\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 388}

\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=388}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0388}
\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=0.6074, line join=round, line cap=round]
\draw[line width=0.5pt] (3.769,6.549) -- (3.947,6.511) -- (4.125,6.477) -- (4.304,6.447) -- (4.484,6.421) -- (4.665,6.4) -- (4.846,6.382) -- (5.027,6.369) -- (5.209,6.361) -- (5.39,6.356) -- (5.572,6.356) -- (5.754,6.36) -- (5.935,6.369) -- (6.116,6.381) -- (6.297,6.398);
\draw[line width=0.5pt] (6.297,6.398) -- (6.495,6.394) -- (6.693,6.385) -- (6.89,6.371) -- (7.088,6.353) -- (7.285,6.33) -- (7.481,6.303) -- (7.676,6.271) -- (7.871,6.234) -- (8.065,6.193) -- (8.258,6.147) -- (8.449,6.097) -- (8.64,6.042) -- (8.829,5.983) -- (9.016,5.919);
\draw[line width=0.5pt] (6.297,6.398) -- (6.256,6.218) -- (6.22,6.037) -- (6.188,5.855) -- (6.16,5.672) -- (6.136,5.489) -- (6.117,5.305) -- (6.102,5.121) -- (6.091,4.936) -- (6.085,4.752) -- (6.083,4.567) -- (6.085,4.382) -- (6.092,4.197) -- (6.103,4.013) -- (6.119,3.829);
\draw[line width=0.5pt] (2.027,6.155) -- (2.146,6.202) -- (2.267,6.246) -- (2.388,6.287) -- (2.51,6.326) -- (2.633,6.361) -- (2.757,6.394) -- (2.882,6.424) -- (3.007,6.451) -- (3.133,6.475) -- (3.26,6.496) -- (3.387,6.514) -- (3.514,6.529) -- (3.642,6.54) -- (3.769,6.549);
\draw[line width=0.5pt] (2.027,6.155) -- (1.994,6.015) -- (1.963,5.874) -- (1.936,5.733) -- (1.913,5.591) -- (1.892,5.449) -- (1.875,5.306) -- (1.862,5.163) -- (1.851,5.02) -- (1.844,4.876) -- (1.841,4.732) -- (1.841,4.588) -- (1.844,4.445) -- (1.851,4.301) -- (1.861,4.158);
\draw[line width=0.5pt] (9.016,5.919) -- (8.998,5.754) -- (8.977,5.59) -- (8.951,5.426) -- (8.922,5.263) -- (8.888,5.101) -- (8.851,4.939) -- (8.81,4.779) -- (8.766,4.619) -- (8.717,4.46) -- (8.665,4.303) -- (8.61,4.147) -- (8.55,3.992) -- (8.487,3.839) -- (8.421,3.687);
\draw[line width=0.5pt] (1.861,4.158) -- (2.017,4.121) -- (2.175,4.088) -- (2.333,4.059) -- (2.492,4.033) -- (2.651,4.012) -- (2.811,3.994) -- (2.972,3.979) -- (3.133,3.969) -- (3.293,3.962) -- (3.454,3.959) -- (3.615,3.96) -- (3.776,3.965) -- (3.937,3.973) -- (4.097,3.986);
\draw[line width=0.5pt] (1.861,4.158) -- (1.848,3.976) -- (1.831,3.795) -- (1.809,3.615) -- (1.783,3.435) -- (1.753,3.255) -- (1.719,3.076) -- (1.681,2.898) -- (1.638,2.721) -- (1.591,2.545) -- (1.541,2.371) -- (1.486,2.197) -- (1.427,2.025) -- (1.364,1.854) -- (1.297,1.685);
\draw[line width=0.5pt] (4.097,3.986) -- (4.239,3.952) -- (4.381,3.923) -- (4.524,3.896) -- (4.668,3.873) -- (4.812,3.853) -- (4.956,3.837) -- (5.101,3.824) -- (5.247,3.814) -- (5.392,3.808) -- (5.538,3.806) -- (5.683,3.806) -- (5.828,3.81) -- (5.974,3.818) -- (6.119,3.829);
\draw[line width=0.5pt] (4.097,3.986) -- (4.099,3.805) -- (4.097,3.625) -- (4.091,3.445) -- (4.08,3.265) -- (4.065,3.085) -- (4.046,2.906) -- (4.023,2.727) -- (3.995,2.549) -- (3.963,2.372) -- (3.928,2.195) -- (3.888,2.019) -- (3.843,1.844) -- (3.795,1.671) -- (3.743,1.498);
\draw[line width=0.5pt] (6.119,3.829) -- (6.28,3.794) -- (6.443,3.762) -- (6.606,3.735) -- (6.77,3.711) -- (6.934,3.691) -- (7.099,3.675) -- (7.264,3.663) -- (7.429,3.655) -- (7.595,3.651) -- (7.76,3.65) -- (7.926,3.653) -- (8.091,3.661) -- (8.256,3.672) -- (8.421,3.687);
\draw[line width=0.5pt] (6.119,3.829) -- (6.151,3.668) -- (6.179,3.506) -- (6.204,3.343) -- (6.225,3.18) -- (6.241,3.016) -- (6.254,2.852) -- (6.264,2.687) -- (6.269,2.523) -- (6.27,2.358) -- (6.268,2.193) -- (6.262,2.029) -- (6.252,1.865) -- (6.238,1.701) -- (6.22,1.537);
\draw[line width=0.5pt] (8.421,3.687) -- (8.628,3.677) -- (8.835,3.671) -- (9.042,3.67) -- (9.25,3.674) -- (9.457,3.683) -- (9.664,3.697) -- (9.871,3.716) -- (10.077,3.74) -- (10.283,3.768) -- (10.487,3.801) -- (10.691,3.84) -- (10.894,3.882) -- (11.096,3.93) -- (11.297,3.982);
\draw[line width=0.5pt] (1.297,1.685) -- (1.472,1.699) -- (1.648,1.708) -- (1.824,1.713) -- (2.0,1.714) -- (2.176,1.711) -- (2.352,1.704) -- (2.527,1.693) -- (2.703,1.677) -- (2.878,1.658) -- (3.052,1.634) -- (3.226,1.606) -- (3.399,1.574) -- (3.572,1.538) -- (3.743,1.498);
\draw[line width=0.5pt] (3.743,1.498) -- (3.919,1.474) -- (4.095,1.454) -- (4.272,1.438) -- (4.45,1.426) -- (4.627,1.418) -- (4.805,1.415) -- (4.983,1.416) -- (5.161,1.42) -- (5.338,1.43) -- (5.515,1.443) -- (5.692,1.46) -- (5.869,1.482) -- (6.045,1.507) -- (6.22,1.537);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.929,6.549) -- (3.908,6.629) -- (3.849,6.688) -- (3.769,6.709) -- (3.689,6.688) -- (3.631,6.629) -- (3.609,6.549) -- (3.631,6.469) -- (3.689,6.411) -- (3.769,6.389) -- (3.849,6.411) -- (3.908,6.469) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.457,6.398) -- (6.436,6.478) -- (6.377,6.537) -- (6.297,6.558) -- (6.217,6.537) -- (6.159,6.478) -- (6.137,6.398) -- (6.159,6.318) -- (6.217,6.26) -- (6.297,6.238) -- (6.377,6.26) -- (6.436,6.318) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.187,6.155) -- (2.166,6.235) -- (2.107,6.293) -- (2.027,6.315) -- (1.947,6.293) -- (1.889,6.235) -- (1.867,6.155) -- (1.889,6.075) -- (1.947,6.016) -- (2.027,5.995) -- (2.107,6.016) -- (2.166,6.075) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.176,5.919) -- (9.155,5.999) -- (9.096,6.058) -- (9.016,6.079) -- (8.936,6.058) -- (8.878,5.999) -- (8.856,5.919) -- (8.878,5.839) -- (8.936,5.78) -- (9.016,5.759) -- (9.096,5.78) -- (9.155,5.839) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.021,4.158) -- (1.999,4.238) -- (1.941,4.296) -- (1.861,4.318) -- (1.781,4.296) -- (1.722,4.238) -- (1.701,4.158) -- (1.722,4.078) -- (1.781,4.019) -- (1.861,3.998) -- (1.941,4.019) -- (1.999,4.078) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.257,3.986) -- (4.236,4.066) -- (4.177,4.124) -- (4.097,4.146) -- (4.017,4.124) -- (3.959,4.066) -- (3.937,3.986) -- (3.959,3.906) -- (4.017,3.847) -- (4.097,3.826) -- (4.177,3.847) -- (4.236,3.906) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.457,3.982) -- (11.435,4.062) -- (11.377,4.121) -- (11.297,4.142) -- (11.217,4.121) -- (11.158,4.062) -- (11.137,3.982) -- (11.158,3.902) -- (11.217,3.844) -- (11.297,3.822) -- (11.377,3.844) -- (11.435,3.902) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.279,3.829) -- (6.257,3.909) -- (6.199,3.968) -- (6.119,3.989) -- (6.039,3.968) -- (5.98,3.909) -- (5.959,3.829) -- (5.98,3.749) -- (6.039,3.69) -- (6.119,3.669) -- (6.199,3.69) -- (6.257,3.749) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.581,3.687) -- (8.559,3.767) -- (8.501,3.826) -- (8.421,3.847) -- (8.341,3.826) -- (8.282,3.767) -- (8.261,3.687) -- (8.282,3.607) -- (8.341,3.549) -- (8.421,3.527) -- (8.501,3.549) -- (8.559,3.607) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.457,1.685) -- (1.435,1.765) -- (1.377,1.824) -- (1.297,1.845) -- (1.217,1.824) -- (1.158,1.765) -- (1.137,1.685) -- (1.158,1.605) -- (1.217,1.547) -- (1.297,1.525) -- (1.377,1.547) -- (1.435,1.605) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.38,1.537) -- (6.358,1.617) -- (6.3,1.676) -- (6.22,1.697) -- (6.14,1.676) -- (6.081,1.617) -- (6.06,1.537) -- (6.081,1.457) -- (6.14,1.399) -- (6.22,1.377) -- (6.3,1.399) -- (6.358,1.457) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.903,1.498) -- (3.881,1.578) -- (3.823,1.637) -- (3.743,1.658) -- (3.663,1.637) -- (3.604,1.578) -- (3.583,1.498) -- (3.604,1.418) -- (3.663,1.36) -- (3.743,1.338) -- (3.823,1.36) -- (3.881,1.418) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.051,6.369) -- (5.172,6.388) -- (5.282,6.444) -- (5.369,6.531) -- (5.425,6.641) -- (5.444,6.763) -- (5.425,6.884) -- (5.369,6.994) -- (5.282,7.081) -- (5.172,7.137) -- (5.051,7.157) -- (5.051,7.157) -- (4.929,7.137) -- (4.819,7.081) -- (4.732,6.994) -- (4.676,6.884) -- (4.657,6.763) -- (4.676,6.641) -- (4.732,6.531) -- (4.819,6.444) -- (4.929,6.388) -- (5.051,6.369) -- cycle;
\node at (5.051,6.763) {\scalebox{1.646}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.745,6.265) -- (7.866,6.284) -- (7.976,6.34) -- (8.063,6.427) -- (8.119,6.537) -- (8.139,6.659) -- (8.119,6.78) -- (8.063,6.89) -- (7.976,6.977) -- (7.866,7.033) -- (7.745,7.053) -- (7.745,7.053) -- (7.623,7.033) -- (7.513,6.977) -- (7.426,6.89) -- (7.37,6.78) -- (7.351,6.659) -- (7.37,6.537) -- (7.426,6.427) -- (7.513,6.34) -- (7.623,6.284) -- (7.745,6.265) -- cycle;
\node at (7.745,6.659) {\scalebox{1.646}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.495,4.7) -- (6.617,4.719) -- (6.726,4.775) -- (6.814,4.862) -- (6.869,4.972) -- (6.889,5.094) -- (6.869,5.215) -- (6.814,5.325) -- (6.726,5.412) -- (6.617,5.468) -- (6.495,5.488) -- (6.495,5.488) -- (6.373,5.468) -- (6.263,5.412) -- (6.176,5.325) -- (6.12,5.215) -- (6.101,5.094) -- (6.12,4.972) -- (6.176,4.862) -- (6.263,4.775) -- (6.373,4.719) -- (6.495,4.7) -- cycle;
\node at (6.495,5.094) {\scalebox{1.646}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.969,5.646) -- (3.091,5.665) -- (3.201,5.721) -- (3.288,5.808) -- (3.344,5.918) -- (3.363,6.04) -- (3.344,6.161) -- (3.288,6.271) -- (3.201,6.358) -- (3.091,6.414) -- (2.969,6.434) -- (2.969,6.434) -- (2.847,6.414) -- (2.737,6.358) -- (2.65,6.271) -- (2.594,6.161) -- (2.575,6.04) -- (2.594,5.918) -- (2.65,5.808) -- (2.737,5.721) -- (2.847,5.665) -- (2.969,5.646) -- cycle;
\node at (2.969,6.04) {\scalebox{1.646}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.254,4.736) -- (2.376,4.756) -- (2.486,4.812) -- (2.573,4.899) -- (2.629,5.009) -- (2.648,5.13) -- (2.629,5.252) -- (2.573,5.362) -- (2.486,5.449) -- (2.376,5.505) -- (2.254,5.524) -- (2.254,5.524) -- (2.132,5.505) -- (2.023,5.449) -- (1.935,5.362) -- (1.88,5.252) -- (1.86,5.13) -- (1.88,5.009) -- (1.935,4.899) -- (2.023,4.812) -- (2.132,4.756) -- (2.254,4.736) -- cycle;
\node at (2.254,5.13) {\scalebox{1.646}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.43,4.486) -- (8.552,4.505) -- (8.661,4.561) -- (8.748,4.649) -- (8.804,4.758) -- (8.824,4.88) -- (8.804,5.002) -- (8.748,5.112) -- (8.661,5.199) -- (8.552,5.255) -- (8.43,5.274) -- (8.43,5.274) -- (8.308,5.255) -- (8.198,5.199) -- (8.111,5.112) -- (8.055,5.002) -- (8.036,4.88) -- (8.055,4.758) -- (8.111,4.649) -- (8.198,4.561) -- (8.308,4.505) -- (8.43,4.486) -- cycle;
\node at (8.43,4.88) {\scalebox{1.646}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.915,2.846) -- (3.037,2.866) -- (3.146,2.921) -- (3.234,3.009) -- (3.29,3.118) -- (3.309,3.24) -- (3.29,3.362) -- (3.234,3.472) -- (3.146,3.559) -- (3.037,3.615) -- (2.915,3.634) -- (2.915,3.634) -- (2.793,3.615) -- (2.683,3.559) -- (2.596,3.472) -- (2.54,3.362) -- (2.521,3.24) -- (2.54,3.118) -- (2.596,3.009) -- (2.683,2.921) -- (2.793,2.866) -- (2.915,2.846) -- cycle;
\node at (2.915,3.24) {\scalebox{1.646}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.065,2.417) -- (2.186,2.436) -- (2.296,2.492) -- (2.383,2.579) -- (2.439,2.689) -- (2.459,2.811) -- (2.439,2.932) -- (2.383,3.042) -- (2.296,3.129) -- (2.186,3.185) -- (2.065,3.205) -- (2.065,3.205) -- (1.943,3.185) -- (1.833,3.129) -- (1.746,3.042) -- (1.69,2.932) -- (1.671,2.811) -- (1.69,2.689) -- (1.746,2.579) -- (1.833,2.492) -- (1.943,2.436) -- (2.065,2.417) -- cycle;
\node at (2.065,2.811) {\scalebox{1.646}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.132,3.823) -- (5.254,3.842) -- (5.363,3.898) -- (5.451,3.985) -- (5.506,4.095) -- (5.526,4.217) -- (5.506,4.338) -- (5.451,4.448) -- (5.363,4.535) -- (5.254,4.591) -- (5.132,4.611) -- (5.132,4.611) -- (5.01,4.591) -- (4.9,4.535) -- (4.813,4.448) -- (4.757,4.338) -- (4.738,4.217) -- (4.757,4.095) -- (4.813,3.985) -- (4.9,3.898) -- (5.01,3.842) -- (5.132,3.823) -- cycle;
\node at (5.132,4.217) {\scalebox{1.646}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.598,2.278) -- (4.72,2.297) -- (4.83,2.353) -- (4.917,2.44) -- (4.973,2.55) -- (4.992,2.672) -- (4.973,2.793) -- (4.917,2.903) -- (4.83,2.99) -- (4.72,3.046) -- (4.598,3.066) -- (4.227,3.066) -- (4.106,3.046) -- (3.996,2.99) -- (3.909,2.903) -- (3.853,2.793) -- (3.833,2.672) -- (3.853,2.55) -- (3.909,2.44) -- (3.996,2.353) -- (4.106,2.297) -- (4.227,2.278) -- cycle;
\node at (4.413,2.672) {\scalebox{1.646}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.469,3.589) -- (7.59,3.608) -- (7.7,3.664) -- (7.787,3.751) -- (7.843,3.861) -- (7.863,3.983) -- (7.843,4.104) -- (7.787,4.214) -- (7.7,4.301) -- (7.59,4.357) -- (7.469,4.376) -- (7.098,4.376) -- (6.976,4.357) -- (6.867,4.301) -- (6.779,4.214) -- (6.723,4.104) -- (6.704,3.983) -- (6.723,3.861) -- (6.779,3.751) -- (6.867,3.664) -- (6.976,3.608) -- (7.098,3.589) -- cycle;
\node at (7.283,3.983) {\scalebox{1.646}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.055,2.276) -- (6.177,2.295) -- (6.287,2.351) -- (6.374,2.438) -- (6.43,2.548) -- (6.449,2.67) -- (6.43,2.792) -- (6.374,2.902) -- (6.287,2.989) -- (6.177,3.045) -- (6.055,3.064) -- (5.685,3.064) -- (5.563,3.045) -- (5.453,2.989) -- (5.366,2.902) -- (5.31,2.792) -- (5.291,2.67) -- (5.31,2.548) -- (5.366,2.438) -- (5.453,2.351) -- (5.563,2.295) -- (5.685,2.276) -- cycle;
\node at (5.87,2.67) {\scalebox{1.646}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.526,2.985) -- (10.648,3.004) -- (10.758,3.06) -- (10.845,3.147) -- (10.901,3.257) -- (10.92,3.379) -- (10.901,3.5) -- (10.845,3.61) -- (10.758,3.697) -- (10.648,3.753) -- (10.526,3.773) -- (10.156,3.773) -- (10.034,3.753) -- (9.924,3.697) -- (9.837,3.61) -- (9.781,3.5) -- (9.762,3.379) -- (9.781,3.257) -- (9.837,3.147) -- (9.924,3.06) -- (10.034,3.004) -- (10.156,2.985) -- cycle;
\node at (10.341,3.379) {\scalebox{1.646}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.516,0.917) -- (2.638,0.936) -- (2.748,0.992) -- (2.835,1.079) -- (2.891,1.189) -- (2.91,1.311) -- (2.891,1.432) -- (2.835,1.542) -- (2.748,1.629) -- (2.638,1.685) -- (2.516,1.705) -- (2.146,1.705) -- (2.024,1.685) -- (1.914,1.629) -- (1.827,1.542) -- (1.771,1.432) -- (1.752,1.311) -- (1.771,1.189) -- (1.827,1.079) -- (1.914,0.992) -- (2.024,0.936) -- (2.146,0.917) -- cycle;
\node at (2.331,1.311) {\scalebox{1.646}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.174,0.628) -- (5.296,0.647) -- (5.406,0.703) -- (5.493,0.79) -- (5.549,0.9) -- (5.568,1.022) -- (5.549,1.143) -- (5.493,1.253) -- (5.406,1.34) -- (5.296,1.396) -- (5.174,1.416) -- (4.804,1.416) -- (4.682,1.396) -- (4.572,1.34) -- (4.485,1.253) -- (4.429,1.143) -- (4.41,1.022) -- (4.429,0.9) -- (4.485,0.79) -- (4.572,0.703) -- (4.682,0.647) -- (4.804,0.628) -- cycle;
\node at (4.989,1.022) {\scalebox{1.646}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (3.797,3.686) -- (4.397,3.686) -- (4.397,4.286) -- (3.797,4.286) -- cycle;
\node at (4.097,3.986) {\scalebox{1.646}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (7.379,2.119) -- (7.979,2.119) -- (7.979,2.719) -- (7.379,2.719) -- cycle;
\node at (7.679,2.419) {\scalebox{1.646}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (9.664,4.799) -- (10.264,4.799) -- (10.264,5.399) -- (9.664,5.399) -- cycle;
\node at (9.964,5.099) {\scalebox{1.646}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.775,5.51) -- (7.375,5.51) -- (7.375,6.11) -- (6.775,6.11) -- cycle;
\node at (7.075,5.81) {\scalebox{1.646}{\small d}};
\node at (3.432,7.07) {\scalebox{1.646}{\small A}};
\node at (6.535,6.97) {\scalebox{1.646}{\small B}};
\node at (1.499,6.479) {\scalebox{1.646}{\small C}};
\node at (9.573,6.192) {\scalebox{1.646}{\small D}};
\node at (1.241,4.165) {\scalebox{1.646}{\small E}};
\node at (3.481,3.922) {\scalebox{1.646}{\small F}};
\node at (11.916,3.969) {\scalebox{1.646}{\small G}};
\node at (6.7,3.613) {\scalebox{1.646}{\small H}};
\node at (9.035,3.602) {\scalebox{1.646}{\small I}};
\node at (0.765,1.366) {\scalebox{1.646}{\small J}};
\node at (6.418,0.95) {\scalebox{1.646}{\small K}};
\node at (3.419,0.97) {\scalebox{1.646}{\small L}};
\draw[line width=0.5pt] (12.597,5.599) -- (12.597,6.699);
\draw[line width=0.5pt] (12.366,6.314) -- (12.597,6.699) -- (12.828,6.314);
\draw[line width=0.5pt] (12.294,6.149) -- (12.899,6.149);
\node at (12.597,7.084) {\scalebox{1.646}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour F, 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 de l'Église & Ⓑ Carrefour Saint-Roch & Ⓚ Place du Vieux-Puits \\ \hline
① rue du Pont & ⑩ rue des Écoles & Ⓒ Place de la Fontaine & Ⓛ Carrefour du Moulin \\ \hline
② avenue de la Gare & ⑪ quai aux Fleurs & Ⓓ Place du Marché & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ chemin de Ronde & ⑫ rue du Four & Ⓔ Carrefour de l'Hôpital & \textbf{a} l'école \\ \hline
④ rue des Lilas & ⑬ impasse des Roses & Ⓕ Carrefour du Lavoir & \textbf{b} la salle des fêtes \\ \hline
⑤ rue du Puits & ⑭ allée des Tilleuls & Ⓖ Place de l'Église & \textbf{c} le stade \\ \hline
⑥ sentier des Prés & ⑮ chemin des Vignes & Ⓗ Carrefour des Tilleuls & \textbf{d} la boulangerie \\ \hline
⑦ venelle du Chat & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour de la Mairie &  \\ \hline
⑧ rue du Stade & Ⓐ Place aux Herbes & Ⓙ Place des Tanneurs &  \\ \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=0.6223, line join=round, line cap=round]
\draw[line width=0.5pt] (3.769,6.549) -- (3.947,6.511) -- (4.125,6.477) -- (4.304,6.447) -- (4.484,6.421) -- (4.665,6.4) -- (4.846,6.382) -- (5.027,6.369) -- (5.209,6.361) -- (5.39,6.356) -- (5.572,6.356) -- (5.754,6.36) -- (5.935,6.369) -- (6.116,6.381) -- (6.297,6.398);
\draw[line width=0.5pt] (6.297,6.398) -- (6.495,6.394) -- (6.693,6.385) -- (6.89,6.371) -- (7.088,6.353) -- (7.285,6.33) -- (7.481,6.303) -- (7.676,6.271) -- (7.871,6.234) -- (8.065,6.193) -- (8.258,6.147) -- (8.449,6.097) -- (8.64,6.042) -- (8.829,5.983) -- (9.016,5.919);
\draw[line width=0.5pt] (6.297,6.398) -- (6.256,6.218) -- (6.22,6.037) -- (6.188,5.855) -- (6.16,5.672) -- (6.136,5.489) -- (6.117,5.305) -- (6.102,5.121) -- (6.091,4.936) -- (6.085,4.752) -- (6.083,4.567) -- (6.085,4.382) -- (6.092,4.197) -- (6.103,4.013) -- (6.119,3.829);
\draw[line width=0.5pt] (2.027,6.155) -- (2.146,6.202) -- (2.267,6.246) -- (2.388,6.287) -- (2.51,6.326) -- (2.633,6.361) -- (2.757,6.394) -- (2.882,6.424) -- (3.007,6.451) -- (3.133,6.475) -- (3.26,6.496) -- (3.387,6.514) -- (3.514,6.529) -- (3.642,6.54) -- (3.769,6.549);
\draw[line width=0.5pt] (2.027,6.155) -- (1.994,6.015) -- (1.963,5.874) -- (1.936,5.733) -- (1.913,5.591) -- (1.892,5.449) -- (1.875,5.306) -- (1.862,5.163) -- (1.851,5.02) -- (1.844,4.876) -- (1.841,4.732) -- (1.841,4.588) -- (1.844,4.445) -- (1.851,4.301) -- (1.861,4.158);
\draw[line width=0.5pt] (9.016,5.919) -- (8.998,5.754) -- (8.977,5.59) -- (8.951,5.426) -- (8.922,5.263) -- (8.888,5.101) -- (8.851,4.939) -- (8.81,4.779) -- (8.766,4.619) -- (8.717,4.46) -- (8.665,4.303) -- (8.61,4.147) -- (8.55,3.992) -- (8.487,3.839) -- (8.421,3.687);
\draw[line width=0.5pt] (1.861,4.158) -- (2.017,4.121) -- (2.175,4.088) -- (2.333,4.059) -- (2.492,4.033) -- (2.651,4.012) -- (2.811,3.994) -- (2.972,3.979) -- (3.133,3.969) -- (3.293,3.962) -- (3.454,3.959) -- (3.615,3.96) -- (3.776,3.965) -- (3.937,3.973) -- (4.097,3.986);
\draw[line width=0.5pt] (1.861,4.158) -- (1.848,3.976) -- (1.831,3.795) -- (1.809,3.615) -- (1.783,3.435) -- (1.753,3.255) -- (1.719,3.076) -- (1.681,2.898) -- (1.638,2.721) -- (1.591,2.545) -- (1.541,2.371) -- (1.486,2.197) -- (1.427,2.025) -- (1.364,1.854) -- (1.297,1.685);
\draw[line width=0.5pt] (4.097,3.986) -- (4.239,3.952) -- (4.381,3.923) -- (4.524,3.896) -- (4.668,3.873) -- (4.812,3.853) -- (4.956,3.837) -- (5.101,3.824) -- (5.247,3.814) -- (5.392,3.808) -- (5.538,3.806) -- (5.683,3.806) -- (5.828,3.81) -- (5.974,3.818) -- (6.119,3.829);
\draw[line width=0.5pt] (4.097,3.986) -- (4.099,3.805) -- (4.097,3.625) -- (4.091,3.445) -- (4.08,3.265) -- (4.065,3.085) -- (4.046,2.906) -- (4.023,2.727) -- (3.995,2.549) -- (3.963,2.372) -- (3.928,2.195) -- (3.888,2.019) -- (3.843,1.844) -- (3.795,1.671) -- (3.743,1.498);
\draw[line width=0.5pt] (6.119,3.829) -- (6.28,3.794) -- (6.443,3.762) -- (6.606,3.735) -- (6.77,3.711) -- (6.934,3.691) -- (7.099,3.675) -- (7.264,3.663) -- (7.429,3.655) -- (7.595,3.651) -- (7.76,3.65) -- (7.926,3.653) -- (8.091,3.661) -- (8.256,3.672) -- (8.421,3.687);
\draw[line width=0.5pt] (6.119,3.829) -- (6.151,3.668) -- (6.179,3.506) -- (6.204,3.343) -- (6.225,3.18) -- (6.241,3.016) -- (6.254,2.852) -- (6.264,2.687) -- (6.269,2.523) -- (6.27,2.358) -- (6.268,2.193) -- (6.262,2.029) -- (6.252,1.865) -- (6.238,1.701) -- (6.22,1.537);
\draw[line width=0.5pt] (8.421,3.687) -- (8.628,3.677) -- (8.835,3.671) -- (9.042,3.67) -- (9.25,3.674) -- (9.457,3.683) -- (9.664,3.697) -- (9.871,3.716) -- (10.077,3.74) -- (10.283,3.768) -- (10.487,3.801) -- (10.691,3.84) -- (10.894,3.882) -- (11.096,3.93) -- (11.297,3.982);
\draw[line width=0.5pt] (1.297,1.685) -- (1.472,1.699) -- (1.648,1.708) -- (1.824,1.713) -- (2.0,1.714) -- (2.176,1.711) -- (2.352,1.704) -- (2.527,1.693) -- (2.703,1.677) -- (2.878,1.658) -- (3.052,1.634) -- (3.226,1.606) -- (3.399,1.574) -- (3.572,1.538) -- (3.743,1.498);
\draw[line width=0.5pt] (3.743,1.498) -- (3.919,1.474) -- (4.095,1.454) -- (4.272,1.438) -- (4.45,1.426) -- (4.627,1.418) -- (4.805,1.415) -- (4.983,1.416) -- (5.161,1.42) -- (5.338,1.43) -- (5.515,1.443) -- (5.692,1.46) -- (5.869,1.482) -- (6.045,1.507) -- (6.22,1.537);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.142,6.549) -- (4.114,6.692) -- (4.033,6.813) -- (3.912,6.894) -- (3.769,6.923) -- (3.626,6.894) -- (3.505,6.813) -- (3.424,6.692) -- (3.396,6.549) -- (3.424,6.407) -- (3.505,6.285) -- (3.626,6.205) -- (3.769,6.176) -- (3.912,6.205) -- (4.033,6.285) -- (4.114,6.407) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.67,6.398) -- (6.642,6.541) -- (6.561,6.662) -- (6.44,6.743) -- (6.297,6.771) -- (6.154,6.743) -- (6.033,6.662) -- (5.952,6.541) -- (5.924,6.398) -- (5.952,6.255) -- (6.033,6.134) -- (6.154,6.053) -- (6.297,6.025) -- (6.44,6.053) -- (6.561,6.134) -- (6.642,6.255) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.4,6.155) -- (2.372,6.297) -- (2.291,6.418) -- (2.17,6.499) -- (2.027,6.528) -- (1.884,6.499) -- (1.763,6.418) -- (1.682,6.297) -- (1.654,6.155) -- (1.682,6.012) -- (1.763,5.891) -- (1.884,5.81) -- (2.027,5.781) -- (2.17,5.81) -- (2.291,5.891) -- (2.372,6.012) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.389,5.919) -- (9.361,6.062) -- (9.28,6.183) -- (9.159,6.264) -- (9.016,6.292) -- (8.873,6.264) -- (8.752,6.183) -- (8.671,6.062) -- (8.643,5.919) -- (8.671,5.776) -- (8.752,5.655) -- (8.873,5.574) -- (9.016,5.546) -- (9.159,5.574) -- (9.28,5.655) -- (9.361,5.776) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.234,4.158) -- (2.206,4.301) -- (2.125,4.422) -- (2.004,4.503) -- (1.861,4.531) -- (1.718,4.503) -- (1.597,4.422) -- (1.516,4.301) -- (1.488,4.158) -- (1.516,4.015) -- (1.597,3.894) -- (1.718,3.813) -- (1.861,3.785) -- (2.004,3.813) -- (2.125,3.894) -- (2.206,4.015) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.47,3.986) -- (4.442,4.128) -- (4.361,4.249) -- (4.24,4.33) -- (4.097,4.359) -- (3.954,4.33) -- (3.833,4.249) -- (3.752,4.128) -- (3.724,3.986) -- (3.752,3.843) -- (3.833,3.722) -- (3.954,3.641) -- (4.097,3.612) -- (4.24,3.641) -- (4.361,3.722) -- (4.442,3.843) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.67,3.982) -- (11.641,4.125) -- (11.561,4.246) -- (11.439,4.327) -- (11.297,4.356) -- (11.154,4.327) -- (11.033,4.246) -- (10.952,4.125) -- (10.923,3.982) -- (10.952,3.84) -- (11.033,3.718) -- (11.154,3.638) -- (11.297,3.609) -- (11.439,3.638) -- (11.561,3.718) -- (11.641,3.84) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.492,3.829) -- (6.463,3.972) -- (6.383,4.093) -- (6.261,4.174) -- (6.119,4.202) -- (5.976,4.174) -- (5.855,4.093) -- (5.774,3.972) -- (5.745,3.829) -- (5.774,3.686) -- (5.855,3.565) -- (5.976,3.484) -- (6.119,3.456) -- (6.261,3.484) -- (6.383,3.565) -- (6.463,3.686) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.794,3.687) -- (8.765,3.83) -- (8.684,3.951) -- (8.563,4.032) -- (8.421,4.06) -- (8.278,4.032) -- (8.157,3.951) -- (8.076,3.83) -- (8.047,3.687) -- (8.076,3.544) -- (8.157,3.423) -- (8.278,3.342) -- (8.421,3.314) -- (8.563,3.342) -- (8.684,3.423) -- (8.765,3.544) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.67,1.685) -- (1.641,1.828) -- (1.561,1.949) -- (1.439,2.03) -- (1.297,2.058) -- (1.154,2.03) -- (1.033,1.949) -- (0.952,1.828) -- (0.923,1.685) -- (0.952,1.542) -- (1.033,1.421) -- (1.154,1.34) -- (1.297,1.312) -- (1.439,1.34) -- (1.561,1.421) -- (1.641,1.542) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.593,1.537) -- (6.564,1.68) -- (6.484,1.801) -- (6.362,1.882) -- (6.22,1.91) -- (6.077,1.882) -- (5.956,1.801) -- (5.875,1.68) -- (5.846,1.537) -- (5.875,1.394) -- (5.956,1.273) -- (6.077,1.192) -- (6.22,1.164) -- (6.362,1.192) -- (6.484,1.273) -- (6.564,1.394) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.116,1.498) -- (4.088,1.641) -- (4.007,1.762) -- (3.886,1.843) -- (3.743,1.871) -- (3.6,1.843) -- (3.479,1.762) -- (3.398,1.641) -- (3.37,1.498) -- (3.398,1.355) -- (3.479,1.234) -- (3.6,1.153) -- (3.743,1.125) -- (3.886,1.153) -- (4.007,1.234) -- (4.088,1.355) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.05,6.369) -- (5.169,6.388) -- (5.276,6.442) -- (5.361,6.527) -- (5.416,6.634) -- (5.434,6.753) -- (5.416,6.872) -- (5.361,6.979) -- (5.276,7.064) -- (5.169,7.119) -- (5.05,7.138) -- (5.05,7.138) -- (4.931,7.119) -- (4.824,7.064) -- (4.739,6.979) -- (4.684,6.872) -- (4.665,6.753) -- (4.684,6.634) -- (4.739,6.527) -- (4.824,6.442) -- (4.931,6.388) -- (5.05,6.369) -- cycle;
\node at (5.05,6.753) {\scalebox{1.607}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.743,6.265) -- (7.862,6.284) -- (7.969,6.338) -- (8.054,6.423) -- (8.109,6.531) -- (8.128,6.649) -- (8.109,6.768) -- (8.054,6.875) -- (7.969,6.96) -- (7.862,7.015) -- (7.743,7.034) -- (7.743,7.034) -- (7.624,7.015) -- (7.517,6.96) -- (7.432,6.875) -- (7.377,6.768) -- (7.359,6.649) -- (7.377,6.531) -- (7.432,6.423) -- (7.517,6.338) -- (7.624,6.284) -- (7.743,6.265) -- cycle;
\node at (7.743,6.649) {\scalebox{1.607}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.486,4.71) -- (6.604,4.729) -- (6.712,4.783) -- (6.797,4.868) -- (6.851,4.975) -- (6.87,5.094) -- (6.851,5.213) -- (6.797,5.32) -- (6.712,5.405) -- (6.604,5.46) -- (6.486,5.479) -- (6.486,5.479) -- (6.367,5.46) -- (6.259,5.405) -- (6.174,5.32) -- (6.12,5.213) -- (6.101,5.094) -- (6.12,4.975) -- (6.174,4.868) -- (6.259,4.783) -- (6.367,4.729) -- (6.486,4.71) -- cycle;
\node at (6.486,5.094) {\scalebox{1.607}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.967,5.664) -- (3.086,5.683) -- (3.193,5.738) -- (3.278,5.823) -- (3.333,5.93) -- (3.351,6.049) -- (3.333,6.168) -- (3.278,6.275) -- (3.193,6.36) -- (3.086,6.415) -- (2.967,6.433) -- (2.967,6.433) -- (2.848,6.415) -- (2.741,6.36) -- (2.656,6.275) -- (2.601,6.168) -- (2.582,6.049) -- (2.601,5.93) -- (2.656,5.823) -- (2.741,5.738) -- (2.848,5.683) -- (2.967,5.664) -- cycle;
\node at (2.967,6.049) {\scalebox{1.607}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.245,4.747) -- (2.364,4.765) -- (2.471,4.82) -- (2.556,4.905) -- (2.61,5.012) -- (2.629,5.131) -- (2.61,5.25) -- (2.556,5.357) -- (2.471,5.442) -- (2.364,5.497) -- (2.245,5.516) -- (2.245,5.516) -- (2.126,5.497) -- (2.019,5.442) -- (1.934,5.357) -- (1.879,5.25) -- (1.86,5.131) -- (1.879,5.012) -- (1.934,4.905) -- (2.019,4.82) -- (2.126,4.765) -- (2.245,4.747) -- cycle;
\node at (2.245,5.131) {\scalebox{1.607}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.182,4.295) -- (9.301,4.314) -- (9.408,4.368) -- (9.493,4.453) -- (9.548,4.561) -- (9.566,4.679) -- (9.548,4.798) -- (9.493,4.905) -- (9.408,4.99) -- (9.301,5.045) -- (9.182,5.064) -- (9.182,5.064) -- (9.063,5.045) -- (8.956,4.99) -- (8.871,4.905) -- (8.816,4.798) -- (8.797,4.679) -- (8.816,4.561) -- (8.871,4.453) -- (8.956,4.368) -- (9.063,4.314) -- (9.182,4.295) -- cycle;
\node at (9.182,4.679) {\scalebox{1.607}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.757,3.069) -- (2.876,3.088) -- (2.983,3.142) -- (3.068,3.228) -- (3.123,3.335) -- (3.142,3.454) -- (3.123,3.572) -- (3.068,3.68) -- (2.983,3.765) -- (2.876,3.819) -- (2.757,3.838) -- (2.757,3.838) -- (2.638,3.819) -- (2.531,3.765) -- (2.446,3.68) -- (2.391,3.572) -- (2.373,3.454) -- (2.391,3.335) -- (2.446,3.228) -- (2.531,3.142) -- (2.638,3.088) -- (2.757,3.069) -- cycle;
\node at (2.757,3.454) {\scalebox{1.607}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.056,2.428) -- (2.174,2.447) -- (2.282,2.502) -- (2.367,2.587) -- (2.421,2.694) -- (2.44,2.813) -- (2.421,2.932) -- (2.367,3.039) -- (2.282,3.124) -- (2.174,3.178) -- (2.056,3.197) -- (2.056,3.197) -- (1.937,3.178) -- (1.83,3.124) -- (1.744,3.039) -- (1.69,2.932) -- (1.671,2.813) -- (1.69,2.694) -- (1.744,2.587) -- (1.83,2.502) -- (1.937,2.447) -- (2.056,2.428) -- cycle;
\node at (2.056,2.813) {\scalebox{1.607}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.131,3.823) -- (5.25,3.842) -- (5.357,3.896) -- (5.442,3.981) -- (5.497,4.088) -- (5.516,4.207) -- (5.497,4.326) -- (5.442,4.433) -- (5.357,4.518) -- (5.25,4.573) -- (5.131,4.592) -- (5.131,4.592) -- (5.012,4.573) -- (4.905,4.518) -- (4.82,4.433) -- (4.765,4.326) -- (4.747,4.207) -- (4.765,4.088) -- (4.82,3.981) -- (4.905,3.896) -- (5.012,3.842) -- (5.131,3.823) -- cycle;
\node at (5.131,4.207) {\scalebox{1.607}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.584,2.289) -- (4.703,2.307) -- (4.81,2.362) -- (4.895,2.447) -- (4.95,2.554) -- (4.969,2.673) -- (4.95,2.792) -- (4.895,2.899) -- (4.81,2.984) -- (4.703,3.039) -- (4.584,3.058) -- (4.222,3.058) -- (4.104,3.039) -- (3.996,2.984) -- (3.911,2.899) -- (3.857,2.792) -- (3.838,2.673) -- (3.857,2.554) -- (3.911,2.447) -- (3.996,2.362) -- (4.104,2.307) -- (4.222,2.289) -- cycle;
\node at (4.403,2.673) {\scalebox{1.607}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.786,3.99) -- (7.905,4.009) -- (8.012,4.063) -- (8.097,4.148) -- (8.152,4.255) -- (8.171,4.374) -- (8.152,4.493) -- (8.097,4.6) -- (8.012,4.685) -- (7.905,4.74) -- (7.786,4.759) -- (7.424,4.759) -- (7.305,4.74) -- (7.198,4.685) -- (7.113,4.6) -- (7.058,4.493) -- (7.04,4.374) -- (7.058,4.255) -- (7.113,4.148) -- (7.198,4.063) -- (7.305,4.009) -- (7.424,3.99) -- cycle;
\node at (7.605,4.374) {\scalebox{1.607}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.06,2.286) -- (6.179,2.305) -- (6.286,2.359) -- (6.372,2.444) -- (6.426,2.552) -- (6.445,2.67) -- (6.426,2.789) -- (6.372,2.896) -- (6.286,2.981) -- (6.179,3.036) -- (6.06,3.055) -- (5.699,3.055) -- (5.58,3.036) -- (5.473,2.981) -- (5.387,2.896) -- (5.333,2.789) -- (5.314,2.67) -- (5.333,2.552) -- (5.387,2.444) -- (5.473,2.359) -- (5.58,2.305) -- (5.699,2.286) -- cycle;
\node at (5.879,2.67) {\scalebox{1.607}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.521,3.003) -- (10.639,3.022) -- (10.747,3.077) -- (10.832,3.162) -- (10.886,3.269) -- (10.905,3.388) -- (10.886,3.507) -- (10.832,3.614) -- (10.747,3.699) -- (10.639,3.754) -- (10.521,3.772) -- (10.159,3.772) -- (10.04,3.754) -- (9.933,3.699) -- (9.848,3.614) -- (9.793,3.507) -- (9.774,3.388) -- (9.793,3.269) -- (9.848,3.162) -- (9.933,3.077) -- (10.04,3.022) -- (10.159,3.003) -- cycle;
\node at (10.34,3.388) {\scalebox{1.607}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.512,0.936) -- (2.631,0.954) -- (2.738,1.009) -- (2.823,1.094) -- (2.878,1.201) -- (2.897,1.32) -- (2.878,1.439) -- (2.823,1.546) -- (2.738,1.631) -- (2.631,1.686) -- (2.512,1.705) -- (2.15,1.705) -- (2.032,1.686) -- (1.924,1.631) -- (1.839,1.546) -- (1.785,1.439) -- (1.766,1.32) -- (1.785,1.201) -- (1.839,1.094) -- (1.924,1.009) -- (2.032,0.954) -- (2.15,0.936) -- cycle;
\node at (2.331,1.32) {\scalebox{1.607}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.17,0.647) -- (5.289,0.665) -- (5.396,0.72) -- (5.481,0.805) -- (5.536,0.912) -- (5.554,1.031) -- (5.536,1.15) -- (5.481,1.257) -- (5.396,1.342) -- (5.289,1.397) -- (5.17,1.416) -- (4.808,1.416) -- (4.689,1.397) -- (4.582,1.342) -- (4.497,1.257) -- (4.442,1.15) -- (4.423,1.031) -- (4.442,0.912) -- (4.497,0.805) -- (4.582,0.72) -- (4.689,0.665) -- (4.808,0.647) -- cycle;
\node at (4.989,1.031) {\scalebox{1.607}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (3.797,3.686) -- (4.397,3.686) -- (4.397,4.286) -- (3.797,4.286) -- cycle;
\node at (4.097,3.986) {\scalebox{1.607}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (7.379,2.119) -- (7.979,2.119) -- (7.979,2.719) -- (7.379,2.719) -- cycle;
\node at (7.679,2.419) {\scalebox{1.607}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (9.664,4.799) -- (10.264,4.799) -- (10.264,5.399) -- (9.664,5.399) -- cycle;
\node at (9.964,5.099) {\scalebox{1.607}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.775,5.51) -- (7.375,5.51) -- (7.375,6.11) -- (6.775,6.11) -- cycle;
\node at (7.075,5.81) {\scalebox{1.607}{\small d}};
\node at (3.432,7.07) {\scalebox{1.607}{\small A}};
\node at (6.535,6.97) {\scalebox{1.607}{\small B}};
\node at (1.499,6.479) {\scalebox{1.607}{\small C}};
\node at (9.573,6.192) {\scalebox{1.607}{\small D}};
\node at (1.241,4.165) {\scalebox{1.607}{\small E}};
\node at (3.481,3.922) {\scalebox{1.607}{\small F}};
\node at (11.916,3.969) {\scalebox{1.607}{\small G}};
\node at (6.7,3.613) {\scalebox{1.607}{\small H}};
\node at (9.035,3.602) {\scalebox{1.607}{\small I}};
\node at (0.765,1.366) {\scalebox{1.607}{\small J}};
\node at (6.418,0.95) {\scalebox{1.607}{\small K}};
\node at (3.419,0.97) {\scalebox{1.607}{\small L}};
\draw[line width=0.5pt] (12.597,5.599) -- (12.597,6.699);
\draw[line width=0.5pt] (12.366,6.314) -- (12.597,6.699) -- (12.828,6.314);
\draw[line width=0.5pt] (12.294,6.149) -- (12.899,6.149);
\node at (12.597,7.084) {\scalebox{1.607}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, D, E, G, H, L & 6 \\ \hline
clairs & B, C, F, I, J, K & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{6} clairs. Il y en a \textbf{autant} de chaque côté.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 12 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Si l'on retire le carrefour \textbf{I}, le quartier tombe en 2 morceaux : toute route de l'un à l'autre passe par lui. Un circuit qui reviendrait à l'école devrait donc s'y arrêter au moins \textbf{deux fois} — c'est interdit.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{F} :}\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 de l'Église ⑨} vers l'est jusqu'à Carrefour des Tilleuls \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Four ⑫}, vers le sud, jusqu'à Place du Vieux-Puits \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin des Vignes ⑮}, vers l'ouest, jusqu'à Carrefour du Moulin \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Tilleuls ⑭}, vers l'ouest, jusqu'à Place des Tanneurs \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Stade ⑧}, vers le nord, jusqu'à Carrefour de l'Hôpital \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Puits ⑤}, vers le nord, jusqu'à Place de la Fontaine \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Lilas ④}, vers l'est, jusqu'à Place aux Herbes \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Pont ①}, vers l'est, jusqu'à Carrefour Saint-Roch \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{avenue de la Gare ②}, vers l'est, jusqu'à Place du Marché \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ⑥}, vers le sud, jusqu'à Carrefour de la Mairie \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse des Roses ⑬}, vers l'est, jusqu'à Place de l'Église \textbf{(G)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 12 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
