (3.996,8.409) -- (3.974,8.329) -- (3.996,8.249) -- (4.054,8.191) -- (4.134,8.169) -- (4.214,8.191) -- (4.273,8.249) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.53,7.946) -- (1.509,8.026) -- (1.45,8.085) -- (1.37,8.106) -- (1.29,8.085) -- (1.232,8.026) -- (1.21,7.946) -- (1.232,7.866) -- (1.29,7.808) -- (1.37,7.786) -- (1.45,7.808) -- (1.509,7.866) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.512,7.715) -- (7.49,7.795) -- (7.432,7.853) -- (7.352,7.875) -- (7.272,7.853) -- (7.213,7.795) -- (7.192,7.715) -- (7.213,7.635) -- (7.272,7.576) -- (7.352,7.555) -- (7.432,7.576) -- (7.49,7.635) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.918,5.303) -- (4.897,5.383) -- (4.838,5.442) -- (4.758,5.463) -- (4.678,5.442) -- (4.62,5.383) -- (4.598,5.303) -- (4.62,5.223) -- (4.678,5.164) -- (4.758,5.143) -- (4.838,5.164) -- (4.897,5.223) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.894,5.28) -- (10.873,5.36) -- (10.814,5.418) -- (10.734,5.44) -- (10.654,5.418) -- (10.596,5.36) -- (10.574,5.28) -- (10.596,5.2) -- (10.654,5.141) -- (10.734,5.12) -- (10.814,5.141) -- (10.873,5.2) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.388,5.18) -- (7.367,5.26) -- (7.308,5.319) -- (7.228,5.34) -- (7.148,5.319) -- (7.09,5.26) -- (7.068,5.18) -- (7.09,5.1) -- (7.148,5.042) -- (7.228,5.02) -- (7.308,5.042) -- (7.367,5.1) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.01,1.635) -- (1.989,1.715) -- (1.93,1.774) -- (1.85,1.795) -- (1.77,1.774) -- (1.712,1.715) -- (1.69,1.635) -- (1.712,1.555) -- (1.77,1.497) -- (1.85,1.475) -- (1.93,1.497) -- (1.989,1.555) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.462,1.558) -- (4.441,1.638) -- (4.382,1.696) -- (4.302,1.718) -- (4.222,1.696) -- (4.164,1.638) -- (4.142,1.558) -- (4.164,1.478) -- (4.222,1.419) -- (4.302,1.398) -- (4.382,1.419) -- (4.441,1.478) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.454,9.259) -- (7.575,9.278) -- (7.685,9.334) -- (7.772,9.421) -- (7.828,9.531) -- (7.848,9.652) -- (7.828,9.774) -- (7.772,9.884) -- (7.685,9.971) -- (7.575,10.027) -- (7.454,10.046) -- (7.454,10.046) -- (7.332,10.027) -- (7.222,9.971) -- (7.135,9.884) -- (7.079,9.774) -- (7.06,9.652) -- (7.079,9.531) -- (7.135,9.421) -- (7.222,9.334) -- (7.332,9.278) -- (7.454,9.259) -- cycle;
\node at (7.454,9.652) {\scalebox{1.578}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.866,10.937) -- (2.987,10.957) -- (3.097,11.013) -- (3.184,11.1) -- (3.24,11.21) -- (3.26,11.331) -- (3.24,11.453) -- (3.184,11.563) -- (3.097,11.65) -- (2.987,11.706) -- (2.866,11.725) -- (2.866,11.725) -- (2.744,11.706) -- (2.634,11.65) -- (2.547,11.563) -- (2.491,11.453) -- (2.472,11.331) -- (2.491,11.21) -- (2.547,11.1) -- (2.634,11.013) -- (2.744,10.957) -- (2.866,10.937) -- cycle;
\node at (2.866,11.331) {\scalebox{1.578}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.722,8.991) -- (1.844,9.011) -- (1.954,9.066) -- (2.041,9.154) -- (2.097,9.263) -- (2.116,9.385) -- (2.097,9.507) -- (2.041,9.617) -- (1.954,9.704) -- (1.844,9.76) -- (1.722,9.779) -- (1.722,9.779) -- (1.6,9.76) -- (1.491,9.704) -- (1.403,9.617) -- (1.347,9.507) -- (1.328,9.385) -- (1.347,9.263) -- (1.403,9.154) -- (1.491,9.066) -- (1.6,9.011) -- (1.722,8.991) -- cycle;
\node at (1.722,9.385) {\scalebox{1.578}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.506,11.287) -- (5.627,11.306) -- (5.737,11.362) -- (5.824,11.449) -- (5.88,11.559) -- (5.899,11.681) -- (5.88,11.803) -- (5.824,11.912) -- (5.737,12.0) -- (5.627,12.056) -- (5.506,12.075) -- (5.506,12.075) -- (5.384,12.056) -- (5.274,12.0) -- (5.187,11.912) -- (5.131,11.803) -- (5.112,11.681) -- (5.131,11.559) -- (5.187,11.449) -- (5.274,11.362) -- (5.384,11.306) -- (5.506,11.287) -- cycle;
\node at (5.506,11.681) {\scalebox{1.578}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.451,9.166) -- (4.573,9.185) -- (4.683,9.241) -- (4.77,9.328) -- (4.826,9.438) -- (4.845,9.56) -- (4.826,9.682) -- (4.77,9.791) -- (4.683,9.879) -- (4.573,9.935) -- (4.451,9.954) -- (4.451,9.954) -- (4.33,9.935) -- (4.22,9.879) -- (4.133,9.791) -- (4.077,9.682) -- (4.057,9.56) -- (4.077,9.438) -- (4.133,9.328) -- (4.22,9.241) -- (4.33,9.185) -- (4.451,9.166) -- cycle;
\node at (4.451,9.56) {\scalebox{1.578}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.13,6.499) -- (11.251,6.519) -- (11.361,6.575) -- (11.448,6.662) -- (11.504,6.772) -- (11.524,6.893) -- (11.504,7.015) -- (11.448,7.125) -- (11.361,7.212) -- (11.251,7.268) -- (11.13,7.287) -- (11.13,7.287) -- (11.008,7.268) -- (10.898,7.212) -- (10.811,7.125) -- (10.755,7.015) -- (10.736,6.893) -- (10.755,6.772) -- (10.811,6.662) -- (10.898,6.575) -- (11.008,6.519) -- (11.13,6.499) -- cycle;
\node at (11.13,6.893) {\scalebox{1.578}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.791,7.882) -- (5.913,7.902) -- (6.023,7.958) -- (6.11,8.045) -- (6.166,8.155) -- (6.185,8.276) -- (6.166,8.398) -- (6.11,8.508) -- (6.023,8.595) -- (5.913,8.651) -- (5.791,8.67) -- (5.791,8.67) -- (5.67,8.651) -- (5.56,8.595) -- (5.473,8.508) -- (5.417,8.398) -- (5.398,8.276) -- (5.417,8.155) -- (5.473,8.045) -- (5.56,7.958) -- (5.67,7.902) -- (5.791,7.882) -- cycle;
\node at (5.791,8.276) {\scalebox{1.578}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.957,6.528) -- (5.078,6.547) -- (5.188,6.603) -- (5.275,6.69) -- (5.331,6.8) -- (5.351,6.922) -- (5.331,7.043) -- (5.275,7.153) -- (5.188,7.24) -- (5.078,7.296) -- (4.957,7.315) -- (4.957,7.315) -- (4.835,7.296) -- (4.725,7.24) -- (4.638,7.153) -- (4.582,7.043) -- (4.563,6.922) -- (4.582,6.8) -- (4.638,6.69) -- (4.725,6.603) -- (4.835,6.547) -- (4.957,6.528) -- cycle;
\node at (4.957,6.922) {\scalebox{1.578}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.714,8.02) -- (2.836,8.039) -- (2.945,8.095) -- (3.033,8.183) -- (3.089,8.292) -- (3.108,8.414) -- (3.089,8.536) -- (3.033,8.646) -- (2.945,8.733) -- (2.836,8.789) -- (2.714,8.808) -- (2.714,8.808) -- (2.592,8.789) -- (2.482,8.733) -- (2.395,8.646) -- (2.339,8.536) -- (2.32,8.414) -- (2.339,8.292) -- (2.395,8.183) -- (2.482,8.095) -- (2.592,8.039) -- (2.714,8.02) -- cycle;
\node at (2.714,8.414) {\scalebox{1.578}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.985,8.188) -- (9.107,8.207) -- (9.217,8.263) -- (9.304,8.351) -- (9.36,8.46) -- (9.379,8.582) -- (9.36,8.704) -- (9.304,8.814) -- (9.217,8.901) -- (9.107,8.957) -- (8.985,8.976) -- (8.615,8.976) -- (8.493,8.957) -- (8.383,8.901) -- (8.296,8.814) -- (8.24,8.704) -- (8.221,8.582) -- (8.24,8.46) -- (8.296,8.351) -- (8.383,8.263) -- (8.493,8.207) -- (8.615,8.188) -- cycle;
\node at (8.8,8.582) {\scalebox{1.578}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.764,6.04) -- (7.886,6.059) -- (7.996,6.115) -- (8.083,6.202) -- (8.139,6.312) -- (8.158,6.433) -- (8.139,6.555) -- (8.083,6.665) -- (7.996,6.752) -- (7.886,6.808) -- (7.764,6.827) -- (7.394,6.827) -- (7.272,6.808) -- (7.162,6.752) -- (7.075,6.665) -- (7.019,6.555) -- (7.0,6.433) -- (7.019,6.312) -- (7.075,6.202) -- (7.162,6.115) -- (7.272,6.059) -- (7.394,6.04) -- cycle;
\node at (7.579,6.433) {\scalebox{1.578}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.203,5.343) -- (6.325,5.362) -- (6.435,5.418) -- (6.522,5.505) -- (6.578,5.615) -- (6.597,5.737) -- (6.578,5.859) -- (6.522,5.968) -- (6.435,6.056) -- (6.325,6.112) -- (6.203,6.131) -- (5.833,6.131) -- (5.711,6.112) -- (5.601,6.056) -- (5.514,5.968) -- (5.458,5.859) -- (5.439,5.737) -- (5.458,5.615) -- (5.514,5.505) -- (5.601,5.418) -- (5.711,5.362) -- (5.833,5.343) -- cycle;
\node at (6.018,5.737) {\scalebox{1.578}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.952,3.008) -- (5.074,3.027) -- (5.184,3.083) -- (5.271,3.17) -- (5.327,3.28) -- (5.346,3.402) -- (5.327,3.523) -- (5.271,3.633) -- (5.184,3.72) -- (5.074,3.776) -- (4.952,3.795) -- (4.581,3.795) -- (4.46,3.776) -- (4.35,3.72) -- (4.263,3.633) -- (4.207,3.523) -- (4.188,3.402) -- (4.207,3.28) -- (4.263,3.17) -- (4.35,3.083) -- (4.46,3.027) -- (4.581,3.008) -- cycle;
\node at (4.767,3.402) {\scalebox{1.578}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.151,5.374) -- (9.273,5.394) -- (9.383,5.45) -- (9.47,5.537) -- (9.526,5.647) -- (9.545,5.768) -- (9.526,5.89) -- (9.47,6.0) -- (9.383,6.087) -- (9.273,6.143) -- (9.151,6.162) -- (8.781,6.162) -- (8.659,6.143) -- (8.549,6.087) -- (8.462,6.0) -- (8.406,5.89) -- (8.387,5.768) -- (8.406,5.647) -- (8.462,5.537) -- (8.549,5.45) -- (8.659,5.394) -- (8.781,5.374) -- cycle;
\node at (8.966,5.768) {\scalebox{1.578}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.271,1.495) -- (3.393,1.514) -- (3.503,1.57) -- (3.59,1.658) -- (3.646,1.767) -- (3.665,1.889) -- (3.646,2.011) -- (3.59,2.121) -- (3.503,2.208) -- (3.393,2.264) -- (3.271,2.283) -- (2.9,2.283) -- (2.779,2.264) -- (2.669,2.208) -- (2.582,2.121) -- (2.526,2.011) -- (2.506,1.889) -- (2.526,1.767) -- (2.582,1.658) -- (2.669,1.57) -- (2.779,1.514) -- (2.9,1.495) -- cycle;
\node at (3.086,1.889) {\scalebox{1.578}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.227,10.553) -- (1.827,10.553) -- (1.827,11.153) -- (1.227,11.153) -- cycle;
\node at (1.527,10.853) {\scalebox{1.578}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.487,9.129) -- (10.087,9.129) -- (10.087,9.729) -- (9.487,9.729) -- cycle;
\node at (9.787,9.429) {\scalebox{1.578}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (9.721,4.99) -- (10.321,4.99) -- (10.321,5.59) -- (9.721,5.59) -- cycle;
\node at (10.021,5.29) {\scalebox{1.578}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.948,0.792) -- (2.548,0.792) -- (2.548,1.392) -- (1.948,1.392) -- cycle;
\node at (2.248,1.092) {\scalebox{1.578}{\small d}};
\node at (7.302,12.139) {\scalebox{1.578}{\small A}};
\node at (1.084,11.287) {\scalebox{1.578}{\small B}};
\node at (3.992,11.391) {\scalebox{1.578}{\small C}};
\node at (11.079,8.585) {\scalebox{1.578}{\small D}};
\node at (3.695,8.767) {\scalebox{1.578}{\small E}};
\node at (0.765,8.081) {\scalebox{1.578}{\small F}};
\node at (7.938,7.917) {\scalebox{1.578}{\small G}};
\node at (4.539,4.723) {\scalebox{1.578}{\small H}};
\node at (11.323,5.084) {\scalebox{1.578}{\small I}};
\node at (7.66,4.735) {\scalebox{1.578}{\small J}};
\node at (1.509,1.118) {\scalebox{1.578}{\small K}};
\node at (4.179,0.95) {\scalebox{1.578}{\small L}};
\draw[line width=0.5pt] (12.034,10.608) -- (12.034,11.708);
\draw[line width=0.5pt] (11.803,11.323) -- (12.034,11.708) -- (12.265,11.323);
\draw[line width=0.5pt] (11.732,11.158) -- (12.337,11.158);
\node at (12.034,12.093) {\scalebox{1.578}{\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 du Pont & Ⓑ Place de l'Église & Ⓚ Place du Marché \\ \hline
① chemin des Vignes & ⑩ rue du Moulin & Ⓒ Rond-point des Vaches & Ⓛ Fontaine Notre-Dame \\ \hline
② allée des Tilleuls & ⑪ rue du Four & Ⓓ Place du Vieux-Puits & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ passage du Marché & ⑫ quai aux Fleurs & Ⓔ Rond-point du Pont & \textbf{a} l'école \\ \hline
④ sentier des Prés & ⑬ rue Verte & Ⓕ Coin du Lavoir & \textbf{b} le marché couvert \\ \hline
⑤ chemin de Ronde & ⑭ rue des Jardins & Ⓖ Carrefour de l'Hôpital & \textbf{c} le stade \\ \hline
⑥ rue de l'Église & ⑮ impasse du Puits & Ⓗ Carrefour Saint-Roch & \textbf{d} la boulangerie \\ \hline
⑦ venelle du Chat & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Placette des Ormes &  \\ \hline
⑧ rue des Lilas & Ⓐ Place Sainte-Croix & Ⓙ Carrefour du Moulin &  \\ \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=0.6493, line join=round, line cap=round]
\draw[line width=0.5pt] (7.087,11.558) -- (7.063,11.282) -- (7.047,11.007) -- (7.037,10.731) -- (7.033,10.454) -- (7.036,10.178) -- (7.045,9.901) -- (7.061,9.625) -- (7.083,9.35) -- (7.112,9.075) -- (7.147,8.801) -- (7.188,8.527) -- (7.236,8.255) -- (7.291,7.984) -- (7.352,7.715);
\draw[line width=0.5pt] (1.527,10.853) -- (1.716,10.879) -- (1.905,10.9) -- (2.095,10.916) -- (2.286,10.928) -- (2.476,10.936) -- (2.667,10.939) -- (2.858,10.938) -- (3.049,10.932) -- (3.239,10.921) -- (3.43,10.906) -- (3.619,10.887) -- (3.809,10.863) -- (3.997,10.835) -- (4.185,10.802);
\draw[line width=0.5pt] (1.527,10.853) -- (1.484,10.649) -- (1.446,10.444) -- (1.413,10.238) -- (1.384,10.031) -- (1.361,9.823) -- (1.342,9.615) -- (1.329,9.406) -- (1.32,9.198) -- (1.316,8.989) -- (1.317,8.78) -- (1.323,8.571) -- (1.334,8.362) -- (1.35,8.154) -- (1.37,7.946);
\draw[line width=0.5pt] (4.185,10.802) -- (4.382,10.888) -- (4.582,10.968) -- (4.783,11.044) -- (4.986,11.115) -- (5.191,11.182) -- (5.397,11.243) -- (5.605,11.3) -- (5.814,11.351) -- (6.024,11.398) -- (6.235,11.44) -- (6.447,11.477) -- (6.66,11.509) -- (6.873,11.536) -- (7.087,11.558);
\draw[line width=0.5pt] (4.185,10.802) -- (4.154,10.628) -- (4.128,10.452) -- (4.105,10.276) -- (4.087,10.1) -- (4.073,9.923) -- (4.063,9.745) -- (4.058,9.568) -- (4.056,9.39) -- (4.059,9.213) -- (4.065,9.036) -- (4.076,8.858) -- (4.091,8.682) -- (4.111,8.505) -- (4.134,8.329);
\draw[line width=0.5pt] (10.481,8.423) -- (10.533,8.203) -- (10.58,7.982) -- (10.622,7.759) -- (10.659,7.536) -- (10.69,7.312) -- (10.716,7.087) -- (10.737,6.862) -- (10.753,6.636) -- (10.763,6.41) -- (10.768,6.183) -- (10.768,5.957) -- (10.762,5.731) -- (10.751,5.505) -- (10.734,5.28);
\draw[line width=0.5pt] (4.134,8.329) -- (4.355,8.251) -- (4.579,8.177) -- (4.803,8.109) -- (5.03,8.046) -- (5.258,7.988) -- (5.487,7.936) -- (5.718,7.889) -- (5.949,7.848) -- (6.181,7.812) -- (6.414,7.782) -- (6.648,7.757) -- (6.882,7.737) -- (7.117,7.723) -- (7.352,7.715);
\draw[line width=0.5pt] (4.134,8.329) -- (4.211,8.122) -- (4.284,7.912) -- (4.351,7.701) -- (4.414,7.488) -- (4.471,7.274) -- (4.524,7.059) -- (4.571,6.842) -- (4.613,6.624) -- (4.65,6.406) -- (4.682,6.186) -- (4.709,5.966) -- (4.731,5.745) -- (4.747,5.524) -- (4.758,5.303);
\draw[line width=0.5pt] (1.37,7.946) -- (1.57,7.943) -- (1.77,7.945) -- (1.971,7.951) -- (2.17,7.962) -- (2.37,7.978) -- (2.569,7.999) -- (2.768,8.024) -- (2.966,8.054) -- (3.163,8.088) -- (3.36,8.127) -- (3.555,8.171) -- (3.749,8.219) -- (3.942,8.272) -- (4.134,8.329);
\draw[line width=0.5pt] (7.352,7.715) -- (7.566,7.799) -- (7.782,7.879) -- (7.999,7.953) -- (8.219,8.022) -- (8.44,8.086) -- (8.663,8.144) -- (8.887,8.198) -- (9.112,8.246) -- (9.338,8.289) -- (9.565,8.326) -- (9.793,8.359) -- (10.022,8.385) -- (10.251,8.407) -- (10.481,8.423);
\draw[line width=0.5pt] (7.352,7.715) -- (7.315,7.537) -- (7.283,7.358) -- (7.255,7.178) -- (7.231,6.997) -- (7.212,6.816) -- (7.196,6.634) -- (7.185,6.453) -- (7.179,6.271) -- (7.176,6.088) -- (7.178,5.906) -- (7.184,5.724) -- (7.195,5.543) -- (7.209,5.361) -- (7.228,5.18);
\draw[line width=0.5pt] (4.758,5.303) -- (4.935,5.321) -- (5.111,5.335) -- (5.289,5.345) -- (5.466,5.351) -- (5.643,5.353) -- (5.821,5.35) -- (5.998,5.343) -- (6.176,5.333) -- (6.352,5.318) -- (6.529,5.298) -- (6.705,5.275) -- (6.88,5.248) -- (7.055,5.216) -- (7.228,5.18);
\draw[line width=0.5pt] (4.758,5.303) -- (4.685,5.043) -- (4.618,4.78) -- (4.557,4.517) -- (4.502,4.252) -- (4.454,3.985) -- (4.412,3.718) -- (4.376,3.449) -- (4.346,3.18) -- (4.323,2.91) -- (4.306,2.64) -- (4.296,2.369) -- (4.291,2.099) -- (4.294,1.828) -- (4.302,1.558);
\draw[line width=0.5pt] (7.228,5.18) -- (7.476,5.226) -- (7.724,5.265) -- (7.974,5.299) -- (8.224,5.327) -- (8.474,5.348) -- (8.726,5.364) -- (8.977,5.375) -- (9.229,5.379) -- (9.481,5.377) -- (9.732,5.369) -- (9.984,5.356) -- (10.235,5.336) -- (10.485,5.311) -- (10.734,5.28);
\draw[line width=0.5pt] (1.85,1.635) -- (2.023,1.603) -- (2.197,1.575) -- (2.371,1.551) -- (2.546,1.531) -- (2.722,1.515) -- (2.897,1.503) -- (3.073,1.495) -- (3.249,1.492) -- (3.425,1.492) -- (3.601,1.497) -- (3.777,1.506) -- (3.953,1.519) -- (4.128,1.536) -- (4.302,1.558);
\filldraw[fill=white, draw=black, line width=0.5pt] (7.46,11.558) -- (7.432,11.7) -- (7.351,11.822) -- (7.23,11.902) -- (7.087,11.931) -- (6.944,11.902) -- (6.823,11.822) -- (6.742,11.7) -- (6.714,11.558) -- (6.742,11.415) -- (6.823,11.294) -- (6.944,11.213) -- (7.087,11.184) -- (7.23,11.213) -- (7.351,11.294) -- (7.432,11.415) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.9,10.853) -- (1.872,10.996) -- (1.791,11.117) -- (1.67,11.198) -- (1.527,11.227) -- (1.384,11.198) -- (1.263,11.117) -- (1.182,10.996) -- (1.154,10.853) -- (1.182,10.711) -- (1.263,10.59) -- (1.384,10.509) -- (1.527,10.48) -- (1.67,10.509) -- (1.791,10.59) -- (1.872,10.711) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.558,10.802) -- (4.53,10.945) -- (4.449,11.066) -- (4.328,11.147) -- (4.185,11.176) -- (4.042,11.147) -- (3.921,11.066) -- (3.84,10.945) -- (3.812,10.802) -- (3.84,10.66) -- (3.921,10.538) -- (4.042,10.458) -- (4.185,10.429) -- (4.328,10.458) -- (4.449,10.538) -- (4.53,10.66) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.854,8.423) -- (10.825,8.565) -- (10.744,8.687) -- (10.623,8.767) -- (10.481,8.796) -- (10.338,8.767) -- (10.217,8.687) -- (10.136,8.565) -- (10.107,8.423) -- (10.136,8.28) -- (10.217,8.159) -- (10.338,8.078) -- (10.481,8.049) -- (10.623,8.078) -- (10.744,8.159) -- (10.825,8.28) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.507,8.329) -- (4.479,8.472) -- (4.398,8.593) -- (4.277,8.674) -- (4.134,8.703) -- (3.991,8.674) -- (3.87,8.593) -- (3.789,8.472) -- (3.761,8.329) -- (3.789,8.187) -- (3.87,8.066) -- (3.991,7.985) -- (4.134,7.956) -- (4.277,7.985) -- (4.398,8.066) -- (4.479,8.187) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.743,7.946) -- (1.715,8.089) -- (1.634,8.21) -- (1.513,8.291) -- (1.37,8.32) -- (1.227,8.291) -- (1.106,8.21) -- (1.025,8.089) -- (0.997,7.946) -- (1.025,7.804) -- (1.106,7.682) -- (1.227,7.602) -- (1.37,7.573) -- (1.513,7.602) -- (1.634,7.682) -- (1.715,7.804) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.725,7.715) -- (7.696,7.858) -- (7.616,7.979) -- (7.494,8.06) -- (7.352,8.088) -- (7.209,8.06) -- (7.088,7.979) -- (7.007,7.858) -- (6.978,7.715) -- (7.007,7.572) -- (7.088,7.451) -- (7.209,7.37) -- (7.352,7.342) -- (7.494,7.37) -- (7.616,7.451) -- (7.696,7.572) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.131,5.303) -- (5.103,5.446) -- (5.022,5.567) -- (4.901,5.648) -- (4.758,5.676) -- (4.615,5.648) -- (4.494,5.567) -- (4.413,5.446) -- (4.385,5.303) -- (4.413,5.16) -- (4.494,5.039) -- (4.615,4.958) -- (4.758,4.93) -- (4.901,4.958) -- (5.022,5.039) -- (5.103,5.16) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.108,5.28) -- (11.079,5.422) -- (10.998,5.544) -- (10.877,5.624) -- (10.734,5.653) -- (10.592,5.624) -- (10.47,5.544) -- (10.39,5.422) -- (10.361,5.28) -- (10.39,5.137) -- (10.47,5.016) -- (10.592,4.935) -- (10.734,4.906) -- (10.877,4.935) -- (10.998,5.016) -- (11.079,5.137) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.602,5.18) -- (7.573,5.323) -- (7.492,5.444) -- (7.371,5.525) -- (7.228,5.553) -- (7.086,5.525) -- (6.965,5.444) -- (6.884,5.323) -- (6.855,5.18) -- (6.884,5.037) -- (6.965,4.916) -- (7.086,4.835) -- (7.228,4.807) -- (7.371,4.835) -- (7.492,4.916) -- (7.573,5.037) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.224,1.635) -- (2.195,1.778) -- (2.114,1.899) -- (1.993,1.98) -- (1.85,2.009) -- (1.708,1.98) -- (1.587,1.899) -- (1.506,1.778) -- (1.477,1.635) -- (1.506,1.493) -- (1.587,1.372) -- (1.708,1.291) -- (1.85,1.262) -- (1.993,1.291) -- (2.114,1.372) -- (2.195,1.493) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.676,1.558) -- (4.647,1.7) -- (4.566,1.822) -- (4.445,1.902) -- (4.302,1.931) -- (4.16,1.902) -- (4.038,1.822) -- (3.958,1.7) -- (3.929,1.558) -- (3.958,1.415) -- (4.038,1.294) -- (4.16,1.213) -- (4.302,1.184) -- (4.445,1.213) -- (4.566,1.294) -- (4.647,1.415) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.444,9.267) -- (7.563,9.286) -- (7.67,9.341) -- (7.755,9.426) -- (7.81,9.533) -- (7.829,9.652) -- (7.81,9.771) -- (7.755,9.878) -- (7.67,9.963) -- (7.563,10.017) -- (7.444,10.036) -- (7.444,10.036) -- (7.325,10.017) -- (7.218,9.963) -- (7.133,9.878) -- (7.079,9.771) -- (7.06,9.652) -- (7.079,9.533) -- (7.133,9.426) -- (7.218,9.341) -- (7.325,9.286) -- (7.444,9.267) -- cycle;
\node at (7.444,9.652) {\scalebox{1.54}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.866,10.937) -- (2.984,10.956) -- (3.092,11.011) -- (3.177,11.096) -- (3.231,11.203) -- (3.25,11.322) -- (3.231,11.441) -- (3.177,11.548) -- (3.092,11.633) -- (2.984,11.688) -- (2.866,11.707) -- (2.866,11.707) -- (2.747,11.688) -- (2.64,11.633) -- (2.554,11.548) -- (2.5,11.441) -- (2.481,11.322) -- (2.5,11.203) -- (2.554,11.096) -- (2.64,11.011) -- (2.747,10.956) -- (2.866,10.937) -- cycle;
\node at (2.866,11.322) {\scalebox{1.54}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.713,9.001) -- (1.832,9.02) -- (1.939,9.075) -- (2.024,9.16) -- (2.078,9.267) -- (2.097,9.386) -- (2.078,9.505) -- (2.024,9.612) -- (1.939,9.697) -- (1.832,9.751) -- (1.713,9.77) -- (1.713,9.77) -- (1.594,9.751) -- (1.487,9.697) -- (1.402,9.612) -- (1.347,9.505) -- (1.328,9.386) -- (1.347,9.267) -- (1.402,9.16) -- (1.487,9.075) -- (1.594,9.02) -- (1.713,9.001) -- cycle;
\node at (1.713,9.386) {\scalebox{1.54}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.508,11.287) -- (5.627,11.306) -- (5.734,11.361) -- (5.819,11.446) -- (5.874,11.553) -- (5.892,11.672) -- (5.874,11.791) -- (5.819,11.898) -- (5.734,11.983) -- (5.627,12.038) -- (5.508,12.056) -- (5.508,12.056) -- (5.389,12.038) -- (5.282,11.983) -- (5.197,11.898) -- (5.142,11.791) -- (5.123,11.672) -- (5.142,11.553) -- (5.197,11.446) -- (5.282,11.361) -- (5.389,11.306) -- (5.508,11.287) -- cycle;
\node at (5.508,11.672) {\scalebox{1.54}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.442,9.176) -- (4.561,9.194) -- (4.668,9.249) -- (4.753,9.334) -- (4.808,9.441) -- (4.827,9.56) -- (4.808,9.679) -- (4.753,9.786) -- (4.668,9.871) -- (4.561,9.926) -- (4.442,9.945) -- (4.442,9.945) -- (4.323,9.926) -- (4.216,9.871) -- (4.131,9.786) -- (4.076,9.679) -- (4.057,9.56) -- (4.076,9.441) -- (4.131,9.334) -- (4.216,9.249) -- (4.323,9.194) -- (4.442,9.176) -- cycle;
\node at (4.442,9.56) {\scalebox{1.54}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.12,6.508) -- (11.239,6.527) -- (11.346,6.581) -- (11.431,6.667) -- (11.486,6.774) -- (11.505,6.893) -- (11.486,7.011) -- (11.431,7.119) -- (11.346,7.204) -- (11.239,7.258) -- (11.12,7.277) -- (11.12,7.277) -- (11.002,7.258) -- (10.894,7.204) -- (10.809,7.119) -- (10.755,7.011) -- (10.736,6.893) -- (10.755,6.774) -- (10.809,6.667) -- (10.894,6.581) -- (11.002,6.527) -- (11.12,6.508) -- cycle;
\node at (11.12,6.893) {\scalebox{1.54}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.79,7.883) -- (5.908,7.901) -- (6.016,7.956) -- (6.101,8.041) -- (6.155,8.148) -- (6.174,8.267) -- (6.155,8.386) -- (6.101,8.493) -- (6.016,8.578) -- (5.908,8.633) -- (5.79,8.652) -- (5.79,8.652) -- (5.671,8.633) -- (5.564,8.578) -- (5.479,8.493) -- (5.424,8.386) -- (5.405,8.267) -- (5.424,8.148) -- (5.479,8.041) -- (5.564,7.956) -- (5.671,7.901) -- (5.79,7.883) -- cycle;
\node at (5.79,8.267) {\scalebox{1.54}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.948,6.535) -- (5.066,6.554) -- (5.174,6.609) -- (5.259,6.694) -- (5.313,6.801) -- (5.332,6.92) -- (5.313,7.038) -- (5.259,7.146) -- (5.174,7.231) -- (5.066,7.285) -- (4.948,7.304) -- (4.948,7.304) -- (4.829,7.285) -- (4.722,7.231) -- (4.636,7.146) -- (4.582,7.038) -- (4.563,6.92) -- (4.582,6.801) -- (4.636,6.694) -- (4.722,6.609) -- (4.829,6.554) -- (4.948,6.535) -- cycle;
\node at (4.948,6.92) {\scalebox{1.54}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.715,8.02) -- (2.834,8.039) -- (2.941,8.094) -- (3.026,8.179) -- (3.081,8.286) -- (3.1,8.405) -- (3.081,8.524) -- (3.026,8.631) -- (2.941,8.716) -- (2.834,8.77) -- (2.715,8.789) -- (2.715,8.789) -- (2.596,8.77) -- (2.489,8.716) -- (2.404,8.631) -- (2.349,8.524) -- (2.331,8.405) -- (2.349,8.286) -- (2.404,8.179) -- (2.489,8.094) -- (2.596,8.039) -- (2.715,8.02) -- cycle;
\node at (2.715,8.405) {\scalebox{1.54}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.983,8.188) -- (9.102,8.207) -- (9.209,8.262) -- (9.294,8.347) -- (9.349,8.454) -- (9.368,8.573) -- (9.349,8.692) -- (9.294,8.799) -- (9.209,8.884) -- (9.102,8.939) -- (8.983,8.957) -- (8.621,8.957) -- (8.502,8.939) -- (8.395,8.884) -- (8.31,8.799) -- (8.255,8.692) -- (8.237,8.573) -- (8.255,8.454) -- (8.31,8.347) -- (8.395,8.262) -- (8.502,8.207) -- (8.621,8.188) -- cycle;
\node at (8.802,8.573) {\scalebox{1.54}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.751,6.049) -- (7.869,6.068) -- (7.977,6.123) -- (8.062,6.208) -- (8.116,6.315) -- (8.135,6.434) -- (8.116,6.553) -- (8.062,6.66) -- (7.977,6.745) -- (7.869,6.8) -- (7.751,6.818) -- (7.389,6.818) -- (7.27,6.8) -- (7.163,6.745) -- (7.078,6.66) -- (7.023,6.553) -- (7.004,6.434) -- (7.023,6.315) -- (7.078,6.208) -- (7.163,6.123) -- (7.27,6.068) -- (7.389,6.049) -- cycle;
\node at (7.57,6.434) {\scalebox{1.54}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.198,5.343) -- (6.317,5.362) -- (6.424,5.416) -- (6.51,5.502) -- (6.564,5.609) -- (6.583,5.728) -- (6.564,5.846) -- (6.51,5.954) -- (6.424,6.039) -- (6.317,6.093) -- (6.198,6.112) -- (5.837,6.112) -- (5.718,6.093) -- (5.611,6.039) -- (5.525,5.954) -- (5.471,5.846) -- (5.452,5.728) -- (5.471,5.609) -- (5.525,5.502) -- (5.611,5.416) -- (5.718,5.362) -- (5.837,5.343) -- cycle;
\node at (6.017,5.728) {\scalebox{1.54}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.938,3.018) -- (5.057,3.037) -- (5.164,3.092) -- (5.25,3.177) -- (5.304,3.284) -- (5.323,3.403) -- (5.304,3.521) -- (5.25,3.629) -- (5.164,3.714) -- (5.057,3.768) -- (4.938,3.787) -- (4.577,3.787) -- (4.458,3.768) -- (4.351,3.714) -- (4.265,3.629) -- (4.211,3.521) -- (4.192,3.403) -- (4.211,3.284) -- (4.265,3.177) -- (4.351,3.092) -- (4.458,3.037) -- (4.577,3.018) -- cycle;
\node at (4.757,3.403) {\scalebox{1.54}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.147,5.374) -- (9.266,5.393) -- (9.373,5.448) -- (9.458,5.533) -- (9.513,5.64) -- (9.532,5.759) -- (9.513,5.878) -- (9.458,5.985) -- (9.373,6.07) -- (9.266,6.125) -- (9.147,6.143) -- (8.785,6.143) -- (8.667,6.125) -- (8.559,6.07) -- (8.474,5.985) -- (8.42,5.878) -- (8.401,5.759) -- (8.42,5.64) -- (8.474,5.533) -- (8.559,5.448) -- (8.667,5.393) -- (8.785,5.374) -- cycle;
\node at (8.966,5.759) {\scalebox{1.54}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.266,1.495) -- (3.385,1.514) -- (3.492,1.569) -- (3.577,1.654) -- (3.632,1.761) -- (3.651,1.88) -- (3.632,1.999) -- (3.577,2.106) -- (3.492,2.191) -- (3.385,2.245) -- (3.266,2.264) -- (2.904,2.264) -- (2.786,2.245) -- (2.678,2.191) -- (2.593,2.106) -- (2.539,1.999) -- (2.52,1.88) -- (2.539,1.761) -- (2.593,1.654) -- (2.678,1.569) -- (2.786,1.514) -- (2.904,1.495) -- cycle;
\node at (3.085,1.88) {\scalebox{1.54}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.227,10.553) -- (1.827,10.553) -- (1.827,11.153) -- (1.227,11.153) -- cycle;
\node at (1.527,10.853) {\scalebox{1.54}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.487,9.129) -- (10.087,9.129) -- (10.087,9.729) -- (9.487,9.729) -- cycle;
\node at (9.787,9.429) {\scalebox{1.54}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (9.721,4.99) -- (10.321,4.99) -- (10.321,5.59) -- (9.721,5.59) -- cycle;
\node at (10.021,5.29) {\scalebox{1.54}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.948,0.792) -- (2.548,0.792) -- (2.548,1.392) -- (1.948,1.392) -- cycle;
\node at (2.248,1.092) {\scalebox{1.54}{\small d}};
\node at (7.302,12.139) {\scalebox{1.54}{\small A}};
\node at (1.084,11.287) {\scalebox{1.54}{\small B}};
\node at (3.992,11.391) {\scalebox{1.54}{\small C}};
\node at (11.079,8.585) {\scalebox{1.54}{\small D}};
\node at (3.695,8.767) {\scalebox{1.54}{\small E}};
\node at (0.765,8.081) {\scalebox{1.54}{\small F}};
\node at (7.938,7.917) {\scalebox{1.54}{\small G}};
\node at (4.539,4.723) {\scalebox{1.54}{\small H}};
\node at (11.323,5.084) {\scalebox{1.54}{\small I}};
\node at (7.66,4.735) {\scalebox{1.54}{\small J}};
\node at (1.509,1.118) {\scalebox{1.54}{\small K}};
\node at (4.179,0.95) {\scalebox{1.54}{\small L}};
\draw[line width=0.5pt] (12.034,10.608) -- (12.034,11.708);
\draw[line width=0.5pt] (11.803,11.323) -- (12.034,11.708) -- (12.265,11.323);
\draw[line width=0.5pt] (11.732,11.158) -- (12.337,11.158);
\node at (12.034,12.093) {\scalebox{1.54}{\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 & C, F, G, H, I, K & 6 \\ \hline
clairs & A, B, D, E, J, 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{H}, 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 le \textbf{passage du Marché ③} vers le sud jusqu'à Coin du Lavoir \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Pont ⑨}, vers l'est, jusqu'à Rond-point du Pont \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin de Ronde ⑤}, vers le nord, jusqu'à Rond-point des Vaches \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ④}, vers l'est, jusqu'à Place Sainte-Croix \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin des Vignes ①}, vers le sud, jusqu'à Carrefour de l'Hôpital \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Moulin ⑩}, vers l'est, jusqu'à Place du Vieux-Puits \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de l'Église ⑥}, vers le sud, jusqu'à Placette des Ormes \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Jardins ⑭}, vers l'ouest, jusqu'à Carrefour du Moulin \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{quai aux Fleurs ⑫}, vers l'ouest, jusqu'à Carrefour Saint-Roch \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Verte ⑬}, vers le sud, jusqu'à Fontaine Notre-Dame \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{impasse du Puits ⑮}, vers l'ouest, jusqu'à Place du Marché \textbf{(K)}.\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}
