\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 201}

\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=201}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0201}
\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=0.6162, line join=round, line cap=round]
\draw[line width=0.5pt] (7.187,11.567) -- (7.456,11.51) -- (7.726,11.46) -- (7.997,11.416) -- (8.269,11.378) -- (8.542,11.347) -- (8.816,11.322) -- (9.09,11.304) -- (9.365,11.292) -- (9.64,11.287) -- (9.915,11.288) -- (10.189,11.295) -- (10.464,11.309) -- (10.738,11.329) -- (11.011,11.356);
\draw[line width=0.5pt] (7.187,11.567) -- (7.27,11.355) -- (7.347,11.142) -- (7.42,10.926) -- (7.487,10.709) -- (7.55,10.49) -- (7.607,10.27) -- (7.659,10.049) -- (7.705,9.826) -- (7.747,9.602) -- (7.783,9.378) -- (7.814,9.153) -- (7.84,8.927) -- (7.86,8.7) -- (7.875,8.473);
\draw[line width=0.5pt] (1.288,11.541) -- (1.558,11.497) -- (1.829,11.459) -- (2.101,11.428) -- (2.373,11.403) -- (2.646,11.385) -- (2.92,11.373) -- (3.193,11.367) -- (3.467,11.368) -- (3.74,11.375) -- (4.014,11.389) -- (4.287,11.409) -- (4.559,11.435) -- (4.83,11.468) -- (5.101,11.508);
\draw[line width=0.5pt] (1.288,11.541) -- (1.342,11.33) -- (1.391,11.117) -- (1.435,10.903) -- (1.474,10.688) -- (1.508,10.472) -- (1.537,10.255) -- (1.561,10.038) -- (1.58,9.82) -- (1.593,9.602) -- (1.601,9.383) -- (1.605,9.165) -- (1.603,8.946) -- (1.596,8.728) -- (1.583,8.51);
\draw[line width=0.5pt] (5.101,11.508) -- (5.249,11.489) -- (5.398,11.474) -- (5.548,11.462) -- (5.697,11.454) -- (5.847,11.45) -- (5.997,11.449) -- (6.147,11.451) -- (6.296,11.457) -- (6.446,11.467) -- (6.595,11.48) -- (6.744,11.496) -- (6.892,11.517) -- (7.04,11.54) -- (7.187,11.567);
\draw[line width=0.5pt] (5.101,11.508) -- (5.09,11.243) -- (5.072,10.979) -- (5.048,10.715) -- (5.018,10.451) -- (4.982,10.189) -- (4.939,9.927) -- (4.89,9.666) -- (4.836,9.407) -- (4.775,9.149) -- (4.708,8.892) -- (4.635,8.637) -- (4.556,8.384) -- (4.472,8.133) -- (4.381,7.884);
\draw[line width=0.5pt] (11.011,11.356) -- (10.95,11.12) -- (10.895,10.882) -- (10.845,10.643) -- (10.801,10.402) -- (10.762,10.161) -- (10.73,9.918) -- (10.703,9.675) -- (10.681,9.432) -- (10.666,9.188) -- (10.656,8.944) -- (10.652,8.699) -- (10.653,8.455) -- (10.661,8.211) -- (10.674,7.967);
\draw[line width=0.5pt] (1.583,8.51) -- (1.789,8.496) -- (1.993,8.478) -- (2.198,8.454) -- (2.401,8.426) -- (2.604,8.393) -- (2.807,8.355) -- (3.008,8.312) -- (3.208,8.265) -- (3.407,8.213) -- (3.605,8.157) -- (3.802,8.095) -- (3.996,8.029) -- (4.19,7.959) -- (4.381,7.884);
\draw[line width=0.5pt] (7.875,8.473) -- (8.068,8.407) -- (8.262,8.345) -- (8.458,8.288) -- (8.655,8.235) -- (8.854,8.187) -- (9.053,8.143) -- (9.254,8.105) -- (9.455,8.071) -- (9.657,8.041) -- (9.86,8.017) -- (10.063,7.997) -- (10.266,7.982) -- (10.47,7.972) -- (10.674,7.967);
\draw[line width=0.5pt] (7.875,8.473) -- (7.928,8.235) -- (7.977,7.996) -- (8.019,7.756) -- (8.056,7.514) -- (8.087,7.272) -- (8.112,7.029) -- (8.132,6.786) -- (8.146,6.542) -- (8.154,6.298) -- (8.156,6.054) -- (8.153,5.81) -- (8.144,5.566) -- (8.129,5.322) -- (8.109,5.079);
\draw[line width=0.5pt] (10.674,7.967) -- (10.678,7.79) -- (10.686,7.613) -- (10.699,7.437) -- (10.715,7.26) -- (10.736,7.084) -- (10.76,6.909) -- (10.789,6.734) -- (10.822,6.56) -- (10.859,6.387) -- (10.9,6.215) -- (10.945,6.044) -- (10.994,5.873) -- (11.048,5.705) -- (11.105,5.537);
\draw[line width=0.5pt] (4.381,7.884) -- (4.622,7.964) -- (4.865,8.038) -- (5.11,8.107) -- (5.356,8.17) -- (5.604,8.227) -- (5.853,8.278) -- (6.104,8.323) -- (6.355,8.362) -- (6.607,8.396) -- (6.86,8.423) -- (7.113,8.445) -- (7.367,8.46) -- (7.621,8.47) -- (7.875,8.473);
\draw[line width=0.5pt] (11.105,5.537) -- (11.01,5.328) -- (10.92,5.117) -- (10.835,4.903) -- (10.755,4.688) -- (10.68,4.471) -- (10.611,4.252) -- (10.546,4.031) -- (10.487,3.809) -- (10.433,3.586) -- (10.384,3.361) -- (10.341,3.135) -- (10.302,2.909) -- (10.27,2.682) -- (10.242,2.454);
\draw[line width=0.5pt] (8.109,5.079) -- (8.316,5.144) -- (8.525,5.205) -- (8.735,5.26) -- (8.947,5.31) -- (9.159,5.356) -- (9.373,5.396) -- (9.588,5.432) -- (9.803,5.462) -- (10.019,5.487) -- (10.236,5.508) -- (10.453,5.523) -- (10.67,5.533) -- (10.887,5.538) -- (11.105,5.537);
\draw[line width=0.5pt] (8.109,5.079) -- (8.048,4.833) -- (7.993,4.587) -- (7.944,4.338) -- (7.901,4.089) -- (7.864,3.839) -- (7.833,3.588) -- (7.808,3.336) -- (7.788,3.084) -- (7.775,2.831) -- (7.767,2.578) -- (7.765,2.325) -- (7.77,2.072) -- (7.78,1.819) -- (7.796,1.567);
\draw[line width=0.5pt] (7.796,1.567) -- (7.96,1.657) -- (8.126,1.742) -- (8.294,1.824) -- (8.463,1.902) -- (8.635,1.976) -- (8.808,2.046) -- (8.983,2.111) -- (9.159,2.173) -- (9.337,2.23) -- (9.516,2.283) -- (9.696,2.332) -- (9.877,2.377) -- (10.059,2.418) -- (10.242,2.454);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.347,11.567) -- (7.326,11.647) -- (7.267,11.706) -- (7.187,11.727) -- (7.107,11.706) -- (7.049,11.647) -- (7.027,11.567) -- (7.049,11.487) -- (7.107,11.429) -- (7.187,11.407) -- (7.267,11.429) -- (7.326,11.487) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.448,11.541) -- (1.427,11.621) -- (1.368,11.68) -- (1.288,11.701) -- (1.208,11.68) -- (1.15,11.621) -- (1.128,11.541) -- (1.15,11.461) -- (1.208,11.403) -- (1.288,11.381) -- (1.368,11.403) -- (1.427,11.461) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.261,11.508) -- (5.239,11.588) -- (5.181,11.646) -- (5.101,11.668) -- (5.021,11.646) -- (4.962,11.588) -- (4.941,11.508) -- (4.962,11.428) -- (5.021,11.369) -- (5.101,11.348) -- (5.181,11.369) -- (5.239,11.428) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.171,11.356) -- (11.149,11.436) -- (11.091,11.495) -- (11.011,11.516) -- (10.931,11.495) -- (10.872,11.436) -- (10.851,11.356) -- (10.872,11.276) -- (10.931,11.218) -- (11.011,11.196) -- (11.091,11.218) -- (11.149,11.276) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.743,8.51) -- (1.722,8.59) -- (1.663,8.649) -- (1.583,8.67) -- (1.503,8.649) -- (1.445,8.59) -- (1.423,8.51) -- (1.445,8.43) -- (1.503,8.371) -- (1.583,8.35) -- (1.663,8.371) -- (1.722,8.43) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.035,8.473) -- (8.013,8.553) -- (7.955,8.612) -- (7.875,8.633) -- (7.795,8.612) -- (7.736,8.553) -- (7.715,8.473) -- (7.736,8.393) -- (7.795,8.335) -- (7.875,8.313) -- (7.955,8.335) -- (8.013,8.393) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.834,7.967) -- (10.813,8.047) -- (10.754,8.105) -- (10.674,8.127) -- (10.594,8.105) -- (10.536,8.047) -- (10.514,7.967) -- (10.536,7.887) -- (10.594,7.828) -- (10.674,7.807) -- (10.754,7.828) -- (10.813,7.887) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.541,7.884) -- (4.519,7.964) -- (4.461,8.023) -- (4.381,8.044) -- (4.301,8.023) -- (4.242,7.964) -- (4.221,7.884) -- (4.242,7.804) -- (4.301,7.745) -- (4.381,7.724) -- (4.461,7.745) -- (4.519,7.804) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.265,5.537) -- (11.243,5.617) -- (11.185,5.676) -- (11.105,5.697) -- (11.025,5.676) -- (10.966,5.617) -- (10.945,5.537) -- (10.966,5.457) -- (11.025,5.399) -- (11.105,5.377) -- (11.185,5.399) -- (11.243,5.457) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.269,5.079) -- (8.247,5.159) -- (8.189,5.217) -- (8.109,5.239) -- (8.029,5.217) -- (7.97,5.159) -- (7.949,5.079) -- (7.97,4.999) -- (8.029,4.94) -- (8.109,4.919) -- (8.189,4.94) -- (8.247,4.999) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.402,2.454) -- (10.381,2.534) -- (10.322,2.592) -- (10.242,2.614) -- (10.162,2.592) -- (10.104,2.534) -- (10.082,2.454) -- (10.104,2.374) -- (10.162,2.315) -- (10.242,2.294) -- (10.322,2.315) -- (10.381,2.374) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.956,1.567) -- (7.935,1.647) -- (7.876,1.706) -- (7.796,1.727) -- (7.716,1.706) -- (7.658,1.647) -- (7.636,1.567) -- (7.658,1.487) -- (7.716,1.429) -- (7.796,1.407) -- (7.876,1.429) -- (7.935,1.487) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.112,11.303) -- (9.234,11.323) -- (9.344,11.379) -- (9.431,11.466) -- (9.487,11.576) -- (9.506,11.697) -- (9.487,11.819) -- (9.431,11.929) -- (9.344,12.016) -- (9.234,12.072) -- (9.112,12.091) -- (9.112,12.091) -- (8.99,12.072) -- (8.88,12.016) -- (8.793,11.929) -- (8.737,11.819) -- (8.718,11.697) -- (8.737,11.576) -- (8.793,11.466) -- (8.88,11.379) -- (8.99,11.323) -- (9.112,11.303) -- cycle;
\node at (9.112,11.697) {\scalebox{1.623}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.043,9.74) -- (8.165,9.759) -- (8.275,9.815) -- (8.362,9.903) -- (8.418,10.012) -- (8.437,10.134) -- (8.418,10.256) -- (8.362,10.366) -- (8.275,10.453) -- (8.165,10.509) -- (8.043,10.528) -- (8.043,10.528) -- (7.921,10.509) -- (7.812,10.453) -- (7.724,10.366) -- (7.668,10.256) -- (7.649,10.134) -- (7.668,10.012) -- (7.724,9.903) -- (7.812,9.815) -- (7.921,9.759) -- (8.043,9.74) -- cycle;
\node at (8.043,10.134) {\scalebox{1.623}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.197,11.367) -- (3.318,11.386) -- (3.428,11.442) -- (3.515,11.53) -- (3.571,11.639) -- (3.591,11.761) -- (3.571,11.883) -- (3.515,11.993) -- (3.428,12.08) -- (3.318,12.136) -- (3.197,12.155) -- (3.197,12.155) -- (3.075,12.136) -- (2.965,12.08) -- (2.878,11.993) -- (2.822,11.883) -- (2.803,11.761) -- (2.822,11.639) -- (2.878,11.53) -- (2.965,11.442) -- (3.075,11.386) -- (3.197,11.367) -- cycle;
\node at (3.197,11.761) {\scalebox{1.623}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.953,9.682) -- (2.075,9.701) -- (2.184,9.757) -- (2.272,9.844) -- (2.328,9.954) -- (2.347,10.076) -- (2.328,10.198) -- (2.272,10.308) -- (2.184,10.395) -- (2.075,10.451) -- (1.953,10.47) -- (1.953,10.47) -- (1.831,10.451) -- (1.721,10.395) -- (1.634,10.308) -- (1.578,10.198) -- (1.559,10.076) -- (1.578,9.954) -- (1.634,9.844) -- (1.721,9.757) -- (1.831,9.701) -- (1.953,9.682) -- cycle;
\node at (1.953,10.076) {\scalebox{1.623}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.135,11.451) -- (6.257,11.47) -- (6.367,11.526) -- (6.454,11.614) -- (6.51,11.723) -- (6.529,11.845) -- (6.51,11.967) -- (6.454,12.077) -- (6.367,12.164) -- (6.257,12.22) -- (6.135,12.239) -- (6.135,12.239) -- (6.014,12.22) -- (5.904,12.164) -- (5.817,12.077) -- (5.761,11.967) -- (5.741,11.845) -- (5.761,11.723) -- (5.817,11.614) -- (5.904,11.526) -- (6.014,11.47) -- (6.135,11.451) -- cycle;
\node at (6.135,11.845) {\scalebox{1.623}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.277,9.195) -- (5.398,9.215) -- (5.508,9.271) -- (5.595,9.358) -- (5.651,9.468) -- (5.671,9.589) -- (5.651,9.711) -- (5.595,9.821) -- (5.508,9.908) -- (5.398,9.964) -- (5.277,9.983) -- (5.277,9.983) -- (5.155,9.964) -- (5.045,9.908) -- (4.958,9.821) -- (4.902,9.711) -- (4.883,9.589) -- (4.902,9.468) -- (4.958,9.358) -- (5.045,9.271) -- (5.155,9.215) -- (5.277,9.195) -- cycle;
\node at (5.277,9.589) {\scalebox{1.623}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.095,9.243) -- (11.216,9.262) -- (11.326,9.318) -- (11.413,9.405) -- (11.469,9.515) -- (11.489,9.637) -- (11.469,9.758) -- (11.413,9.868) -- (11.326,9.955) -- (11.216,10.011) -- (11.095,10.03) -- (11.095,10.03) -- (10.973,10.011) -- (10.863,9.955) -- (10.776,9.868) -- (10.72,9.758) -- (10.701,9.637) -- (10.72,9.515) -- (10.776,9.405) -- (10.863,9.318) -- (10.973,9.262) -- (11.095,9.243) -- cycle;
\node at (11.095,9.637) {\scalebox{1.623}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.094,8.303) -- (3.216,8.322) -- (3.326,8.378) -- (3.413,8.465) -- (3.469,8.575) -- (3.488,8.697) -- (3.469,8.819) -- (3.413,8.928) -- (3.326,9.016) -- (3.216,9.071) -- (3.094,9.091) -- (3.094,9.091) -- (2.972,9.071) -- (2.863,9.016) -- (2.775,8.928) -- (2.719,8.819) -- (2.7,8.697) -- (2.719,8.575) -- (2.775,8.465) -- (2.863,8.378) -- (2.972,8.322) -- (3.094,8.303) -- cycle;
\node at (3.094,8.697) {\scalebox{1.623}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.324,8.098) -- (9.445,8.118) -- (9.555,8.174) -- (9.642,8.261) -- (9.698,8.371) -- (9.718,8.492) -- (9.698,8.614) -- (9.642,8.724) -- (9.555,8.811) -- (9.445,8.867) -- (9.324,8.886) -- (9.324,8.886) -- (9.202,8.867) -- (9.092,8.811) -- (9.005,8.724) -- (8.949,8.614) -- (8.93,8.492) -- (8.949,8.371) -- (9.005,8.261) -- (9.092,8.174) -- (9.202,8.118) -- (9.324,8.098) -- cycle;
\node at (9.324,8.492) {\scalebox{1.623}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.71,6.419) -- (8.832,6.438) -- (8.942,6.494) -- (9.029,6.581) -- (9.085,6.691) -- (9.104,6.813) -- (9.085,6.935) -- (9.029,7.044) -- (8.942,7.132) -- (8.832,7.188) -- (8.71,7.207) -- (8.339,7.207) -- (8.218,7.188) -- (8.108,7.132) -- (8.021,7.044) -- (7.965,6.935) -- (7.945,6.813) -- (7.965,6.691) -- (8.021,6.581) -- (8.108,6.494) -- (8.218,6.438) -- (8.339,6.419) -- cycle;
\node at (8.525,6.813) {\scalebox{1.623}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.362,6.409) -- (11.484,6.428) -- (11.594,6.484) -- (11.681,6.571) -- (11.737,6.681) -- (11.756,6.803) -- (11.737,6.925) -- (11.681,7.035) -- (11.594,7.122) -- (11.484,7.178) -- (11.362,7.197) -- (10.992,7.197) -- (10.87,7.178) -- (10.76,7.122) -- (10.673,7.035) -- (10.617,6.925) -- (10.598,6.803) -- (10.617,6.681) -- (10.673,6.571) -- (10.76,6.484) -- (10.87,6.428) -- (10.992,6.409) -- cycle;
\node at (11.177,6.803) {\scalebox{1.623}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.223,8.317) -- (6.345,8.337) -- (6.455,8.393) -- (6.542,8.48) -- (6.598,8.59) -- (6.617,8.711) -- (6.598,8.833) -- (6.542,8.943) -- (6.455,9.03) -- (6.345,9.086) -- (6.223,9.105) -- (5.853,9.105) -- (5.731,9.086) -- (5.621,9.03) -- (5.534,8.943) -- (5.478,8.833) -- (5.459,8.711) -- (5.478,8.59) -- (5.534,8.48) -- (5.621,8.393) -- (5.731,8.337) -- (5.853,8.317) -- cycle;
\node at (6.038,8.711) {\scalebox{1.623}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.111,3.531) -- (11.233,3.55) -- (11.342,3.606) -- (11.43,3.693) -- (11.486,3.803) -- (11.505,3.925) -- (11.486,4.047) -- (11.43,4.156) -- (11.342,4.244) -- (11.233,4.3) -- (11.111,4.319) -- (10.74,4.319) -- (10.619,4.3) -- (10.509,4.244) -- (10.422,4.156) -- (10.366,4.047) -- (10.346,3.925) -- (10.366,3.803) -- (10.422,3.693) -- (10.509,3.606) -- (10.619,3.55) -- (10.74,3.531) -- cycle;
\node at (10.926,3.925) {\scalebox{1.623}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.714,5.427) -- (9.835,5.446) -- (9.945,5.502) -- (10.032,5.59) -- (10.088,5.699) -- (10.107,5.821) -- (10.088,5.943) -- (10.032,6.053) -- (9.945,6.14) -- (9.835,6.196) -- (9.714,6.215) -- (9.343,6.215) -- (9.221,6.196) -- (9.111,6.14) -- (9.024,6.053) -- (8.968,5.943) -- (8.949,5.821) -- (8.968,5.699) -- (9.024,5.59) -- (9.111,5.502) -- (9.221,5.446) -- (9.343,5.427) -- cycle;
\node at (9.528,5.821) {\scalebox{1.623}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.385,2.907) -- (8.507,2.926) -- (8.617,2.982) -- (8.704,3.07) -- (8.76,3.179) -- (8.779,3.301) -- (8.76,3.423) -- (8.704,3.533) -- (8.617,3.62) -- (8.507,3.676) -- (8.385,3.695) -- (8.015,3.695) -- (7.893,3.676) -- (7.783,3.62) -- (7.696,3.533) -- (7.64,3.423) -- (7.621,3.301) -- (7.64,3.179) -- (7.696,3.07) -- (7.783,2.982) -- (7.893,2.926) -- (8.015,2.907) -- cycle;
\node at (8.2,3.301) {\scalebox{1.623}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.302,1.347) -- (9.424,1.366) -- (9.534,1.422) -- (9.621,1.509) -- (9.677,1.619) -- (9.696,1.741) -- (9.677,1.863) -- (9.621,1.972) -- (9.534,2.06) -- (9.424,2.116) -- (9.302,2.135) -- (8.932,2.135) -- (8.81,2.116) -- (8.7,2.06) -- (8.613,1.972) -- (8.557,1.863) -- (8.538,1.741) -- (8.557,1.619) -- (8.613,1.509) -- (8.7,1.422) -- (8.81,1.366) -- (8.932,1.347) -- cycle;
\node at (9.117,1.741) {\scalebox{1.623}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (10.711,11.056) -- (11.311,11.056) -- (11.311,11.656) -- (10.711,11.656) -- cycle;
\node at (11.011,11.356) {\scalebox{1.623}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.904,10.253) -- (5.504,10.253) -- (5.504,10.853) -- (4.904,10.853) -- cycle;
\node at (5.204,10.553) {\scalebox{1.623}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.654,12.635) -- (9.254,12.635) -- (9.254,13.235) -- (8.654,13.235) -- cycle;
\node at (8.954,12.935) {\scalebox{1.623}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.943,10.86) -- (8.543,10.86) -- (8.543,11.46) -- (7.943,11.46) -- cycle;
\node at (8.243,11.16) {\scalebox{1.623}{\small d}};
\node at (7.186,12.187) {\scalebox{1.623}{\small A}};
\node at (0.765,11.874) {\scalebox{1.623}{\small B}};
\node at (4.797,12.048) {\scalebox{1.623}{\small C}};
\node at (11.464,11.78) {\scalebox{1.623}{\small D}};
\node at (0.969,8.589) {\scalebox{1.623}{\small E}};
\node at (8.311,8.914) {\scalebox{1.623}{\small F}};
\node at (11.293,7.999) {\scalebox{1.623}{\small G}};
\node at (3.761,7.905) {\scalebox{1.623}{\small H}};
\node at (11.642,5.228) {\scalebox{1.623}{\small I}};
\node at (8.307,4.491) {\scalebox{1.623}{\small J}};
\node at (10.55,1.915) {\scalebox{1.623}{\small K}};
\node at (7.856,0.95) {\scalebox{1.623}{\small L}};
\draw[line width=0.5pt] (12.405,10.617) -- (12.405,11.717);
\draw[line width=0.5pt] (12.174,11.332) -- (12.405,11.717) -- (12.636,11.332);
\draw[line width=0.5pt] (12.102,11.167) -- (12.707,11.167);
\node at (12.405,12.102) {\scalebox{1.623}{\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 & ⑨ rue de la Poste & Ⓐ Carrefour de l'Hôpital & Ⓙ Rond-point du Pont \\ \hline
① rue des Jardins & ⑩ rue Neuve & Ⓑ Fontaine Notre-Dame & Ⓚ Place du Vieux-Puits \\ \hline
② quai aux Fleurs & ⑪ rue du Puits & Ⓒ Rond-point de la Poste & Ⓛ Coin du Lavoir \\ \hline
③ sentier des Prés & ⑫ rue Haute & Ⓓ Placette des Ormes & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ chemin du Lavoir & ⑬ rue des Lilas & Ⓔ Place du Marché & \textbf{a} l'école \\ \hline
⑤ rue de la Mairie & ⑭ allée des Peupliers & Ⓕ Carrefour des Tilleuls & \textbf{b} le marché couvert \\ \hline
⑥ rue du Stade & ⑮ rue Verte & Ⓖ Carrefour de la Mairie & \textbf{c} le cimetière \\ \hline
⑦ chemin Creux & ⑯ avenue de la Gare & Ⓗ Carrefour du Lavoir & \textbf{d} le gymnase \\ \hline
⑧ rue de l'Église & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Rond-point de la Gare &  \\ \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=0.6312, line join=round, line cap=round]
\draw[line width=0.5pt] (7.187,11.567) -- (7.456,11.51) -- (7.726,11.46) -- (7.997,11.416) -- (8.269,11.378) -- (8.542,11.347) -- (8.816,11.322) -- (9.09,11.304) -- (9.365,11.292) -- (9.64,11.287) -- (9.915,11.288) -- (10.189,11.295) -- (10.464,11.309) -- (10.738,11.329) -- (11.011,11.356);
\draw[line width=0.5pt] (7.187,11.567) -- (7.27,11.355) -- (7.347,11.142) -- (7.42,10.926) -- (7.487,10.709) -- (7.55,10.49) -- (7.607,10.27) -- (7.659,10.049) -- (7.705,9.826) -- (7.747,9.602) -- (7.783,9.378) -- (7.814,9.153) -- (7.84,8.927) -- (7.86,8.7) -- (7.875,8.473);
\draw[line width=0.5pt] (1.288,11.541) -- (1.558,11.497) -- (1.829,11.459) -- (2.101,11.428) -- (2.373,11.403) -- (2.646,11.385) -- (2.92,11.373) -- (3.193,11.367) -- (3.467,11.368) -- (3.74,11.375) -- (4.014,11.389) -- (4.287,11.409) -- (4.559,11.435) -- (4.83,11.468) -- (5.101,11.508);
\draw[line width=0.5pt] (1.288,11.541) -- (1.342,11.33) -- (1.391,11.117) -- (1.435,10.903) -- (1.474,10.688) -- (1.508,10.472) -- (1.537,10.255) -- (1.561,10.038) -- (1.58,9.82) -- (1.593,9.602) -- (1.601,9.383) -- (1.605,9.165) -- (1.603,8.946) -- (1.596,8.728) -- (1.583,8.51);
\draw[line width=0.5pt] (5.101,11.508) -- (5.249,11.489) -- (5.398,11.474) -- (5.548,11.462) -- (5.697,11.454) -- (5.847,11.45) -- (5.997,11.449) -- (6.147,11.451) -- (6.296,11.457) -- (6.446,11.467) -- (6.595,11.48) -- (6.744,11.496) -- (6.892,11.517) -- (7.04,11.54) -- (7.187,11.567);
\draw[line width=0.5pt] (5.101,11.508) -- (5.09,11.243) -- (5.072,10.979) -- (5.048,10.715) -- (5.018,10.451) -- (4.982,10.189) -- (4.939,9.927) -- (4.89,9.666) -- (4.836,9.407) -- (4.775,9.149) -- (4.708,8.892) -- (4.635,8.637) -- (4.556,8.384) -- (4.472,8.133) -- (4.381,7.884);
\draw[line width=0.5pt] (11.011,11.356) -- (10.95,11.12) -- (10.895,10.882) -- (10.845,10.643) -- (10.801,10.402) -- (10.762,10.161) -- (10.73,9.918) -- (10.703,9.675) -- (10.681,9.432) -- (10.666,9.188) -- (10.656,8.944) -- (10.652,8.699) -- (10.653,8.455) -- (10.661,8.211) -- (10.674,7.967);
\draw[line width=0.5pt] (1.583,8.51) -- (1.789,8.496) -- (1.993,8.478) -- (2.198,8.454) -- (2.401,8.426) -- (2.604,8.393) -- (2.807,8.355) -- (3.008,8.312) -- (3.208,8.265) -- (3.407,8.213) -- (3.605,8.157) -- (3.802,8.095) -- (3.996,8.029) -- (4.19,7.959) -- (4.381,7.884);
\draw[line width=0.5pt] (7.875,8.473) -- (8.068,8.407) -- (8.262,8.345) -- (8.458,8.288) -- (8.655,8.235) -- (8.854,8.187) -- (9.053,8.143) -- (9.254,8.105) -- (9.455,8.071) -- (9.657,8.041) -- (9.86,8.017) -- (10.063,7.997) -- (10.266,7.982) -- (10.47,7.972) -- (10.674,7.967);
\draw[line width=0.5pt] (7.875,8.473) -- (7.928,8.235) -- (7.977,7.996) -- (8.019,7.756) -- (8.056,7.514) -- (8.087,7.272) -- (8.112,7.029) -- (8.132,6.786) -- (8.146,6.542) -- (8.154,6.298) -- (8.156,6.054) -- (8.153,5.81) -- (8.144,5.566) -- (8.129,5.322) -- (8.109,5.079);
\draw[line width=0.5pt] (10.674,7.967) -- (10.678,7.79) -- (10.686,7.613) -- (10.699,7.437) -- (10.715,7.26) -- (10.736,7.084) -- (10.76,6.909) -- (10.789,6.734) -- (10.822,6.56) -- (10.859,6.387) -- (10.9,6.215) -- (10.945,6.044) -- (10.994,5.873) -- (11.048,5.705) -- (11.105,5.537);
\draw[line width=0.5pt] (4.381,7.884) -- (4.622,7.964) -- (4.865,8.038) -- (5.11,8.107) -- (5.356,8.17) -- (5.604,8.227) -- (5.853,8.278) -- (6.104,8.323) -- (6.355,8.362) -- (6.607,8.396) -- (6.86,8.423) -- (7.113,8.445) -- (7.367,8.46) -- (7.621,8.47) -- (7.875,8.473);
\draw[line width=0.5pt] (11.105,5.537) -- (11.01,5.328) -- (10.92,5.117) -- (10.835,4.903) -- (10.755,4.688) -- (10.68,4.471) -- (10.611,4.252) -- (10.546,4.031) -- (10.487,3.809) -- (10.433,3.586) -- (10.384,3.361) -- (10.341,3.135) -- (10.302,2.909) -- (10.27,2.682) -- (10.242,2.454);
\draw[line width=0.5pt] (8.109,5.079) -- (8.316,5.144) -- (8.525,5.205) -- (8.735,5.26) -- (8.947,5.31) -- (9.159,5.356) -- (9.373,5.396) -- (9.588,5.432) -- (9.803,5.462) -- (10.019,5.487) -- (10.236,5.508) -- (10.453,5.523) -- (10.67,5.533) -- (10.887,5.538) -- (11.105,5.537);
\draw[line width=0.5pt] (8.109,5.079) -- (8.048,4.833) -- (7.993,4.587) -- (7.944,4.338) -- (7.901,4.089) -- (7.864,3.839) -- (7.833,3.588) -- (7.808,3.336) -- (7.788,3.084) -- (7.775,2.831) -- (7.767,2.578) -- (7.765,2.325) -- (7.77,2.072) -- (7.78,1.819) -- (7.796,1.567);
\draw[line width=0.5pt] (7.796,1.567) -- (7.96,1.657) -- (8.126,1.742) -- (8.294,1.824) -- (8.463,1.902) -- (8.635,1.976) -- (8.808,2.046) -- (8.983,2.111) -- (9.159,2.173) -- (9.337,2.23) -- (9.516,2.283) -- (9.696,2.332) -- (9.877,2.377) -- (10.059,2.418) -- (10.242,2.454);
\filldraw[fill=white, draw=black, line width=0.5pt] (7.56,11.567) -- (7.532,11.71) -- (7.451,11.831) -- (7.33,11.912) -- (7.187,11.94) -- (7.044,11.912) -- (6.923,11.831) -- (6.842,11.71) -- (6.814,11.567) -- (6.842,11.424) -- (6.923,11.303) -- (7.044,11.222) -- (7.187,11.194) -- (7.33,11.222) -- (7.451,11.303) -- (7.532,11.424) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.661,11.541) -- (1.633,11.684) -- (1.552,11.805) -- (1.431,11.886) -- (1.288,11.914) -- (1.145,11.886) -- (1.024,11.805) -- (0.943,11.684) -- (0.915,11.541) -- (0.943,11.398) -- (1.024,11.277) -- (1.145,11.196) -- (1.288,11.168) -- (1.431,11.196) -- (1.552,11.277) -- (1.633,11.398) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.474,11.508) -- (5.446,11.65) -- (5.365,11.772) -- (5.244,11.852) -- (5.101,11.881) -- (4.958,11.852) -- (4.837,11.772) -- (4.756,11.65) -- (4.728,11.508) -- (4.756,11.365) -- (4.837,11.244) -- (4.958,11.163) -- (5.101,11.134) -- (5.244,11.163) -- (5.365,11.244) -- (5.446,11.365) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.384,11.356) -- (11.356,11.499) -- (11.275,11.62) -- (11.154,11.701) -- (11.011,11.729) -- (10.868,11.701) -- (10.747,11.62) -- (10.666,11.499) -- (10.638,11.356) -- (10.666,11.213) -- (10.747,11.092) -- (10.868,11.011) -- (11.011,10.983) -- (11.154,11.011) -- (11.275,11.092) -- (11.356,11.213) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.957,8.51) -- (1.928,8.653) -- (1.847,8.774) -- (1.726,8.855) -- (1.583,8.883) -- (1.441,8.855) -- (1.32,8.774) -- (1.239,8.653) -- (1.21,8.51) -- (1.239,8.367) -- (1.32,8.246) -- (1.441,8.165) -- (1.583,8.137) -- (1.726,8.165) -- (1.847,8.246) -- (1.928,8.367) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.248,8.473) -- (8.22,8.616) -- (8.139,8.737) -- (8.018,8.818) -- (7.875,8.847) -- (7.732,8.818) -- (7.611,8.737) -- (7.53,8.616) -- (7.502,8.473) -- (7.53,8.331) -- (7.611,8.21) -- (7.732,8.129) -- (7.875,8.1) -- (8.018,8.129) -- (8.139,8.21) -- (8.22,8.331) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.047,7.967) -- (11.019,8.11) -- (10.938,8.231) -- (10.817,8.312) -- (10.674,8.34) -- (10.531,8.312) -- (10.41,8.231) -- (10.329,8.11) -- (10.301,7.967) -- (10.329,7.824) -- (10.41,7.703) -- (10.531,7.622) -- (10.674,7.594) -- (10.817,7.622) -- (10.938,7.703) -- (11.019,7.824) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.754,7.884) -- (4.726,8.027) -- (4.645,8.148) -- (4.524,8.229) -- (4.381,8.257) -- (4.238,8.229) -- (4.117,8.148) -- (4.036,8.027) -- (4.008,7.884) -- (4.036,7.741) -- (4.117,7.62) -- (4.238,7.539) -- (4.381,7.511) -- (4.524,7.539) -- (4.645,7.62) -- (4.726,7.741) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.478,5.537) -- (11.449,5.68) -- (11.369,5.801) -- (11.247,5.882) -- (11.105,5.91) -- (10.962,5.882) -- (10.841,5.801) -- (10.76,5.68) -- (10.731,5.537) -- (10.76,5.394) -- (10.841,5.273) -- (10.962,5.192) -- (11.105,5.164) -- (11.247,5.192) -- (11.369,5.273) -- (11.449,5.394) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.482,5.079) -- (8.453,5.222) -- (8.372,5.343) -- (8.251,5.424) -- (8.109,5.452) -- (7.966,5.424) -- (7.845,5.343) -- (7.764,5.222) -- (7.735,5.079) -- (7.764,4.936) -- (7.845,4.815) -- (7.966,4.734) -- (8.109,4.706) -- (8.251,4.734) -- (8.372,4.815) -- (8.453,4.936) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.615,2.454) -- (10.587,2.596) -- (10.506,2.718) -- (10.385,2.798) -- (10.242,2.827) -- (10.099,2.798) -- (9.978,2.718) -- (9.897,2.596) -- (9.869,2.454) -- (9.897,2.311) -- (9.978,2.19) -- (10.099,2.109) -- (10.242,2.08) -- (10.385,2.109) -- (10.506,2.19) -- (10.587,2.311) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.17,1.567) -- (8.141,1.71) -- (8.06,1.831) -- (7.939,1.912) -- (7.796,1.94) -- (7.654,1.912) -- (7.533,1.831) -- (7.452,1.71) -- (7.423,1.567) -- (7.452,1.424) -- (7.533,1.303) -- (7.654,1.222) -- (7.796,1.194) -- (7.939,1.222) -- (8.06,1.303) -- (8.141,1.424) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.111,11.303) -- (9.23,11.322) -- (9.338,11.377) -- (9.423,11.462) -- (9.477,11.569) -- (9.496,11.688) -- (9.477,11.807) -- (9.423,11.914) -- (9.338,11.999) -- (9.23,12.054) -- (9.111,12.072) -- (9.111,12.072) -- (8.993,12.054) -- (8.885,11.999) -- (8.8,11.914) -- (8.746,11.807) -- (8.727,11.688) -- (8.746,11.569) -- (8.8,11.462) -- (8.885,11.377) -- (8.993,11.322) -- (9.111,11.303) -- cycle;
\node at (9.111,11.688) {\scalebox{1.584}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.034,9.748) -- (8.153,9.766) -- (8.26,9.821) -- (8.345,9.906) -- (8.4,10.013) -- (8.418,10.132) -- (8.4,10.251) -- (8.345,10.358) -- (8.26,10.443) -- (8.153,10.498) -- (8.034,10.517) -- (8.034,10.517) -- (7.915,10.498) -- (7.808,10.443) -- (7.723,10.358) -- (7.668,10.251) -- (7.649,10.132) -- (7.668,10.013) -- (7.723,9.906) -- (7.808,9.821) -- (7.915,9.766) -- (8.034,9.748) -- cycle;
\node at (8.034,10.132) {\scalebox{1.584}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.197,11.367) -- (3.315,11.386) -- (3.423,11.441) -- (3.508,11.526) -- (3.562,11.633) -- (3.581,11.752) -- (3.562,11.871) -- (3.508,11.978) -- (3.423,12.063) -- (3.315,12.117) -- (3.197,12.136) -- (3.197,12.136) -- (3.078,12.117) -- (2.971,12.063) -- (2.886,11.978) -- (2.831,11.871) -- (2.812,11.752) -- (2.831,11.633) -- (2.886,11.526) -- (2.971,11.441) -- (3.078,11.386) -- (3.197,11.367) -- cycle;
\node at (3.197,11.752) {\scalebox{1.584}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.944,9.691) -- (2.062,9.709) -- (2.17,9.764) -- (2.255,9.849) -- (2.309,9.956) -- (2.328,10.075) -- (2.309,10.194) -- (2.255,10.301) -- (2.17,10.386) -- (2.062,10.441) -- (1.944,10.46) -- (1.944,10.46) -- (1.825,10.441) -- (1.718,10.386) -- (1.633,10.301) -- (1.578,10.194) -- (1.559,10.075) -- (1.578,9.956) -- (1.633,9.849) -- (1.718,9.764) -- (1.825,9.709) -- (1.944,9.691) -- cycle;
\node at (1.944,10.075) {\scalebox{1.584}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.136,11.451) -- (6.254,11.47) -- (6.362,11.525) -- (6.447,11.61) -- (6.501,11.717) -- (6.52,11.836) -- (6.501,11.955) -- (6.447,12.062) -- (6.362,12.147) -- (6.254,12.201) -- (6.136,12.22) -- (6.136,12.22) -- (6.017,12.201) -- (5.91,12.147) -- (5.824,12.062) -- (5.77,11.955) -- (5.751,11.836) -- (5.77,11.717) -- (5.824,11.61) -- (5.91,11.525) -- (6.017,11.47) -- (6.136,11.451) -- cycle;
\node at (6.136,11.836) {\scalebox{1.584}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.268,9.207) -- (5.386,9.225) -- (5.494,9.28) -- (5.579,9.365) -- (5.633,9.472) -- (5.652,9.591) -- (5.633,9.71) -- (5.579,9.817) -- (5.494,9.902) -- (5.386,9.957) -- (5.268,9.976) -- (5.268,9.976) -- (5.149,9.957) -- (5.042,9.902) -- (4.956,9.817) -- (4.902,9.71) -- (4.883,9.591) -- (4.902,9.472) -- (4.956,9.365) -- (5.042,9.28) -- (5.149,9.225) -- (5.268,9.207) -- cycle;
\node at (5.268,9.591) {\scalebox{1.584}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.085,9.253) -- (11.204,9.272) -- (11.311,9.326) -- (11.396,9.411) -- (11.451,9.519) -- (11.47,9.637) -- (11.451,9.756) -- (11.396,9.863) -- (11.311,9.949) -- (11.204,10.003) -- (11.085,10.022) -- (11.085,10.022) -- (10.966,10.003) -- (10.859,9.949) -- (10.774,9.863) -- (10.72,9.756) -- (10.701,9.637) -- (10.72,9.519) -- (10.774,9.411) -- (10.859,9.326) -- (10.966,9.272) -- (11.085,9.253) -- cycle;
\node at (11.085,9.637) {\scalebox{1.584}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.092,8.303) -- (3.211,8.322) -- (3.318,8.377) -- (3.403,8.462) -- (3.458,8.569) -- (3.477,8.688) -- (3.458,8.807) -- (3.403,8.914) -- (3.318,8.999) -- (3.211,9.053) -- (3.092,9.072) -- (3.092,9.072) -- (2.973,9.053) -- (2.866,8.999) -- (2.781,8.914) -- (2.726,8.807) -- (2.707,8.688) -- (2.726,8.569) -- (2.781,8.462) -- (2.866,8.377) -- (2.973,8.322) -- (3.092,8.303) -- cycle;
\node at (3.092,8.688) {\scalebox{1.584}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.322,8.099) -- (9.441,8.117) -- (9.548,8.172) -- (9.633,8.257) -- (9.688,8.364) -- (9.707,8.483) -- (9.688,8.602) -- (9.633,8.709) -- (9.548,8.794) -- (9.441,8.849) -- (9.322,8.868) -- (9.322,8.868) -- (9.203,8.849) -- (9.096,8.794) -- (9.011,8.709) -- (8.956,8.602) -- (8.937,8.483) -- (8.956,8.364) -- (9.011,8.257) -- (9.096,8.172) -- (9.203,8.117) -- (9.322,8.099) -- cycle;
\node at (9.322,8.483) {\scalebox{1.584}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.696,6.428) -- (8.815,6.447) -- (8.922,6.501) -- (9.007,6.586) -- (9.062,6.693) -- (9.081,6.812) -- (9.062,6.931) -- (9.007,7.038) -- (8.922,7.123) -- (8.815,7.178) -- (8.696,7.197) -- (8.334,7.197) -- (8.216,7.178) -- (8.108,7.123) -- (8.023,7.038) -- (7.969,6.931) -- (7.95,6.812) -- (7.969,6.693) -- (8.023,6.586) -- (8.108,6.501) -- (8.216,6.447) -- (8.334,6.428) -- cycle;
\node at (8.515,6.812) {\scalebox{1.584}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.349,6.417) -- (11.468,6.436) -- (11.575,6.49) -- (11.66,6.575) -- (11.714,6.683) -- (11.733,6.801) -- (11.714,6.92) -- (11.66,7.027) -- (11.575,7.112) -- (11.468,7.167) -- (11.349,7.186) -- (10.987,7.186) -- (10.868,7.167) -- (10.761,7.112) -- (10.676,7.027) -- (10.621,6.92) -- (10.602,6.801) -- (10.621,6.683) -- (10.676,6.575) -- (10.761,6.49) -- (10.868,6.436) -- (10.987,6.417) -- cycle;
\node at (11.168,6.801) {\scalebox{1.584}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.221,8.318) -- (6.339,8.336) -- (6.447,8.391) -- (6.532,8.476) -- (6.586,8.583) -- (6.605,8.702) -- (6.586,8.821) -- (6.532,8.928) -- (6.447,9.013) -- (6.339,9.068) -- (6.221,9.087) -- (5.859,9.087) -- (5.74,9.068) -- (5.633,9.013) -- (5.548,8.928) -- (5.493,8.821) -- (5.474,8.702) -- (5.493,8.583) -- (5.548,8.476) -- (5.633,8.391) -- (5.74,8.336) -- (5.859,8.318) -- cycle;
\node at (6.04,8.702) {\scalebox{1.584}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.098,3.543) -- (11.216,3.562) -- (11.324,3.616) -- (11.409,3.701) -- (11.463,3.809) -- (11.482,3.927) -- (11.463,4.046) -- (11.409,4.153) -- (11.324,4.239) -- (11.216,4.293) -- (11.098,4.312) -- (10.736,4.312) -- (10.617,4.293) -- (10.51,4.239) -- (10.425,4.153) -- (10.37,4.046) -- (10.351,3.927) -- (10.37,3.809) -- (10.425,3.701) -- (10.51,3.616) -- (10.617,3.562) -- (10.736,3.543) -- cycle;
\node at (10.917,3.927) {\scalebox{1.584}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.711,5.427) -- (9.829,5.446) -- (9.937,5.501) -- (10.022,5.586) -- (10.076,5.693) -- (10.095,5.812) -- (10.076,5.931) -- (10.022,6.038) -- (9.937,6.123) -- (9.829,6.177) -- (9.711,6.196) -- (9.349,6.196) -- (9.23,6.177) -- (9.123,6.123) -- (9.038,6.038) -- (8.983,5.931) -- (8.964,5.812) -- (8.983,5.693) -- (9.038,5.586) -- (9.123,5.501) -- (9.23,5.446) -- (9.349,5.427) -- cycle;
\node at (9.53,5.812) {\scalebox{1.584}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.372,2.917) -- (8.49,2.936) -- (8.598,2.991) -- (8.683,3.076) -- (8.737,3.183) -- (8.756,3.302) -- (8.737,3.421) -- (8.683,3.528) -- (8.598,3.613) -- (8.49,3.668) -- (8.372,3.686) -- (8.01,3.686) -- (7.891,3.668) -- (7.784,3.613) -- (7.699,3.528) -- (7.644,3.421) -- (7.625,3.302) -- (7.644,3.183) -- (7.699,3.076) -- (7.784,2.991) -- (7.891,2.936) -- (8.01,2.917) -- cycle;
\node at (8.191,3.302) {\scalebox{1.584}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.295,1.365) -- (9.414,1.384) -- (9.521,1.439) -- (9.606,1.524) -- (9.66,1.631) -- (9.679,1.75) -- (9.66,1.869) -- (9.606,1.976) -- (9.521,2.061) -- (9.414,2.115) -- (9.295,2.134) -- (8.933,2.134) -- (8.814,2.115) -- (8.707,2.061) -- (8.622,1.976) -- (8.567,1.869) -- (8.548,1.75) -- (8.567,1.631) -- (8.622,1.524) -- (8.707,1.439) -- (8.814,1.384) -- (8.933,1.365) -- cycle;
\node at (9.114,1.75) {\scalebox{1.584}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (10.711,11.056) -- (11.311,11.056) -- (11.311,11.656) -- (10.711,11.656) -- cycle;
\node at (11.011,11.356) {\scalebox{1.584}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.904,10.253) -- (5.504,10.253) -- (5.504,10.853) -- (4.904,10.853) -- cycle;
\node at (5.204,10.553) {\scalebox{1.584}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.654,12.635) -- (9.254,12.635) -- (9.254,13.235) -- (8.654,13.235) -- cycle;
\node at (8.954,12.935) {\scalebox{1.584}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.943,10.86) -- (8.543,10.86) -- (8.543,11.46) -- (7.943,11.46) -- cycle;
\node at (8.243,11.16) {\scalebox{1.584}{\small d}};
\node at (7.186,12.187) {\scalebox{1.584}{\small A}};
\node at (0.765,11.874) {\scalebox{1.584}{\small B}};
\node at (4.797,12.048) {\scalebox{1.584}{\small C}};
\node at (11.464,11.78) {\scalebox{1.584}{\small D}};
\node at (0.969,8.589) {\scalebox{1.584}{\small E}};
\node at (8.311,8.914) {\scalebox{1.584}{\small F}};
\node at (11.293,7.999) {\scalebox{1.584}{\small G}};
\node at (3.761,7.905) {\scalebox{1.584}{\small H}};
\node at (11.642,5.228) {\scalebox{1.584}{\small I}};
\node at (8.307,4.491) {\scalebox{1.584}{\small J}};
\node at (10.55,1.915) {\scalebox{1.584}{\small K}};
\node at (7.856,0.95) {\scalebox{1.584}{\small L}};
\draw[line width=0.5pt] (12.405,10.617) -- (12.405,11.717);
\draw[line width=0.5pt] (12.174,11.332) -- (12.405,11.717) -- (12.636,11.332);
\draw[line width=0.5pt] (12.102,11.167) -- (12.707,11.167);
\node at (12.405,12.102) {\scalebox{1.584}{\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, D, E, F, I, L & 6 \\ \hline
clairs & A, B, G, H, 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{Oui.} Un circuit existe :}\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 des Jardins ①} vers l'ouest jusqu'à Carrefour de l'Hôpital \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de la Mairie ⑤}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{sentier des Prés ③}, vers l'ouest, jusqu'à Fontaine Notre-Dame \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin du Lavoir ④}, vers le sud, jusqu'à Place du Marché \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de l'Église ⑧}, vers l'est, jusqu'à Carrefour du Lavoir \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Haute ⑫}, vers l'est, jusqu'à Carrefour des Tilleuls \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Neuve ⑩}, vers le sud, jusqu'à Rond-point du Pont \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Verte ⑮}, vers le sud, jusqu'à Coin du Lavoir \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{avenue de la Gare ⑯}, vers l'est, jusqu'à Place du Vieux-Puits \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ⑬}, vers le nord, jusqu'à Rond-point de la Gare \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ⑪}, vers le nord, jusqu'à Carrefour de la Mairie \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin Creux ⑦}, vers le nord, jusqu'à Placette des Ormes \textbf{(D)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Le car est passé une fois par chaque carrefour et se retrouve à l'école.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, et c'est déjà fait : le trajet ci-dessus dessert tous les carrefours. Il suffit au car de s'arrêter avant le dernier trajet de retour.}\par\addvspace{0.4em}
\end{document}
