\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 241}

\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=241}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0241}
\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=0.604, line join=round, line cap=round]
\draw[line width=0.5pt] (4.484,10.711) -- (4.727,10.617) -- (4.971,10.53) -- (5.218,10.448) -- (5.466,10.372) -- (5.716,10.302) -- (5.968,10.237) -- (6.221,10.179) -- (6.476,10.127) -- (6.732,10.08) -- (6.988,10.04) -- (7.246,10.006) -- (7.504,9.977) -- (7.763,9.955) -- (8.022,9.939);
\draw[line width=0.5pt] (4.484,10.711) -- (4.515,10.49) -- (4.539,10.269) -- (4.559,10.046) -- (4.574,9.824) -- (4.583,9.601) -- (4.587,9.378) -- (4.586,9.155) -- (4.579,8.932) -- (4.568,8.709) -- (4.551,8.487) -- (4.528,8.265) -- (4.501,8.044) -- (4.469,7.824) -- (4.431,7.604);
\draw[line width=0.5pt] (11.375,10.269) -- (11.344,10.077) -- (11.319,9.885) -- (11.298,9.691) -- (11.281,9.497) -- (11.269,9.303) -- (11.261,9.109) -- (11.259,8.914) -- (11.26,8.719) -- (11.266,8.525) -- (11.277,8.331) -- (11.293,8.137) -- (11.313,7.943) -- (11.337,7.75) -- (11.366,7.558);
\draw[line width=0.5pt] (8.022,9.939) -- (8.256,9.999) -- (8.491,10.054) -- (8.728,10.103) -- (8.966,10.146) -- (9.205,10.184) -- (9.444,10.216) -- (9.685,10.243) -- (9.926,10.263) -- (10.167,10.279) -- (10.409,10.288) -- (10.65,10.292) -- (10.892,10.29) -- (11.133,10.283) -- (11.375,10.269);
\draw[line width=0.5pt] (8.022,9.939) -- (7.999,9.769) -- (7.98,9.599) -- (7.965,9.428) -- (7.954,9.257) -- (7.947,9.085) -- (7.944,8.914) -- (7.946,8.742) -- (7.951,8.57) -- (7.96,8.399) -- (7.973,8.228) -- (7.991,8.057) -- (8.012,7.887) -- (8.037,7.717) -- (8.066,7.548);
\draw[line width=0.5pt] (4.431,7.604) -- (4.689,7.64) -- (4.948,7.669) -- (5.208,7.693) -- (5.468,7.71) -- (5.729,7.722) -- (5.99,7.727) -- (6.251,7.726) -- (6.512,7.719) -- (6.772,7.706) -- (7.033,7.686) -- (7.292,7.661) -- (7.551,7.63) -- (7.809,7.592) -- (8.066,7.548);
\draw[line width=0.5pt] (4.431,7.604) -- (4.395,7.341) -- (4.366,7.077) -- (4.343,6.813) -- (4.326,6.547) -- (4.315,6.282) -- (4.311,6.016) -- (4.312,5.751) -- (4.32,5.485) -- (4.335,5.22) -- (4.355,4.955) -- (4.382,4.691) -- (4.415,4.427) -- (4.454,4.164) -- (4.499,3.903);
\draw[line width=0.5pt] (8.066,7.548) -- (8.3,7.585) -- (8.535,7.616) -- (8.77,7.642) -- (9.006,7.662) -- (9.242,7.677) -- (9.479,7.686) -- (9.716,7.689) -- (9.953,7.687) -- (10.189,7.679) -- (10.426,7.666) -- (10.662,7.648) -- (10.897,7.623) -- (11.132,7.593) -- (11.366,7.558);
\draw[line width=0.5pt] (8.066,7.548) -- (8.018,7.281) -- (7.976,7.013) -- (7.94,6.744) -- (7.911,6.474) -- (7.888,6.204) -- (7.872,5.933) -- (7.861,5.661) -- (7.858,5.39) -- (7.86,5.118) -- (7.869,4.847) -- (7.884,4.576) -- (7.906,4.305) -- (7.934,4.035) -- (7.969,3.766);
\draw[line width=0.5pt] (1.375,7.176) -- (1.587,7.24) -- (1.8,7.298) -- (2.015,7.352) -- (2.231,7.401) -- (2.448,7.444) -- (2.666,7.483) -- (2.885,7.516) -- (3.105,7.544) -- (3.325,7.567) -- (3.546,7.585) -- (3.767,7.598) -- (3.988,7.605) -- (4.209,7.607) -- (4.431,7.604);
\draw[line width=0.5pt] (1.375,7.176) -- (1.382,6.979) -- (1.394,6.782) -- (1.411,6.585) -- (1.432,6.388) -- (1.458,6.193) -- (1.488,5.997) -- (1.523,5.803) -- (1.563,5.609) -- (1.607,5.417) -- (1.656,5.225) -- (1.709,5.035) -- (1.767,4.846) -- (1.829,4.659) -- (1.895,4.473);
\draw[line width=0.5pt] (1.895,4.473) -- (2.086,4.461) -- (2.276,4.445) -- (2.467,4.424) -- (2.656,4.398) -- (2.845,4.368) -- (3.033,4.334) -- (3.221,4.295) -- (3.407,4.252) -- (3.593,4.205) -- (3.777,4.153) -- (3.96,4.097) -- (4.141,4.036) -- (4.321,3.972) -- (4.499,3.903);
\draw[line width=0.5pt] (1.895,4.473) -- (1.841,4.277) -- (1.791,4.08) -- (1.745,3.882) -- (1.705,3.682) -- (1.669,3.482) -- (1.638,3.281) -- (1.611,3.079) -- (1.589,2.877) -- (1.573,2.674) -- (1.56,2.471) -- (1.553,2.268) -- (1.55,2.064) -- (1.553,1.861) -- (1.56,1.658);
\draw[line width=0.5pt] (4.499,3.903) -- (4.747,3.931) -- (4.995,3.954) -- (5.243,3.97) -- (5.492,3.981) -- (5.741,3.986) -- (5.991,3.984) -- (6.24,3.978) -- (6.489,3.965) -- (6.737,3.946) -- (6.985,3.922) -- (7.232,3.892) -- (7.479,3.856) -- (7.724,3.814) -- (7.969,3.766);
\draw[line width=0.5pt] (4.499,3.903) -- (4.522,3.73) -- (4.549,3.557) -- (4.581,3.385) -- (4.616,3.214) -- (4.655,3.043) -- (4.698,2.874) -- (4.745,2.705) -- (4.796,2.538) -- (4.851,2.372) -- (4.91,2.207) -- (4.973,2.044) -- (5.04,1.882) -- (5.11,1.722) -- (5.184,1.564);
\draw[line width=0.5pt] (1.56,1.658) -- (1.817,1.691) -- (2.076,1.718) -- (2.335,1.739) -- (2.595,1.753) -- (2.855,1.762) -- (3.115,1.764) -- (3.376,1.761) -- (3.636,1.751) -- (3.896,1.735) -- (4.155,1.713) -- (4.414,1.685) -- (4.671,1.651) -- (4.928,1.61) -- (5.184,1.564);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.644,10.711) -- (4.623,10.791) -- (4.564,10.849) -- (4.484,10.871) -- (4.404,10.849) -- (4.346,10.791) -- (4.324,10.711) -- (4.346,10.631) -- (4.404,10.572) -- (4.484,10.551) -- (4.564,10.572) -- (4.623,10.631) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.535,10.269) -- (11.513,10.349) -- (11.455,10.408) -- (11.375,10.429) -- (11.295,10.408) -- (11.236,10.349) -- (11.215,10.269) -- (11.236,10.189) -- (11.295,10.131) -- (11.375,10.109) -- (11.455,10.131) -- (11.513,10.189) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.182,9.939) -- (8.161,10.019) -- (8.102,10.078) -- (8.022,10.099) -- (7.942,10.078) -- (7.884,10.019) -- (7.862,9.939) -- (7.884,9.859) -- (7.942,9.801) -- (8.022,9.779) -- (8.102,9.801) -- (8.161,9.859) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.591,7.604) -- (4.569,7.684) -- (4.511,7.743) -- (4.431,7.764) -- (4.351,7.743) -- (4.292,7.684) -- (4.271,7.604) -- (4.292,7.524) -- (4.351,7.465) -- (4.431,7.444) -- (4.511,7.465) -- (4.569,7.524) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.526,7.558) -- (11.505,7.638) -- (11.446,7.697) -- (11.366,7.718) -- (11.286,7.697) -- (11.228,7.638) -- (11.206,7.558) -- (11.228,7.478) -- (11.286,7.419) -- (11.366,7.398) -- (11.446,7.419) -- (11.505,7.478) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.226,7.548) -- (8.205,7.628) -- (8.146,7.687) -- (8.066,7.708) -- (7.986,7.687) -- (7.928,7.628) -- (7.906,7.548) -- (7.928,7.468) -- (7.986,7.41) -- (8.066,7.388) -- (8.146,7.41) -- (8.205,7.468) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.535,7.176) -- (1.513,7.256) -- (1.455,7.314) -- (1.375,7.336) -- (1.295,7.314) -- (1.236,7.256) -- (1.215,7.176) -- (1.236,7.096) -- (1.295,7.037) -- (1.375,7.016) -- (1.455,7.037) -- (1.513,7.096) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.055,4.473) -- (2.034,4.553) -- (1.975,4.611) -- (1.895,4.633) -- (1.815,4.611) -- (1.757,4.553) -- (1.735,4.473) -- (1.757,4.393) -- (1.815,4.334) -- (1.895,4.313) -- (1.975,4.334) -- (2.034,4.393) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.659,3.903) -- (4.638,3.983) -- (4.579,4.042) -- (4.499,4.063) -- (4.419,4.042) -- (4.361,3.983) -- (4.339,3.903) -- (4.361,3.823) -- (4.419,3.764) -- (4.499,3.743) -- (4.579,3.764) -- (4.638,3.823) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.129,3.766) -- (8.107,3.846) -- (8.049,3.905) -- (7.969,3.926) -- (7.889,3.905) -- (7.83,3.846) -- (7.809,3.766) -- (7.83,3.686) -- (7.889,3.627) -- (7.969,3.606) -- (8.049,3.627) -- (8.107,3.686) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.72,1.658) -- (1.698,1.738) -- (1.64,1.797) -- (1.56,1.818) -- (1.48,1.797) -- (1.421,1.738) -- (1.4,1.658) -- (1.421,1.578) -- (1.48,1.519) -- (1.56,1.498) -- (1.64,1.519) -- (1.698,1.578) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.344,1.564) -- (5.323,1.644) -- (5.264,1.703) -- (5.184,1.724) -- (5.104,1.703) -- (5.046,1.644) -- (5.024,1.564) -- (5.046,1.484) -- (5.104,1.425) -- (5.184,1.404) -- (5.264,1.425) -- (5.323,1.484) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.305,10.17) -- (6.427,10.189) -- (6.537,10.245) -- (6.624,10.332) -- (6.68,10.442) -- (6.699,10.564) -- (6.68,10.686) -- (6.624,10.795) -- (6.537,10.883) -- (6.427,10.939) -- (6.305,10.958) -- (6.305,10.958) -- (6.184,10.939) -- (6.074,10.883) -- (5.987,10.795) -- (5.931,10.686) -- (5.911,10.564) -- (5.931,10.442) -- (5.987,10.332) -- (6.074,10.245) -- (6.184,10.189) -- (6.305,10.17) -- cycle;
\node at (6.305,10.564) {\scalebox{1.656}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.98,8.754) -- (5.101,8.774) -- (5.211,8.83) -- (5.298,8.917) -- (5.354,9.027) -- (5.373,9.148) -- (5.354,9.27) -- (5.298,9.38) -- (5.211,9.467) -- (5.101,9.523) -- (4.98,9.542) -- (4.98,9.542) -- (4.858,9.523) -- (4.748,9.467) -- (4.661,9.38) -- (4.605,9.27) -- (4.586,9.148) -- (4.605,9.027) -- (4.661,8.917) -- (4.748,8.83) -- (4.858,8.774) -- (4.98,8.754) -- cycle;
\node at (4.98,9.148) {\scalebox{1.656}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.652,8.519) -- (11.774,8.538) -- (11.884,8.594) -- (11.971,8.681) -- (12.027,8.791) -- (12.046,8.913) -- (12.027,9.034) -- (11.971,9.144) -- (11.884,9.231) -- (11.774,9.287) -- (11.652,9.307) -- (11.652,9.307) -- (11.531,9.287) -- (11.421,9.231) -- (11.334,9.144) -- (11.278,9.034) -- (11.259,8.913) -- (11.278,8.791) -- (11.334,8.681) -- (11.421,8.594) -- (11.531,8.538) -- (11.652,8.519) -- cycle;
\node at (11.652,8.913) {\scalebox{1.656}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.646,10.241) -- (9.768,10.26) -- (9.878,10.316) -- (9.965,10.403) -- (10.021,10.513) -- (10.04,10.635) -- (10.021,10.756) -- (9.965,10.866) -- (9.878,10.953) -- (9.768,11.009) -- (9.646,11.029) -- (9.646,11.029) -- (9.524,11.009) -- (9.415,10.953) -- (9.328,10.866) -- (9.272,10.756) -- (9.252,10.635) -- (9.272,10.513) -- (9.328,10.403) -- (9.415,10.316) -- (9.524,10.26) -- (9.646,10.241) -- cycle;
\node at (9.646,10.635) {\scalebox{1.656}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.339,8.355) -- (8.461,8.375) -- (8.571,8.431) -- (8.658,8.518) -- (8.714,8.627) -- (8.733,8.749) -- (8.714,8.871) -- (8.658,8.981) -- (8.571,9.068) -- (8.461,9.124) -- (8.339,9.143) -- (8.339,9.143) -- (8.218,9.124) -- (8.108,9.068) -- (8.021,8.981) -- (7.965,8.871) -- (7.946,8.749) -- (7.965,8.627) -- (8.021,8.518) -- (8.108,8.431) -- (8.218,8.375) -- (8.339,8.355) -- cycle;
\node at (8.339,8.749) {\scalebox{1.656}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.257,7.726) -- (6.379,7.745) -- (6.488,7.801) -- (6.576,7.888) -- (6.632,7.998) -- (6.651,8.12) -- (6.632,8.242) -- (6.576,8.351) -- (6.488,8.439) -- (6.379,8.495) -- (6.257,8.514) -- (6.257,8.514) -- (6.135,8.495) -- (6.025,8.439) -- (5.938,8.351) -- (5.882,8.242) -- (5.863,8.12) -- (5.882,7.998) -- (5.938,7.888) -- (6.025,7.801) -- (6.135,7.745) -- (6.257,7.726) -- cycle;
\node at (6.257,8.12) {\scalebox{1.656}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.706,5.364) -- (4.828,5.383) -- (4.938,5.439) -- (5.025,5.526) -- (5.081,5.636) -- (5.1,5.758) -- (5.081,5.88) -- (5.025,5.989) -- (4.938,6.077) -- (4.828,6.133) -- (4.706,6.152) -- (4.706,6.152) -- (4.585,6.133) -- (4.475,6.077) -- (4.388,5.989) -- (4.332,5.88) -- (4.312,5.758) -- (4.332,5.636) -- (4.388,5.526) -- (4.475,5.439) -- (4.585,5.383) -- (4.706,5.364) -- cycle;
\node at (4.706,5.758) {\scalebox{1.656}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.717,6.901) -- (9.839,6.921) -- (9.949,6.977) -- (10.036,7.064) -- (10.092,7.174) -- (10.111,7.295) -- (10.092,7.417) -- (10.036,7.527) -- (9.949,7.614) -- (9.839,7.67) -- (9.717,7.689) -- (9.717,7.689) -- (9.595,7.67) -- (9.486,7.614) -- (9.398,7.527) -- (9.342,7.417) -- (9.323,7.295) -- (9.342,7.174) -- (9.398,7.064) -- (9.486,6.977) -- (9.595,6.921) -- (9.717,6.901) -- cycle;
\node at (9.717,7.295) {\scalebox{1.656}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.468,5.277) -- (7.589,5.297) -- (7.699,5.353) -- (7.786,5.44) -- (7.842,5.55) -- (7.862,5.671) -- (7.842,5.793) -- (7.786,5.903) -- (7.699,5.99) -- (7.589,6.046) -- (7.468,6.065) -- (7.468,6.065) -- (7.346,6.046) -- (7.236,5.99) -- (7.149,5.903) -- (7.093,5.793) -- (7.074,5.671) -- (7.093,5.55) -- (7.149,5.44) -- (7.236,5.353) -- (7.346,5.297) -- (7.468,5.277) -- cycle;
\node at (7.468,5.671) {\scalebox{1.656}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.125,6.732) -- (3.247,6.751) -- (3.357,6.807) -- (3.444,6.894) -- (3.5,7.004) -- (3.519,7.126) -- (3.5,7.248) -- (3.444,7.357) -- (3.357,7.445) -- (3.247,7.5) -- (3.125,7.52) -- (2.754,7.52) -- (2.633,7.5) -- (2.523,7.445) -- (2.436,7.357) -- (2.38,7.248) -- (2.36,7.126) -- (2.38,7.004) -- (2.436,6.894) -- (2.523,6.807) -- (2.633,6.751) -- (2.754,6.732) -- cycle;
\node at (2.94,7.126) {\scalebox{1.656}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.096,5.483) -- (2.217,5.503) -- (2.327,5.559) -- (2.414,5.646) -- (2.47,5.756) -- (2.489,5.877) -- (2.47,5.999) -- (2.414,6.109) -- (2.327,6.196) -- (2.217,6.252) -- (2.096,6.271) -- (1.725,6.271) -- (1.603,6.252) -- (1.493,6.196) -- (1.406,6.109) -- (1.35,5.999) -- (1.331,5.877) -- (1.35,5.756) -- (1.406,5.646) -- (1.493,5.559) -- (1.603,5.503) -- (1.725,5.483) -- cycle;
\node at (1.91,5.877) {\scalebox{1.656}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.49,4.286) -- (3.612,4.306) -- (3.722,4.361) -- (3.809,4.449) -- (3.865,4.558) -- (3.884,4.68) -- (3.865,4.802) -- (3.809,4.912) -- (3.722,4.999) -- (3.612,5.055) -- (3.49,5.074) -- (3.12,5.074) -- (2.998,5.055) -- (2.888,4.999) -- (2.801,4.912) -- (2.745,4.802) -- (2.726,4.68) -- (2.745,4.558) -- (2.801,4.449) -- (2.888,4.361) -- (2.998,4.306) -- (3.12,4.286) -- cycle;
\node at (3.305,4.68) {\scalebox{1.656}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.188,2.639) -- (2.309,2.658) -- (2.419,2.714) -- (2.506,2.801) -- (2.562,2.911) -- (2.582,3.033) -- (2.562,3.154) -- (2.506,3.264) -- (2.419,3.351) -- (2.309,3.407) -- (2.188,3.427) -- (1.817,3.427) -- (1.695,3.407) -- (1.585,3.351) -- (1.498,3.264) -- (1.442,3.154) -- (1.423,3.033) -- (1.442,2.911) -- (1.498,2.801) -- (1.585,2.714) -- (1.695,2.658) -- (1.817,2.639) -- cycle;
\node at (2.002,3.033) {\scalebox{1.656}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.441,3.977) -- (6.562,3.997) -- (6.672,4.052) -- (6.759,4.14) -- (6.815,4.249) -- (6.834,4.371) -- (6.815,4.493) -- (6.759,4.603) -- (6.672,4.69) -- (6.562,4.746) -- (6.441,4.765) -- (6.07,4.765) -- (5.948,4.746) -- (5.838,4.69) -- (5.751,4.603) -- (5.695,4.493) -- (5.676,4.371) -- (5.695,4.249) -- (5.751,4.14) -- (5.838,4.052) -- (5.948,3.997) -- (6.07,3.977) -- cycle;
\node at (6.255,4.371) {\scalebox{1.656}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.309,2.422) -- (5.43,2.441) -- (5.54,2.497) -- (5.627,2.584) -- (5.683,2.694) -- (5.703,2.816) -- (5.683,2.938) -- (5.627,3.047) -- (5.54,3.135) -- (5.43,3.191) -- (5.309,3.21) -- (4.938,3.21) -- (4.816,3.191) -- (4.706,3.135) -- (4.619,3.047) -- (4.563,2.938) -- (4.544,2.816) -- (4.563,2.694) -- (4.619,2.584) -- (4.706,2.497) -- (4.816,2.441) -- (4.938,2.422) -- cycle;
\node at (5.123,2.816) {\scalebox{1.656}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.571,1.76) -- (3.693,1.78) -- (3.803,1.836) -- (3.89,1.923) -- (3.946,2.033) -- (3.965,2.154) -- (3.946,2.276) -- (3.89,2.386) -- (3.803,2.473) -- (3.693,2.529) -- (3.571,2.548) -- (3.201,2.548) -- (3.079,2.529) -- (2.969,2.473) -- (2.882,2.386) -- (2.826,2.276) -- (2.807,2.154) -- (2.826,2.033) -- (2.882,1.923) -- (2.969,1.836) -- (3.079,1.78) -- (3.201,1.76) -- cycle;
\node at (3.386,2.154) {\scalebox{1.656}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.766,7.248) -- (8.366,7.248) -- (8.366,7.848) -- (7.766,7.848) -- cycle;
\node at (8.066,7.548) {\scalebox{1.656}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.875,8.462) -- (10.475,8.462) -- (10.475,9.062) -- (9.875,9.062) -- cycle;
\node at (10.175,8.762) {\scalebox{1.656}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.685,5.825) -- (9.285,5.825) -- (9.285,6.425) -- (8.685,6.425) -- cycle;
\node at (8.985,6.125) {\scalebox{1.656}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.548,7.917) -- (2.148,7.917) -- (2.148,8.517) -- (1.548,8.517) -- cycle;
\node at (1.848,8.217) {\scalebox{1.656}{\small d}};
\node at (4.299,11.302) {\scalebox{1.656}{\small A}};
\node at (11.88,10.628) {\scalebox{1.656}{\small B}};
\node at (8.343,10.47) {\scalebox{1.656}{\small C}};
\node at (3.966,8.015) {\scalebox{1.656}{\small D}};
\node at (11.972,7.691) {\scalebox{1.656}{\small E}};
\node at (8.611,7.844) {\scalebox{1.656}{\small F}};
\node at (0.765,7.289) {\scalebox{1.656}{\small G}};
\node at (1.335,4.207) {\scalebox{1.656}{\small H}};
\node at (4.199,3.36) {\scalebox{1.656}{\small I}};
\node at (8.362,3.287) {\scalebox{1.656}{\small J}};
\node at (1.141,1.201) {\scalebox{1.656}{\small K}};
\node at (5.098,0.95) {\scalebox{1.656}{\small L}};
\draw[line width=0.5pt] (12.675,9.761) -- (12.675,10.861);
\draw[line width=0.5pt] (12.444,10.476) -- (12.675,10.861) -- (12.906,10.476);
\draw[line width=0.5pt] (12.372,10.311) -- (12.977,10.311);
\node at (12.675,11.246) {\scalebox{1.656}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour F, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ rue Verte & Ⓐ Place Sainte-Croix & Ⓙ Place aux Herbes \\ \hline
① chemin des Vignes & ⑩ rue du Moulin & Ⓑ Placette des Ormes & Ⓚ Place du Marché \\ \hline
② rue Basse & ⑪ rue des Lilas & Ⓒ Carrefour des Tilleuls & Ⓛ Place de la Fontaine \\ \hline
③ rue du Stade & ⑫ rue du Pont & Ⓓ Rond-point de la Poste & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ chemin de Ronde & ⑬ rue du Four & Ⓔ Coin du Lavoir & \textbf{a} l'école \\ \hline
⑤ chemin du Lavoir & ⑭ chemin Creux & Ⓕ Rond-point du Pont & \textbf{b} la salle des fêtes \\ \hline
⑥ avenue de la Gare & ⑮ venelle du Chat & Ⓖ Place de l'Église & \textbf{c} la bibliothèque \\ \hline
⑦ rue de la Mairie & ⑯ rue du Puits & Ⓗ Carrefour de l'Hôpital & \textbf{d} l'église \\ \hline
⑧ rue de l'Église & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Rond-point du Stade &  \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\begin{enumerate}
  \item Colorier les carrefours du plan avec \textbf{deux couleurs}, de sorte que deux carrefours \textbf{reliés par une rue} ne soient jamais de la même couleur. Combien y en a-t-il de chaque couleur ?
  \item Le car roule d'un carrefour au suivant. Que peut-on dire de la \textbf{couleur} des arrêts successifs ? Qu'en déduire sur les couleurs des arrêts d'une tournée qui les dessert tous ?
  \item \textbf{Le car peut-il desservir tous les carrefours, une fois chacun, et revenir à l'école ?} Si oui, décrire un trajet ; sinon, démontrer que c'est impossible.
  \item \textbf{Et s'il n'est pas obligé de revenir à l'école ?} Même consigne.
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage — Correction}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6187, line join=round, line cap=round]
\draw[line width=0.5pt] (4.484,10.711) -- (4.727,10.617) -- (4.971,10.53) -- (5.218,10.448) -- (5.466,10.372) -- (5.716,10.302) -- (5.968,10.237) -- (6.221,10.179) -- (6.476,10.127) -- (6.732,10.08) -- (6.988,10.04) -- (7.246,10.006) -- (7.504,9.977) -- (7.763,9.955) -- (8.022,9.939);
\draw[line width=0.5pt] (4.484,10.711) -- (4.515,10.49) -- (4.539,10.269) -- (4.559,10.046) -- (4.574,9.824) -- (4.583,9.601) -- (4.587,9.378) -- (4.586,9.155) -- (4.579,8.932) -- (4.568,8.709) -- (4.551,8.487) -- (4.528,8.265) -- (4.501,8.044) -- (4.469,7.824) -- (4.431,7.604);
\draw[line width=0.5pt] (11.375,10.269) -- (11.344,10.077) -- (11.319,9.885) -- (11.298,9.691) -- (11.281,9.497) -- (11.269,9.303) -- (11.261,9.109) -- (11.259,8.914) -- (11.26,8.719) -- (11.266,8.525) -- (11.277,8.331) -- (11.293,8.137) -- (11.313,7.943) -- (11.337,7.75) -- (11.366,7.558);
\draw[line width=0.5pt] (8.022,9.939) -- (8.256,9.999) -- (8.491,10.054) -- (8.728,10.103) -- (8.966,10.146) -- (9.205,10.184) -- (9.444,10.216) -- (9.685,10.243) -- (9.926,10.263) -- (10.167,10.279) -- (10.409,10.288) -- (10.65,10.292) -- (10.892,10.29) -- (11.133,10.283) -- (11.375,10.269);
\draw[line width=0.5pt] (8.022,9.939) -- (7.999,9.769) -- (7.98,9.599) -- (7.965,9.428) -- (7.954,9.257) -- (7.947,9.085) -- (7.944,8.914) -- (7.946,8.742) -- (7.951,8.57) -- (7.96,8.399) -- (7.973,8.228) -- (7.991,8.057) -- (8.012,7.887) -- (8.037,7.717) -- (8.066,7.548);
\draw[line width=0.5pt] (4.431,7.604) -- (4.689,7.64) -- (4.948,7.669) -- (5.208,7.693) -- (5.468,7.71) -- (5.729,7.722) -- (5.99,7.727) -- (6.251,7.726) -- (6.512,7.719) -- (6.772,7.706) -- (7.033,7.686) -- (7.292,7.661) -- (7.551,7.63) -- (7.809,7.592) -- (8.066,7.548);
\draw[line width=0.5pt] (4.431,7.604) -- (4.395,7.341) -- (4.366,7.077) -- (4.343,6.813) -- (4.326,6.547) -- (4.315,6.282) -- (4.311,6.016) -- (4.312,5.751) -- (4.32,5.485) -- (4.335,5.22) -- (4.355,4.955) -- (4.382,4.691) -- (4.415,4.427) -- (4.454,4.164) -- (4.499,3.903);
\draw[line width=0.5pt] (8.066,7.548) -- (8.3,7.585) -- (8.535,7.616) -- (8.77,7.642) -- (9.006,7.662) -- (9.242,7.677) -- (9.479,7.686) -- (9.716,7.689) -- (9.953,7.687) -- (10.189,7.679) -- (10.426,7.666) -- (10.662,7.648) -- (10.897,7.623) -- (11.132,7.593) -- (11.366,7.558);
\draw[line width=0.5pt] (8.066,7.548) -- (8.018,7.281) -- (7.976,7.013) -- (7.94,6.744) -- (7.911,6.474) -- (7.888,6.204) -- (7.872,5.933) -- (7.861,5.661) -- (7.858,5.39) -- (7.86,5.118) -- (7.869,4.847) -- (7.884,4.576) -- (7.906,4.305) -- (7.934,4.035) -- (7.969,3.766);
\draw[line width=0.5pt] (1.375,7.176) -- (1.587,7.24) -- (1.8,7.298) -- (2.015,7.352) -- (2.231,7.401) -- (2.448,7.444) -- (2.666,7.483) -- (2.885,7.516) -- (3.105,7.544) -- (3.325,7.567) -- (3.546,7.585) -- (3.767,7.598) -- (3.988,7.605) -- (4.209,7.607) -- (4.431,7.604);
\draw[line width=0.5pt] (1.375,7.176) -- (1.382,6.979) -- (1.394,6.782) -- (1.411,6.585) -- (1.432,6.388) -- (1.458,6.193) -- (1.488,5.997) -- (1.523,5.803) -- (1.563,5.609) -- (1.607,5.417) -- (1.656,5.225) -- (1.709,5.035) -- (1.767,4.846) -- (1.829,4.659) -- (1.895,4.473);
\draw[line width=0.5pt] (1.895,4.473) -- (2.086,4.461) -- (2.276,4.445) -- (2.467,4.424) -- (2.656,4.398) -- (2.845,4.368) -- (3.033,4.334) -- (3.221,4.295) -- (3.407,4.252) -- (3.593,4.205) -- (3.777,4.153) -- (3.96,4.097) -- (4.141,4.036) -- (4.321,3.972) -- (4.499,3.903);
\draw[line width=0.5pt] (1.895,4.473) -- (1.841,4.277) -- (1.791,4.08) -- (1.745,3.882) -- (1.705,3.682) -- (1.669,3.482) -- (1.638,3.281) -- (1.611,3.079) -- (1.589,2.877) -- (1.573,2.674) -- (1.56,2.471) -- (1.553,2.268) -- (1.55,2.064) -- (1.553,1.861) -- (1.56,1.658);
\draw[line width=0.5pt] (4.499,3.903) -- (4.747,3.931) -- (4.995,3.954) -- (5.243,3.97) -- (5.492,3.981) -- (5.741,3.986) -- (5.991,3.984) -- (6.24,3.978) -- (6.489,3.965) -- (6.737,3.946) -- (6.985,3.922) -- (7.232,3.892) -- (7.479,3.856) -- (7.724,3.814) -- (7.969,3.766);
\draw[line width=0.5pt] (4.499,3.903) -- (4.522,3.73) -- (4.549,3.557) -- (4.581,3.385) -- (4.616,3.214) -- (4.655,3.043) -- (4.698,2.874) -- (4.745,2.705) -- (4.796,2.538) -- (4.851,2.372) -- (4.91,2.207) -- (4.973,2.044) -- (5.04,1.882) -- (5.11,1.722) -- (5.184,1.564);
\draw[line width=0.5pt] (1.56,1.658) -- (1.817,1.691) -- (2.076,1.718) -- (2.335,1.739) -- (2.595,1.753) -- (2.855,1.762) -- (3.115,1.764) -- (3.376,1.761) -- (3.636,1.751) -- (3.896,1.735) -- (4.155,1.713) -- (4.414,1.685) -- (4.671,1.651) -- (4.928,1.61) -- (5.184,1.564);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.858,10.711) -- (4.829,10.854) -- (4.748,10.975) -- (4.627,11.056) -- (4.484,11.084) -- (4.342,11.056) -- (4.22,10.975) -- (4.14,10.854) -- (4.111,10.711) -- (4.14,10.568) -- (4.22,10.447) -- (4.342,10.366) -- (4.484,10.337) -- (4.627,10.366) -- (4.748,10.447) -- (4.829,10.568) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.748,10.269) -- (11.719,10.412) -- (11.639,10.533) -- (11.517,10.614) -- (11.375,10.643) -- (11.232,10.614) -- (11.111,10.533) -- (11.03,10.412) -- (11.001,10.269) -- (11.03,10.127) -- (11.111,10.005) -- (11.232,9.925) -- (11.375,9.896) -- (11.517,9.925) -- (11.639,10.005) -- (11.719,10.127) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.395,9.939) -- (8.367,10.082) -- (8.286,10.203) -- (8.165,10.284) -- (8.022,10.313) -- (7.879,10.284) -- (7.758,10.203) -- (7.677,10.082) -- (7.649,9.939) -- (7.677,9.797) -- (7.758,9.675) -- (7.879,9.595) -- (8.022,9.566) -- (8.165,9.595) -- (8.286,9.675) -- (8.367,9.797) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.804,7.604) -- (4.776,7.747) -- (4.695,7.868) -- (4.574,7.949) -- (4.431,7.977) -- (4.288,7.949) -- (4.167,7.868) -- (4.086,7.747) -- (4.057,7.604) -- (4.086,7.461) -- (4.167,7.34) -- (4.288,7.259) -- (4.431,7.231) -- (4.574,7.259) -- (4.695,7.34) -- (4.776,7.461) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.739,7.558) -- (11.711,7.701) -- (11.63,7.822) -- (11.509,7.903) -- (11.366,7.931) -- (11.223,7.903) -- (11.102,7.822) -- (11.021,7.701) -- (10.993,7.558) -- (11.021,7.415) -- (11.102,7.294) -- (11.223,7.213) -- (11.366,7.185) -- (11.509,7.213) -- (11.63,7.294) -- (11.711,7.415) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.44,7.548) -- (8.411,7.691) -- (8.33,7.812) -- (8.209,7.893) -- (8.066,7.921) -- (7.924,7.893) -- (7.802,7.812) -- (7.722,7.691) -- (7.693,7.548) -- (7.722,7.405) -- (7.802,7.284) -- (7.924,7.203) -- (8.066,7.175) -- (8.209,7.203) -- (8.33,7.284) -- (8.411,7.405) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.748,7.176) -- (1.719,7.319) -- (1.639,7.44) -- (1.517,7.521) -- (1.375,7.549) -- (1.232,7.521) -- (1.111,7.44) -- (1.03,7.319) -- (1.001,7.176) -- (1.03,7.033) -- (1.111,6.912) -- (1.232,6.831) -- (1.375,6.803) -- (1.517,6.831) -- (1.639,6.912) -- (1.719,7.033) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.268,4.473) -- (2.24,4.616) -- (2.159,4.737) -- (2.038,4.818) -- (1.895,4.846) -- (1.752,4.818) -- (1.631,4.737) -- (1.55,4.616) -- (1.522,4.473) -- (1.55,4.33) -- (1.631,4.209) -- (1.752,4.128) -- (1.895,4.1) -- (2.038,4.128) -- (2.159,4.209) -- (2.24,4.33) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.873,3.903) -- (4.844,4.046) -- (4.763,4.167) -- (4.642,4.248) -- (4.499,4.276) -- (4.357,4.248) -- (4.236,4.167) -- (4.155,4.046) -- (4.126,3.903) -- (4.155,3.76) -- (4.236,3.639) -- (4.357,3.558) -- (4.499,3.53) -- (4.642,3.558) -- (4.763,3.639) -- (4.844,3.76) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.342,3.766) -- (8.313,3.909) -- (8.233,4.03) -- (8.111,4.111) -- (7.969,4.139) -- (7.826,4.111) -- (7.705,4.03) -- (7.624,3.909) -- (7.595,3.766) -- (7.624,3.623) -- (7.705,3.502) -- (7.826,3.421) -- (7.969,3.393) -- (8.111,3.421) -- (8.233,3.502) -- (8.313,3.623) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.933,1.658) -- (1.904,1.801) -- (1.823,1.922) -- (1.702,2.003) -- (1.56,2.031) -- (1.417,2.003) -- (1.296,1.922) -- (1.215,1.801) -- (1.186,1.658) -- (1.215,1.515) -- (1.296,1.394) -- (1.417,1.313) -- (1.56,1.285) -- (1.702,1.313) -- (1.823,1.394) -- (1.904,1.515) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.557,1.564) -- (5.529,1.707) -- (5.448,1.828) -- (5.327,1.909) -- (5.184,1.937) -- (5.041,1.909) -- (4.92,1.828) -- (4.839,1.707) -- (4.811,1.564) -- (4.839,1.421) -- (4.92,1.3) -- (5.041,1.219) -- (5.184,1.191) -- (5.327,1.219) -- (5.448,1.3) -- (5.529,1.421) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.303,10.17) -- (6.422,10.189) -- (6.529,10.244) -- (6.614,10.329) -- (6.669,10.436) -- (6.688,10.555) -- (6.669,10.674) -- (6.614,10.781) -- (6.529,10.866) -- (6.422,10.92) -- (6.303,10.939) -- (6.303,10.939) -- (6.185,10.92) -- (6.077,10.866) -- (5.992,10.781) -- (5.938,10.674) -- (5.919,10.555) -- (5.938,10.436) -- (5.992,10.329) -- (6.077,10.244) -- (6.185,10.189) -- (6.303,10.17) -- cycle;
\node at (6.303,10.555) {\scalebox{1.616}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.97,8.764) -- (5.089,8.783) -- (5.196,8.837) -- (5.281,8.922) -- (5.336,9.03) -- (5.355,9.148) -- (5.336,9.267) -- (5.281,9.374) -- (5.196,9.46) -- (5.089,9.514) -- (4.97,9.533) -- (4.97,9.533) -- (4.851,9.514) -- (4.744,9.46) -- (4.659,9.374) -- (4.604,9.267) -- (4.586,9.148) -- (4.604,9.03) -- (4.659,8.922) -- (4.744,8.837) -- (4.851,8.783) -- (4.97,8.764) -- cycle;
\node at (4.97,9.148) {\scalebox{1.616}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.643,8.528) -- (11.762,8.547) -- (11.869,8.602) -- (11.954,8.687) -- (12.009,8.794) -- (12.028,8.913) -- (12.009,9.032) -- (11.954,9.139) -- (11.869,9.224) -- (11.762,9.279) -- (11.643,9.297) -- (11.643,9.297) -- (11.524,9.279) -- (11.417,9.224) -- (11.332,9.139) -- (11.277,9.032) -- (11.259,8.913) -- (11.277,8.794) -- (11.332,8.687) -- (11.417,8.602) -- (11.524,8.547) -- (11.643,8.528) -- cycle;
\node at (11.643,8.913) {\scalebox{1.616}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.647,10.241) -- (9.766,10.26) -- (9.873,10.314) -- (9.958,10.399) -- (10.013,10.506) -- (10.032,10.625) -- (10.013,10.744) -- (9.958,10.851) -- (9.873,10.936) -- (9.766,10.991) -- (9.647,11.01) -- (9.647,11.01) -- (9.528,10.991) -- (9.421,10.936) -- (9.336,10.851) -- (9.281,10.744) -- (9.263,10.625) -- (9.281,10.506) -- (9.336,10.399) -- (9.421,10.314) -- (9.528,10.26) -- (9.647,10.241) -- cycle;
\node at (9.647,10.625) {\scalebox{1.616}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.33,8.365) -- (8.449,8.383) -- (8.556,8.438) -- (8.641,8.523) -- (8.696,8.63) -- (8.715,8.749) -- (8.696,8.868) -- (8.641,8.975) -- (8.556,9.06) -- (8.449,9.115) -- (8.33,9.134) -- (8.33,9.134) -- (8.211,9.115) -- (8.104,9.06) -- (8.019,8.975) -- (7.964,8.868) -- (7.946,8.749) -- (7.964,8.63) -- (8.019,8.523) -- (8.104,8.438) -- (8.211,8.383) -- (8.33,8.365) -- cycle;
\node at (8.33,8.749) {\scalebox{1.616}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.257,7.726) -- (6.376,7.745) -- (6.483,7.799) -- (6.568,7.884) -- (6.622,7.992) -- (6.641,8.111) -- (6.622,8.229) -- (6.568,8.337) -- (6.483,8.422) -- (6.376,8.476) -- (6.257,8.495) -- (6.257,8.495) -- (6.138,8.476) -- (6.031,8.422) -- (5.946,8.337) -- (5.891,8.229) -- (5.872,8.111) -- (5.891,7.992) -- (5.946,7.884) -- (6.031,7.799) -- (6.138,7.745) -- (6.257,7.726) -- cycle;
\node at (6.257,8.111) {\scalebox{1.616}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.697,5.373) -- (4.816,5.392) -- (4.923,5.447) -- (5.008,5.532) -- (5.063,5.639) -- (5.081,5.758) -- (5.063,5.877) -- (5.008,5.984) -- (4.923,6.069) -- (4.816,6.123) -- (4.697,6.142) -- (4.697,6.142) -- (4.578,6.123) -- (4.471,6.069) -- (4.386,5.984) -- (4.331,5.877) -- (4.312,5.758) -- (4.331,5.639) -- (4.386,5.532) -- (4.471,5.447) -- (4.578,5.392) -- (4.697,5.373) -- cycle;
\node at (4.697,5.758) {\scalebox{1.616}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.717,6.92) -- (9.836,6.939) -- (9.943,6.994) -- (10.028,7.079) -- (10.083,7.186) -- (10.102,7.305) -- (10.083,7.423) -- (10.028,7.531) -- (9.943,7.616) -- (9.836,7.67) -- (9.717,7.689) -- (9.717,7.689) -- (9.598,7.67) -- (9.491,7.616) -- (9.406,7.531) -- (9.351,7.423) -- (9.332,7.305) -- (9.351,7.186) -- (9.406,7.079) -- (9.491,6.994) -- (9.598,6.939) -- (9.717,6.92) -- cycle;
\node at (9.717,7.305) {\scalebox{1.616}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.246,5.267) -- (8.365,5.285) -- (8.472,5.34) -- (8.557,5.425) -- (8.612,5.532) -- (8.63,5.651) -- (8.612,5.77) -- (8.557,5.877) -- (8.472,5.962) -- (8.365,6.017) -- (8.246,6.036) -- (8.246,6.036) -- (8.127,6.017) -- (8.02,5.962) -- (7.935,5.877) -- (7.88,5.77) -- (7.861,5.651) -- (7.88,5.532) -- (7.935,5.425) -- (8.02,5.34) -- (8.127,5.285) -- (8.246,5.267) -- cycle;
\node at (8.246,5.651) {\scalebox{1.616}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.119,6.751) -- (3.238,6.769) -- (3.345,6.824) -- (3.43,6.909) -- (3.485,7.016) -- (3.504,7.135) -- (3.485,7.254) -- (3.43,7.361) -- (3.345,7.446) -- (3.238,7.501) -- (3.119,7.52) -- (2.757,7.52) -- (2.639,7.501) -- (2.531,7.446) -- (2.446,7.361) -- (2.392,7.254) -- (2.373,7.135) -- (2.392,7.016) -- (2.446,6.909) -- (2.531,6.824) -- (2.639,6.769) -- (2.757,6.751) -- cycle;
\node at (2.938,7.135) {\scalebox{1.616}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.082,5.491) -- (2.201,5.51) -- (2.308,5.565) -- (2.393,5.65) -- (2.448,5.757) -- (2.466,5.876) -- (2.448,5.994) -- (2.393,6.102) -- (2.308,6.187) -- (2.201,6.241) -- (2.082,6.26) -- (1.72,6.26) -- (1.601,6.241) -- (1.494,6.187) -- (1.409,6.102) -- (1.354,5.994) -- (1.335,5.876) -- (1.354,5.757) -- (1.409,5.65) -- (1.494,5.565) -- (1.601,5.51) -- (1.72,5.491) -- cycle;
\node at (1.901,5.876) {\scalebox{1.616}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.484,4.286) -- (3.603,4.305) -- (3.71,4.36) -- (3.795,4.445) -- (3.85,4.552) -- (3.868,4.671) -- (3.85,4.79) -- (3.795,4.897) -- (3.71,4.982) -- (3.603,5.037) -- (3.484,5.056) -- (3.122,5.056) -- (3.003,5.037) -- (2.896,4.982) -- (2.811,4.897) -- (2.756,4.79) -- (2.738,4.671) -- (2.756,4.552) -- (2.811,4.445) -- (2.896,4.36) -- (3.003,4.305) -- (3.122,4.286) -- cycle;
\node at (3.303,4.671) {\scalebox{1.616}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.174,2.649) -- (2.293,2.668) -- (2.4,2.723) -- (2.485,2.808) -- (2.54,2.915) -- (2.559,3.034) -- (2.54,3.153) -- (2.485,3.26) -- (2.4,3.345) -- (2.293,3.399) -- (2.174,3.418) -- (1.812,3.418) -- (1.693,3.399) -- (1.586,3.345) -- (1.501,3.26) -- (1.446,3.153) -- (1.428,3.034) -- (1.446,2.915) -- (1.501,2.808) -- (1.586,2.723) -- (1.693,2.668) -- (1.812,2.649) -- cycle;
\node at (1.993,3.034) {\scalebox{1.616}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.436,3.977) -- (6.555,3.996) -- (6.662,4.051) -- (6.747,4.136) -- (6.802,4.243) -- (6.82,4.362) -- (6.802,4.481) -- (6.747,4.588) -- (6.662,4.673) -- (6.555,4.728) -- (6.436,4.746) -- (6.074,4.746) -- (5.955,4.728) -- (5.848,4.673) -- (5.763,4.588) -- (5.708,4.481) -- (5.689,4.362) -- (5.708,4.243) -- (5.763,4.136) -- (5.848,4.051) -- (5.955,3.996) -- (6.074,3.977) -- cycle;
\node at (6.255,4.362) {\scalebox{1.616}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.295,2.429) -- (5.414,2.448) -- (5.521,2.502) -- (5.606,2.587) -- (5.661,2.694) -- (5.68,2.813) -- (5.661,2.932) -- (5.606,3.039) -- (5.521,3.124) -- (5.414,3.179) -- (5.295,3.198) -- (4.933,3.198) -- (4.815,3.179) -- (4.707,3.124) -- (4.622,3.039) -- (4.568,2.932) -- (4.549,2.813) -- (4.568,2.694) -- (4.622,2.587) -- (4.707,2.502) -- (4.815,2.448) -- (4.933,2.429) -- cycle;
\node at (5.114,2.813) {\scalebox{1.616}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.567,1.76) -- (3.685,1.779) -- (3.793,1.834) -- (3.878,1.919) -- (3.932,2.026) -- (3.951,2.145) -- (3.932,2.264) -- (3.878,2.371) -- (3.793,2.456) -- (3.685,2.511) -- (3.567,2.529) -- (3.205,2.529) -- (3.086,2.511) -- (2.979,2.456) -- (2.894,2.371) -- (2.839,2.264) -- (2.82,2.145) -- (2.839,2.026) -- (2.894,1.919) -- (2.979,1.834) -- (3.086,1.779) -- (3.205,1.76) -- cycle;
\node at (3.386,2.145) {\scalebox{1.616}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.766,7.248) -- (8.366,7.248) -- (8.366,7.848) -- (7.766,7.848) -- cycle;
\node at (8.066,7.548) {\scalebox{1.616}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.875,8.462) -- (10.475,8.462) -- (10.475,9.062) -- (9.875,9.062) -- cycle;
\node at (10.175,8.762) {\scalebox{1.616}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.685,5.825) -- (9.285,5.825) -- (9.285,6.425) -- (8.685,6.425) -- cycle;
\node at (8.985,6.125) {\scalebox{1.616}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.548,7.917) -- (2.148,7.917) -- (2.148,8.517) -- (1.548,8.517) -- cycle;
\node at (1.848,8.217) {\scalebox{1.616}{\small d}};
\node at (4.299,11.302) {\scalebox{1.616}{\small A}};
\node at (11.88,10.628) {\scalebox{1.616}{\small B}};
\node at (8.343,10.47) {\scalebox{1.616}{\small C}};
\node at (3.966,8.015) {\scalebox{1.616}{\small D}};
\node at (11.972,7.691) {\scalebox{1.616}{\small E}};
\node at (8.611,7.844) {\scalebox{1.616}{\small F}};
\node at (0.765,7.289) {\scalebox{1.616}{\small G}};
\node at (1.335,4.207) {\scalebox{1.616}{\small H}};
\node at (4.199,3.36) {\scalebox{1.616}{\small I}};
\node at (8.362,3.287) {\scalebox{1.616}{\small J}};
\node at (1.141,1.201) {\scalebox{1.616}{\small K}};
\node at (5.098,0.95) {\scalebox{1.616}{\small L}};
\draw[line width=0.5pt] (12.675,9.761) -- (12.675,10.861);
\draw[line width=0.5pt] (12.444,10.476) -- (12.675,10.861) -- (12.906,10.476);
\draw[line width=0.5pt] (12.372,10.311) -- (12.977,10.311);
\node at (12.675,11.246) {\scalebox{1.616}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, B, F, G, I, K & 6 \\ \hline
clairs & C, D, E, H, J, L & 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 de l'Église ⑧} vers l'est jusqu'à Coin du Lavoir \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Stade ③}, vers le nord, jusqu'à Placette des Ormes \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin de Ronde ④}, vers l'ouest, jusqu'à Carrefour des Tilleuls \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin des Vignes ①}, vers l'ouest, jusqu'à Place Sainte-Croix \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ②}, vers le sud, jusqu'à Rond-point de la Poste \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Moulin ⑩}, vers l'ouest, jusqu'à Place de l'Église \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ⑪}, vers le sud, jusqu'à Carrefour de l'Hôpital \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Four ⑬}, vers le sud, jusqu'à Place du Marché \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ⑯}, vers l'est, jusqu'à Place de la Fontaine \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{venelle du Chat ⑮}, vers le nord, jusqu'à Rond-point du Stade \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin Creux ⑭}, vers l'est, jusqu'à Place aux Herbes \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Verte ⑨}, vers le nord, jusqu'à Rond-point du Pont \textbf{(F)}.\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}
