\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 184}

\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=184}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0184}
\end{minipage}%
}
}

\begin{document}

\begin{center}
    \small Un car de ramassage doit s'arrêter à chaque carrefour d'un quartier, une fois et une seule. Selon le plan c'est possible ou non — et un simple coloriage permet souvent de le dire d'avance, sans essayer.
\end{center}
\vspace{0.5em}
\sloppy
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6035, line join=round, line cap=round]
\draw[line width=0.5pt] (8.621,6.841) -- (8.814,6.807) -- (9.009,6.777) -- (9.203,6.753) -- (9.399,6.733) -- (9.595,6.717) -- (9.791,6.706) -- (9.987,6.7) -- (10.184,6.698) -- (10.38,6.701) -- (10.577,6.709) -- (10.773,6.721) -- (10.968,6.738) -- (11.164,6.759) -- (11.358,6.785);
\draw[line width=0.5pt] (8.621,6.841) -- (8.664,6.701) -- (8.705,6.561) -- (8.741,6.42) -- (8.775,6.278) -- (8.805,6.135) -- (8.832,5.992) -- (8.855,5.848) -- (8.875,5.703) -- (8.892,5.558) -- (8.905,5.413) -- (8.915,5.267) -- (8.921,5.121) -- (8.924,4.975) -- (8.924,4.83);
\draw[line width=0.5pt] (1.991,6.824) -- (2.126,6.791) -- (2.262,6.762) -- (2.399,6.735) -- (2.536,6.712) -- (2.674,6.692) -- (2.813,6.675) -- (2.952,6.662) -- (3.091,6.651) -- (3.23,6.644) -- (3.369,6.64) -- (3.508,6.64) -- (3.648,6.643) -- (3.787,6.649) -- (3.926,6.658);
\draw[line width=0.5pt] (1.991,6.824) -- (1.972,6.663) -- (1.949,6.501) -- (1.922,6.341) -- (1.892,6.181) -- (1.858,6.022) -- (1.82,5.863) -- (1.778,5.706) -- (1.733,5.549) -- (1.684,5.394) -- (1.631,5.24) -- (1.575,5.087) -- (1.515,4.935) -- (1.452,4.785) -- (1.385,4.637);
\draw[line width=0.5pt] (11.358,6.785) -- (11.338,6.643) -- (11.321,6.499) -- (11.308,6.356) -- (11.298,6.212) -- (11.292,6.067) -- (11.288,5.923) -- (11.289,5.779) -- (11.292,5.634) -- (11.299,5.49) -- (11.31,5.346) -- (11.323,5.202) -- (11.341,5.059) -- (11.361,4.916) -- (11.385,4.774);
\draw[line width=0.5pt] (6.527,6.737) -- (6.675,6.767) -- (6.823,6.794) -- (6.971,6.817) -- (7.12,6.837) -- (7.27,6.853) -- (7.42,6.866) -- (7.57,6.875) -- (7.72,6.881) -- (7.871,6.883) -- (8.021,6.882) -- (8.172,6.877) -- (8.322,6.868) -- (8.472,6.856) -- (8.621,6.841);
\draw[line width=0.5pt] (6.527,6.737) -- (6.516,6.577) -- (6.509,6.417) -- (6.506,6.256) -- (6.506,6.095) -- (6.51,5.935) -- (6.518,5.774) -- (6.529,5.614) -- (6.545,5.454) -- (6.564,5.294) -- (6.587,5.135) -- (6.613,4.976) -- (6.644,4.819) -- (6.678,4.662) -- (6.715,4.506);
\draw[line width=0.5pt] (3.926,6.658) -- (4.109,6.692) -- (4.294,6.722) -- (4.479,6.748) -- (4.664,6.768) -- (4.85,6.785) -- (5.037,6.797) -- (5.223,6.805) -- (5.41,6.809) -- (5.597,6.808) -- (5.784,6.802) -- (5.97,6.793) -- (6.156,6.779) -- (6.342,6.76) -- (6.527,6.737);
\draw[line width=0.5pt] (8.924,4.83) -- (9.097,4.799) -- (9.272,4.772) -- (9.447,4.749) -- (9.623,4.731) -- (9.799,4.717) -- (9.975,4.706) -- (10.152,4.7) -- (10.329,4.698) -- (10.505,4.701) -- (10.682,4.707) -- (10.858,4.717) -- (11.034,4.732) -- (11.21,4.751) -- (11.385,4.774);
\draw[line width=0.5pt] (8.924,4.83) -- (8.948,4.614) -- (8.968,4.398) -- (8.983,4.181) -- (8.992,3.965) -- (8.997,3.747) -- (8.996,3.53) -- (8.99,3.313) -- (8.979,3.096) -- (8.963,2.88) -- (8.942,2.664) -- (8.916,2.448) -- (8.885,2.233) -- (8.848,2.019) -- (8.807,1.806);
\draw[line width=0.5pt] (11.385,4.774) -- (11.345,4.605) -- (11.309,4.436) -- (11.277,4.266) -- (11.249,4.094) -- (11.225,3.923) -- (11.205,3.75) -- (11.19,3.578) -- (11.178,3.405) -- (11.17,3.232) -- (11.166,3.058) -- (11.167,2.885) -- (11.171,2.712) -- (11.18,2.539) -- (11.193,2.366);
\draw[line width=0.5pt] (1.385,4.637) -- (1.421,4.458) -- (1.454,4.278) -- (1.482,4.098) -- (1.505,3.917) -- (1.525,3.735) -- (1.54,3.553) -- (1.551,3.37) -- (1.558,3.188) -- (1.56,3.005) -- (1.558,2.822) -- (1.552,2.64) -- (1.541,2.457) -- (1.527,2.275) -- (1.507,2.094);
\draw[line width=0.5pt] (6.715,4.506) -- (6.875,4.504) -- (7.036,4.507) -- (7.196,4.513) -- (7.355,4.524) -- (7.515,4.537) -- (7.674,4.555) -- (7.833,4.576) -- (7.991,4.602) -- (8.149,4.63) -- (8.306,4.663) -- (8.462,4.699) -- (8.617,4.739) -- (8.771,4.782) -- (8.924,4.83);
\draw[line width=0.5pt] (6.715,4.506) -- (6.737,4.348) -- (6.755,4.189) -- (6.77,4.03) -- (6.78,3.871) -- (6.787,3.711) -- (6.79,3.552) -- (6.789,3.392) -- (6.785,3.232) -- (6.776,3.073) -- (6.765,2.914) -- (6.749,2.755) -- (6.729,2.596) -- (6.706,2.439) -- (6.679,2.281);
\draw[line width=0.5pt] (6.679,2.281) -- (6.825,2.224) -- (6.972,2.171) -- (7.12,2.121) -- (7.269,2.074) -- (7.42,2.031) -- (7.571,1.992) -- (7.724,1.956) -- (7.877,1.924) -- (8.031,1.895) -- (8.185,1.87) -- (8.34,1.849) -- (8.495,1.831) -- (8.651,1.817) -- (8.807,1.806);
\draw[line width=0.5pt] (8.807,1.806) -- (8.97,1.872) -- (9.134,1.934) -- (9.3,1.992) -- (9.468,2.046) -- (9.636,2.096) -- (9.806,2.142) -- (9.977,2.184) -- (10.149,2.223) -- (10.321,2.257) -- (10.494,2.287) -- (10.668,2.313) -- (10.843,2.335) -- (11.018,2.352) -- (11.193,2.366);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.781,6.841) -- (8.76,6.921) -- (8.701,6.979) -- (8.621,7.001) -- (8.541,6.979) -- (8.482,6.921) -- (8.461,6.841) -- (8.482,6.761) -- (8.541,6.702) -- (8.621,6.681) -- (8.701,6.702) -- (8.76,6.761) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.151,6.824) -- (2.129,6.904) -- (2.071,6.963) -- (1.991,6.984) -- (1.911,6.963) -- (1.852,6.904) -- (1.831,6.824) -- (1.852,6.744) -- (1.911,6.686) -- (1.991,6.664) -- (2.071,6.686) -- (2.129,6.744) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.518,6.785) -- (11.497,6.865) -- (11.438,6.924) -- (11.358,6.945) -- (11.278,6.924) -- (11.22,6.865) -- (11.198,6.785) -- (11.22,6.705) -- (11.278,6.647) -- (11.358,6.625) -- (11.438,6.647) -- (11.497,6.705) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.687,6.737) -- (6.666,6.817) -- (6.607,6.876) -- (6.527,6.897) -- (6.447,6.876) -- (6.389,6.817) -- (6.367,6.737) -- (6.389,6.657) -- (6.447,6.599) -- (6.527,6.577) -- (6.607,6.599) -- (6.666,6.657) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.086,6.658) -- (4.064,6.738) -- (4.006,6.797) -- (3.926,6.818) -- (3.846,6.797) -- (3.787,6.738) -- (3.766,6.658) -- (3.787,6.578) -- (3.846,6.52) -- (3.926,6.498) -- (4.006,6.52) -- (4.064,6.578) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.084,4.83) -- (9.062,4.91) -- (9.004,4.968) -- (8.924,4.99) -- (8.844,4.968) -- (8.785,4.91) -- (8.764,4.83) -- (8.785,4.75) -- (8.844,4.691) -- (8.924,4.67) -- (9.004,4.691) -- (9.062,4.75) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.545,4.774) -- (11.524,4.854) -- (11.465,4.913) -- (11.385,4.934) -- (11.305,4.913) -- (11.246,4.854) -- (11.225,4.774) -- (11.246,4.694) -- (11.305,4.635) -- (11.385,4.614) -- (11.465,4.635) -- (11.524,4.694) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.545,4.637) -- (1.524,4.717) -- (1.465,4.775) -- (1.385,4.797) -- (1.305,4.775) -- (1.246,4.717) -- (1.225,4.637) -- (1.246,4.557) -- (1.305,4.498) -- (1.385,4.477) -- (1.465,4.498) -- (1.524,4.557) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.875,4.506) -- (6.854,4.586) -- (6.795,4.644) -- (6.715,4.666) -- (6.635,4.644) -- (6.577,4.586) -- (6.555,4.506) -- (6.577,4.426) -- (6.635,4.367) -- (6.715,4.346) -- (6.795,4.367) -- (6.854,4.426) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.353,2.366) -- (11.331,2.446) -- (11.273,2.504) -- (11.193,2.526) -- (11.113,2.504) -- (11.054,2.446) -- (11.033,2.366) -- (11.054,2.286) -- (11.113,2.227) -- (11.193,2.206) -- (11.273,2.227) -- (11.331,2.286) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.839,2.281) -- (6.818,2.361) -- (6.759,2.42) -- (6.679,2.441) -- (6.599,2.42) -- (6.541,2.361) -- (6.519,2.281) -- (6.541,2.201) -- (6.599,2.143) -- (6.679,2.121) -- (6.759,2.143) -- (6.818,2.201) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.667,2.094) -- (1.646,2.174) -- (1.587,2.232) -- (1.507,2.254) -- (1.427,2.232) -- (1.369,2.174) -- (1.347,2.094) -- (1.369,2.014) -- (1.427,1.955) -- (1.507,1.934) -- (1.587,1.955) -- (1.646,2.014) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.967,1.806) -- (8.946,1.886) -- (8.887,1.945) -- (8.807,1.966) -- (8.727,1.945) -- (8.668,1.886) -- (8.647,1.806) -- (8.668,1.726) -- (8.727,1.668) -- (8.807,1.646) -- (8.887,1.668) -- (8.946,1.726) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.995,6.7) -- (10.117,6.719) -- (10.227,6.775) -- (10.314,6.862) -- (10.37,6.972) -- (10.389,7.094) -- (10.37,7.216) -- (10.314,7.325) -- (10.227,7.413) -- (10.117,7.469) -- (9.995,7.488) -- (9.995,7.488) -- (9.874,7.469) -- (9.764,7.413) -- (9.677,7.325) -- (9.621,7.216) -- (9.601,7.094) -- (9.621,6.972) -- (9.677,6.862) -- (9.764,6.775) -- (9.874,6.719) -- (9.995,6.7) -- cycle;
\node at (9.995,7.094) {\scalebox{1.657}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.466,5.395) -- (8.588,5.414) -- (8.697,5.47) -- (8.784,5.557) -- (8.84,5.667) -- (8.86,5.789) -- (8.84,5.911) -- (8.784,6.02) -- (8.697,6.108) -- (8.588,6.164) -- (8.466,6.183) -- (8.466,6.183) -- (8.344,6.164) -- (8.234,6.108) -- (8.147,6.02) -- (8.091,5.911) -- (8.072,5.789) -- (8.091,5.667) -- (8.147,5.557) -- (8.234,5.47) -- (8.344,5.414) -- (8.466,5.395) -- cycle;
\node at (8.466,5.789) {\scalebox{1.657}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.918,5.875) -- (3.04,5.894) -- (3.149,5.95) -- (3.237,6.038) -- (3.293,6.147) -- (3.312,6.269) -- (3.293,6.391) -- (3.237,6.501) -- (3.149,6.588) -- (3.04,6.644) -- (2.918,6.663) -- (2.918,6.663) -- (2.796,6.644) -- (2.686,6.588) -- (2.599,6.501) -- (2.543,6.391) -- (2.524,6.269) -- (2.543,6.147) -- (2.599,6.038) -- (2.686,5.95) -- (2.796,5.894) -- (2.918,5.875) -- cycle;
\node at (2.918,6.269) {\scalebox{1.657}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.158,5.207) -- (2.279,5.226) -- (2.389,5.282) -- (2.476,5.369) -- (2.532,5.479) -- (2.552,5.6) -- (2.532,5.722) -- (2.476,5.832) -- (2.389,5.919) -- (2.279,5.975) -- (2.158,5.994) -- (2.158,5.994) -- (2.036,5.975) -- (1.926,5.919) -- (1.839,5.832) -- (1.783,5.722) -- (1.764,5.6) -- (1.783,5.479) -- (1.839,5.369) -- (1.926,5.282) -- (2.036,5.226) -- (2.158,5.207) -- cycle;
\node at (2.158,5.6) {\scalebox{1.657}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.682,5.39) -- (11.804,5.409) -- (11.914,5.465) -- (12.001,5.552) -- (12.057,5.662) -- (12.076,5.784) -- (12.057,5.906) -- (12.001,6.015) -- (11.914,6.102) -- (11.804,6.158) -- (11.682,6.178) -- (11.682,6.178) -- (11.561,6.158) -- (11.451,6.102) -- (11.364,6.015) -- (11.308,5.906) -- (11.289,5.784) -- (11.308,5.662) -- (11.364,5.552) -- (11.451,5.465) -- (11.561,5.409) -- (11.682,5.39) -- cycle;
\node at (11.682,5.784) {\scalebox{1.657}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.551,6.875) -- (7.672,6.894) -- (7.782,6.95) -- (7.869,7.037) -- (7.925,7.147) -- (7.944,7.269) -- (7.925,7.39) -- (7.869,7.5) -- (7.782,7.587) -- (7.672,7.643) -- (7.551,7.663) -- (7.551,7.663) -- (7.429,7.643) -- (7.319,7.587) -- (7.232,7.5) -- (7.176,7.39) -- (7.157,7.269) -- (7.176,7.147) -- (7.232,7.037) -- (7.319,6.95) -- (7.429,6.894) -- (7.551,6.875) -- cycle;
\node at (7.551,7.269) {\scalebox{1.657}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.922,5.253) -- (7.044,5.272) -- (7.153,5.328) -- (7.241,5.415) -- (7.297,5.525) -- (7.316,5.647) -- (7.297,5.768) -- (7.241,5.878) -- (7.153,5.965) -- (7.044,6.021) -- (6.922,6.041) -- (6.922,6.041) -- (6.8,6.021) -- (6.69,5.965) -- (6.603,5.878) -- (6.547,5.768) -- (6.528,5.647) -- (6.547,5.525) -- (6.603,5.415) -- (6.69,5.328) -- (6.8,5.272) -- (6.922,5.253) -- cycle;
\node at (6.922,5.647) {\scalebox{1.657}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.212,6.805) -- (5.333,6.824) -- (5.443,6.88) -- (5.53,6.967) -- (5.586,7.077) -- (5.605,7.199) -- (5.586,7.321) -- (5.53,7.43) -- (5.443,7.518) -- (5.333,7.573) -- (5.212,7.593) -- (5.212,7.593) -- (5.09,7.573) -- (4.98,7.518) -- (4.893,7.43) -- (4.837,7.321) -- (4.818,7.199) -- (4.837,7.077) -- (4.893,6.967) -- (4.98,6.88) -- (5.09,6.824) -- (5.212,6.805) -- cycle;
\node at (5.212,7.199) {\scalebox{1.657}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.515,3.913) -- (10.637,3.932) -- (10.747,3.988) -- (10.834,4.075) -- (10.89,4.185) -- (10.909,4.307) -- (10.89,4.428) -- (10.834,4.538) -- (10.747,4.625) -- (10.637,4.681) -- (10.515,4.701) -- (10.515,4.701) -- (10.393,4.681) -- (10.283,4.625) -- (10.196,4.538) -- (10.14,4.428) -- (10.121,4.307) -- (10.14,4.185) -- (10.196,4.075) -- (10.283,3.988) -- (10.393,3.932) -- (10.515,3.913) -- cycle;
\node at (10.515,4.307) {\scalebox{1.657}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.569,2.904) -- (9.691,2.923) -- (9.801,2.979) -- (9.888,3.066) -- (9.944,3.176) -- (9.963,3.298) -- (9.944,3.42) -- (9.888,3.53) -- (9.801,3.617) -- (9.691,3.673) -- (9.569,3.692) -- (9.198,3.692) -- (9.077,3.673) -- (8.967,3.617) -- (8.88,3.53) -- (8.824,3.42) -- (8.804,3.298) -- (8.824,3.176) -- (8.88,3.066) -- (8.967,2.979) -- (9.077,2.923) -- (9.198,2.904) -- cycle;
\node at (9.384,3.298) {\scalebox{1.657}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.768,3.153) -- (11.889,3.172) -- (11.999,3.228) -- (12.086,3.315) -- (12.142,3.425) -- (12.161,3.546) -- (12.142,3.668) -- (12.086,3.778) -- (11.999,3.865) -- (11.889,3.921) -- (11.768,3.94) -- (11.397,3.94) -- (11.275,3.921) -- (11.165,3.865) -- (11.078,3.778) -- (11.022,3.668) -- (11.003,3.546) -- (11.022,3.425) -- (11.078,3.315) -- (11.165,3.228) -- (11.275,3.172) -- (11.397,3.153) -- cycle;
\node at (11.582,3.546) {\scalebox{1.657}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.13,2.995) -- (2.252,3.015) -- (2.361,3.071) -- (2.449,3.158) -- (2.505,3.268) -- (2.524,3.389) -- (2.505,3.511) -- (2.449,3.621) -- (2.361,3.708) -- (2.252,3.764) -- (2.13,3.783) -- (1.759,3.783) -- (1.637,3.764) -- (1.528,3.708) -- (1.44,3.621) -- (1.385,3.511) -- (1.365,3.389) -- (1.385,3.268) -- (1.44,3.158) -- (1.528,3.071) -- (1.637,3.015) -- (1.759,2.995) -- cycle;
\node at (1.945,3.389) {\scalebox{1.657}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.961,4.572) -- (8.083,4.592) -- (8.193,4.648) -- (8.28,4.735) -- (8.336,4.845) -- (8.355,4.966) -- (8.336,5.088) -- (8.28,5.198) -- (8.193,5.285) -- (8.083,5.341) -- (7.961,5.36) -- (7.59,5.36) -- (7.469,5.341) -- (7.359,5.285) -- (7.272,5.198) -- (7.216,5.088) -- (7.196,4.966) -- (7.216,4.845) -- (7.272,4.735) -- (7.359,4.648) -- (7.469,4.592) -- (7.59,4.572) -- cycle;
\node at (7.776,4.966) {\scalebox{1.657}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.003,2.602) -- (6.125,2.621) -- (6.235,2.677) -- (6.322,2.764) -- (6.378,2.874) -- (6.397,2.996) -- (6.378,3.118) -- (6.322,3.227) -- (6.235,3.315) -- (6.125,3.371) -- (6.003,3.39) -- (5.633,3.39) -- (5.511,3.371) -- (5.401,3.315) -- (5.314,3.227) -- (5.258,3.118) -- (5.239,2.996) -- (5.258,2.874) -- (5.314,2.764) -- (5.401,2.677) -- (5.511,2.621) -- (5.633,2.602) -- cycle;
\node at (5.818,2.996) {\scalebox{1.657}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.995,1.947) -- (8.116,1.966) -- (8.226,2.022) -- (8.313,2.109) -- (8.369,2.219) -- (8.389,2.341) -- (8.369,2.462) -- (8.313,2.572) -- (8.226,2.659) -- (8.116,2.715) -- (7.995,2.734) -- (7.624,2.734) -- (7.502,2.715) -- (7.392,2.659) -- (7.305,2.572) -- (7.249,2.462) -- (7.23,2.341) -- (7.249,2.219) -- (7.305,2.109) -- (7.392,2.022) -- (7.502,1.966) -- (7.624,1.947) -- cycle;
\node at (7.809,2.341) {\scalebox{1.657}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.642,1.134) -- (10.763,1.153) -- (10.873,1.209) -- (10.96,1.296) -- (11.016,1.406) -- (11.036,1.528) -- (11.016,1.649) -- (10.96,1.759) -- (10.873,1.846) -- (10.763,1.902) -- (10.642,1.922) -- (10.271,1.922) -- (10.149,1.902) -- (10.039,1.846) -- (9.952,1.759) -- (9.896,1.649) -- (9.877,1.528) -- (9.896,1.406) -- (9.952,1.296) -- (10.039,1.209) -- (10.149,1.153) -- (10.271,1.134) -- cycle;
\node at (10.456,1.528) {\scalebox{1.657}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (8.507,1.506) -- (9.107,1.506) -- (9.107,2.106) -- (8.507,2.106) -- cycle;
\node at (8.807,1.806) {\scalebox{1.657}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.805,0.65) -- (8.405,0.65) -- (8.405,1.25) -- (7.805,1.25) -- cycle;
\node at (8.105,0.95) {\scalebox{1.657}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.037,3.428) -- (7.637,3.428) -- (7.637,4.028) -- (7.037,4.028) -- cycle;
\node at (7.337,3.728) {\scalebox{1.657}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.537,4.965) -- (10.137,4.965) -- (10.137,5.565) -- (9.537,5.565) -- cycle;
\node at (9.837,5.265) {\scalebox{1.657}{\small d}};
\node at (9.017,7.318) {\scalebox{1.657}{\small A}};
\node at (1.423,7.072) {\scalebox{1.657}{\small B}};
\node at (11.921,7.045) {\scalebox{1.657}{\small C}};
\node at (6.431,7.35) {\scalebox{1.657}{\small D}};
\node at (3.411,7.003) {\scalebox{1.657}{\small E}};
\node at (9.543,4.867) {\scalebox{1.657}{\small F}};
\node at (12.005,4.784) {\scalebox{1.657}{\small G}};
\node at (0.765,4.629) {\scalebox{1.657}{\small H}};
\node at (6.371,3.99) {\scalebox{1.657}{\small I}};
\node at (11.739,2.072) {\scalebox{1.657}{\small J}};
\node at (6.636,1.663) {\scalebox{1.657}{\small K}};
\node at (0.95,1.822) {\scalebox{1.657}{\small L}};
\node at (9.154,1.293) {\scalebox{1.657}{\small M}};
\draw[line width=0.5pt] (12.685,5.891) -- (12.685,6.991);
\draw[line width=0.5pt] (12.454,6.606) -- (12.685,6.991) -- (12.916,6.606);
\draw[line width=0.5pt] (12.382,6.441) -- (12.987,6.441);
\node at (12.685,7.376) {\scalebox{1.657}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour M, 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 Neuve & Ⓐ Carrefour de l'Hôpital & Ⓙ Placette des Ormes \\ \hline
① sentier des Prés & ⑩ allée des Peupliers & Ⓑ Place des Tanneurs & Ⓚ Place du Vieux-Puits \\ \hline
② rue du Puits & ⑪ rue du Stade & Ⓒ Place du Marché & Ⓛ Place de la Fontaine \\ \hline
③ rue Haute & ⑫ impasse des Cerisiers & Ⓓ Carrefour du Moulin & Ⓜ Rond-point du Stade \\ \hline
④ rue Verte & ⑬ rue de la Mairie & Ⓔ Place Sainte-Croix & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ rue des Écoles & ⑭ rue du Pont & Ⓕ Carrefour des Tilleuls & \textbf{a} l'école \\ \hline
⑥ rue du Moulin & ⑮ chemin des Vignes & Ⓖ Carrefour du Lavoir & \textbf{b} le cimetière \\ \hline
⑦ boulevard du Parc & ⑯ venelle du Chat & Ⓗ Fontaine Notre-Dame & \textbf{c} la boulangerie \\ \hline
⑧ rue Basse & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour Saint-Roch & \textbf{d} la mairie \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\begin{enumerate}
  \item Colorier les carrefours du plan avec \textbf{deux couleurs}, de sorte que deux carrefours \textbf{reliés par une rue} ne soient jamais de la même couleur. Combien y en a-t-il de chaque couleur ?
  \item Le car roule d'un carrefour au suivant. Que peut-on dire de la \textbf{couleur} des arrêts successifs ? Qu'en déduire sur les couleurs des arrêts d'une tournée qui les dessert tous ?
  \item \textbf{Le car peut-il desservir tous les carrefours, une fois chacun, et revenir à l'école ?} Si oui, décrire un trajet ; sinon, démontrer que c'est impossible.
  \item \textbf{Et s'il n'est pas obligé de revenir à l'école ?} Même consigne.
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage — Correction}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6182, line join=round, line cap=round]
\draw[line width=0.5pt] (8.621,6.841) -- (8.814,6.807) -- (9.009,6.777) -- (9.203,6.753) -- (9.399,6.733) -- (9.595,6.717) -- (9.791,6.706) -- (9.987,6.7) -- (10.184,6.698) -- (10.38,6.701) -- (10.577,6.709) -- (10.773,6.721) -- (10.968,6.738) -- (11.164,6.759) -- (11.358,6.785);
\draw[line width=0.5pt] (8.621,6.841) -- (8.664,6.701) -- (8.705,6.561) -- (8.741,6.42) -- (8.775,6.278) -- (8.805,6.135) -- (8.832,5.992) -- (8.855,5.848) -- (8.875,5.703) -- (8.892,5.558) -- (8.905,5.413) -- (8.915,5.267) -- (8.921,5.121) -- (8.924,4.975) -- (8.924,4.83);
\draw[line width=0.5pt] (1.991,6.824) -- (2.126,6.791) -- (2.262,6.762) -- (2.399,6.735) -- (2.536,6.712) -- (2.674,6.692) -- (2.813,6.675) -- (2.952,6.662) -- (3.091,6.651) -- (3.23,6.644) -- (3.369,6.64) -- (3.508,6.64) -- (3.648,6.643) -- (3.787,6.649) -- (3.926,6.658);
\draw[line width=0.5pt] (1.991,6.824) -- (1.972,6.663) -- (1.949,6.501) -- (1.922,6.341) -- (1.892,6.181) -- (1.858,6.022) -- (1.82,5.863) -- (1.778,5.706) -- (1.733,5.549) -- (1.684,5.394) -- (1.631,5.24) -- (1.575,5.087) -- (1.515,4.935) -- (1.452,4.785) -- (1.385,4.637);
\draw[line width=0.5pt] (11.358,6.785) -- (11.338,6.643) -- (11.321,6.499) -- (11.308,6.356) -- (11.298,6.212) -- (11.292,6.067) -- (11.288,5.923) -- (11.289,5.779) -- (11.292,5.634) -- (11.299,5.49) -- (11.31,5.346) -- (11.323,5.202) -- (11.341,5.059) -- (11.361,4.916) -- (11.385,4.774);
\draw[line width=0.5pt] (6.527,6.737) -- (6.675,6.767) -- (6.823,6.794) -- (6.971,6.817) -- (7.12,6.837) -- (7.27,6.853) -- (7.42,6.866) -- (7.57,6.875) -- (7.72,6.881) -- (7.871,6.883) -- (8.021,6.882) -- (8.172,6.877) -- (8.322,6.868) -- (8.472,6.856) -- (8.621,6.841);
\draw[line width=0.5pt] (6.527,6.737) -- (6.516,6.577) -- (6.509,6.417) -- (6.506,6.256) -- (6.506,6.095) -- (6.51,5.935) -- (6.518,5.774) -- (6.529,5.614) -- (6.545,5.454) -- (6.564,5.294) -- (6.587,5.135) -- (6.613,4.976) -- (6.644,4.819) -- (6.678,4.662) -- (6.715,4.506);
\draw[line width=0.5pt] (3.926,6.658) -- (4.109,6.692) -- (4.294,6.722) -- (4.479,6.748) -- (4.664,6.768) -- (4.85,6.785) -- (5.037,6.797) -- (5.223,6.805) -- (5.41,6.809) -- (5.597,6.808) -- (5.784,6.802) -- (5.97,6.793) -- (6.156,6.779) -- (6.342,6.76) -- (6.527,6.737);
\draw[line width=0.5pt] (8.924,4.83) -- (9.097,4.799) -- (9.272,4.772) -- (9.447,4.749) -- (9.623,4.731) -- (9.799,4.717) -- (9.975,4.706) -- (10.152,4.7) -- (10.329,4.698) -- (10.505,4.701) -- (10.682,4.707) -- (10.858,4.717) -- (11.034,4.732) -- (11.21,4.751) -- (11.385,4.774);
\draw[line width=0.5pt] (8.924,4.83) -- (8.948,4.614) -- (8.968,4.398) -- (8.983,4.181) -- (8.992,3.965) -- (8.997,3.747) -- (8.996,3.53) -- (8.99,3.313) -- (8.979,3.096) -- (8.963,2.88) -- (8.942,2.664) -- (8.916,2.448) -- (8.885,2.233) -- (8.848,2.019) -- (8.807,1.806);
\draw[line width=0.5pt] (11.385,4.774) -- (11.345,4.605) -- (11.309,4.436) -- (11.277,4.266) -- (11.249,4.094) -- (11.225,3.923) -- (11.205,3.75) -- (11.19,3.578) -- (11.178,3.405) -- (11.17,3.232) -- (11.166,3.058) -- (11.167,2.885) -- (11.171,2.712) -- (11.18,2.539) -- (11.193,2.366);
\draw[line width=0.5pt] (1.385,4.637) -- (1.421,4.458) -- (1.454,4.278) -- (1.482,4.098) -- (1.505,3.917) -- (1.525,3.735) -- (1.54,3.553) -- (1.551,3.37) -- (1.558,3.188) -- (1.56,3.005) -- (1.558,2.822) -- (1.552,2.64) -- (1.541,2.457) -- (1.527,2.275) -- (1.507,2.094);
\draw[line width=0.5pt] (6.715,4.506) -- (6.875,4.504) -- (7.036,4.507) -- (7.196,4.513) -- (7.355,4.524) -- (7.515,4.537) -- (7.674,4.555) -- (7.833,4.576) -- (7.991,4.602) -- (8.149,4.63) -- (8.306,4.663) -- (8.462,4.699) -- (8.617,4.739) -- (8.771,4.782) -- (8.924,4.83);
\draw[line width=0.5pt] (6.715,4.506) -- (6.737,4.348) -- (6.755,4.189) -- (6.77,4.03) -- (6.78,3.871) -- (6.787,3.711) -- (6.79,3.552) -- (6.789,3.392) -- (6.785,3.232) -- (6.776,3.073) -- (6.765,2.914) -- (6.749,2.755) -- (6.729,2.596) -- (6.706,2.439) -- (6.679,2.281);
\draw[line width=0.5pt] (6.679,2.281) -- (6.825,2.224) -- (6.972,2.171) -- (7.12,2.121) -- (7.269,2.074) -- (7.42,2.031) -- (7.571,1.992) -- (7.724,1.956) -- (7.877,1.924) -- (8.031,1.895) -- (8.185,1.87) -- (8.34,1.849) -- (8.495,1.831) -- (8.651,1.817) -- (8.807,1.806);
\draw[line width=0.5pt] (8.807,1.806) -- (8.97,1.872) -- (9.134,1.934) -- (9.3,1.992) -- (9.468,2.046) -- (9.636,2.096) -- (9.806,2.142) -- (9.977,2.184) -- (10.149,2.223) -- (10.321,2.257) -- (10.494,2.287) -- (10.668,2.313) -- (10.843,2.335) -- (11.018,2.352) -- (11.193,2.366);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.994,6.841) -- (8.966,6.983) -- (8.885,7.104) -- (8.764,7.185) -- (8.621,7.214) -- (8.478,7.185) -- (8.357,7.104) -- (8.276,6.983) -- (8.248,6.841) -- (8.276,6.698) -- (8.357,6.577) -- (8.478,6.496) -- (8.621,6.467) -- (8.764,6.496) -- (8.885,6.577) -- (8.966,6.698) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.364,6.824) -- (2.336,6.967) -- (2.255,7.088) -- (2.134,7.169) -- (1.991,7.197) -- (1.848,7.169) -- (1.727,7.088) -- (1.646,6.967) -- (1.618,6.824) -- (1.646,6.681) -- (1.727,6.56) -- (1.848,6.479) -- (1.991,6.451) -- (2.134,6.479) -- (2.255,6.56) -- (2.336,6.681) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.731,6.785) -- (11.703,6.928) -- (11.622,7.049) -- (11.501,7.13) -- (11.358,7.159) -- (11.215,7.13) -- (11.094,7.049) -- (11.013,6.928) -- (10.985,6.785) -- (11.013,6.643) -- (11.094,6.522) -- (11.215,6.441) -- (11.358,6.412) -- (11.501,6.441) -- (11.622,6.522) -- (11.703,6.643) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.901,6.737) -- (6.872,6.88) -- (6.791,7.001) -- (6.67,7.082) -- (6.527,7.11) -- (6.385,7.082) -- (6.264,7.001) -- (6.183,6.88) -- (6.154,6.737) -- (6.183,6.594) -- (6.264,6.473) -- (6.385,6.392) -- (6.527,6.364) -- (6.67,6.392) -- (6.791,6.473) -- (6.872,6.594) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.299,6.658) -- (4.271,6.801) -- (4.19,6.922) -- (4.069,7.003) -- (3.926,7.032) -- (3.783,7.003) -- (3.662,6.922) -- (3.581,6.801) -- (3.553,6.658) -- (3.581,6.516) -- (3.662,6.395) -- (3.783,6.314) -- (3.926,6.285) -- (4.069,6.314) -- (4.19,6.395) -- (4.271,6.516) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.297,4.83) -- (9.268,4.972) -- (9.188,5.094) -- (9.066,5.174) -- (8.924,5.203) -- (8.781,5.174) -- (8.66,5.094) -- (8.579,4.972) -- (8.55,4.83) -- (8.579,4.687) -- (8.66,4.566) -- (8.781,4.485) -- (8.924,4.456) -- (9.066,4.485) -- (9.188,4.566) -- (9.268,4.687) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.758,4.774) -- (11.73,4.917) -- (11.649,5.038) -- (11.528,5.119) -- (11.385,5.147) -- (11.242,5.119) -- (11.121,5.038) -- (11.04,4.917) -- (11.012,4.774) -- (11.04,4.631) -- (11.121,4.51) -- (11.242,4.429) -- (11.385,4.401) -- (11.528,4.429) -- (11.649,4.51) -- (11.73,4.631) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.758,4.637) -- (1.73,4.78) -- (1.649,4.901) -- (1.528,4.982) -- (1.385,5.01) -- (1.242,4.982) -- (1.121,4.901) -- (1.04,4.78) -- (1.012,4.637) -- (1.04,4.494) -- (1.121,4.373) -- (1.242,4.292) -- (1.385,4.264) -- (1.528,4.292) -- (1.649,4.373) -- (1.73,4.494) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.089,4.506) -- (7.06,4.648) -- (6.979,4.769) -- (6.858,4.85) -- (6.715,4.879) -- (6.573,4.85) -- (6.452,4.769) -- (6.371,4.648) -- (6.342,4.506) -- (6.371,4.363) -- (6.452,4.242) -- (6.573,4.161) -- (6.715,4.132) -- (6.858,4.161) -- (6.979,4.242) -- (7.06,4.363) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.566,2.366) -- (11.538,2.509) -- (11.457,2.63) -- (11.336,2.711) -- (11.193,2.739) -- (11.05,2.711) -- (10.929,2.63) -- (10.848,2.509) -- (10.82,2.366) -- (10.848,2.223) -- (10.929,2.102) -- (11.05,2.021) -- (11.193,1.993) -- (11.336,2.021) -- (11.457,2.102) -- (11.538,2.223) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.053,2.281) -- (7.024,2.424) -- (6.943,2.545) -- (6.822,2.626) -- (6.679,2.655) -- (6.537,2.626) -- (6.415,2.545) -- (6.335,2.424) -- (6.306,2.281) -- (6.335,2.138) -- (6.415,2.017) -- (6.537,1.937) -- (6.679,1.908) -- (6.822,1.937) -- (6.943,2.017) -- (7.024,2.138) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.881,2.094) -- (1.852,2.237) -- (1.771,2.358) -- (1.65,2.439) -- (1.507,2.467) -- (1.365,2.439) -- (1.243,2.358) -- (1.163,2.237) -- (1.134,2.094) -- (1.163,1.951) -- (1.243,1.83) -- (1.365,1.749) -- (1.507,1.721) -- (1.65,1.749) -- (1.771,1.83) -- (1.852,1.951) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.18,1.806) -- (9.152,1.949) -- (9.071,2.07) -- (8.95,2.151) -- (8.807,2.18) -- (8.664,2.151) -- (8.543,2.07) -- (8.462,1.949) -- (8.434,1.806) -- (8.462,1.664) -- (8.543,1.542) -- (8.664,1.462) -- (8.807,1.433) -- (8.95,1.462) -- (9.071,1.542) -- (9.152,1.664) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.995,6.7) -- (10.114,6.719) -- (10.221,6.773) -- (10.306,6.858) -- (10.361,6.966) -- (10.38,7.085) -- (10.361,7.203) -- (10.306,7.311) -- (10.221,7.396) -- (10.114,7.45) -- (9.995,7.469) -- (9.995,7.469) -- (9.876,7.45) -- (9.769,7.396) -- (9.684,7.311) -- (9.629,7.203) -- (9.611,7.085) -- (9.629,6.966) -- (9.684,6.858) -- (9.769,6.773) -- (9.876,6.719) -- (9.995,6.7) -- cycle;
\node at (9.995,7.085) {\scalebox{1.618}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.236,5.52) -- (9.354,5.539) -- (9.462,5.594) -- (9.547,5.679) -- (9.601,5.786) -- (9.62,5.905) -- (9.601,6.024) -- (9.547,6.131) -- (9.462,6.216) -- (9.354,6.27) -- (9.236,6.289) -- (9.236,6.289) -- (9.117,6.27) -- (9.01,6.216) -- (8.924,6.131) -- (8.87,6.024) -- (8.851,5.905) -- (8.87,5.786) -- (8.924,5.679) -- (9.01,5.594) -- (9.117,5.539) -- (9.236,5.52) -- cycle;
\node at (9.236,5.905) {\scalebox{1.618}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.919,5.894) -- (3.038,5.913) -- (3.145,5.967) -- (3.23,6.052) -- (3.284,6.16) -- (3.303,6.278) -- (3.284,6.397) -- (3.23,6.504) -- (3.145,6.589) -- (3.038,6.644) -- (2.919,6.663) -- (2.919,6.663) -- (2.8,6.644) -- (2.693,6.589) -- (2.608,6.504) -- (2.553,6.397) -- (2.534,6.278) -- (2.553,6.16) -- (2.608,6.052) -- (2.693,5.967) -- (2.8,5.913) -- (2.919,5.894) -- cycle;
\node at (2.919,6.278) {\scalebox{1.618}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.149,5.218) -- (2.268,5.237) -- (2.375,5.292) -- (2.46,5.377) -- (2.514,5.484) -- (2.533,5.603) -- (2.514,5.722) -- (2.46,5.829) -- (2.375,5.914) -- (2.268,5.969) -- (2.149,5.987) -- (2.149,5.987) -- (2.03,5.969) -- (1.923,5.914) -- (1.838,5.829) -- (1.783,5.722) -- (1.764,5.603) -- (1.783,5.484) -- (1.838,5.377) -- (1.923,5.292) -- (2.03,5.237) -- (2.149,5.218) -- cycle;
\node at (2.149,5.603) {\scalebox{1.618}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.673,5.399) -- (11.792,5.418) -- (11.899,5.473) -- (11.984,5.558) -- (12.039,5.665) -- (12.058,5.784) -- (12.039,5.903) -- (11.984,6.01) -- (11.899,6.095) -- (11.792,6.149) -- (11.673,6.168) -- (11.673,6.168) -- (11.554,6.149) -- (11.447,6.095) -- (11.362,6.01) -- (11.307,5.903) -- (11.289,5.784) -- (11.307,5.665) -- (11.362,5.558) -- (11.447,5.473) -- (11.554,5.418) -- (11.673,5.399) -- cycle;
\node at (11.673,5.784) {\scalebox{1.618}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.551,6.875) -- (7.67,6.894) -- (7.777,6.948) -- (7.862,7.033) -- (7.917,7.14) -- (7.936,7.259) -- (7.917,7.378) -- (7.862,7.485) -- (7.777,7.57) -- (7.67,7.625) -- (7.551,7.644) -- (7.551,7.644) -- (7.432,7.625) -- (7.325,7.57) -- (7.24,7.485) -- (7.185,7.378) -- (7.167,7.259) -- (7.185,7.14) -- (7.24,7.033) -- (7.325,6.948) -- (7.432,6.894) -- (7.551,6.875) -- cycle;
\node at (7.551,7.259) {\scalebox{1.618}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.913,5.261) -- (7.031,5.28) -- (7.139,5.335) -- (7.224,5.42) -- (7.278,5.527) -- (7.297,5.646) -- (7.278,5.765) -- (7.224,5.872) -- (7.139,5.957) -- (7.031,6.012) -- (6.913,6.03) -- (6.913,6.03) -- (6.794,6.012) -- (6.687,5.957) -- (6.601,5.872) -- (6.547,5.765) -- (6.528,5.646) -- (6.547,5.527) -- (6.601,5.42) -- (6.687,5.335) -- (6.794,5.28) -- (6.913,5.261) -- cycle;
\node at (6.913,5.646) {\scalebox{1.618}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.212,6.805) -- (5.331,6.824) -- (5.438,6.878) -- (5.523,6.963) -- (5.577,7.071) -- (5.596,7.189) -- (5.577,7.308) -- (5.523,7.415) -- (5.438,7.501) -- (5.331,7.555) -- (5.212,7.574) -- (5.212,7.574) -- (5.093,7.555) -- (4.986,7.501) -- (4.901,7.415) -- (4.846,7.308) -- (4.827,7.189) -- (4.846,7.071) -- (4.901,6.963) -- (4.986,6.878) -- (5.093,6.824) -- (5.212,6.805) -- cycle;
\node at (5.212,7.189) {\scalebox{1.618}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.515,3.932) -- (10.634,3.95) -- (10.741,4.005) -- (10.826,4.09) -- (10.88,4.197) -- (10.899,4.316) -- (10.88,4.435) -- (10.826,4.542) -- (10.741,4.627) -- (10.634,4.682) -- (10.515,4.701) -- (10.515,4.701) -- (10.396,4.682) -- (10.289,4.627) -- (10.204,4.542) -- (10.149,4.435) -- (10.13,4.316) -- (10.149,4.197) -- (10.204,4.09) -- (10.289,4.005) -- (10.396,3.95) -- (10.515,3.932) -- cycle;
\node at (10.515,4.316) {\scalebox{1.618}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.555,2.914) -- (9.674,2.933) -- (9.781,2.987) -- (9.866,3.072) -- (9.921,3.18) -- (9.94,3.298) -- (9.921,3.417) -- (9.866,3.524) -- (9.781,3.609) -- (9.674,3.664) -- (9.555,3.683) -- (9.193,3.683) -- (9.074,3.664) -- (8.967,3.609) -- (8.882,3.524) -- (8.828,3.417) -- (8.809,3.298) -- (8.828,3.18) -- (8.882,3.072) -- (8.967,2.987) -- (9.074,2.933) -- (9.193,2.914) -- cycle;
\node at (9.374,3.298) {\scalebox{1.618}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.754,3.163) -- (11.873,3.182) -- (11.98,3.236) -- (12.065,3.321) -- (12.12,3.428) -- (12.138,3.547) -- (12.12,3.666) -- (12.065,3.773) -- (11.98,3.858) -- (11.873,3.913) -- (11.754,3.932) -- (11.392,3.932) -- (11.273,3.913) -- (11.166,3.858) -- (11.081,3.773) -- (11.026,3.666) -- (11.007,3.547) -- (11.026,3.428) -- (11.081,3.321) -- (11.166,3.236) -- (11.273,3.182) -- (11.392,3.163) -- cycle;
\node at (11.573,3.547) {\scalebox{1.618}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.116,3.004) -- (2.235,3.023) -- (2.342,3.078) -- (2.427,3.163) -- (2.482,3.27) -- (2.501,3.389) -- (2.482,3.508) -- (2.427,3.615) -- (2.342,3.7) -- (2.235,3.755) -- (2.116,3.773) -- (1.754,3.773) -- (1.635,3.755) -- (1.528,3.7) -- (1.443,3.615) -- (1.388,3.508) -- (1.37,3.389) -- (1.388,3.27) -- (1.443,3.163) -- (1.528,3.078) -- (1.635,3.023) -- (1.754,3.004) -- cycle;
\node at (1.935,3.389) {\scalebox{1.618}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.958,4.572) -- (8.077,4.591) -- (8.184,4.646) -- (8.269,4.731) -- (8.324,4.838) -- (8.343,4.957) -- (8.324,5.076) -- (8.269,5.183) -- (8.184,5.268) -- (8.077,5.323) -- (7.958,5.341) -- (7.596,5.341) -- (7.477,5.323) -- (7.37,5.268) -- (7.285,5.183) -- (7.23,5.076) -- (7.212,4.957) -- (7.23,4.838) -- (7.285,4.731) -- (7.37,4.646) -- (7.477,4.591) -- (7.596,4.572) -- cycle;
\node at (7.777,4.957) {\scalebox{1.618}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.022,2.609) -- (6.14,2.628) -- (6.248,2.683) -- (6.333,2.768) -- (6.387,2.875) -- (6.406,2.994) -- (6.387,3.113) -- (6.333,3.22) -- (6.248,3.305) -- (6.14,3.36) -- (6.022,3.378) -- (5.66,3.378) -- (5.541,3.36) -- (5.434,3.305) -- (5.349,3.22) -- (5.294,3.113) -- (5.275,2.994) -- (5.294,2.875) -- (5.349,2.768) -- (5.434,2.683) -- (5.541,2.628) -- (5.66,2.609) -- cycle;
\node at (5.841,2.994) {\scalebox{1.618}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.988,1.947) -- (8.107,1.966) -- (8.214,2.02) -- (8.299,2.105) -- (8.354,2.213) -- (8.373,2.331) -- (8.354,2.45) -- (8.299,2.557) -- (8.214,2.642) -- (8.107,2.697) -- (7.988,2.716) -- (7.626,2.716) -- (7.508,2.697) -- (7.4,2.642) -- (7.315,2.557) -- (7.261,2.45) -- (7.242,2.331) -- (7.261,2.213) -- (7.315,2.105) -- (7.4,2.02) -- (7.508,1.966) -- (7.626,1.947) -- cycle;
\node at (7.807,2.331) {\scalebox{1.618}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.246,1.426) -- (10.364,1.444) -- (10.472,1.499) -- (10.557,1.584) -- (10.611,1.691) -- (10.63,1.81) -- (10.611,1.929) -- (10.557,2.036) -- (10.472,2.121) -- (10.364,2.176) -- (10.246,2.195) -- (9.884,2.195) -- (9.765,2.176) -- (9.658,2.121) -- (9.573,2.036) -- (9.518,1.929) -- (9.499,1.81) -- (9.518,1.691) -- (9.573,1.584) -- (9.658,1.499) -- (9.765,1.444) -- (9.884,1.426) -- cycle;
\node at (10.065,1.81) {\scalebox{1.618}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (8.507,1.506) -- (9.107,1.506) -- (9.107,2.106) -- (8.507,2.106) -- cycle;
\node at (8.807,1.806) {\scalebox{1.618}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.805,0.65) -- (8.405,0.65) -- (8.405,1.25) -- (7.805,1.25) -- cycle;
\node at (8.105,0.95) {\scalebox{1.618}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.037,3.428) -- (7.637,3.428) -- (7.637,4.028) -- (7.037,4.028) -- cycle;
\node at (7.337,3.728) {\scalebox{1.618}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.537,4.965) -- (10.137,4.965) -- (10.137,5.565) -- (9.537,5.565) -- cycle;
\node at (9.837,5.265) {\scalebox{1.618}{\small d}};
\node at (9.017,7.318) {\scalebox{1.618}{\small A}};
\node at (1.423,7.072) {\scalebox{1.618}{\small B}};
\node at (11.921,7.045) {\scalebox{1.618}{\small C}};
\node at (6.431,7.35) {\scalebox{1.618}{\small D}};
\node at (3.411,7.003) {\scalebox{1.618}{\small E}};
\node at (9.543,4.867) {\scalebox{1.618}{\small F}};
\node at (12.005,4.784) {\scalebox{1.618}{\small G}};
\node at (0.765,4.629) {\scalebox{1.618}{\small H}};
\node at (6.371,3.99) {\scalebox{1.618}{\small I}};
\node at (11.739,2.072) {\scalebox{1.618}{\small J}};
\node at (6.636,1.663) {\scalebox{1.618}{\small K}};
\node at (0.95,1.822) {\scalebox{1.618}{\small L}};
\node at (9.154,1.293) {\scalebox{1.618}{\small M}};
\draw[line width=0.5pt] (12.685,5.891) -- (12.685,6.991);
\draw[line width=0.5pt] (12.454,6.606) -- (12.685,6.991) -- (12.916,6.606);
\draw[line width=0.5pt] (12.382,6.441) -- (12.987,6.441);
\node at (12.685,7.376) {\scalebox{1.618}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, E, G, H, I, M & 6 \\ \hline
clairs & B, C, D, F, J, K, L & 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{Non plus.} Avec un carrefour clair de plus, la tournée doit \textbf{partir et arriver} sur un carrefour clair. Or l'école est en \textbf{M}, qui est sombre.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Une tournée existe pourtant dans ce quartier : il suffirait d'installer l'école sur un carrefour clair. Ce n'est pas le plan qui est en cause, c'est l'endroit d'où l'on part.}\par\addvspace{0.4em}
\end{document}
