\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 158}

\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=158}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0158}
\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=0.714, line join=round, line cap=round]
\draw[line width=0.5pt] (4.195,11.544) -- (4.339,11.506) -- (4.484,11.473) -- (4.63,11.442) -- (4.777,11.415) -- (4.924,11.392) -- (5.072,11.372) -- (5.22,11.355) -- (5.368,11.342) -- (5.517,11.333) -- (5.666,11.327) -- (5.815,11.324) -- (5.964,11.325) -- (6.113,11.33) -- (6.262,11.338);
\draw[line width=0.5pt] (4.195,11.544) -- (4.223,11.371) -- (4.248,11.197) -- (4.268,11.023) -- (4.284,10.848) -- (4.296,10.673) -- (4.304,10.498) -- (4.307,10.323) -- (4.307,10.147) -- (4.303,9.972) -- (4.294,9.797) -- (4.281,9.622) -- (4.264,9.448) -- (4.243,9.274) -- (4.218,9.1);
\draw[line width=0.5pt] (6.262,11.338) -- (6.271,11.212) -- (6.284,11.087) -- (6.3,10.962) -- (6.318,10.837) -- (6.34,10.713) -- (6.364,10.589) -- (6.392,10.466) -- (6.422,10.343) -- (6.455,10.221) -- (6.491,10.101) -- (6.53,9.98) -- (6.571,9.861) -- (6.616,9.743) -- (6.663,9.627);
\draw[line width=0.5pt] (6.663,9.627) -- (6.656,9.429) -- (6.645,9.231) -- (6.628,9.033) -- (6.607,8.836) -- (6.582,8.639) -- (6.552,8.443) -- (6.517,8.248) -- (6.478,8.053) -- (6.434,7.86) -- (6.386,7.667) -- (6.333,7.476) -- (6.275,7.287) -- (6.214,7.098) -- (6.147,6.911);
\draw[line width=0.5pt] (4.218,9.1) -- (4.397,9.111) -- (4.576,9.126) -- (4.755,9.145) -- (4.932,9.168) -- (5.11,9.195) -- (5.286,9.227) -- (5.462,9.263) -- (5.637,9.302) -- (5.811,9.346) -- (5.984,9.394) -- (6.156,9.446) -- (6.326,9.502) -- (6.495,9.563) -- (6.663,9.627);
\draw[line width=0.5pt] (4.218,9.1) -- (4.244,8.942) -- (4.265,8.783) -- (4.282,8.624) -- (4.296,8.465) -- (4.306,8.305) -- (4.313,8.145) -- (4.315,7.985) -- (4.314,7.824) -- (4.309,7.664) -- (4.3,7.504) -- (4.288,7.344) -- (4.272,7.185) -- (4.252,7.026) -- (4.228,6.868);
\draw[line width=0.5pt] (6.147,6.911) -- (6.139,6.709) -- (6.135,6.507) -- (6.135,6.304) -- (6.141,6.102) -- (6.151,5.9) -- (6.166,5.698) -- (6.186,5.496) -- (6.211,5.295) -- (6.24,5.094) -- (6.274,4.895) -- (6.312,4.696) -- (6.356,4.498) -- (6.404,4.302) -- (6.456,4.106);
\draw[line width=0.5pt] (4.228,6.868) -- (4.363,6.892) -- (4.5,6.913) -- (4.636,6.931) -- (4.773,6.945) -- (4.911,6.956) -- (5.048,6.964) -- (5.186,6.969) -- (5.324,6.97) -- (5.461,6.969) -- (5.599,6.964) -- (5.737,6.956) -- (5.874,6.944) -- (6.011,6.929) -- (6.147,6.911);
\draw[line width=0.5pt] (4.228,6.868) -- (4.176,6.664) -- (4.128,6.458) -- (4.086,6.251) -- (4.048,6.044) -- (4.015,5.835) -- (3.987,5.626) -- (3.964,5.416) -- (3.946,5.205) -- (3.933,4.995) -- (3.925,4.784) -- (3.922,4.572) -- (3.923,4.361) -- (3.93,4.15) -- (3.942,3.94);
\draw[line width=0.5pt] (6.456,4.106) -- (6.616,4.125) -- (6.776,4.14) -- (6.936,4.152) -- (7.097,4.159) -- (7.257,4.163) -- (7.418,4.163) -- (7.579,4.159) -- (7.74,4.152) -- (7.9,4.14) -- (8.06,4.125) -- (8.22,4.107) -- (8.379,4.084) -- (8.537,4.058) -- (8.695,4.028);
\draw[line width=0.5pt] (6.456,4.106) -- (6.444,3.922) -- (6.437,3.738) -- (6.434,3.553) -- (6.435,3.369) -- (6.441,3.184) -- (6.451,3.0) -- (6.465,2.815) -- (6.483,2.632) -- (6.506,2.448) -- (6.534,2.266) -- (6.565,2.084) -- (6.601,1.903) -- (6.641,1.723) -- (6.685,1.544);
\draw[line width=0.5pt] (8.695,4.028) -- (8.762,3.872) -- (8.825,3.715) -- (8.884,3.556) -- (8.939,3.396) -- (8.991,3.235) -- (9.039,3.073) -- (9.083,2.909) -- (9.123,2.744) -- (9.159,2.579) -- (9.192,2.413) -- (9.22,2.246) -- (9.245,2.078) -- (9.265,1.91) -- (9.282,1.742);
\draw[line width=0.5pt] (1.326,4.005) -- (1.51,3.972) -- (1.696,3.943) -- (1.882,3.918) -- (2.069,3.898) -- (2.256,3.883) -- (2.443,3.871) -- (2.631,3.864) -- (2.819,3.862) -- (3.007,3.864) -- (3.195,3.87) -- (3.382,3.881) -- (3.569,3.896) -- (3.756,3.916) -- (3.942,3.94);
\draw[line width=0.5pt] (1.326,4.005) -- (1.371,3.875) -- (1.414,3.745) -- (1.453,3.613) -- (1.489,3.481) -- (1.522,3.348) -- (1.552,3.214) -- (1.579,3.079) -- (1.603,2.944) -- (1.623,2.808) -- (1.641,2.672) -- (1.655,2.535) -- (1.666,2.398) -- (1.673,2.261) -- (1.678,2.124);
\draw[line width=0.5pt] (3.942,3.94) -- (4.122,3.924) -- (4.302,3.913) -- (4.483,3.905) -- (4.664,3.903) -- (4.845,3.904) -- (5.025,3.909) -- (5.206,3.919) -- (5.386,3.933) -- (5.566,3.952) -- (5.746,3.974) -- (5.925,4.001) -- (6.103,4.032) -- (6.28,4.067) -- (6.456,4.106);
\draw[line width=0.5pt] (6.685,1.544) -- (6.867,1.586) -- (7.05,1.624) -- (7.233,1.658) -- (7.418,1.688) -- (7.603,1.713) -- (7.789,1.733) -- (7.975,1.75) -- (8.162,1.762) -- (8.349,1.77) -- (8.535,1.773) -- (8.722,1.772) -- (8.909,1.766) -- (9.096,1.756) -- (9.282,1.742);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.355,11.544) -- (4.334,11.624) -- (4.275,11.682) -- (4.195,11.704) -- (4.115,11.682) -- (4.056,11.624) -- (4.035,11.544) -- (4.056,11.464) -- (4.115,11.405) -- (4.195,11.384) -- (4.275,11.405) -- (4.334,11.464) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.422,11.338) -- (6.4,11.418) -- (6.342,11.476) -- (6.262,11.498) -- (6.182,11.476) -- (6.123,11.418) -- (6.102,11.338) -- (6.123,11.258) -- (6.182,11.199) -- (6.262,11.178) -- (6.342,11.199) -- (6.4,11.258) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.823,9.627) -- (6.801,9.707) -- (6.743,9.765) -- (6.663,9.787) -- (6.583,9.765) -- (6.524,9.707) -- (6.503,9.627) -- (6.524,9.547) -- (6.583,9.488) -- (6.663,9.467) -- (6.743,9.488) -- (6.801,9.547) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.378,9.1) -- (4.357,9.18) -- (4.298,9.239) -- (4.218,9.26) -- (4.138,9.239) -- (4.08,9.18) -- (4.058,9.1) -- (4.08,9.02) -- (4.138,8.962) -- (4.218,8.94) -- (4.298,8.962) -- (4.357,9.02) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.307,6.911) -- (6.286,6.991) -- (6.227,7.05) -- (6.147,7.071) -- (6.067,7.05) -- (6.009,6.991) -- (5.987,6.911) -- (6.009,6.831) -- (6.067,6.773) -- (6.147,6.751) -- (6.227,6.773) -- (6.286,6.831) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.388,6.868) -- (4.367,6.948) -- (4.308,7.007) -- (4.228,7.028) -- (4.148,7.007) -- (4.089,6.948) -- (4.068,6.868) -- (4.089,6.788) -- (4.148,6.729) -- (4.228,6.708) -- (4.308,6.729) -- (4.367,6.788) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.616,4.106) -- (6.595,4.186) -- (6.536,4.245) -- (6.456,4.266) -- (6.376,4.245) -- (6.318,4.186) -- (6.296,4.106) -- (6.318,4.026) -- (6.376,3.968) -- (6.456,3.946) -- (6.536,3.968) -- (6.595,4.026) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.855,4.028) -- (8.834,4.108) -- (8.775,4.166) -- (8.695,4.188) -- (8.615,4.166) -- (8.557,4.108) -- (8.535,4.028) -- (8.557,3.948) -- (8.615,3.889) -- (8.695,3.868) -- (8.775,3.889) -- (8.834,3.948) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.486,4.005) -- (1.464,4.085) -- (1.406,4.143) -- (1.326,4.165) -- (1.246,4.143) -- (1.187,4.085) -- (1.166,4.005) -- (1.187,3.925) -- (1.246,3.866) -- (1.326,3.845) -- (1.406,3.866) -- (1.464,3.925) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.102,3.94) -- (4.081,4.02) -- (4.022,4.078) -- (3.942,4.1) -- (3.862,4.078) -- (3.803,4.02) -- (3.782,3.94) -- (3.803,3.86) -- (3.862,3.801) -- (3.942,3.78) -- (4.022,3.801) -- (4.081,3.86) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.838,2.124) -- (1.816,2.204) -- (1.758,2.263) -- (1.678,2.284) -- (1.598,2.263) -- (1.539,2.204) -- (1.518,2.124) -- (1.539,2.044) -- (1.598,1.986) -- (1.678,1.964) -- (1.758,1.986) -- (1.816,2.044) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.442,1.742) -- (9.42,1.822) -- (9.362,1.881) -- (9.282,1.902) -- (9.202,1.881) -- (9.143,1.822) -- (9.122,1.742) -- (9.143,1.662) -- (9.202,1.603) -- (9.282,1.582) -- (9.362,1.603) -- (9.42,1.662) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.845,1.544) -- (6.823,1.624) -- (6.765,1.682) -- (6.685,1.704) -- (6.605,1.682) -- (6.546,1.624) -- (6.525,1.544) -- (6.546,1.464) -- (6.605,1.405) -- (6.685,1.384) -- (6.765,1.405) -- (6.823,1.464) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.259,11.353) -- (5.381,11.373) -- (5.49,11.429) -- (5.577,11.516) -- (5.633,11.626) -- (5.653,11.747) -- (5.633,11.869) -- (5.577,11.979) -- (5.49,12.066) -- (5.381,12.122) -- (5.259,12.141) -- (5.259,12.141) -- (5.137,12.122) -- (5.027,12.066) -- (4.94,11.979) -- (4.884,11.869) -- (4.865,11.747) -- (4.884,11.626) -- (4.94,11.516) -- (5.027,11.429) -- (5.137,11.373) -- (5.259,11.353) -- cycle;
\node at (5.259,11.747) {\scalebox{1.401}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.701,9.933) -- (4.823,9.952) -- (4.933,10.008) -- (5.02,10.095) -- (5.076,10.205) -- (5.095,10.327) -- (5.076,10.448) -- (5.02,10.558) -- (4.933,10.645) -- (4.823,10.701) -- (4.701,10.721) -- (4.701,10.721) -- (4.58,10.701) -- (4.47,10.645) -- (4.383,10.558) -- (4.327,10.448) -- (4.307,10.327) -- (4.327,10.205) -- (4.383,10.095) -- (4.47,10.008) -- (4.58,9.952) -- (4.701,9.933) -- cycle;
\node at (4.701,10.327) {\scalebox{1.401}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.85,9.945) -- (5.972,9.964) -- (6.082,10.02) -- (6.169,10.107) -- (6.225,10.217) -- (6.244,10.339) -- (6.225,10.46) -- (6.169,10.57) -- (6.082,10.657) -- (5.972,10.713) -- (5.85,10.733) -- (5.85,10.733) -- (5.729,10.713) -- (5.619,10.657) -- (5.532,10.57) -- (5.476,10.46) -- (5.456,10.339) -- (5.476,10.217) -- (5.532,10.107) -- (5.619,10.02) -- (5.729,9.964) -- (5.85,9.945) -- cycle;
\node at (5.85,10.339) {\scalebox{1.401}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.13,7.927) -- (6.252,7.947) -- (6.362,8.003) -- (6.449,8.09) -- (6.505,8.2) -- (6.524,8.321) -- (6.505,8.443) -- (6.449,8.553) -- (6.362,8.64) -- (6.252,8.696) -- (6.13,8.715) -- (6.13,8.715) -- (6.008,8.696) -- (5.899,8.64) -- (5.811,8.553) -- (5.756,8.443) -- (5.736,8.321) -- (5.756,8.2) -- (5.811,8.09) -- (5.899,8.003) -- (6.008,7.947) -- (6.13,7.927) -- cycle;
\node at (6.13,8.321) {\scalebox{1.401}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.395,9.182) -- (5.517,9.201) -- (5.627,9.257) -- (5.714,9.344) -- (5.77,9.454) -- (5.789,9.575) -- (5.77,9.697) -- (5.714,9.807) -- (5.627,9.894) -- (5.517,9.95) -- (5.395,9.969) -- (5.395,9.969) -- (5.273,9.95) -- (5.163,9.894) -- (5.076,9.807) -- (5.02,9.697) -- (5.001,9.575) -- (5.02,9.454) -- (5.076,9.344) -- (5.163,9.257) -- (5.273,9.201) -- (5.395,9.182) -- cycle;
\node at (5.395,9.575) {\scalebox{1.401}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.709,7.592) -- (4.831,7.612) -- (4.941,7.668) -- (5.028,7.755) -- (5.084,7.864) -- (5.103,7.986) -- (5.084,8.108) -- (5.028,8.218) -- (4.941,8.305) -- (4.831,8.361) -- (4.709,8.38) -- (4.709,8.38) -- (4.587,8.361) -- (4.478,8.305) -- (4.39,8.218) -- (4.335,8.108) -- (4.315,7.986) -- (4.335,7.864) -- (4.39,7.755) -- (4.478,7.668) -- (4.587,7.612) -- (4.709,7.592) -- cycle;
\node at (4.709,7.986) {\scalebox{1.401}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.578,5.145) -- (6.699,5.165) -- (6.809,5.22) -- (6.896,5.308) -- (6.952,5.417) -- (6.972,5.539) -- (6.952,5.661) -- (6.896,5.771) -- (6.809,5.858) -- (6.699,5.914) -- (6.578,5.933) -- (6.578,5.933) -- (6.456,5.914) -- (6.346,5.858) -- (6.259,5.771) -- (6.203,5.661) -- (6.184,5.539) -- (6.203,5.417) -- (6.259,5.308) -- (6.346,5.22) -- (6.456,5.165) -- (6.578,5.145) -- cycle;
\node at (6.578,5.539) {\scalebox{1.401}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.195,6.181) -- (5.316,6.2) -- (5.426,6.256) -- (5.513,6.344) -- (5.569,6.453) -- (5.589,6.575) -- (5.569,6.697) -- (5.513,6.807) -- (5.426,6.894) -- (5.316,6.95) -- (5.195,6.969) -- (5.195,6.969) -- (5.073,6.95) -- (4.963,6.894) -- (4.876,6.807) -- (4.82,6.697) -- (4.801,6.575) -- (4.82,6.453) -- (4.876,6.344) -- (4.963,6.256) -- (5.073,6.2) -- (5.195,6.181) -- cycle;
\node at (5.195,6.575) {\scalebox{1.401}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.356,4.984) -- (4.478,5.003) -- (4.588,5.059) -- (4.675,5.146) -- (4.731,5.256) -- (4.75,5.377) -- (4.731,5.499) -- (4.675,5.609) -- (4.588,5.696) -- (4.478,5.752) -- (4.356,5.771) -- (4.356,5.771) -- (4.235,5.752) -- (4.125,5.696) -- (4.038,5.609) -- (3.982,5.499) -- (3.962,5.377) -- (3.982,5.256) -- (4.038,5.146) -- (4.125,5.059) -- (4.235,5.003) -- (4.356,4.984) -- cycle;
\node at (4.356,5.377) {\scalebox{1.401}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.778,4.159) -- (7.9,4.178) -- (8.01,4.234) -- (8.097,4.321) -- (8.153,4.431) -- (8.172,4.553) -- (8.153,4.675) -- (8.097,4.784) -- (8.01,4.872) -- (7.9,4.928) -- (7.778,4.947) -- (7.407,4.947) -- (7.286,4.928) -- (7.176,4.872) -- (7.089,4.784) -- (7.033,4.675) -- (7.014,4.553) -- (7.033,4.431) -- (7.089,4.321) -- (7.176,4.234) -- (7.286,4.178) -- (7.407,4.159) -- cycle;
\node at (7.593,4.553) {\scalebox{1.401}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.096,2.372) -- (6.218,2.391) -- (6.328,2.447) -- (6.415,2.534) -- (6.471,2.644) -- (6.49,2.766) -- (6.471,2.888) -- (6.415,2.998) -- (6.328,3.085) -- (6.218,3.141) -- (6.096,3.16) -- (5.726,3.16) -- (5.604,3.141) -- (5.494,3.085) -- (5.407,2.998) -- (5.351,2.888) -- (5.332,2.766) -- (5.351,2.644) -- (5.407,2.534) -- (5.494,2.447) -- (5.604,2.391) -- (5.726,2.372) -- cycle;
\node at (5.911,2.766) {\scalebox{1.401}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.73,2.377) -- (8.851,2.396) -- (8.961,2.452) -- (9.048,2.539) -- (9.104,2.649) -- (9.123,2.771) -- (9.104,2.893) -- (9.048,3.002) -- (8.961,3.089) -- (8.851,3.145) -- (8.73,3.165) -- (8.359,3.165) -- (8.237,3.145) -- (8.127,3.089) -- (8.04,3.002) -- (7.984,2.893) -- (7.965,2.771) -- (7.984,2.649) -- (8.04,2.539) -- (8.127,2.452) -- (8.237,2.396) -- (8.359,2.377) -- cycle;
\node at (8.544,2.771) {\scalebox{1.401}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.826,3.864) -- (2.948,3.884) -- (3.058,3.94) -- (3.145,4.027) -- (3.201,4.136) -- (3.22,4.258) -- (3.201,4.38) -- (3.145,4.49) -- (3.058,4.577) -- (2.948,4.633) -- (2.826,4.652) -- (2.456,4.652) -- (2.334,4.633) -- (2.224,4.577) -- (2.137,4.49) -- (2.081,4.38) -- (2.062,4.258) -- (2.081,4.136) -- (2.137,4.027) -- (2.224,3.94) -- (2.334,3.884) -- (2.456,3.864) -- cycle;
\node at (2.641,4.258) {\scalebox{1.401}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.152,2.758) -- (2.273,2.777) -- (2.383,2.833) -- (2.47,2.92) -- (2.526,3.03) -- (2.546,3.151) -- (2.526,3.273) -- (2.47,3.383) -- (2.383,3.47) -- (2.273,3.526) -- (2.152,3.545) -- (1.781,3.545) -- (1.659,3.526) -- (1.549,3.47) -- (1.462,3.383) -- (1.406,3.273) -- (1.387,3.151) -- (1.406,3.03) -- (1.462,2.92) -- (1.549,2.833) -- (1.659,2.777) -- (1.781,2.758) -- cycle;
\node at (1.966,3.151) {\scalebox{1.401}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.365,3.918) -- (5.487,3.938) -- (5.597,3.994) -- (5.684,4.081) -- (5.74,4.191) -- (5.759,4.312) -- (5.74,4.434) -- (5.684,4.544) -- (5.597,4.631) -- (5.487,4.687) -- (5.365,4.706) -- (4.995,4.706) -- (4.873,4.687) -- (4.763,4.631) -- (4.676,4.544) -- (4.62,4.434) -- (4.601,4.312) -- (4.62,4.191) -- (4.676,4.081) -- (4.763,3.994) -- (4.873,3.938) -- (4.995,3.918) -- cycle;
\node at (5.18,4.312) {\scalebox{1.401}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.191,0.963) -- (8.312,0.983) -- (8.422,1.038) -- (8.509,1.126) -- (8.565,1.235) -- (8.584,1.357) -- (8.565,1.479) -- (8.509,1.589) -- (8.422,1.676) -- (8.312,1.732) -- (8.191,1.751) -- (7.82,1.751) -- (7.698,1.732) -- (7.588,1.676) -- (7.501,1.589) -- (7.445,1.479) -- (7.426,1.357) -- (7.445,1.235) -- (7.501,1.126) -- (7.588,1.038) -- (7.698,0.983) -- (7.82,0.963) -- cycle;
\node at (8.005,1.357) {\scalebox{1.401}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (8.395,3.728) -- (8.995,3.728) -- (8.995,4.328) -- (8.395,4.328) -- cycle;
\node at (8.695,4.028) {\scalebox{1.401}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.157,3.121) -- (4.757,3.121) -- (4.757,3.721) -- (4.157,3.721) -- cycle;
\node at (4.457,3.421) {\scalebox{1.401}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.442,8.571) -- (5.042,8.571) -- (5.042,9.171) -- (4.442,9.171) -- cycle;
\node at (4.742,8.871) {\scalebox{1.401}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.5,5.651) -- (8.1,5.651) -- (8.1,6.251) -- (7.5,6.251) -- cycle;
\node at (7.8,5.951) {\scalebox{1.401}{\small d}};
\node at (4.069,12.151) {\scalebox{1.401}{\small A}};
\node at (6.362,11.95) {\scalebox{1.401}{\small B}};
\node at (6.867,10.212) {\scalebox{1.401}{\small C}};
\node at (4.008,9.683) {\scalebox{1.401}{\small D}};
\node at (6.529,7.4) {\scalebox{1.401}{\small E}};
\node at (3.753,7.266) {\scalebox{1.401}{\small F}};
\node at (6.776,3.575) {\scalebox{1.401}{\small G}};
\node at (9.235,3.722) {\scalebox{1.401}{\small H}};
\node at (0.765,3.74) {\scalebox{1.401}{\small I}};
\node at (3.579,3.437) {\scalebox{1.401}{\small J}};
\node at (1.245,1.68) {\scalebox{1.401}{\small K}};
\node at (9.706,1.29) {\scalebox{1.401}{\small L}};
\node at (6.864,0.95) {\scalebox{1.401}{\small M}};
\draw[line width=0.5pt] (10.582,10.594) -- (10.582,11.694);
\draw[line width=0.5pt] (10.351,11.309) -- (10.582,11.694) -- (10.813,11.309);
\draw[line width=0.5pt] (10.279,11.144) -- (10.884,11.144);
\node at (10.582,12.079) {\scalebox{1.401}{\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 du Stade & Ⓐ Coin du Lavoir & Ⓙ Rond-point du Stade \\ \hline
① chemin du Lavoir & ⑩ quai aux Fleurs & Ⓑ Placette des Ormes & Ⓚ Place de la Fontaine \\ \hline
② rue de la Poste & ⑪ venelle du Chat & Ⓒ Carrefour Saint-Roch & Ⓛ Place Sainte-Croix \\ \hline
③ rue des Lilas & ⑫ rue Neuve & Ⓓ Carrefour de la Mairie & Ⓜ Place de l'Église \\ \hline
④ avenue de la Gare & ⑬ allée des Peupliers & Ⓔ Carrefour de l'Hôpital & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ rue Verte & ⑭ impasse du Vieux-Four & Ⓕ Rond-point de la Gare & \textbf{a} l'école \\ \hline
⑥ chemin Creux & ⑮ rue Haute & Ⓖ Rond-point de la Poste & \textbf{b} la mairie \\ \hline
⑦ passage du Marché & ⑯ allée des Tilleuls & Ⓗ Place du Marché & \textbf{c} la pharmacie \\ \hline
⑧ rue du Puits & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Place des Tanneurs & \textbf{d} le stade \\ \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=0.7314, line join=round, line cap=round]
\draw[line width=0.5pt] (4.195,11.544) -- (4.339,11.506) -- (4.484,11.473) -- (4.63,11.442) -- (4.777,11.415) -- (4.924,11.392) -- (5.072,11.372) -- (5.22,11.355) -- (5.368,11.342) -- (5.517,11.333) -- (5.666,11.327) -- (5.815,11.324) -- (5.964,11.325) -- (6.113,11.33) -- (6.262,11.338);
\draw[line width=0.5pt] (4.195,11.544) -- (4.223,11.371) -- (4.248,11.197) -- (4.268,11.023) -- (4.284,10.848) -- (4.296,10.673) -- (4.304,10.498) -- (4.307,10.323) -- (4.307,10.147) -- (4.303,9.972) -- (4.294,9.797) -- (4.281,9.622) -- (4.264,9.448) -- (4.243,9.274) -- (4.218,9.1);
\draw[line width=0.5pt] (6.262,11.338) -- (6.271,11.212) -- (6.284,11.087) -- (6.3,10.962) -- (6.318,10.837) -- (6.34,10.713) -- (6.364,10.589) -- (6.392,10.466) -- (6.422,10.343) -- (6.455,10.221) -- (6.491,10.101) -- (6.53,9.98) -- (6.571,9.861) -- (6.616,9.743) -- (6.663,9.627);
\draw[line width=0.5pt] (6.663,9.627) -- (6.656,9.429) -- (6.645,9.231) -- (6.628,9.033) -- (6.607,8.836) -- (6.582,8.639) -- (6.552,8.443) -- (6.517,8.248) -- (6.478,8.053) -- (6.434,7.86) -- (6.386,7.667) -- (6.333,7.476) -- (6.275,7.287) -- (6.214,7.098) -- (6.147,6.911);
\draw[line width=0.5pt] (4.218,9.1) -- (4.397,9.111) -- (4.576,9.126) -- (4.755,9.145) -- (4.932,9.168) -- (5.11,9.195) -- (5.286,9.227) -- (5.462,9.263) -- (5.637,9.302) -- (5.811,9.346) -- (5.984,9.394) -- (6.156,9.446) -- (6.326,9.502) -- (6.495,9.563) -- (6.663,9.627);
\draw[line width=0.5pt] (4.218,9.1) -- (4.244,8.942) -- (4.265,8.783) -- (4.282,8.624) -- (4.296,8.465) -- (4.306,8.305) -- (4.313,8.145) -- (4.315,7.985) -- (4.314,7.824) -- (4.309,7.664) -- (4.3,7.504) -- (4.288,7.344) -- (4.272,7.185) -- (4.252,7.026) -- (4.228,6.868);
\draw[line width=0.5pt] (6.147,6.911) -- (6.139,6.709) -- (6.135,6.507) -- (6.135,6.304) -- (6.141,6.102) -- (6.151,5.9) -- (6.166,5.698) -- (6.186,5.496) -- (6.211,5.295) -- (6.24,5.094) -- (6.274,4.895) -- (6.312,4.696) -- (6.356,4.498) -- (6.404,4.302) -- (6.456,4.106);
\draw[line width=0.5pt] (4.228,6.868) -- (4.363,6.892) -- (4.5,6.913) -- (4.636,6.931) -- (4.773,6.945) -- (4.911,6.956) -- (5.048,6.964) -- (5.186,6.969) -- (5.324,6.97) -- (5.461,6.969) -- (5.599,6.964) -- (5.737,6.956) -- (5.874,6.944) -- (6.011,6.929) -- (6.147,6.911);
\draw[line width=0.5pt] (4.228,6.868) -- (4.176,6.664) -- (4.128,6.458) -- (4.086,6.251) -- (4.048,6.044) -- (4.015,5.835) -- (3.987,5.626) -- (3.964,5.416) -- (3.946,5.205) -- (3.933,4.995) -- (3.925,4.784) -- (3.922,4.572) -- (3.923,4.361) -- (3.93,4.15) -- (3.942,3.94);
\draw[line width=0.5pt] (6.456,4.106) -- (6.616,4.125) -- (6.776,4.14) -- (6.936,4.152) -- (7.097,4.159) -- (7.257,4.163) -- (7.418,4.163) -- (7.579,4.159) -- (7.74,4.152) -- (7.9,4.14) -- (8.06,4.125) -- (8.22,4.107) -- (8.379,4.084) -- (8.537,4.058) -- (8.695,4.028);
\draw[line width=0.5pt] (6.456,4.106) -- (6.444,3.922) -- (6.437,3.738) -- (6.434,3.553) -- (6.435,3.369) -- (6.441,3.184) -- (6.451,3.0) -- (6.465,2.815) -- (6.483,2.632) -- (6.506,2.448) -- (6.534,2.266) -- (6.565,2.084) -- (6.601,1.903) -- (6.641,1.723) -- (6.685,1.544);
\draw[line width=0.5pt] (8.695,4.028) -- (8.762,3.872) -- (8.825,3.715) -- (8.884,3.556) -- (8.939,3.396) -- (8.991,3.235) -- (9.039,3.073) -- (9.083,2.909) -- (9.123,2.744) -- (9.159,2.579) -- (9.192,2.413) -- (9.22,2.246) -- (9.245,2.078) -- (9.265,1.91) -- (9.282,1.742);
\draw[line width=0.5pt] (1.326,4.005) -- (1.51,3.972) -- (1.696,3.943) -- (1.882,3.918) -- (2.069,3.898) -- (2.256,3.883) -- (2.443,3.871) -- (2.631,3.864) -- (2.819,3.862) -- (3.007,3.864) -- (3.195,3.87) -- (3.382,3.881) -- (3.569,3.896) -- (3.756,3.916) -- (3.942,3.94);
\draw[line width=0.5pt] (1.326,4.005) -- (1.371,3.875) -- (1.414,3.745) -- (1.453,3.613) -- (1.489,3.481) -- (1.522,3.348) -- (1.552,3.214) -- (1.579,3.079) -- (1.603,2.944) -- (1.623,2.808) -- (1.641,2.672) -- (1.655,2.535) -- (1.666,2.398) -- (1.673,2.261) -- (1.678,2.124);
\draw[line width=0.5pt] (3.942,3.94) -- (4.122,3.924) -- (4.302,3.913) -- (4.483,3.905) -- (4.664,3.903) -- (4.845,3.904) -- (5.025,3.909) -- (5.206,3.919) -- (5.386,3.933) -- (5.566,3.952) -- (5.746,3.974) -- (5.925,4.001) -- (6.103,4.032) -- (6.28,4.067) -- (6.456,4.106);
\draw[line width=0.5pt] (6.685,1.544) -- (6.867,1.586) -- (7.05,1.624) -- (7.233,1.658) -- (7.418,1.688) -- (7.603,1.713) -- (7.789,1.733) -- (7.975,1.75) -- (8.162,1.762) -- (8.349,1.77) -- (8.535,1.773) -- (8.722,1.772) -- (8.909,1.766) -- (9.096,1.756) -- (9.282,1.742);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.568,11.544) -- (4.54,11.686) -- (4.459,11.808) -- (4.338,11.888) -- (4.195,11.917) -- (4.052,11.888) -- (3.931,11.808) -- (3.85,11.686) -- (3.822,11.544) -- (3.85,11.401) -- (3.931,11.28) -- (4.052,11.199) -- (4.195,11.17) -- (4.338,11.199) -- (4.459,11.28) -- (4.54,11.401) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.635,11.338) -- (6.606,11.481) -- (6.525,11.602) -- (6.404,11.683) -- (6.262,11.711) -- (6.119,11.683) -- (5.998,11.602) -- (5.917,11.481) -- (5.888,11.338) -- (5.917,11.195) -- (5.998,11.074) -- (6.119,10.993) -- (6.262,10.965) -- (6.404,10.993) -- (6.525,11.074) -- (6.606,11.195) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.036,9.627) -- (7.008,9.769) -- (6.927,9.891) -- (6.806,9.971) -- (6.663,10.0) -- (6.52,9.971) -- (6.399,9.891) -- (6.318,9.769) -- (6.29,9.627) -- (6.318,9.484) -- (6.399,9.363) -- (6.52,9.282) -- (6.663,9.253) -- (6.806,9.282) -- (6.927,9.363) -- (7.008,9.484) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.592,9.1) -- (4.563,9.243) -- (4.482,9.364) -- (4.361,9.445) -- (4.218,9.473) -- (4.076,9.445) -- (3.955,9.364) -- (3.874,9.243) -- (3.845,9.1) -- (3.874,8.957) -- (3.955,8.836) -- (4.076,8.755) -- (4.218,8.727) -- (4.361,8.755) -- (4.482,8.836) -- (4.563,8.957) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.521,6.911) -- (6.492,7.054) -- (6.411,7.175) -- (6.29,7.256) -- (6.147,7.285) -- (6.005,7.256) -- (5.883,7.175) -- (5.803,7.054) -- (5.774,6.911) -- (5.803,6.769) -- (5.883,6.648) -- (6.005,6.567) -- (6.147,6.538) -- (6.29,6.567) -- (6.411,6.648) -- (6.492,6.769) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.601,6.868) -- (4.573,7.011) -- (4.492,7.132) -- (4.371,7.213) -- (4.228,7.241) -- (4.085,7.213) -- (3.964,7.132) -- (3.883,7.011) -- (3.855,6.868) -- (3.883,6.725) -- (3.964,6.604) -- (4.085,6.523) -- (4.228,6.495) -- (4.371,6.523) -- (4.492,6.604) -- (4.573,6.725) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.83,4.106) -- (6.801,4.249) -- (6.72,4.37) -- (6.599,4.451) -- (6.456,4.479) -- (6.314,4.451) -- (6.192,4.37) -- (6.112,4.249) -- (6.083,4.106) -- (6.112,3.963) -- (6.192,3.842) -- (6.314,3.761) -- (6.456,3.733) -- (6.599,3.761) -- (6.72,3.842) -- (6.801,3.963) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.068,4.028) -- (9.04,4.171) -- (8.959,4.292) -- (8.838,4.373) -- (8.695,4.401) -- (8.552,4.373) -- (8.431,4.292) -- (8.35,4.171) -- (8.322,4.028) -- (8.35,3.885) -- (8.431,3.764) -- (8.552,3.683) -- (8.695,3.655) -- (8.838,3.683) -- (8.959,3.764) -- (9.04,3.885) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.699,4.005) -- (1.67,4.148) -- (1.59,4.269) -- (1.468,4.35) -- (1.326,4.378) -- (1.183,4.35) -- (1.062,4.269) -- (0.981,4.148) -- (0.952,4.005) -- (0.981,3.862) -- (1.062,3.741) -- (1.183,3.66) -- (1.326,3.632) -- (1.468,3.66) -- (1.59,3.741) -- (1.67,3.862) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.315,3.94) -- (4.287,4.083) -- (4.206,4.204) -- (4.085,4.285) -- (3.942,4.313) -- (3.799,4.285) -- (3.678,4.204) -- (3.597,4.083) -- (3.569,3.94) -- (3.597,3.797) -- (3.678,3.676) -- (3.799,3.595) -- (3.942,3.567) -- (4.085,3.595) -- (4.206,3.676) -- (4.287,3.797) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.051,2.124) -- (2.022,2.267) -- (1.941,2.388) -- (1.82,2.469) -- (1.678,2.497) -- (1.535,2.469) -- (1.414,2.388) -- (1.333,2.267) -- (1.304,2.124) -- (1.333,1.981) -- (1.414,1.86) -- (1.535,1.779) -- (1.678,1.751) -- (1.82,1.779) -- (1.941,1.86) -- (2.022,1.981) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.655,1.742) -- (9.627,1.885) -- (9.546,2.006) -- (9.425,2.087) -- (9.282,2.115) -- (9.139,2.087) -- (9.018,2.006) -- (8.937,1.885) -- (8.909,1.742) -- (8.937,1.599) -- (9.018,1.478) -- (9.139,1.397) -- (9.282,1.369) -- (9.425,1.397) -- (9.546,1.478) -- (9.627,1.599) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.058,1.544) -- (7.03,1.686) -- (6.949,1.808) -- (6.828,1.888) -- (6.685,1.917) -- (6.542,1.888) -- (6.421,1.808) -- (6.34,1.686) -- (6.312,1.544) -- (6.34,1.401) -- (6.421,1.28) -- (6.542,1.199) -- (6.685,1.17) -- (6.828,1.199) -- (6.949,1.28) -- (7.03,1.401) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.258,11.354) -- (5.377,11.372) -- (5.484,11.427) -- (5.569,11.512) -- (5.624,11.619) -- (5.642,11.738) -- (5.624,11.857) -- (5.569,11.964) -- (5.484,12.049) -- (5.377,12.104) -- (5.258,12.123) -- (5.258,12.123) -- (5.139,12.104) -- (5.032,12.049) -- (4.947,11.964) -- (4.892,11.857) -- (4.873,11.738) -- (4.892,11.619) -- (4.947,11.512) -- (5.032,11.427) -- (5.139,11.372) -- (5.258,11.354) -- cycle;
\node at (5.258,11.738) {\scalebox{1.367}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.692,9.942) -- (4.811,9.961) -- (4.918,10.015) -- (5.003,10.101) -- (5.058,10.208) -- (5.077,10.327) -- (5.058,10.445) -- (5.003,10.553) -- (4.918,10.638) -- (4.811,10.692) -- (4.692,10.711) -- (4.692,10.711) -- (4.573,10.692) -- (4.466,10.638) -- (4.381,10.553) -- (4.326,10.445) -- (4.307,10.327) -- (4.326,10.208) -- (4.381,10.101) -- (4.466,10.015) -- (4.573,9.961) -- (4.692,9.942) -- cycle;
\node at (4.692,10.327) {\scalebox{1.367}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.017,9.993) -- (6.136,10.012) -- (6.243,10.067) -- (6.328,10.152) -- (6.383,10.259) -- (6.402,10.378) -- (6.383,10.497) -- (6.328,10.604) -- (6.243,10.689) -- (6.136,10.744) -- (6.017,10.762) -- (6.017,10.762) -- (5.898,10.744) -- (5.791,10.689) -- (5.706,10.604) -- (5.652,10.497) -- (5.633,10.378) -- (5.652,10.259) -- (5.706,10.152) -- (5.791,10.067) -- (5.898,10.012) -- (6.017,9.993) -- cycle;
\node at (6.017,10.378) {\scalebox{1.367}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.139,7.935) -- (6.258,7.954) -- (6.365,8.008) -- (6.45,8.094) -- (6.505,8.201) -- (6.524,8.32) -- (6.505,8.438) -- (6.45,8.546) -- (6.365,8.631) -- (6.258,8.685) -- (6.139,8.704) -- (6.139,8.704) -- (6.021,8.685) -- (5.913,8.631) -- (5.828,8.546) -- (5.774,8.438) -- (5.755,8.32) -- (5.774,8.201) -- (5.828,8.094) -- (5.913,8.008) -- (6.021,7.954) -- (6.139,7.935) -- cycle;
\node at (6.139,8.32) {\scalebox{1.367}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.381,9.254) -- (5.5,9.273) -- (5.607,9.327) -- (5.693,9.412) -- (5.747,9.52) -- (5.766,9.638) -- (5.747,9.757) -- (5.693,9.865) -- (5.607,9.95) -- (5.5,10.004) -- (5.381,10.023) -- (5.381,10.023) -- (5.263,10.004) -- (5.155,9.95) -- (5.07,9.865) -- (5.016,9.757) -- (4.997,9.638) -- (5.016,9.52) -- (5.07,9.412) -- (5.155,9.327) -- (5.263,9.273) -- (5.381,9.254) -- cycle;
\node at (5.381,9.638) {\scalebox{1.367}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.7,7.602) -- (4.819,7.62) -- (4.926,7.675) -- (5.011,7.76) -- (5.065,7.867) -- (5.084,7.986) -- (5.065,8.105) -- (5.011,8.212) -- (4.926,8.297) -- (4.819,8.352) -- (4.7,8.371) -- (4.7,8.371) -- (4.581,8.352) -- (4.474,8.297) -- (4.389,8.212) -- (4.334,8.105) -- (4.315,7.986) -- (4.334,7.867) -- (4.389,7.76) -- (4.474,7.675) -- (4.581,7.62) -- (4.7,7.602) -- cycle;
\node at (4.7,7.986) {\scalebox{1.367}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.568,5.154) -- (6.687,5.172) -- (6.794,5.227) -- (6.879,5.312) -- (6.934,5.419) -- (6.953,5.538) -- (6.934,5.657) -- (6.879,5.764) -- (6.794,5.849) -- (6.687,5.904) -- (6.568,5.923) -- (6.568,5.923) -- (6.449,5.904) -- (6.342,5.849) -- (6.257,5.764) -- (6.203,5.657) -- (6.184,5.538) -- (6.203,5.419) -- (6.257,5.312) -- (6.342,5.227) -- (6.449,5.172) -- (6.568,5.154) -- cycle;
\node at (6.568,5.538) {\scalebox{1.367}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.195,6.2) -- (5.313,6.219) -- (5.421,6.273) -- (5.506,6.358) -- (5.56,6.466) -- (5.579,6.584) -- (5.56,6.703) -- (5.506,6.811) -- (5.421,6.896) -- (5.313,6.95) -- (5.195,6.969) -- (5.195,6.969) -- (5.076,6.95) -- (4.969,6.896) -- (4.883,6.811) -- (4.829,6.703) -- (4.81,6.584) -- (4.829,6.466) -- (4.883,6.358) -- (4.969,6.273) -- (5.076,6.219) -- (5.195,6.2) -- cycle;
\node at (5.195,6.584) {\scalebox{1.367}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.347,4.994) -- (4.466,5.013) -- (4.573,5.067) -- (4.658,5.152) -- (4.713,5.26) -- (4.731,5.378) -- (4.713,5.497) -- (4.658,5.604) -- (4.573,5.689) -- (4.466,5.744) -- (4.347,5.763) -- (4.347,5.763) -- (4.228,5.744) -- (4.121,5.689) -- (4.036,5.604) -- (3.981,5.497) -- (3.962,5.378) -- (3.981,5.26) -- (4.036,5.152) -- (4.121,5.067) -- (4.228,5.013) -- (4.347,4.994) -- cycle;
\node at (4.347,5.378) {\scalebox{1.367}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.773,4.159) -- (7.892,4.178) -- (7.999,4.232) -- (8.085,4.318) -- (8.139,4.425) -- (8.158,4.544) -- (8.139,4.662) -- (8.085,4.77) -- (7.999,4.855) -- (7.892,4.909) -- (7.773,4.928) -- (7.412,4.928) -- (7.293,4.909) -- (7.186,4.855) -- (7.1,4.77) -- (7.046,4.662) -- (7.027,4.544) -- (7.046,4.425) -- (7.1,4.318) -- (7.186,4.232) -- (7.293,4.178) -- (7.412,4.159) -- cycle;
\node at (7.592,4.544) {\scalebox{1.367}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.263,2.397) -- (6.382,2.416) -- (6.489,2.47) -- (6.574,2.555) -- (6.629,2.662) -- (6.647,2.781) -- (6.629,2.9) -- (6.574,3.007) -- (6.489,3.092) -- (6.382,3.147) -- (6.263,3.166) -- (5.901,3.166) -- (5.782,3.147) -- (5.675,3.092) -- (5.59,3.007) -- (5.535,2.9) -- (5.516,2.781) -- (5.535,2.662) -- (5.59,2.555) -- (5.675,2.47) -- (5.782,2.416) -- (5.901,2.397) -- cycle;
\node at (6.082,2.781) {\scalebox{1.367}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.891,2.429) -- (9.01,2.448) -- (9.117,2.502) -- (9.202,2.587) -- (9.257,2.695) -- (9.276,2.813) -- (9.257,2.932) -- (9.202,3.039) -- (9.117,3.125) -- (9.01,3.179) -- (8.891,3.198) -- (8.529,3.198) -- (8.411,3.179) -- (8.303,3.125) -- (8.218,3.039) -- (8.164,2.932) -- (8.145,2.813) -- (8.164,2.695) -- (8.218,2.587) -- (8.303,2.502) -- (8.411,2.448) -- (8.529,2.429) -- cycle;
\node at (8.71,2.813) {\scalebox{1.367}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.822,3.864) -- (2.94,3.883) -- (3.048,3.938) -- (3.133,4.023) -- (3.187,4.13) -- (3.206,4.249) -- (3.187,4.368) -- (3.133,4.475) -- (3.048,4.56) -- (2.94,4.615) -- (2.822,4.633) -- (2.46,4.633) -- (2.341,4.615) -- (2.234,4.56) -- (2.149,4.475) -- (2.094,4.368) -- (2.075,4.249) -- (2.094,4.13) -- (2.149,4.023) -- (2.234,3.938) -- (2.341,3.883) -- (2.46,3.864) -- cycle;
\node at (2.641,4.249) {\scalebox{1.367}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.138,2.765) -- (2.257,2.784) -- (2.364,2.839) -- (2.449,2.924) -- (2.504,3.031) -- (2.523,3.15) -- (2.504,3.269) -- (2.449,3.376) -- (2.364,3.461) -- (2.257,3.515) -- (2.138,3.534) -- (1.776,3.534) -- (1.657,3.515) -- (1.55,3.461) -- (1.465,3.376) -- (1.41,3.269) -- (1.392,3.15) -- (1.41,3.031) -- (1.465,2.924) -- (1.55,2.839) -- (1.657,2.784) -- (1.776,2.765) -- cycle;
\node at (1.957,3.15) {\scalebox{1.367}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.362,3.918) -- (5.48,3.937) -- (5.588,3.992) -- (5.673,4.077) -- (5.727,4.184) -- (5.746,4.303) -- (5.727,4.422) -- (5.673,4.529) -- (5.588,4.614) -- (5.48,4.669) -- (5.362,4.687) -- (5.0,4.687) -- (4.881,4.669) -- (4.774,4.614) -- (4.689,4.529) -- (4.634,4.422) -- (4.615,4.303) -- (4.634,4.184) -- (4.689,4.077) -- (4.774,3.992) -- (4.881,3.937) -- (5.0,3.918) -- cycle;
\node at (5.181,4.303) {\scalebox{1.367}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.185,0.982) -- (8.304,1.001) -- (8.411,1.055) -- (8.497,1.14) -- (8.551,1.248) -- (8.57,1.366) -- (8.551,1.485) -- (8.497,1.593) -- (8.411,1.678) -- (8.304,1.732) -- (8.185,1.751) -- (7.824,1.751) -- (7.705,1.732) -- (7.598,1.678) -- (7.512,1.593) -- (7.458,1.485) -- (7.439,1.366) -- (7.458,1.248) -- (7.512,1.14) -- (7.598,1.055) -- (7.705,1.001) -- (7.824,0.982) -- cycle;
\node at (8.004,1.366) {\scalebox{1.367}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (8.395,3.728) -- (8.995,3.728) -- (8.995,4.328) -- (8.395,4.328) -- cycle;
\node at (8.695,4.028) {\scalebox{1.367}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.157,3.121) -- (4.757,3.121) -- (4.757,3.721) -- (4.157,3.721) -- cycle;
\node at (4.457,3.421) {\scalebox{1.367}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.442,8.571) -- (5.042,8.571) -- (5.042,9.171) -- (4.442,9.171) -- cycle;
\node at (4.742,8.871) {\scalebox{1.367}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.5,5.651) -- (8.1,5.651) -- (8.1,6.251) -- (7.5,6.251) -- cycle;
\node at (7.8,5.951) {\scalebox{1.367}{\small d}};
\node at (4.069,12.151) {\scalebox{1.367}{\small A}};
\node at (6.362,11.95) {\scalebox{1.367}{\small B}};
\node at (6.867,10.212) {\scalebox{1.367}{\small C}};
\node at (4.008,9.683) {\scalebox{1.367}{\small D}};
\node at (6.529,7.4) {\scalebox{1.367}{\small E}};
\node at (3.753,7.266) {\scalebox{1.367}{\small F}};
\node at (6.776,3.575) {\scalebox{1.367}{\small G}};
\node at (9.235,3.722) {\scalebox{1.367}{\small H}};
\node at (0.765,3.74) {\scalebox{1.367}{\small I}};
\node at (3.579,3.437) {\scalebox{1.367}{\small J}};
\node at (1.245,1.68) {\scalebox{1.367}{\small K}};
\node at (9.706,1.29) {\scalebox{1.367}{\small L}};
\node at (6.864,0.95) {\scalebox{1.367}{\small M}};
\draw[line width=0.5pt] (10.582,10.594) -- (10.582,11.694);
\draw[line width=0.5pt] (10.351,11.309) -- (10.582,11.694) -- (10.813,11.309);
\draw[line width=0.5pt] (10.279,11.144) -- (10.884,11.144);
\node at (10.582,12.079) {\scalebox{1.367}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, C, F, G, I, L & 6 \\ \hline
clairs & B, D, E, H, J, K, M & 7 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{7} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 13 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 6 carrefours sombres et 7 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{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 Neuve ⑫} vers le sud jusqu'à Place Sainte-Croix \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Tilleuls ⑯}, vers l'ouest, jusqu'à Place de l'Église \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{venelle du Chat ⑪}, vers le nord, jusqu'à Rond-point de la Poste \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{passage du Marché ⑦}, vers le nord, jusqu'à Carrefour de l'Hôpital \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{avenue de la Gare ④}, vers le nord, jusqu'à Carrefour Saint-Roch \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Lilas ③}, vers le nord, jusqu'à Placette des Ormes \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin du Lavoir ①}, vers l'ouest, jusqu'à Coin du Lavoir \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de la Poste ②}, vers le sud, jusqu'à Carrefour de la Mairie \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin Creux ⑥}, vers le sud, jusqu'à Rond-point de la Gare \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Stade ⑨}, vers le sud, jusqu'à Rond-point du Stade \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Peupliers ⑬}, vers l'ouest, jusqu'à Place des Tanneurs \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse du Vieux-Four ⑭}, vers le sud, jusqu'à Place de la Fontaine \textbf{(K)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 13 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
