\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 368}

\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=368}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0368}
\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=0.6046, line join=round, line cap=round]
\draw[line width=0.5pt] (7.884,11.162) -- (7.865,10.936) -- (7.852,10.709) -- (7.843,10.482) -- (7.84,10.255) -- (7.843,10.028) -- (7.851,9.801) -- (7.864,9.574) -- (7.882,9.347) -- (7.906,9.121) -- (7.935,8.896) -- (7.969,8.672) -- (8.008,8.448) -- (8.053,8.225) -- (8.103,8.004);
\draw[line width=0.5pt] (1.707,10.996) -- (1.935,11.02) -- (2.164,11.037) -- (2.393,11.05) -- (2.622,11.057) -- (2.852,11.059) -- (3.081,11.055) -- (3.31,11.046) -- (3.539,11.031) -- (3.768,11.011) -- (3.996,10.986) -- (4.223,10.956) -- (4.45,10.92) -- (4.675,10.878) -- (4.9,10.832);
\draw[line width=0.5pt] (1.707,10.996) -- (1.691,10.798) -- (1.68,10.599) -- (1.673,10.401) -- (1.671,10.202) -- (1.674,10.003) -- (1.682,9.804) -- (1.694,9.605) -- (1.711,9.407) -- (1.732,9.209) -- (1.758,9.012) -- (1.789,8.815) -- (1.824,8.619) -- (1.864,8.425) -- (1.908,8.231);
\draw[line width=0.5pt] (4.9,10.832) -- (5.108,10.888) -- (5.317,10.939) -- (5.527,10.985) -- (5.739,11.026) -- (5.951,11.063) -- (6.165,11.094) -- (6.378,11.12) -- (6.593,11.141) -- (6.808,11.158) -- (7.023,11.169) -- (7.239,11.175) -- (7.454,11.176) -- (7.669,11.171) -- (7.884,11.162);
\draw[line width=0.5pt] (4.9,10.832) -- (4.859,10.668) -- (4.823,10.502) -- (4.79,10.336) -- (4.762,10.169) -- (4.737,10.002) -- (4.716,9.834) -- (4.699,9.665) -- (4.686,9.496) -- (4.678,9.327) -- (4.673,9.158) -- (4.672,8.989) -- (4.675,8.819) -- (4.682,8.65) -- (4.693,8.481);
\draw[line width=0.5pt] (4.693,8.481) -- (4.94,8.485) -- (5.187,8.483) -- (5.434,8.474) -- (5.68,8.46) -- (5.927,8.44) -- (6.173,8.415) -- (6.418,8.383) -- (6.662,8.346) -- (6.906,8.303) -- (7.148,8.255) -- (7.389,8.201) -- (7.629,8.141) -- (7.867,8.075) -- (8.103,8.004);
\draw[line width=0.5pt] (4.693,8.481) -- (4.672,8.21) -- (4.657,7.939) -- (4.649,7.667) -- (4.647,7.395) -- (4.652,7.123) -- (4.663,6.851) -- (4.68,6.579) -- (4.704,6.308) -- (4.734,6.037) -- (4.771,5.768) -- (4.814,5.499) -- (4.863,5.231) -- (4.918,4.965) -- (4.98,4.7);
\draw[line width=0.5pt] (1.908,8.231) -- (2.103,8.279) -- (2.299,8.323) -- (2.495,8.362) -- (2.693,8.396) -- (2.892,8.426) -- (3.091,8.451) -- (3.29,8.471) -- (3.49,8.487) -- (3.691,8.498) -- (3.891,8.504) -- (4.092,8.505) -- (4.292,8.502) -- (4.493,8.494) -- (4.693,8.481);
\draw[line width=0.5pt] (1.908,8.231) -- (1.837,8.027) -- (1.771,7.822) -- (1.709,7.616) -- (1.652,7.407) -- (1.601,7.198) -- (1.554,6.987) -- (1.512,6.776) -- (1.475,6.563) -- (1.443,6.35) -- (1.416,6.135) -- (1.394,5.921) -- (1.377,5.706) -- (1.366,5.491) -- (1.359,5.275);
\draw[line width=0.5pt] (11.245,8.163) -- (11.286,7.948) -- (11.323,7.732) -- (11.354,7.515) -- (11.38,7.298) -- (11.401,7.08) -- (11.417,6.861) -- (11.428,6.642) -- (11.433,6.423) -- (11.434,6.204) -- (11.429,5.985) -- (11.419,5.766) -- (11.405,5.547) -- (11.384,5.329) -- (11.359,5.112);
\draw[line width=0.5pt] (8.103,8.004) -- (8.328,7.981) -- (8.553,7.963) -- (8.778,7.951) -- (9.003,7.943) -- (9.229,7.942) -- (9.455,7.945) -- (9.681,7.954) -- (9.906,7.968) -- (10.131,7.987) -- (10.355,8.012) -- (10.579,8.042) -- (10.802,8.077) -- (11.024,8.118) -- (11.245,8.163);
\draw[line width=0.5pt] (8.103,8.004) -- (8.177,7.766) -- (8.245,7.527) -- (8.307,7.286) -- (8.363,7.044) -- (8.414,6.801) -- (8.459,6.556) -- (8.498,6.31) -- (8.532,6.064) -- (8.559,5.816) -- (8.581,5.569) -- (8.597,5.32) -- (8.608,5.072) -- (8.612,4.823) -- (8.61,4.575);
\draw[line width=0.5pt] (1.359,5.275) -- (1.609,5.195) -- (1.862,5.12) -- (2.116,5.052) -- (2.371,4.989) -- (2.628,4.933) -- (2.886,4.883) -- (3.146,4.838) -- (3.406,4.8) -- (3.667,4.768) -- (3.929,4.742) -- (4.192,4.722) -- (4.454,4.709) -- (4.717,4.701) -- (4.98,4.7);
\draw[line width=0.5pt] (4.98,4.7) -- (5.239,4.731) -- (5.498,4.756) -- (5.758,4.774) -- (6.018,4.787) -- (6.279,4.793) -- (6.54,4.793) -- (6.8,4.787) -- (7.061,4.775) -- (7.321,4.757) -- (7.58,4.733) -- (7.839,4.702) -- (8.097,4.666) -- (8.354,4.623) -- (8.61,4.575);
\draw[line width=0.5pt] (8.61,4.575) -- (8.811,4.583) -- (9.011,4.595) -- (9.212,4.613) -- (9.411,4.635) -- (9.611,4.662) -- (9.809,4.694) -- (10.007,4.73) -- (10.204,4.771) -- (10.4,4.816) -- (10.594,4.866) -- (10.788,4.921) -- (10.98,4.98) -- (11.17,5.043) -- (11.359,5.112);
\draw[line width=0.5pt] (8.61,4.575) -- (8.589,4.375) -- (8.563,4.176) -- (8.533,3.977) -- (8.497,3.78) -- (8.457,3.583) -- (8.413,3.387) -- (8.364,3.192) -- (8.31,2.998) -- (8.252,2.806) -- (8.189,2.615) -- (8.122,2.425) -- (8.05,2.238) -- (7.974,2.052) -- (7.893,1.868);
\draw[line width=0.5pt] (7.893,1.868) -- (8.101,1.804) -- (8.311,1.746) -- (8.522,1.692) -- (8.734,1.643) -- (8.947,1.6) -- (9.162,1.561) -- (9.377,1.527) -- (9.593,1.499) -- (9.809,1.475) -- (10.026,1.457) -- (10.243,1.443) -- (10.461,1.435) -- (10.678,1.432) -- (10.896,1.434);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.044,11.162) -- (8.023,11.242) -- (7.964,11.301) -- (7.884,11.322) -- (7.804,11.301) -- (7.746,11.242) -- (7.724,11.162) -- (7.746,11.082) -- (7.804,11.024) -- (7.884,11.002) -- (7.964,11.024) -- (8.023,11.082) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.867,10.996) -- (1.846,11.076) -- (1.787,11.135) -- (1.707,11.156) -- (1.627,11.135) -- (1.569,11.076) -- (1.547,10.996) -- (1.569,10.916) -- (1.627,10.858) -- (1.707,10.836) -- (1.787,10.858) -- (1.846,10.916) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.06,10.832) -- (5.038,10.912) -- (4.98,10.97) -- (4.9,10.992) -- (4.82,10.97) -- (4.761,10.912) -- (4.74,10.832) -- (4.761,10.752) -- (4.82,10.693) -- (4.9,10.672) -- (4.98,10.693) -- (5.038,10.752) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.853,8.481) -- (4.831,8.561) -- (4.773,8.62) -- (4.693,8.641) -- (4.613,8.62) -- (4.554,8.561) -- (4.533,8.481) -- (4.554,8.401) -- (4.613,8.343) -- (4.693,8.321) -- (4.773,8.343) -- (4.831,8.401) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.068,8.231) -- (2.047,8.311) -- (1.988,8.369) -- (1.908,8.391) -- (1.828,8.369) -- (1.77,8.311) -- (1.748,8.231) -- (1.77,8.151) -- (1.828,8.092) -- (1.908,8.071) -- (1.988,8.092) -- (2.047,8.151) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.405,8.163) -- (11.383,8.243) -- (11.325,8.302) -- (11.245,8.323) -- (11.165,8.302) -- (11.106,8.243) -- (11.085,8.163) -- (11.106,8.083) -- (11.165,8.025) -- (11.245,8.003) -- (11.325,8.025) -- (11.383,8.083) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.263,8.004) -- (8.242,8.084) -- (8.183,8.142) -- (8.103,8.164) -- (8.023,8.142) -- (7.965,8.084) -- (7.943,8.004) -- (7.965,7.924) -- (8.023,7.865) -- (8.103,7.844) -- (8.183,7.865) -- (8.242,7.924) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.519,5.275) -- (1.498,5.355) -- (1.439,5.414) -- (1.359,5.435) -- (1.279,5.414) -- (1.221,5.355) -- (1.199,5.275) -- (1.221,5.195) -- (1.279,5.136) -- (1.359,5.115) -- (1.439,5.136) -- (1.498,5.195) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.519,5.112) -- (11.498,5.192) -- (11.439,5.25) -- (11.359,5.272) -- (11.279,5.25) -- (11.221,5.192) -- (11.199,5.112) -- (11.221,5.032) -- (11.279,4.973) -- (11.359,4.952) -- (11.439,4.973) -- (11.498,5.032) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.14,4.7) -- (5.118,4.78) -- (5.06,4.839) -- (4.98,4.86) -- (4.9,4.839) -- (4.841,4.78) -- (4.82,4.7) -- (4.841,4.62) -- (4.9,4.562) -- (4.98,4.54) -- (5.06,4.562) -- (5.118,4.62) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.77,4.575) -- (8.749,4.655) -- (8.69,4.713) -- (8.61,4.735) -- (8.53,4.713) -- (8.472,4.655) -- (8.45,4.575) -- (8.472,4.495) -- (8.53,4.436) -- (8.61,4.415) -- (8.69,4.436) -- (8.749,4.495) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.053,1.868) -- (8.032,1.948) -- (7.973,2.006) -- (7.893,2.028) -- (7.813,2.006) -- (7.755,1.948) -- (7.733,1.868) -- (7.755,1.788) -- (7.813,1.729) -- (7.893,1.708) -- (7.973,1.729) -- (8.032,1.788) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.056,1.434) -- (11.034,1.514) -- (10.976,1.573) -- (10.896,1.594) -- (10.816,1.573) -- (10.757,1.514) -- (10.736,1.434) -- (10.757,1.354) -- (10.816,1.296) -- (10.896,1.274) -- (10.976,1.296) -- (11.034,1.354) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.257,9.207) -- (8.378,9.227) -- (8.488,9.283) -- (8.575,9.37) -- (8.631,9.48) -- (8.65,9.601) -- (8.631,9.723) -- (8.575,9.833) -- (8.488,9.92) -- (8.378,9.976) -- (8.257,9.995) -- (8.257,9.995) -- (8.135,9.976) -- (8.025,9.92) -- (7.938,9.833) -- (7.882,9.723) -- (7.863,9.601) -- (7.882,9.48) -- (7.938,9.37) -- (8.025,9.283) -- (8.135,9.227) -- (8.257,9.207) -- cycle;
\node at (8.257,9.601) {\scalebox{1.654}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.33,11.045) -- (3.452,11.065) -- (3.562,11.12) -- (3.649,11.208) -- (3.705,11.317) -- (3.724,11.439) -- (3.705,11.561) -- (3.649,11.671) -- (3.562,11.758) -- (3.452,11.814) -- (3.33,11.833) -- (3.33,11.833) -- (3.209,11.814) -- (3.099,11.758) -- (3.012,11.671) -- (2.956,11.561) -- (2.937,11.439) -- (2.956,11.317) -- (3.012,11.208) -- (3.099,11.12) -- (3.209,11.065) -- (3.33,11.045) -- cycle;
\node at (3.33,11.439) {\scalebox{1.654}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.087,9.24) -- (2.208,9.259) -- (2.318,9.315) -- (2.405,9.402) -- (2.461,9.512) -- (2.48,9.634) -- (2.461,9.756) -- (2.405,9.865) -- (2.318,9.952) -- (2.208,10.008) -- (2.087,10.028) -- (2.087,10.028) -- (1.965,10.008) -- (1.855,9.952) -- (1.768,9.865) -- (1.712,9.756) -- (1.693,9.634) -- (1.712,9.512) -- (1.768,9.402) -- (1.855,9.315) -- (1.965,9.259) -- (2.087,9.24) -- cycle;
\node at (2.087,9.634) {\scalebox{1.654}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.335,11.118) -- (6.457,11.137) -- (6.567,11.193) -- (6.654,11.28) -- (6.71,11.39) -- (6.729,11.512) -- (6.71,11.633) -- (6.654,11.743) -- (6.567,11.83) -- (6.457,11.886) -- (6.335,11.906) -- (6.335,11.906) -- (6.213,11.886) -- (6.104,11.83) -- (6.016,11.743) -- (5.961,11.633) -- (5.941,11.512) -- (5.961,11.39) -- (6.016,11.28) -- (6.104,11.193) -- (6.213,11.137) -- (6.335,11.118) -- cycle;
\node at (6.335,11.512) {\scalebox{1.654}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.092,9.237) -- (5.213,9.256) -- (5.323,9.312) -- (5.41,9.399) -- (5.466,9.509) -- (5.486,9.631) -- (5.466,9.752) -- (5.41,9.862) -- (5.323,9.949) -- (5.213,10.005) -- (5.092,10.025) -- (5.092,10.025) -- (4.97,10.005) -- (4.86,9.949) -- (4.773,9.862) -- (4.717,9.752) -- (4.698,9.631) -- (4.717,9.509) -- (4.773,9.399) -- (4.86,9.312) -- (4.97,9.256) -- (5.092,9.237) -- cycle;
\node at (5.092,9.631) {\scalebox{1.654}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.472,8.38) -- (6.594,8.399) -- (6.704,8.455) -- (6.791,8.542) -- (6.847,8.652) -- (6.866,8.773) -- (6.847,8.895) -- (6.791,9.005) -- (6.704,9.092) -- (6.594,9.148) -- (6.472,9.167) -- (6.472,9.167) -- (6.351,9.148) -- (6.241,9.092) -- (6.154,9.005) -- (6.098,8.895) -- (6.078,8.773) -- (6.098,8.652) -- (6.154,8.542) -- (6.241,8.455) -- (6.351,8.399) -- (6.472,8.38) -- cycle;
\node at (6.472,8.773) {\scalebox{1.654}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.073,6.215) -- (5.195,6.234) -- (5.305,6.29) -- (5.392,6.377) -- (5.448,6.487) -- (5.467,6.609) -- (5.448,6.731) -- (5.392,6.84) -- (5.305,6.928) -- (5.195,6.984) -- (5.073,7.003) -- (5.073,7.003) -- (4.951,6.984) -- (4.842,6.928) -- (4.754,6.84) -- (4.699,6.731) -- (4.679,6.609) -- (4.699,6.487) -- (4.754,6.377) -- (4.842,6.29) -- (4.951,6.234) -- (5.073,6.215) -- cycle;
\node at (5.073,6.609) {\scalebox{1.654}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.255,8.469) -- (3.377,8.489) -- (3.486,8.545) -- (3.574,8.632) -- (3.63,8.742) -- (3.649,8.863) -- (3.63,8.985) -- (3.574,9.095) -- (3.486,9.182) -- (3.377,9.238) -- (3.255,9.257) -- (3.255,9.257) -- (3.133,9.238) -- (3.023,9.182) -- (2.936,9.095) -- (2.88,8.985) -- (2.861,8.863) -- (2.88,8.742) -- (2.936,8.632) -- (3.023,8.545) -- (3.133,8.489) -- (3.255,8.469) -- cycle;
\node at (3.255,8.863) {\scalebox{1.654}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.899,6.31) -- (2.021,6.329) -- (2.131,6.385) -- (2.218,6.472) -- (2.274,6.582) -- (2.293,6.704) -- (2.274,6.825) -- (2.218,6.935) -- (2.131,7.022) -- (2.021,7.078) -- (1.899,7.098) -- (1.899,7.098) -- (1.777,7.078) -- (1.668,7.022) -- (1.58,6.935) -- (1.524,6.825) -- (1.505,6.704) -- (1.524,6.582) -- (1.58,6.472) -- (1.668,6.385) -- (1.777,6.329) -- (1.899,6.31) -- cycle;
\node at (1.899,6.704) {\scalebox{1.654}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (12.007,6.263) -- (12.129,6.282) -- (12.238,6.338) -- (12.326,6.425) -- (12.381,6.535) -- (12.401,6.657) -- (12.381,6.779) -- (12.326,6.888) -- (12.238,6.976) -- (12.129,7.031) -- (12.007,7.051) -- (11.636,7.051) -- (11.514,7.031) -- (11.405,6.976) -- (11.317,6.888) -- (11.261,6.779) -- (11.242,6.657) -- (11.261,6.535) -- (11.317,6.425) -- (11.405,6.338) -- (11.514,6.282) -- (11.636,6.263) -- cycle;
\node at (11.821,6.657) {\scalebox{1.654}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.846,7.953) -- (9.968,7.973) -- (10.078,8.029) -- (10.165,8.116) -- (10.221,8.226) -- (10.24,8.347) -- (10.221,8.469) -- (10.165,8.579) -- (10.078,8.666) -- (9.968,8.722) -- (9.846,8.741) -- (9.475,8.741) -- (9.354,8.722) -- (9.244,8.666) -- (9.157,8.579) -- (9.101,8.469) -- (9.081,8.347) -- (9.101,8.226) -- (9.157,8.116) -- (9.244,8.029) -- (9.354,7.973) -- (9.475,7.953) -- cycle;
\node at (9.661,8.347) {\scalebox{1.654}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.073,5.974) -- (9.195,5.993) -- (9.305,6.049) -- (9.392,6.136) -- (9.448,6.246) -- (9.467,6.368) -- (9.448,6.49) -- (9.392,6.599) -- (9.305,6.687) -- (9.195,6.743) -- (9.073,6.762) -- (8.703,6.762) -- (8.581,6.743) -- (8.471,6.687) -- (8.384,6.599) -- (8.328,6.49) -- (8.309,6.368) -- (8.328,6.246) -- (8.384,6.136) -- (8.471,6.049) -- (8.581,5.993) -- (8.703,5.974) -- cycle;
\node at (8.888,6.368) {\scalebox{1.654}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.393,4.833) -- (3.515,4.853) -- (3.624,4.909) -- (3.712,4.996) -- (3.768,5.106) -- (3.787,5.227) -- (3.768,5.349) -- (3.712,5.459) -- (3.624,5.546) -- (3.515,5.602) -- (3.393,5.621) -- (3.022,5.621) -- (2.9,5.602) -- (2.791,5.546) -- (2.703,5.459) -- (2.648,5.349) -- (2.628,5.227) -- (2.648,5.106) -- (2.703,4.996) -- (2.791,4.909) -- (2.9,4.853) -- (3.022,4.833) -- cycle;
\node at (3.208,5.227) {\scalebox{1.654}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.999,4.787) -- (7.121,4.806) -- (7.231,4.862) -- (7.318,4.949) -- (7.374,5.059) -- (7.393,5.181) -- (7.374,5.303) -- (7.318,5.413) -- (7.231,5.5) -- (7.121,5.556) -- (6.999,5.575) -- (6.629,5.575) -- (6.507,5.556) -- (6.397,5.5) -- (6.31,5.413) -- (6.254,5.303) -- (6.235,5.181) -- (6.254,5.059) -- (6.31,4.949) -- (6.397,4.862) -- (6.507,4.806) -- (6.629,4.787) -- cycle;
\node at (6.814,5.181) {\scalebox{1.654}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.117,4.723) -- (10.238,4.742) -- (10.348,4.798) -- (10.435,4.885) -- (10.491,4.995) -- (10.511,5.116) -- (10.491,5.238) -- (10.435,5.348) -- (10.348,5.435) -- (10.238,5.491) -- (10.117,5.51) -- (9.746,5.51) -- (9.624,5.491) -- (9.514,5.435) -- (9.427,5.348) -- (9.371,5.238) -- (9.352,5.116) -- (9.371,4.995) -- (9.427,4.885) -- (9.514,4.798) -- (9.624,4.742) -- (9.746,4.723) -- cycle;
\node at (9.931,5.116) {\scalebox{1.654}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.011,2.94) -- (8.133,2.959) -- (8.243,3.015) -- (8.33,3.103) -- (8.386,3.212) -- (8.405,3.334) -- (8.386,3.456) -- (8.33,3.566) -- (8.243,3.653) -- (8.133,3.709) -- (8.011,3.728) -- (7.641,3.728) -- (7.519,3.709) -- (7.409,3.653) -- (7.322,3.566) -- (7.266,3.456) -- (7.247,3.334) -- (7.266,3.212) -- (7.322,3.103) -- (7.409,3.015) -- (7.519,2.959) -- (7.641,2.94) -- cycle;
\node at (7.826,3.334) {\scalebox{1.654}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.618,1.523) -- (9.74,1.542) -- (9.85,1.598) -- (9.937,1.686) -- (9.993,1.795) -- (10.012,1.917) -- (9.993,2.039) -- (9.937,2.149) -- (9.85,2.236) -- (9.74,2.292) -- (9.618,2.311) -- (9.248,2.311) -- (9.126,2.292) -- (9.016,2.236) -- (8.929,2.149) -- (8.873,2.039) -- (8.854,1.917) -- (8.873,1.795) -- (8.929,1.686) -- (9.016,1.598) -- (9.126,1.542) -- (9.248,1.523) -- cycle;
\node at (9.433,1.917) {\scalebox{1.654}{\small 17}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.393,8.181) -- (4.993,8.181) -- (4.993,8.781) -- (4.393,8.781) -- cycle;
\node at (4.693,8.481) {\scalebox{1.654}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.363,10.219) -- (8.963,10.219) -- (8.963,10.819) -- (8.363,10.819) -- cycle;
\node at (8.663,10.519) {\scalebox{1.654}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.695,3.167) -- (9.295,3.167) -- (9.295,3.767) -- (8.695,3.767) -- cycle;
\node at (8.995,3.467) {\scalebox{1.654}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.75,3.966) -- (6.35,3.966) -- (6.35,4.566) -- (5.75,4.566) -- cycle;
\node at (6.05,4.266) {\scalebox{1.654}{\small d}};
\node at (8.063,11.756) {\scalebox{1.654}{\small A}};
\node at (1.236,11.399) {\scalebox{1.654}{\small B}};
\node at (4.66,11.404) {\scalebox{1.654}{\small C}};
\node at (4.226,8.889) {\scalebox{1.654}{\small D}};
\node at (1.314,8.408) {\scalebox{1.654}{\small E}};
\node at (11.841,8.333) {\scalebox{1.654}{\small F}};
\node at (8.595,8.382) {\scalebox{1.654}{\small G}};
\node at (0.765,5.098) {\scalebox{1.654}{\small H}};
\node at (11.942,4.901) {\scalebox{1.654}{\small I}};
\node at (4.608,4.204) {\scalebox{1.654}{\small J}};
\node at (9.025,4.114) {\scalebox{1.654}{\small K}};
\node at (8.052,1.268) {\scalebox{1.654}{\small L}};
\node at (11.283,0.95) {\scalebox{1.654}{\small M}};
\draw[line width=0.5pt] (12.659,10.212) -- (12.659,11.312);
\draw[line width=0.5pt] (12.428,10.927) -- (12.659,11.312) -- (12.89,10.927);
\draw[line width=0.5pt] (12.357,10.762) -- (12.962,10.762);
\node at (12.659,11.697) {\scalebox{1.654}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour D, 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 & ⑩ rue des Écoles & Ⓑ Place aux Herbes & Ⓛ Place des Tanneurs \\ \hline
① venelle du Chat & ⑪ chemin du Lavoir & Ⓒ Rond-point de la Poste & Ⓜ Place du Marché \\ \hline
② rue du Puits & ⑫ rue Neuve & Ⓓ Rond-point du Stade & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ rue des Lilas & ⑬ rue du Four & Ⓔ Rond-point des Vaches & \textbf{a} l'école \\ \hline
④ allée des Peupliers & ⑭ chemin des Vignes & Ⓕ Placette des Ormes & \textbf{b} la gendarmerie \\ \hline
⑤ avenue de la Gare & ⑮ rue de la Mairie & Ⓖ Carrefour du Lavoir & \textbf{c} le cimetière \\ \hline
⑥ rue du Moulin & ⑯ chemin de Ronde & Ⓗ Place Sainte-Croix & \textbf{d} le stade \\ \hline
⑦ quai aux Fleurs & ⑰ impasse Sainte-Anne & Ⓘ Place de l'Église &  \\ \hline
⑧ passage du Marché & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Carrefour Saint-Roch &  \\ \hline
⑨ rue de la Poste & Ⓐ Place du Vieux-Puits & Ⓚ Carrefour des Tilleuls &  \\ \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=0.6194, line join=round, line cap=round]
\draw[line width=0.5pt] (7.884,11.162) -- (7.865,10.936) -- (7.852,10.709) -- (7.843,10.482) -- (7.84,10.255) -- (7.843,10.028) -- (7.851,9.801) -- (7.864,9.574) -- (7.882,9.347) -- (7.906,9.121) -- (7.935,8.896) -- (7.969,8.672) -- (8.008,8.448) -- (8.053,8.225) -- (8.103,8.004);
\draw[line width=0.5pt] (1.707,10.996) -- (1.935,11.02) -- (2.164,11.037) -- (2.393,11.05) -- (2.622,11.057) -- (2.852,11.059) -- (3.081,11.055) -- (3.31,11.046) -- (3.539,11.031) -- (3.768,11.011) -- (3.996,10.986) -- (4.223,10.956) -- (4.45,10.92) -- (4.675,10.878) -- (4.9,10.832);
\draw[line width=0.5pt] (1.707,10.996) -- (1.691,10.798) -- (1.68,10.599) -- (1.673,10.401) -- (1.671,10.202) -- (1.674,10.003) -- (1.682,9.804) -- (1.694,9.605) -- (1.711,9.407) -- (1.732,9.209) -- (1.758,9.012) -- (1.789,8.815) -- (1.824,8.619) -- (1.864,8.425) -- (1.908,8.231);
\draw[line width=0.5pt] (4.9,10.832) -- (5.108,10.888) -- (5.317,10.939) -- (5.527,10.985) -- (5.739,11.026) -- (5.951,11.063) -- (6.165,11.094) -- (6.378,11.12) -- (6.593,11.141) -- (6.808,11.158) -- (7.023,11.169) -- (7.239,11.175) -- (7.454,11.176) -- (7.669,11.171) -- (7.884,11.162);
\draw[line width=0.5pt] (4.9,10.832) -- (4.859,10.668) -- (4.823,10.502) -- (4.79,10.336) -- (4.762,10.169) -- (4.737,10.002) -- (4.716,9.834) -- (4.699,9.665) -- (4.686,9.496) -- (4.678,9.327) -- (4.673,9.158) -- (4.672,8.989) -- (4.675,8.819) -- (4.682,8.65) -- (4.693,8.481);
\draw[line width=0.5pt] (4.693,8.481) -- (4.94,8.485) -- (5.187,8.483) -- (5.434,8.474) -- (5.68,8.46) -- (5.927,8.44) -- (6.173,8.415) -- (6.418,8.383) -- (6.662,8.346) -- (6.906,8.303) -- (7.148,8.255) -- (7.389,8.201) -- (7.629,8.141) -- (7.867,8.075) -- (8.103,8.004);
\draw[line width=0.5pt] (4.693,8.481) -- (4.672,8.21) -- (4.657,7.939) -- (4.649,7.667) -- (4.647,7.395) -- (4.652,7.123) -- (4.663,6.851) -- (4.68,6.579) -- (4.704,6.308) -- (4.734,6.037) -- (4.771,5.768) -- (4.814,5.499) -- (4.863,5.231) -- (4.918,4.965) -- (4.98,4.7);
\draw[line width=0.5pt] (1.908,8.231) -- (2.103,8.279) -- (2.299,8.323) -- (2.495,8.362) -- (2.693,8.396) -- (2.892,8.426) -- (3.091,8.451) -- (3.29,8.471) -- (3.49,8.487) -- (3.691,8.498) -- (3.891,8.504) -- (4.092,8.505) -- (4.292,8.502) -- (4.493,8.494) -- (4.693,8.481);
\draw[line width=0.5pt] (1.908,8.231) -- (1.837,8.027) -- (1.771,7.822) -- (1.709,7.616) -- (1.652,7.407) -- (1.601,7.198) -- (1.554,6.987) -- (1.512,6.776) -- (1.475,6.563) -- (1.443,6.35) -- (1.416,6.135) -- (1.394,5.921) -- (1.377,5.706) -- (1.366,5.491) -- (1.359,5.275);
\draw[line width=0.5pt] (11.245,8.163) -- (11.286,7.948) -- (11.323,7.732) -- (11.354,7.515) -- (11.38,7.298) -- (11.401,7.08) -- (11.417,6.861) -- (11.428,6.642) -- (11.433,6.423) -- (11.434,6.204) -- (11.429,5.985) -- (11.419,5.766) -- (11.405,5.547) -- (11.384,5.329) -- (11.359,5.112);
\draw[line width=0.5pt] (8.103,8.004) -- (8.328,7.981) -- (8.553,7.963) -- (8.778,7.951) -- (9.003,7.943) -- (9.229,7.942) -- (9.455,7.945) -- (9.681,7.954) -- (9.906,7.968) -- (10.131,7.987) -- (10.355,8.012) -- (10.579,8.042) -- (10.802,8.077) -- (11.024,8.118) -- (11.245,8.163);
\draw[line width=0.5pt] (8.103,8.004) -- (8.177,7.766) -- (8.245,7.527) -- (8.307,7.286) -- (8.363,7.044) -- (8.414,6.801) -- (8.459,6.556) -- (8.498,6.31) -- (8.532,6.064) -- (8.559,5.816) -- (8.581,5.569) -- (8.597,5.32) -- (8.608,5.072) -- (8.612,4.823) -- (8.61,4.575);
\draw[line width=0.5pt] (1.359,5.275) -- (1.609,5.195) -- (1.862,5.12) -- (2.116,5.052) -- (2.371,4.989) -- (2.628,4.933) -- (2.886,4.883) -- (3.146,4.838) -- (3.406,4.8) -- (3.667,4.768) -- (3.929,4.742) -- (4.192,4.722) -- (4.454,4.709) -- (4.717,4.701) -- (4.98,4.7);
\draw[line width=0.5pt] (4.98,4.7) -- (5.239,4.731) -- (5.498,4.756) -- (5.758,4.774) -- (6.018,4.787) -- (6.279,4.793) -- (6.54,4.793) -- (6.8,4.787) -- (7.061,4.775) -- (7.321,4.757) -- (7.58,4.733) -- (7.839,4.702) -- (8.097,4.666) -- (8.354,4.623) -- (8.61,4.575);
\draw[line width=0.5pt] (8.61,4.575) -- (8.811,4.583) -- (9.011,4.595) -- (9.212,4.613) -- (9.411,4.635) -- (9.611,4.662) -- (9.809,4.694) -- (10.007,4.73) -- (10.204,4.771) -- (10.4,4.816) -- (10.594,4.866) -- (10.788,4.921) -- (10.98,4.98) -- (11.17,5.043) -- (11.359,5.112);
\draw[line width=0.5pt] (8.61,4.575) -- (8.589,4.375) -- (8.563,4.176) -- (8.533,3.977) -- (8.497,3.78) -- (8.457,3.583) -- (8.413,3.387) -- (8.364,3.192) -- (8.31,2.998) -- (8.252,2.806) -- (8.189,2.615) -- (8.122,2.425) -- (8.05,2.238) -- (7.974,2.052) -- (7.893,1.868);
\draw[line width=0.5pt] (7.893,1.868) -- (8.101,1.804) -- (8.311,1.746) -- (8.522,1.692) -- (8.734,1.643) -- (8.947,1.6) -- (9.162,1.561) -- (9.377,1.527) -- (9.593,1.499) -- (9.809,1.475) -- (10.026,1.457) -- (10.243,1.443) -- (10.461,1.435) -- (10.678,1.432) -- (10.896,1.434);
\filldraw[fill=white, draw=black, line width=0.5pt] (8.258,11.162) -- (8.229,11.305) -- (8.148,11.426) -- (8.027,11.507) -- (7.884,11.535) -- (7.742,11.507) -- (7.621,11.426) -- (7.54,11.305) -- (7.511,11.162) -- (7.54,11.019) -- (7.621,10.898) -- (7.742,10.817) -- (7.884,10.789) -- (8.027,10.817) -- (8.148,10.898) -- (8.229,11.019) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.08,10.996) -- (2.052,11.139) -- (1.971,11.26) -- (1.85,11.341) -- (1.707,11.369) -- (1.564,11.341) -- (1.443,11.26) -- (1.362,11.139) -- (1.334,10.996) -- (1.362,10.853) -- (1.443,10.732) -- (1.564,10.651) -- (1.707,10.623) -- (1.85,10.651) -- (1.971,10.732) -- (2.052,10.853) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.273,10.832) -- (5.245,10.975) -- (5.164,11.096) -- (5.043,11.177) -- (4.9,11.205) -- (4.757,11.177) -- (4.636,11.096) -- (4.555,10.975) -- (4.527,10.832) -- (4.555,10.689) -- (4.636,10.568) -- (4.757,10.487) -- (4.9,10.459) -- (5.043,10.487) -- (5.164,10.568) -- (5.245,10.689) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.066,8.481) -- (5.038,8.624) -- (4.957,8.745) -- (4.836,8.826) -- (4.693,8.855) -- (4.55,8.826) -- (4.429,8.745) -- (4.348,8.624) -- (4.32,8.481) -- (4.348,8.339) -- (4.429,8.218) -- (4.55,8.137) -- (4.693,8.108) -- (4.836,8.137) -- (4.957,8.218) -- (5.038,8.339) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.282,8.231) -- (2.253,8.374) -- (2.172,8.495) -- (2.051,8.576) -- (1.908,8.604) -- (1.766,8.576) -- (1.645,8.495) -- (1.564,8.374) -- (1.535,8.231) -- (1.564,8.088) -- (1.645,7.967) -- (1.766,7.886) -- (1.908,7.858) -- (2.051,7.886) -- (2.172,7.967) -- (2.253,8.088) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.618,8.163) -- (11.59,8.306) -- (11.509,8.427) -- (11.388,8.508) -- (11.245,8.536) -- (11.102,8.508) -- (10.981,8.427) -- (10.9,8.306) -- (10.872,8.163) -- (10.9,8.02) -- (10.981,7.899) -- (11.102,7.818) -- (11.245,7.79) -- (11.388,7.818) -- (11.509,7.899) -- (11.59,8.02) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.476,8.004) -- (8.448,8.147) -- (8.367,8.268) -- (8.246,8.349) -- (8.103,8.377) -- (7.96,8.349) -- (7.839,8.268) -- (7.758,8.147) -- (7.73,8.004) -- (7.758,7.861) -- (7.839,7.74) -- (7.96,7.659) -- (8.103,7.631) -- (8.246,7.659) -- (8.367,7.74) -- (8.448,7.861) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.732,5.275) -- (1.704,5.418) -- (1.623,5.539) -- (1.502,5.62) -- (1.359,5.648) -- (1.216,5.62) -- (1.095,5.539) -- (1.014,5.418) -- (0.986,5.275) -- (1.014,5.132) -- (1.095,5.011) -- (1.216,4.93) -- (1.359,4.902) -- (1.502,4.93) -- (1.623,5.011) -- (1.704,5.132) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.732,5.112) -- (11.704,5.254) -- (11.623,5.376) -- (11.502,5.456) -- (11.359,5.485) -- (11.216,5.456) -- (11.095,5.376) -- (11.014,5.254) -- (10.986,5.112) -- (11.014,4.969) -- (11.095,4.848) -- (11.216,4.767) -- (11.359,4.738) -- (11.502,4.767) -- (11.623,4.848) -- (11.704,4.969) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.353,4.7) -- (5.325,4.843) -- (5.244,4.964) -- (5.123,5.045) -- (4.98,5.074) -- (4.837,5.045) -- (4.716,4.964) -- (4.635,4.843) -- (4.607,4.7) -- (4.635,4.558) -- (4.716,4.436) -- (4.837,4.356) -- (4.98,4.327) -- (5.123,4.356) -- (5.244,4.436) -- (5.325,4.558) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.984,4.575) -- (8.955,4.718) -- (8.874,4.839) -- (8.753,4.92) -- (8.61,4.948) -- (8.468,4.92) -- (8.346,4.839) -- (8.266,4.718) -- (8.237,4.575) -- (8.266,4.432) -- (8.346,4.311) -- (8.468,4.23) -- (8.61,4.202) -- (8.753,4.23) -- (8.874,4.311) -- (8.955,4.432) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.267,1.868) -- (8.238,2.011) -- (8.157,2.132) -- (8.036,2.213) -- (7.893,2.241) -- (7.751,2.213) -- (7.63,2.132) -- (7.549,2.011) -- (7.52,1.868) -- (7.549,1.725) -- (7.63,1.604) -- (7.751,1.523) -- (7.893,1.495) -- (8.036,1.523) -- (8.157,1.604) -- (8.238,1.725) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.269,1.434) -- (11.241,1.577) -- (11.16,1.698) -- (11.039,1.779) -- (10.896,1.808) -- (10.753,1.779) -- (10.632,1.698) -- (10.551,1.577) -- (10.523,1.434) -- (10.551,1.291) -- (10.632,1.17) -- (10.753,1.089) -- (10.896,1.061) -- (11.039,1.089) -- (11.16,1.17) -- (11.241,1.291) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.247,9.216) -- (8.366,9.235) -- (8.473,9.289) -- (8.558,9.375) -- (8.613,9.482) -- (8.632,9.601) -- (8.613,9.719) -- (8.558,9.827) -- (8.473,9.912) -- (8.366,9.966) -- (8.247,9.985) -- (8.247,9.985) -- (8.128,9.966) -- (8.021,9.912) -- (7.936,9.827) -- (7.881,9.719) -- (7.863,9.601) -- (7.881,9.482) -- (7.936,9.375) -- (8.021,9.289) -- (8.128,9.235) -- (8.247,9.216) -- cycle;
\node at (8.247,9.601) {\scalebox{1.614}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.33,11.045) -- (3.449,11.064) -- (3.556,11.119) -- (3.641,11.204) -- (3.696,11.311) -- (3.715,11.43) -- (3.696,11.549) -- (3.641,11.656) -- (3.556,11.741) -- (3.449,11.795) -- (3.33,11.814) -- (3.33,11.814) -- (3.211,11.795) -- (3.104,11.741) -- (3.019,11.656) -- (2.964,11.549) -- (2.945,11.43) -- (2.964,11.311) -- (3.019,11.204) -- (3.104,11.119) -- (3.211,11.064) -- (3.33,11.045) -- cycle;
\node at (3.33,11.43) {\scalebox{1.614}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.077,9.249) -- (2.196,9.267) -- (2.303,9.322) -- (2.388,9.407) -- (2.443,9.514) -- (2.462,9.633) -- (2.443,9.752) -- (2.388,9.859) -- (2.303,9.944) -- (2.196,9.999) -- (2.077,10.018) -- (2.077,10.018) -- (1.958,9.999) -- (1.851,9.944) -- (1.766,9.859) -- (1.712,9.752) -- (1.693,9.633) -- (1.712,9.514) -- (1.766,9.407) -- (1.851,9.322) -- (1.958,9.267) -- (2.077,9.249) -- cycle;
\node at (2.077,9.633) {\scalebox{1.614}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.336,11.118) -- (6.455,11.137) -- (6.562,11.191) -- (6.647,11.276) -- (6.702,11.384) -- (6.721,11.502) -- (6.702,11.621) -- (6.647,11.728) -- (6.562,11.813) -- (6.455,11.868) -- (6.336,11.887) -- (6.336,11.887) -- (6.217,11.868) -- (6.11,11.813) -- (6.025,11.728) -- (5.97,11.621) -- (5.952,11.502) -- (5.97,11.384) -- (6.025,11.276) -- (6.11,11.191) -- (6.217,11.137) -- (6.336,11.118) -- cycle;
\node at (6.336,11.502) {\scalebox{1.614}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.082,9.247) -- (5.201,9.266) -- (5.308,9.32) -- (5.393,9.405) -- (5.448,9.513) -- (5.467,9.631) -- (5.448,9.75) -- (5.393,9.857) -- (5.308,9.943) -- (5.201,9.997) -- (5.082,10.016) -- (5.082,10.016) -- (4.964,9.997) -- (4.856,9.943) -- (4.771,9.857) -- (4.717,9.75) -- (4.698,9.631) -- (4.717,9.513) -- (4.771,9.405) -- (4.856,9.32) -- (4.964,9.266) -- (5.082,9.247) -- cycle;
\node at (5.082,9.631) {\scalebox{1.614}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.471,8.38) -- (6.59,8.398) -- (6.697,8.453) -- (6.782,8.538) -- (6.837,8.645) -- (6.856,8.764) -- (6.837,8.883) -- (6.782,8.99) -- (6.697,9.075) -- (6.59,9.13) -- (6.471,9.149) -- (6.471,9.149) -- (6.352,9.13) -- (6.245,9.075) -- (6.16,8.99) -- (6.105,8.883) -- (6.087,8.764) -- (6.105,8.645) -- (6.16,8.538) -- (6.245,8.453) -- (6.352,8.398) -- (6.471,8.38) -- cycle;
\node at (6.471,8.764) {\scalebox{1.614}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.064,6.224) -- (5.183,6.242) -- (5.29,6.297) -- (5.375,6.382) -- (5.429,6.489) -- (5.448,6.608) -- (5.429,6.727) -- (5.375,6.834) -- (5.29,6.919) -- (5.183,6.974) -- (5.064,6.993) -- (5.064,6.993) -- (4.945,6.974) -- (4.838,6.919) -- (4.753,6.834) -- (4.698,6.727) -- (4.679,6.608) -- (4.698,6.489) -- (4.753,6.382) -- (4.838,6.297) -- (4.945,6.242) -- (5.064,6.224) -- cycle;
\node at (5.064,6.608) {\scalebox{1.614}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.256,8.469) -- (3.375,8.488) -- (3.482,8.543) -- (3.567,8.628) -- (3.621,8.735) -- (3.64,8.854) -- (3.621,8.973) -- (3.567,9.08) -- (3.482,9.165) -- (3.375,9.22) -- (3.256,9.238) -- (3.256,9.238) -- (3.137,9.22) -- (3.03,9.165) -- (2.945,9.08) -- (2.89,8.973) -- (2.871,8.854) -- (2.89,8.735) -- (2.945,8.628) -- (3.03,8.543) -- (3.137,8.488) -- (3.256,8.469) -- cycle;
\node at (3.256,8.854) {\scalebox{1.614}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.89,6.321) -- (2.009,6.34) -- (2.116,6.394) -- (2.201,6.479) -- (2.256,6.587) -- (2.274,6.705) -- (2.256,6.824) -- (2.201,6.931) -- (2.116,7.016) -- (2.009,7.071) -- (1.89,7.09) -- (1.89,7.09) -- (1.771,7.071) -- (1.664,7.016) -- (1.579,6.931) -- (1.524,6.824) -- (1.505,6.705) -- (1.524,6.587) -- (1.579,6.479) -- (1.664,6.394) -- (1.771,6.34) -- (1.89,6.321) -- cycle;
\node at (1.89,6.705) {\scalebox{1.614}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.993,6.272) -- (12.112,6.291) -- (12.219,6.345) -- (12.304,6.431) -- (12.359,6.538) -- (12.378,6.657) -- (12.359,6.775) -- (12.304,6.883) -- (12.219,6.968) -- (12.112,7.022) -- (11.993,7.041) -- (11.631,7.041) -- (11.512,7.022) -- (11.405,6.968) -- (11.32,6.883) -- (11.265,6.775) -- (11.247,6.657) -- (11.265,6.538) -- (11.32,6.431) -- (11.405,6.345) -- (11.512,6.291) -- (11.631,6.272) -- cycle;
\node at (11.812,6.657) {\scalebox{1.614}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.842,7.953) -- (9.961,7.972) -- (10.068,8.027) -- (10.153,8.112) -- (10.208,8.219) -- (10.227,8.338) -- (10.208,8.457) -- (10.153,8.564) -- (10.068,8.649) -- (9.961,8.704) -- (9.842,8.723) -- (9.48,8.723) -- (9.361,8.704) -- (9.254,8.649) -- (9.169,8.564) -- (9.114,8.457) -- (9.096,8.338) -- (9.114,8.219) -- (9.169,8.112) -- (9.254,8.027) -- (9.361,7.972) -- (9.48,7.953) -- cycle;
\node at (9.661,8.338) {\scalebox{1.614}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.06,5.982) -- (9.178,6.001) -- (9.286,6.055) -- (9.371,6.141) -- (9.425,6.248) -- (9.444,6.367) -- (9.425,6.485) -- (9.371,6.593) -- (9.286,6.678) -- (9.178,6.732) -- (9.06,6.751) -- (8.698,6.751) -- (8.579,6.732) -- (8.472,6.678) -- (8.387,6.593) -- (8.332,6.485) -- (8.313,6.367) -- (8.332,6.248) -- (8.387,6.141) -- (8.472,6.055) -- (8.579,6.001) -- (8.698,5.982) -- cycle;
\node at (8.879,6.367) {\scalebox{1.614}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.387,4.834) -- (3.506,4.852) -- (3.613,4.907) -- (3.698,4.992) -- (3.753,5.099) -- (3.772,5.218) -- (3.753,5.337) -- (3.698,5.444) -- (3.613,5.529) -- (3.506,5.584) -- (3.387,5.603) -- (3.025,5.603) -- (2.906,5.584) -- (2.799,5.529) -- (2.714,5.444) -- (2.659,5.337) -- (2.641,5.218) -- (2.659,5.099) -- (2.714,4.992) -- (2.799,4.907) -- (2.906,4.852) -- (3.025,4.834) -- cycle;
\node at (3.206,5.218) {\scalebox{1.614}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.995,4.787) -- (7.113,4.806) -- (7.221,4.861) -- (7.306,4.946) -- (7.36,5.053) -- (7.379,5.172) -- (7.36,5.29) -- (7.306,5.398) -- (7.221,5.483) -- (7.113,5.537) -- (6.995,5.556) -- (6.633,5.556) -- (6.514,5.537) -- (6.407,5.483) -- (6.322,5.398) -- (6.267,5.29) -- (6.248,5.172) -- (6.267,5.053) -- (6.322,4.946) -- (6.407,4.861) -- (6.514,4.806) -- (6.633,4.787) -- cycle;
\node at (6.814,5.172) {\scalebox{1.614}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.114,4.723) -- (10.233,4.742) -- (10.34,4.796) -- (10.425,4.881) -- (10.48,4.988) -- (10.499,5.107) -- (10.48,5.226) -- (10.425,5.333) -- (10.34,5.418) -- (10.233,5.473) -- (10.114,5.492) -- (9.752,5.492) -- (9.633,5.473) -- (9.526,5.418) -- (9.441,5.333) -- (9.387,5.226) -- (9.368,5.107) -- (9.387,4.988) -- (9.441,4.881) -- (9.526,4.796) -- (9.633,4.742) -- (9.752,4.723) -- cycle;
\node at (9.933,5.107) {\scalebox{1.614}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.066,2.537) -- (8.185,2.556) -- (8.292,2.61) -- (8.377,2.695) -- (8.432,2.803) -- (8.45,2.921) -- (8.432,3.04) -- (8.377,3.147) -- (8.292,3.233) -- (8.185,3.287) -- (8.066,3.306) -- (7.704,3.306) -- (7.585,3.287) -- (7.478,3.233) -- (7.393,3.147) -- (7.338,3.04) -- (7.319,2.921) -- (7.338,2.803) -- (7.393,2.695) -- (7.478,2.61) -- (7.585,2.556) -- (7.704,2.537) -- cycle;
\node at (7.885,2.921) {\scalebox{1.614}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.613,1.523) -- (9.731,1.542) -- (9.839,1.597) -- (9.924,1.682) -- (9.978,1.789) -- (9.997,1.908) -- (9.978,2.027) -- (9.924,2.134) -- (9.839,2.219) -- (9.731,2.273) -- (9.613,2.292) -- (9.251,2.292) -- (9.132,2.273) -- (9.025,2.219) -- (8.94,2.134) -- (8.885,2.027) -- (8.866,1.908) -- (8.885,1.789) -- (8.94,1.682) -- (9.025,1.597) -- (9.132,1.542) -- (9.251,1.523) -- cycle;
\node at (9.432,1.908) {\scalebox{1.614}{\small 17}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.393,8.181) -- (4.993,8.181) -- (4.993,8.781) -- (4.393,8.781) -- cycle;
\node at (4.693,8.481) {\scalebox{1.614}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.363,10.219) -- (8.963,10.219) -- (8.963,10.819) -- (8.363,10.819) -- cycle;
\node at (8.663,10.519) {\scalebox{1.614}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.695,3.167) -- (9.295,3.167) -- (9.295,3.767) -- (8.695,3.767) -- cycle;
\node at (8.995,3.467) {\scalebox{1.614}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.75,3.966) -- (6.35,3.966) -- (6.35,4.566) -- (5.75,4.566) -- cycle;
\node at (6.05,4.266) {\scalebox{1.614}{\small d}};
\node at (8.063,11.756) {\scalebox{1.614}{\small A}};
\node at (1.236,11.399) {\scalebox{1.614}{\small B}};
\node at (4.66,11.404) {\scalebox{1.614}{\small C}};
\node at (4.226,8.889) {\scalebox{1.614}{\small D}};
\node at (1.314,8.408) {\scalebox{1.614}{\small E}};
\node at (11.841,8.333) {\scalebox{1.614}{\small F}};
\node at (8.595,8.382) {\scalebox{1.614}{\small G}};
\node at (0.765,5.098) {\scalebox{1.614}{\small H}};
\node at (11.942,4.901) {\scalebox{1.614}{\small I}};
\node at (4.608,4.204) {\scalebox{1.614}{\small J}};
\node at (9.025,4.114) {\scalebox{1.614}{\small K}};
\node at (8.052,1.268) {\scalebox{1.614}{\small L}};
\node at (11.283,0.95) {\scalebox{1.614}{\small M}};
\draw[line width=0.5pt] (12.659,10.212) -- (12.659,11.312);
\draw[line width=0.5pt] (12.428,10.927) -- (12.659,11.312) -- (12.89,10.927);
\draw[line width=0.5pt] (12.357,10.762) -- (12.962,10.762);
\node at (12.659,11.697) {\scalebox{1.614}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & C, E, G, I, J, L & 6 \\ \hline
clairs & A, B, D, F, H, K, M & 7 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{7} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 13 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 6 carrefours sombres et 7 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{D} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend le \textbf{quai aux Fleurs ⑦} vers le sud jusqu'à Carrefour Saint-Roch \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Four ⑬}, vers l'ouest, jusqu'à Place Sainte-Croix \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Poste ⑨}, vers le nord, jusqu'à Rond-point des Vaches \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Lilas ③}, vers le nord, jusqu'à Place aux Herbes \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Puits ②}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Peupliers ④}, vers l'est, jusqu'à Place du Vieux-Puits \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{venelle du Chat ①}, vers le sud, jusqu'à Carrefour du Lavoir \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin du Lavoir ⑪}, vers l'est, jusqu'à Placette des Ormes \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Écoles ⑩}, vers le sud, jusqu'à Place de l'Église \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Mairie ⑮}, vers l'ouest, jusqu'à Carrefour des Tilleuls \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin de Ronde ⑯}, vers le sud, jusqu'à Place des Tanneurs \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse Sainte-Anne ⑰}, vers l'est, jusqu'à Place du Marché \textbf{(M)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 13 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
