\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 28}

\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=28}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0028}
\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] (9.261,7.147) -- (9.222,6.995) -- (9.186,6.843) -- (9.154,6.689) -- (9.125,6.535) -- (9.1,6.38) -- (9.079,6.224) -- (9.061,6.068) -- (9.047,5.912) -- (9.037,5.755) -- (9.031,5.598) -- (9.028,5.441) -- (9.028,5.284) -- (9.033,5.127) -- (9.041,4.971);
\draw[line width=0.5pt] (6.362,6.863) -- (6.564,6.915) -- (6.767,6.962) -- (6.972,7.004) -- (7.178,7.041) -- (7.384,7.074) -- (7.592,7.102) -- (7.8,7.125) -- (8.008,7.143) -- (8.217,7.156) -- (8.425,7.164) -- (8.634,7.167) -- (8.843,7.165) -- (9.052,7.159) -- (9.261,7.147);
\draw[line width=0.5pt] (6.362,6.863) -- (6.366,6.718) -- (6.373,6.573) -- (6.384,6.429) -- (6.398,6.284) -- (6.416,6.141) -- (6.437,5.997) -- (6.462,5.854) -- (6.49,5.712) -- (6.521,5.57) -- (6.555,5.429) -- (6.593,5.289) -- (6.634,5.15) -- (6.678,5.012) -- (6.726,4.876);
\draw[line width=0.5pt] (1.374,5.015) -- (1.568,5.036) -- (1.762,5.052) -- (1.956,5.063) -- (2.151,5.07) -- (2.346,5.072) -- (2.541,5.07) -- (2.735,5.063) -- (2.93,5.052) -- (3.124,5.036) -- (3.318,5.015) -- (3.511,4.99) -- (3.704,4.96) -- (3.895,4.926) -- (4.086,4.887);
\draw[line width=0.5pt] (1.374,5.015) -- (1.359,4.829) -- (1.348,4.643) -- (1.341,4.456) -- (1.339,4.269) -- (1.341,4.082) -- (1.347,3.895) -- (1.358,3.708) -- (1.373,3.522) -- (1.393,3.336) -- (1.417,3.15) -- (1.445,2.965) -- (1.478,2.781) -- (1.515,2.598) -- (1.556,2.416);
\draw[line width=0.5pt] (9.041,4.971) -- (9.047,4.752) -- (9.047,4.534) -- (9.042,4.315) -- (9.031,4.097) -- (9.016,3.879) -- (8.996,3.661) -- (8.97,3.444) -- (8.939,3.227) -- (8.904,3.012) -- (8.863,2.797) -- (8.817,2.583) -- (8.766,2.37) -- (8.71,2.159) -- (8.649,1.949);
\draw[line width=0.5pt] (4.086,4.887) -- (4.273,4.857) -- (4.461,4.832) -- (4.649,4.811) -- (4.838,4.795) -- (5.027,4.783) -- (5.216,4.775) -- (5.405,4.772) -- (5.595,4.774) -- (5.784,4.78) -- (5.974,4.79) -- (6.163,4.805) -- (6.351,4.824) -- (6.539,4.848) -- (6.726,4.876);
\draw[line width=0.5pt] (4.086,4.887) -- (4.101,4.708) -- (4.111,4.528) -- (4.117,4.348) -- (4.119,4.168) -- (4.117,3.987) -- (4.11,3.807) -- (4.1,3.627) -- (4.085,3.448) -- (4.066,3.268) -- (4.042,3.09) -- (4.015,2.912) -- (3.983,2.734) -- (3.947,2.558) -- (3.907,2.382);
\draw[line width=0.5pt] (6.726,4.876) -- (6.891,4.857) -- (7.057,4.842) -- (7.223,4.832) -- (7.389,4.825) -- (7.555,4.822) -- (7.721,4.823) -- (7.888,4.828) -- (8.054,4.836) -- (8.22,4.849) -- (8.385,4.866) -- (8.55,4.886) -- (8.714,4.91) -- (8.878,4.939) -- (9.041,4.971);
\draw[line width=0.5pt] (6.726,4.876) -- (6.768,4.683) -- (6.805,4.49) -- (6.837,4.296) -- (6.865,4.101) -- (6.889,3.905) -- (6.907,3.709) -- (6.922,3.513) -- (6.931,3.316) -- (6.936,3.119) -- (6.936,2.922) -- (6.932,2.725) -- (6.923,2.529) -- (6.91,2.332) -- (6.891,2.136);
\draw[line width=0.5pt] (1.556,2.416) -- (1.722,2.388) -- (1.889,2.364) -- (2.057,2.343) -- (2.225,2.327) -- (2.393,2.315) -- (2.561,2.307) -- (2.73,2.302) -- (2.899,2.302) -- (3.068,2.305) -- (3.236,2.313) -- (3.404,2.324) -- (3.572,2.34) -- (3.74,2.359) -- (3.907,2.382);
\draw[line width=0.5pt] (3.907,2.382) -- (4.121,2.397) -- (4.336,2.408) -- (4.55,2.413) -- (4.765,2.413) -- (4.98,2.408) -- (5.195,2.398) -- (5.409,2.382) -- (5.623,2.362) -- (5.837,2.337) -- (6.05,2.307) -- (6.261,2.272) -- (6.473,2.232) -- (6.683,2.186) -- (6.891,2.136);
\draw[line width=0.5pt] (6.891,2.136) -- (7.014,2.104) -- (7.137,2.074) -- (7.261,2.048) -- (7.386,2.024) -- (7.511,2.003) -- (7.637,1.985) -- (7.763,1.97) -- (7.889,1.958) -- (8.016,1.949) -- (8.142,1.943) -- (8.269,1.94) -- (8.396,1.94) -- (8.523,1.943) -- (8.649,1.949);
\draw[line width=0.5pt] (8.649,1.949) -- (8.838,2.008) -- (9.028,2.061) -- (9.22,2.111) -- (9.412,2.156) -- (9.606,2.196) -- (9.8,2.232) -- (9.995,2.263) -- (10.191,2.29) -- (10.388,2.312) -- (10.585,2.329) -- (10.782,2.342) -- (10.979,2.35) -- (11.177,2.353) -- (11.374,2.352);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.421,7.147) -- (9.399,7.227) -- (9.341,7.286) -- (9.261,7.307) -- (9.181,7.286) -- (9.122,7.227) -- (9.101,7.147) -- (9.122,7.067) -- (9.181,7.009) -- (9.261,6.987) -- (9.341,7.009) -- (9.399,7.067) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.522,6.863) -- (6.5,6.943) -- (6.442,7.001) -- (6.362,7.023) -- (6.282,7.001) -- (6.223,6.943) -- (6.202,6.863) -- (6.223,6.783) -- (6.282,6.724) -- (6.362,6.703) -- (6.442,6.724) -- (6.5,6.783) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.534,5.015) -- (1.513,5.095) -- (1.454,5.154) -- (1.374,5.175) -- (1.294,5.154) -- (1.236,5.095) -- (1.214,5.015) -- (1.236,4.935) -- (1.294,4.877) -- (1.374,4.855) -- (1.454,4.877) -- (1.513,4.935) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.201,4.971) -- (9.18,5.051) -- (9.121,5.109) -- (9.041,5.131) -- (8.961,5.109) -- (8.903,5.051) -- (8.881,4.971) -- (8.903,4.891) -- (8.961,4.832) -- (9.041,4.811) -- (9.121,4.832) -- (9.18,4.891) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.246,4.887) -- (4.224,4.967) -- (4.166,5.026) -- (4.086,5.047) -- (4.006,5.026) -- (3.947,4.967) -- (3.926,4.887) -- (3.947,4.807) -- (4.006,4.749) -- (4.086,4.727) -- (4.166,4.749) -- (4.224,4.807) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.886,4.876) -- (6.865,4.956) -- (6.806,5.014) -- (6.726,5.036) -- (6.646,5.014) -- (6.587,4.956) -- (6.566,4.876) -- (6.587,4.796) -- (6.646,4.737) -- (6.726,4.716) -- (6.806,4.737) -- (6.865,4.796) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.716,2.416) -- (1.695,2.496) -- (1.636,2.555) -- (1.556,2.576) -- (1.476,2.555) -- (1.418,2.496) -- (1.396,2.416) -- (1.418,2.336) -- (1.476,2.277) -- (1.556,2.256) -- (1.636,2.277) -- (1.695,2.336) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.067,2.382) -- (4.045,2.462) -- (3.987,2.521) -- (3.907,2.542) -- (3.827,2.521) -- (3.768,2.462) -- (3.747,2.382) -- (3.768,2.302) -- (3.827,2.244) -- (3.907,2.222) -- (3.987,2.244) -- (4.045,2.302) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.534,2.352) -- (11.513,2.432) -- (11.454,2.491) -- (11.374,2.512) -- (11.294,2.491) -- (11.236,2.432) -- (11.214,2.352) -- (11.236,2.272) -- (11.294,2.214) -- (11.374,2.192) -- (11.454,2.214) -- (11.513,2.272) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.051,2.136) -- (7.03,2.216) -- (6.971,2.275) -- (6.891,2.296) -- (6.811,2.275) -- (6.753,2.216) -- (6.731,2.136) -- (6.753,2.056) -- (6.811,1.998) -- (6.891,1.976) -- (6.971,1.998) -- (7.03,2.056) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.809,1.949) -- (8.788,2.029) -- (8.729,2.088) -- (8.649,2.109) -- (8.569,2.088) -- (8.511,2.029) -- (8.489,1.949) -- (8.511,1.869) -- (8.569,1.811) -- (8.649,1.789) -- (8.729,1.811) -- (8.788,1.869) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.669,5.714) -- (8.791,5.733) -- (8.901,5.789) -- (8.988,5.876) -- (9.044,5.986) -- (9.063,6.108) -- (9.044,6.229) -- (8.988,6.339) -- (8.901,6.426) -- (8.791,6.482) -- (8.669,6.502) -- (8.669,6.502) -- (8.548,6.482) -- (8.438,6.426) -- (8.351,6.339) -- (8.295,6.229) -- (8.275,6.108) -- (8.295,5.986) -- (8.351,5.876) -- (8.438,5.789) -- (8.548,5.733) -- (8.669,5.714) -- cycle;
\node at (8.669,6.108) {\scalebox{1.656}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.761,7.123) -- (7.883,7.142) -- (7.993,7.198) -- (8.08,7.285) -- (8.136,7.395) -- (8.155,7.517) -- (8.136,7.638) -- (8.08,7.748) -- (7.993,7.835) -- (7.883,7.891) -- (7.761,7.911) -- (7.761,7.911) -- (7.639,7.891) -- (7.53,7.835) -- (7.442,7.748) -- (7.386,7.638) -- (7.367,7.517) -- (7.386,7.395) -- (7.442,7.285) -- (7.53,7.198) -- (7.639,7.142) -- (7.761,7.123) -- cycle;
\node at (7.761,7.517) {\scalebox{1.656}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.074,5.389) -- (6.196,5.408) -- (6.306,5.464) -- (6.393,5.552) -- (6.449,5.661) -- (6.468,5.783) -- (6.449,5.905) -- (6.393,6.015) -- (6.306,6.102) -- (6.196,6.158) -- (6.074,6.177) -- (6.074,6.177) -- (5.953,6.158) -- (5.843,6.102) -- (5.756,6.015) -- (5.7,5.905) -- (5.68,5.783) -- (5.7,5.661) -- (5.756,5.552) -- (5.843,5.464) -- (5.953,5.408) -- (6.074,5.389) -- cycle;
\node at (6.074,5.783) {\scalebox{1.656}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.717,4.276) -- (2.839,4.295) -- (2.948,4.351) -- (3.035,4.438) -- (3.091,4.548) -- (3.111,4.67) -- (3.091,4.791) -- (3.035,4.901) -- (2.948,4.988) -- (2.839,5.044) -- (2.717,5.064) -- (2.717,5.064) -- (2.595,5.044) -- (2.485,4.988) -- (2.398,4.901) -- (2.342,4.791) -- (2.323,4.67) -- (2.342,4.548) -- (2.398,4.438) -- (2.485,4.351) -- (2.595,4.295) -- (2.717,4.276) -- cycle;
\node at (2.717,4.67) {\scalebox{1.656}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.751,3.342) -- (1.873,3.361) -- (1.983,3.417) -- (2.07,3.504) -- (2.126,3.614) -- (2.145,3.736) -- (2.126,3.857) -- (2.07,3.967) -- (1.983,4.054) -- (1.873,4.11) -- (1.751,4.13) -- (1.751,4.13) -- (1.629,4.11) -- (1.519,4.054) -- (1.432,3.967) -- (1.376,3.857) -- (1.357,3.736) -- (1.376,3.614) -- (1.432,3.504) -- (1.519,3.417) -- (1.629,3.361) -- (1.751,3.342) -- cycle;
\node at (1.751,3.736) {\scalebox{1.656}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.361,2.999) -- (9.482,3.019) -- (9.592,3.075) -- (9.679,3.162) -- (9.735,3.271) -- (9.755,3.393) -- (9.735,3.515) -- (9.679,3.625) -- (9.592,3.712) -- (9.482,3.768) -- (9.361,3.787) -- (9.361,3.787) -- (9.239,3.768) -- (9.129,3.712) -- (9.042,3.625) -- (8.986,3.515) -- (8.967,3.393) -- (8.986,3.271) -- (9.042,3.162) -- (9.129,3.075) -- (9.239,3.019) -- (9.361,2.999) -- cycle;
\node at (9.361,3.393) {\scalebox{1.656}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.407,4.772) -- (5.529,4.792) -- (5.639,4.848) -- (5.726,4.935) -- (5.782,5.045) -- (5.801,5.166) -- (5.782,5.288) -- (5.726,5.398) -- (5.639,5.485) -- (5.529,5.541) -- (5.407,5.56) -- (5.407,5.56) -- (5.285,5.541) -- (5.176,5.485) -- (5.089,5.398) -- (5.033,5.288) -- (5.013,5.166) -- (5.033,5.045) -- (5.089,4.935) -- (5.176,4.848) -- (5.285,4.792) -- (5.407,4.772) -- cycle;
\node at (5.407,5.166) {\scalebox{1.656}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.493,3.205) -- (4.614,3.225) -- (4.724,3.281) -- (4.811,3.368) -- (4.867,3.477) -- (4.887,3.599) -- (4.867,3.721) -- (4.811,3.831) -- (4.724,3.918) -- (4.614,3.974) -- (4.493,3.993) -- (4.493,3.993) -- (4.371,3.974) -- (4.261,3.918) -- (4.174,3.831) -- (4.118,3.721) -- (4.099,3.599) -- (4.118,3.477) -- (4.174,3.368) -- (4.261,3.281) -- (4.371,3.225) -- (4.493,3.205) -- cycle;
\node at (4.493,3.599) {\scalebox{1.656}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.904,4.04) -- (8.025,4.059) -- (8.135,4.115) -- (8.222,4.203) -- (8.278,4.312) -- (8.298,4.434) -- (8.278,4.556) -- (8.222,4.666) -- (8.135,4.753) -- (8.025,4.809) -- (7.904,4.828) -- (7.904,4.828) -- (7.782,4.809) -- (7.672,4.753) -- (7.585,4.666) -- (7.529,4.556) -- (7.51,4.434) -- (7.529,4.312) -- (7.585,4.203) -- (7.672,4.115) -- (7.782,4.059) -- (7.904,4.04) -- cycle;
\node at (7.904,4.434) {\scalebox{1.656}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.5,3.143) -- (7.622,3.162) -- (7.732,3.218) -- (7.819,3.305) -- (7.875,3.415) -- (7.894,3.536) -- (7.875,3.658) -- (7.819,3.768) -- (7.732,3.855) -- (7.622,3.911) -- (7.5,3.93) -- (7.129,3.93) -- (7.008,3.911) -- (6.898,3.855) -- (6.811,3.768) -- (6.755,3.658) -- (6.736,3.536) -- (6.755,3.415) -- (6.811,3.305) -- (6.898,3.218) -- (7.008,3.162) -- (7.129,3.143) -- cycle;
\node at (7.315,3.536) {\scalebox{1.656}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.924,2.464) -- (3.045,2.484) -- (3.155,2.54) -- (3.242,2.627) -- (3.298,2.736) -- (3.317,2.858) -- (3.298,2.98) -- (3.242,3.09) -- (3.155,3.177) -- (3.045,3.233) -- (2.924,3.252) -- (2.553,3.252) -- (2.431,3.233) -- (2.321,3.177) -- (2.234,3.09) -- (2.178,2.98) -- (2.159,2.858) -- (2.178,2.736) -- (2.234,2.627) -- (2.321,2.54) -- (2.431,2.484) -- (2.553,2.464) -- cycle;
\node at (2.738,2.858) {\scalebox{1.656}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.627,2.381) -- (5.749,2.4) -- (5.858,2.456) -- (5.946,2.543) -- (6.002,2.653) -- (6.021,2.775) -- (6.002,2.897) -- (5.946,3.006) -- (5.858,3.094) -- (5.749,3.15) -- (5.627,3.169) -- (5.256,3.169) -- (5.135,3.15) -- (5.025,3.094) -- (4.938,3.006) -- (4.882,2.897) -- (4.862,2.775) -- (4.882,2.653) -- (4.938,2.543) -- (5.025,2.456) -- (5.135,2.4) -- (5.256,2.381) -- cycle;
\node at (5.442,2.775) {\scalebox{1.656}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.294,0.6) -- (8.416,0.619) -- (8.525,0.675) -- (8.613,0.762) -- (8.669,0.872) -- (8.688,0.994) -- (8.669,1.116) -- (8.613,1.225) -- (8.525,1.313) -- (8.416,1.369) -- (8.294,1.388) -- (7.923,1.388) -- (7.801,1.369) -- (7.692,1.313) -- (7.605,1.225) -- (7.549,1.116) -- (7.529,0.994) -- (7.549,0.872) -- (7.605,0.762) -- (7.692,0.675) -- (7.801,0.619) -- (7.923,0.6) -- cycle;
\node at (8.109,0.994) {\scalebox{1.656}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.534,2.31) -- (10.655,2.329) -- (10.765,2.385) -- (10.852,2.472) -- (10.908,2.582) -- (10.928,2.704) -- (10.908,2.825) -- (10.852,2.935) -- (10.765,3.022) -- (10.655,3.078) -- (10.534,3.098) -- (10.163,3.098) -- (10.041,3.078) -- (9.931,3.022) -- (9.844,2.935) -- (9.788,2.825) -- (9.769,2.704) -- (9.788,2.582) -- (9.844,2.472) -- (9.931,2.385) -- (10.041,2.329) -- (10.163,2.31) -- cycle;
\node at (10.348,2.704) {\scalebox{1.656}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (8.741,4.671) -- (9.341,4.671) -- (9.341,5.271) -- (8.741,5.271) -- cycle;
\node at (9.041,4.971) {\scalebox{1.656}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.366,5.422) -- (4.966,5.422) -- (4.966,6.022) -- (4.366,6.022) -- cycle;
\node at (4.666,5.722) {\scalebox{1.656}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.078,2.231) -- (1.678,2.231) -- (1.678,2.831) -- (1.078,2.831) -- cycle;
\node at (1.378,2.531) {\scalebox{1.656}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.803,6.302) -- (10.403,6.302) -- (10.403,6.902) -- (9.803,6.902) -- cycle;
\node at (10.103,6.602) {\scalebox{1.656}{\small d}};
\node at (9.693,7.592) {\scalebox{1.656}{\small A}};
\node at (6.377,7.483) {\scalebox{1.656}{\small B}};
\node at (0.765,5.13) {\scalebox{1.656}{\small C}};
\node at (9.632,5.16) {\scalebox{1.656}{\small D}};
\node at (3.503,5.098) {\scalebox{1.656}{\small E}};
\node at (7.025,5.419) {\scalebox{1.656}{\small F}};
\node at (0.972,2.209) {\scalebox{1.656}{\small G}};
\node at (3.403,2.021) {\scalebox{1.656}{\small H}};
\node at (11.961,2.152) {\scalebox{1.656}{\small I}};
\node at (7.073,1.543) {\scalebox{1.656}{\small J}};
\node at (9.108,1.532) {\scalebox{1.656}{\small K}};
\draw[line width=0.5pt] (12.674,6.197) -- (12.674,7.297);
\draw[line width=0.5pt] (12.443,6.912) -- (12.674,7.297) -- (12.905,6.912);
\draw[line width=0.5pt] (12.372,6.747) -- (12.977,6.747);
\node at (12.674,7.682) {\scalebox{1.656}{\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 & ⑧ rue de la Mairie & Ⓐ Coin du Lavoir & Ⓘ Fontaine Notre-Dame \\ \hline
① avenue de la Gare & ⑨ boulevard du Parc & Ⓑ Place de la Fontaine & Ⓙ Rond-point du Pont \\ \hline
② rue des Écoles & ⑩ rue Verte & Ⓒ Place du Vieux-Puits & Ⓚ Rond-point du Stade \\ \hline
③ rue du Moulin & ⑪ rue de l'Église & Ⓓ Rond-point des Vaches & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ chemin des Vignes & ⑫ rue du Stade & Ⓔ Carrefour de la Mairie & \textbf{a} l'école \\ \hline
⑤ rue du Puits & ⑬ rue de la Poste & Ⓕ Carrefour du Lavoir & \textbf{b} le marché couvert \\ \hline
⑥ rue Haute & ⑭ impasse Sainte-Anne & Ⓖ Place Sainte-Croix & \textbf{c} la salle des fêtes \\ \hline
⑦ chemin Creux & \cellcolor{black!32}\bfseries Les carrefours & Ⓗ Rond-point de la Poste & \textbf{d} la mairie \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\begin{enumerate}
  \item Colorier les carrefours du plan avec \textbf{deux couleurs}, de sorte que deux carrefours \textbf{reliés par une rue} ne soient jamais de la même couleur. Combien y en a-t-il de chaque couleur ?
  \item Le car roule d'un carrefour au suivant. Que peut-on dire de la \textbf{couleur} des arrêts successifs ? Qu'en déduire sur les couleurs des arrêts d'une tournée qui les dessert tous ?
  \item \textbf{Le car peut-il desservir tous les carrefours, une fois chacun, et revenir à l'école ?} Si oui, décrire un trajet ; sinon, démontrer que c'est impossible.
  \item \textbf{Et s'il n'est pas obligé de revenir à l'école ?} Même consigne.
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage — Correction}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6187, line join=round, line cap=round]
\draw[line width=0.5pt] (9.261,7.115) -- (9.222,6.963) -- (9.186,6.811) -- (9.154,6.657) -- (9.125,6.503) -- (9.1,6.348) -- (9.079,6.192) -- (9.061,6.036) -- (9.047,5.88) -- (9.037,5.723) -- (9.031,5.566) -- (9.028,5.409) -- (9.028,5.252) -- (9.033,5.095) -- (9.041,4.939);
\draw[line width=0.5pt] (6.362,6.831) -- (6.564,6.883) -- (6.767,6.93) -- (6.972,6.972) -- (7.178,7.009) -- (7.384,7.042) -- (7.592,7.07) -- (7.8,7.093) -- (8.008,7.111) -- (8.217,7.124) -- (8.425,7.132) -- (8.634,7.135) -- (8.843,7.133) -- (9.052,7.127) -- (9.261,7.115);
\draw[line width=0.5pt] (6.362,6.831) -- (6.366,6.686) -- (6.373,6.541) -- (6.384,6.397) -- (6.398,6.252) -- (6.416,6.109) -- (6.437,5.965) -- (6.462,5.822) -- (6.49,5.68) -- (6.521,5.538) -- (6.555,5.397) -- (6.593,5.257) -- (6.634,5.118) -- (6.678,4.98) -- (6.726,4.844);
\draw[line width=0.5pt] (1.374,4.983) -- (1.568,5.004) -- (1.762,5.02) -- (1.956,5.031) -- (2.151,5.038) -- (2.346,5.04) -- (2.541,5.038) -- (2.735,5.031) -- (2.93,5.02) -- (3.124,5.004) -- (3.318,4.983) -- (3.511,4.958) -- (3.704,4.928) -- (3.895,4.894) -- (4.086,4.855);
\draw[line width=0.5pt] (1.374,4.983) -- (1.359,4.797) -- (1.348,4.611) -- (1.341,4.424) -- (1.339,4.237) -- (1.341,4.05) -- (1.347,3.863) -- (1.358,3.676) -- (1.373,3.49) -- (1.393,3.304) -- (1.417,3.118) -- (1.445,2.933) -- (1.478,2.749) -- (1.515,2.566) -- (1.556,2.384);
\draw[line width=0.5pt] (9.041,4.939) -- (9.047,4.72) -- (9.047,4.502) -- (9.042,4.283) -- (9.031,4.065) -- (9.016,3.847) -- (8.996,3.629) -- (8.97,3.412) -- (8.939,3.195) -- (8.904,2.98) -- (8.863,2.765) -- (8.817,2.551) -- (8.766,2.338) -- (8.71,2.127) -- (8.649,1.917);
\draw[line width=0.5pt] (4.086,4.855) -- (4.273,4.825) -- (4.461,4.8) -- (4.649,4.779) -- (4.838,4.763) -- (5.027,4.751) -- (5.216,4.744) -- (5.405,4.74) -- (5.595,4.742) -- (5.784,4.748) -- (5.974,4.758) -- (6.163,4.773) -- (6.351,4.792) -- (6.539,4.816) -- (6.726,4.844);
\draw[line width=0.5pt] (4.086,4.855) -- (4.101,4.676) -- (4.111,4.496) -- (4.117,4.316) -- (4.119,4.136) -- (4.117,3.955) -- (4.11,3.775) -- (4.1,3.595) -- (4.085,3.416) -- (4.066,3.236) -- (4.042,3.058) -- (4.015,2.88) -- (3.983,2.702) -- (3.947,2.526) -- (3.907,2.35);
\draw[line width=0.5pt] (6.726,4.844) -- (6.891,4.825) -- (7.057,4.81) -- (7.223,4.8) -- (7.389,4.793) -- (7.555,4.79) -- (7.721,4.791) -- (7.888,4.796) -- (8.054,4.804) -- (8.22,4.817) -- (8.385,4.834) -- (8.55,4.854) -- (8.714,4.879) -- (8.878,4.907) -- (9.041,4.939);
\draw[line width=0.5pt] (6.726,4.844) -- (6.768,4.651) -- (6.805,4.458) -- (6.837,4.264) -- (6.865,4.069) -- (6.889,3.873) -- (6.907,3.677) -- (6.922,3.481) -- (6.931,3.284) -- (6.936,3.087) -- (6.936,2.89) -- (6.932,2.693) -- (6.923,2.497) -- (6.91,2.3) -- (6.891,2.104);
\draw[line width=0.5pt] (1.556,2.384) -- (1.722,2.356) -- (1.889,2.332) -- (2.057,2.312) -- (2.225,2.295) -- (2.393,2.283) -- (2.561,2.275) -- (2.73,2.27) -- (2.899,2.27) -- (3.068,2.273) -- (3.236,2.281) -- (3.404,2.292) -- (3.572,2.308) -- (3.74,2.327) -- (3.907,2.35);
\draw[line width=0.5pt] (3.907,2.35) -- (4.121,2.365) -- (4.336,2.376) -- (4.55,2.381) -- (4.765,2.381) -- (4.98,2.376) -- (5.195,2.366) -- (5.409,2.35) -- (5.623,2.33) -- (5.837,2.305) -- (6.05,2.275) -- (6.261,2.24) -- (6.473,2.2) -- (6.683,2.154) -- (6.891,2.104);
\draw[line width=0.5pt] (6.891,2.104) -- (7.014,2.072) -- (7.137,2.042) -- (7.261,2.016) -- (7.386,1.992) -- (7.511,1.971) -- (7.637,1.953) -- (7.763,1.938) -- (7.889,1.926) -- (8.016,1.917) -- (8.142,1.911) -- (8.269,1.908) -- (8.396,1.908) -- (8.523,1.911) -- (8.649,1.917);
\draw[line width=0.5pt] (8.649,1.917) -- (8.838,1.976) -- (9.028,2.03) -- (9.22,2.079) -- (9.412,2.124) -- (9.606,2.164) -- (9.8,2.2) -- (9.995,2.231) -- (10.191,2.258) -- (10.388,2.28) -- (10.585,2.297) -- (10.782,2.31) -- (10.979,2.318) -- (11.177,2.321) -- (11.374,2.32);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.634,7.115) -- (9.606,7.258) -- (9.525,7.379) -- (9.404,7.46) -- (9.261,7.488) -- (9.118,7.46) -- (8.997,7.379) -- (8.916,7.258) -- (8.888,7.115) -- (8.916,6.972) -- (8.997,6.851) -- (9.118,6.77) -- (9.261,6.742) -- (9.404,6.77) -- (9.525,6.851) -- (9.606,6.972) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.735,6.831) -- (6.706,6.974) -- (6.625,7.095) -- (6.504,7.176) -- (6.362,7.204) -- (6.219,7.176) -- (6.098,7.095) -- (6.017,6.974) -- (5.988,6.831) -- (6.017,6.688) -- (6.098,6.567) -- (6.219,6.486) -- (6.362,6.458) -- (6.504,6.486) -- (6.625,6.567) -- (6.706,6.688) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.748,4.983) -- (1.719,5.126) -- (1.638,5.247) -- (1.517,5.328) -- (1.374,5.357) -- (1.231,5.328) -- (1.11,5.247) -- (1.03,5.126) -- (1.001,4.983) -- (1.03,4.841) -- (1.11,4.72) -- (1.231,4.639) -- (1.374,4.61) -- (1.517,4.639) -- (1.638,4.72) -- (1.719,4.841) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.414,4.939) -- (9.386,5.082) -- (9.305,5.203) -- (9.184,5.284) -- (9.041,5.312) -- (8.898,5.284) -- (8.777,5.203) -- (8.696,5.082) -- (8.668,4.939) -- (8.696,4.796) -- (8.777,4.675) -- (8.898,4.594) -- (9.041,4.566) -- (9.184,4.594) -- (9.305,4.675) -- (9.386,4.796) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.459,4.855) -- (4.431,4.998) -- (4.35,5.119) -- (4.229,5.2) -- (4.086,5.228) -- (3.943,5.2) -- (3.822,5.119) -- (3.741,4.998) -- (3.713,4.855) -- (3.741,4.712) -- (3.822,4.591) -- (3.943,4.51) -- (4.086,4.482) -- (4.229,4.51) -- (4.35,4.591) -- (4.431,4.712) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.099,4.844) -- (7.071,4.986) -- (6.99,5.108) -- (6.869,5.188) -- (6.726,5.217) -- (6.583,5.188) -- (6.462,5.108) -- (6.381,4.986) -- (6.353,4.844) -- (6.381,4.701) -- (6.462,4.58) -- (6.583,4.499) -- (6.726,4.47) -- (6.869,4.499) -- (6.99,4.58) -- (7.071,4.701) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.929,2.384) -- (1.901,2.527) -- (1.82,2.648) -- (1.699,2.729) -- (1.556,2.757) -- (1.413,2.729) -- (1.292,2.648) -- (1.211,2.527) -- (1.183,2.384) -- (1.211,2.241) -- (1.292,2.12) -- (1.413,2.039) -- (1.556,2.011) -- (1.699,2.039) -- (1.82,2.12) -- (1.901,2.241) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.28,2.35) -- (4.252,2.493) -- (4.171,2.614) -- (4.05,2.695) -- (3.907,2.723) -- (3.764,2.695) -- (3.643,2.614) -- (3.562,2.493) -- (3.534,2.35) -- (3.562,2.207) -- (3.643,2.086) -- (3.764,2.005) -- (3.907,1.977) -- (4.05,2.005) -- (4.171,2.086) -- (4.252,2.207) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.748,2.32) -- (11.719,2.463) -- (11.638,2.584) -- (11.517,2.665) -- (11.374,2.693) -- (11.231,2.665) -- (11.11,2.584) -- (11.03,2.463) -- (11.001,2.32) -- (11.03,2.177) -- (11.11,2.056) -- (11.231,1.975) -- (11.374,1.947) -- (11.517,1.975) -- (11.638,2.056) -- (11.719,2.177) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.265,2.104) -- (7.236,2.247) -- (7.155,2.368) -- (7.034,2.449) -- (6.891,2.478) -- (6.748,2.449) -- (6.627,2.368) -- (6.547,2.247) -- (6.518,2.104) -- (6.547,1.961) -- (6.627,1.84) -- (6.748,1.76) -- (6.891,1.731) -- (7.034,1.76) -- (7.155,1.84) -- (7.236,1.961) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.023,1.917) -- (8.994,2.06) -- (8.913,2.181) -- (8.792,2.262) -- (8.649,2.29) -- (8.507,2.262) -- (8.386,2.181) -- (8.305,2.06) -- (8.276,1.917) -- (8.305,1.774) -- (8.386,1.653) -- (8.507,1.572) -- (8.649,1.544) -- (8.792,1.572) -- (8.913,1.653) -- (8.994,1.774) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.444,5.613) -- (9.563,5.632) -- (9.67,5.686) -- (9.755,5.771) -- (9.81,5.879) -- (9.828,5.997) -- (9.81,6.116) -- (9.755,6.223) -- (9.67,6.309) -- (9.563,6.363) -- (9.444,6.382) -- (9.444,6.382) -- (9.325,6.363) -- (9.218,6.309) -- (9.133,6.223) -- (9.078,6.116) -- (9.059,5.997) -- (9.078,5.879) -- (9.133,5.771) -- (9.218,5.686) -- (9.325,5.632) -- (9.444,5.613) -- cycle;
\node at (9.444,5.997) {\scalebox{1.616}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.762,7.091) -- (7.881,7.11) -- (7.988,7.164) -- (8.073,7.249) -- (8.128,7.356) -- (8.146,7.475) -- (8.128,7.594) -- (8.073,7.701) -- (7.988,7.786) -- (7.881,7.841) -- (7.762,7.86) -- (7.762,7.86) -- (7.643,7.841) -- (7.536,7.786) -- (7.451,7.701) -- (7.396,7.594) -- (7.377,7.475) -- (7.396,7.356) -- (7.451,7.249) -- (7.536,7.164) -- (7.643,7.11) -- (7.762,7.091) -- cycle;
\node at (7.762,7.475) {\scalebox{1.616}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.084,5.368) -- (6.202,5.387) -- (6.31,5.442) -- (6.395,5.527) -- (6.449,5.634) -- (6.468,5.753) -- (6.449,5.872) -- (6.395,5.979) -- (6.31,6.064) -- (6.202,6.118) -- (6.084,6.137) -- (6.084,6.137) -- (5.965,6.118) -- (5.858,6.064) -- (5.773,5.979) -- (5.718,5.872) -- (5.699,5.753) -- (5.718,5.634) -- (5.773,5.527) -- (5.858,5.442) -- (5.965,5.387) -- (6.084,5.368) -- cycle;
\node at (6.084,5.753) {\scalebox{1.616}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.717,4.263) -- (2.836,4.281) -- (2.943,4.336) -- (3.028,4.421) -- (3.083,4.528) -- (3.102,4.647) -- (3.083,4.766) -- (3.028,4.873) -- (2.943,4.958) -- (2.836,5.013) -- (2.717,5.032) -- (2.717,5.032) -- (2.598,5.013) -- (2.491,4.958) -- (2.406,4.873) -- (2.352,4.766) -- (2.333,4.647) -- (2.352,4.528) -- (2.406,4.421) -- (2.491,4.336) -- (2.598,4.281) -- (2.717,4.263) -- cycle;
\node at (2.717,4.647) {\scalebox{1.616}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.742,3.319) -- (1.86,3.337) -- (1.968,3.392) -- (2.053,3.477) -- (2.107,3.584) -- (2.126,3.703) -- (2.107,3.822) -- (2.053,3.929) -- (1.968,4.014) -- (1.86,4.069) -- (1.742,4.088) -- (1.742,4.088) -- (1.623,4.069) -- (1.516,4.014) -- (1.431,3.929) -- (1.376,3.822) -- (1.357,3.703) -- (1.376,3.584) -- (1.431,3.477) -- (1.516,3.392) -- (1.623,3.337) -- (1.742,3.319) -- cycle;
\node at (1.742,3.703) {\scalebox{1.616}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.351,2.978) -- (9.47,2.997) -- (9.577,3.051) -- (9.662,3.136) -- (9.717,3.244) -- (9.736,3.362) -- (9.717,3.481) -- (9.662,3.588) -- (9.577,3.674) -- (9.47,3.728) -- (9.351,3.747) -- (9.351,3.747) -- (9.232,3.728) -- (9.125,3.674) -- (9.04,3.588) -- (8.986,3.481) -- (8.967,3.362) -- (8.986,3.244) -- (9.04,3.136) -- (9.125,3.051) -- (9.232,2.997) -- (9.351,2.978) -- cycle;
\node at (9.351,3.362) {\scalebox{1.616}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.407,4.74) -- (5.526,4.759) -- (5.633,4.814) -- (5.718,4.899) -- (5.773,5.006) -- (5.792,5.125) -- (5.773,5.244) -- (5.718,5.351) -- (5.633,5.436) -- (5.526,5.491) -- (5.407,5.51) -- (5.407,5.51) -- (5.288,5.491) -- (5.181,5.436) -- (5.096,5.351) -- (5.041,5.244) -- (5.023,5.125) -- (5.041,5.006) -- (5.096,4.899) -- (5.181,4.814) -- (5.288,4.759) -- (5.407,4.74) -- cycle;
\node at (5.407,5.125) {\scalebox{1.616}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.483,3.183) -- (4.602,3.202) -- (4.709,3.257) -- (4.794,3.342) -- (4.849,3.449) -- (4.868,3.568) -- (4.849,3.687) -- (4.794,3.794) -- (4.709,3.879) -- (4.602,3.934) -- (4.483,3.952) -- (4.483,3.952) -- (4.364,3.934) -- (4.257,3.879) -- (4.172,3.794) -- (4.118,3.687) -- (4.099,3.568) -- (4.118,3.449) -- (4.172,3.342) -- (4.257,3.257) -- (4.364,3.202) -- (4.483,3.183) -- cycle;
\node at (4.483,3.568) {\scalebox{1.616}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.872,4.795) -- (7.991,4.814) -- (8.098,4.869) -- (8.183,4.954) -- (8.237,5.061) -- (8.256,5.18) -- (8.237,5.299) -- (8.183,5.406) -- (8.098,5.491) -- (7.991,5.546) -- (7.872,5.564) -- (7.872,5.564) -- (7.753,5.546) -- (7.646,5.491) -- (7.561,5.406) -- (7.506,5.299) -- (7.487,5.18) -- (7.506,5.061) -- (7.561,4.954) -- (7.646,4.869) -- (7.753,4.814) -- (7.872,4.795) -- cycle;
\node at (7.872,5.18) {\scalebox{1.616}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.486,3.119) -- (7.605,3.138) -- (7.712,3.193) -- (7.798,3.278) -- (7.852,3.385) -- (7.871,3.504) -- (7.852,3.623) -- (7.798,3.73) -- (7.712,3.815) -- (7.605,3.87) -- (7.486,3.888) -- (7.125,3.888) -- (7.006,3.87) -- (6.899,3.815) -- (6.813,3.73) -- (6.759,3.623) -- (6.74,3.504) -- (6.759,3.385) -- (6.813,3.278) -- (6.899,3.193) -- (7.006,3.138) -- (7.125,3.119) -- cycle;
\node at (7.305,3.504) {\scalebox{1.616}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.757,2.274) -- (2.876,2.293) -- (2.983,2.348) -- (3.068,2.433) -- (3.123,2.54) -- (3.141,2.659) -- (3.123,2.778) -- (3.068,2.885) -- (2.983,2.97) -- (2.876,3.025) -- (2.757,3.043) -- (2.395,3.043) -- (2.276,3.025) -- (2.169,2.97) -- (2.084,2.885) -- (2.029,2.778) -- (2.011,2.659) -- (2.029,2.54) -- (2.084,2.433) -- (2.169,2.348) -- (2.276,2.293) -- (2.395,2.274) -- cycle;
\node at (2.576,2.659) {\scalebox{1.616}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.622,2.349) -- (5.741,2.368) -- (5.848,2.423) -- (5.933,2.508) -- (5.987,2.615) -- (6.006,2.734) -- (5.987,2.852) -- (5.933,2.96) -- (5.848,3.045) -- (5.741,3.099) -- (5.622,3.118) -- (5.26,3.118) -- (5.141,3.099) -- (5.034,3.045) -- (4.949,2.96) -- (4.894,2.852) -- (4.875,2.734) -- (4.894,2.615) -- (4.949,2.508) -- (5.034,2.423) -- (5.141,2.368) -- (5.26,2.349) -- cycle;
\node at (5.441,2.734) {\scalebox{1.616}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.29,0.6) -- (8.409,0.619) -- (8.516,0.673) -- (8.601,0.759) -- (8.656,0.866) -- (8.675,0.985) -- (8.656,1.103) -- (8.601,1.211) -- (8.516,1.296) -- (8.409,1.35) -- (8.29,1.369) -- (7.928,1.369) -- (7.81,1.35) -- (7.702,1.296) -- (7.617,1.211) -- (7.563,1.103) -- (7.544,0.985) -- (7.563,0.866) -- (7.617,0.759) -- (7.702,0.673) -- (7.81,0.619) -- (7.928,0.6) -- cycle;
\node at (8.109,0.985) {\scalebox{1.616}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.53,2.278) -- (10.649,2.297) -- (10.756,2.351) -- (10.841,2.436) -- (10.896,2.544) -- (10.915,2.662) -- (10.896,2.781) -- (10.841,2.888) -- (10.756,2.973) -- (10.649,3.028) -- (10.53,3.047) -- (10.168,3.047) -- (10.05,3.028) -- (9.942,2.973) -- (9.857,2.888) -- (9.803,2.781) -- (9.784,2.662) -- (9.803,2.544) -- (9.857,2.436) -- (9.942,2.351) -- (10.05,2.297) -- (10.168,2.278) -- cycle;
\node at (10.349,2.662) {\scalebox{1.616}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (8.741,4.639) -- (9.341,4.639) -- (9.341,5.239) -- (8.741,5.239) -- cycle;
\node at (9.041,4.939) {\scalebox{1.616}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.366,5.39) -- (4.966,5.39) -- (4.966,5.99) -- (4.366,5.99) -- cycle;
\node at (4.666,5.69) {\scalebox{1.616}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.078,2.199) -- (1.678,2.199) -- (1.678,2.799) -- (1.078,2.799) -- cycle;
\node at (1.378,2.499) {\scalebox{1.616}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.803,6.27) -- (10.403,6.27) -- (10.403,6.87) -- (9.803,6.87) -- cycle;
\node at (10.103,6.57) {\scalebox{1.616}{\small d}};
\node at (9.693,7.56) {\scalebox{1.616}{\small A}};
\node at (6.377,7.451) {\scalebox{1.616}{\small B}};
\node at (0.765,5.098) {\scalebox{1.616}{\small C}};
\node at (9.632,5.128) {\scalebox{1.616}{\small D}};
\node at (3.503,5.066) {\scalebox{1.616}{\small E}};
\node at (7.025,5.387) {\scalebox{1.616}{\small F}};
\node at (0.972,2.177) {\scalebox{1.616}{\small G}};
\node at (3.403,1.989) {\scalebox{1.616}{\small H}};
\node at (11.961,2.12) {\scalebox{1.616}{\small I}};
\node at (7.073,1.511) {\scalebox{1.616}{\small J}};
\node at (9.108,1.5) {\scalebox{1.616}{\small K}};
\draw[line width=0.5pt] (12.674,6.165) -- (12.674,7.265);
\draw[line width=0.5pt] (12.443,6.88) -- (12.674,7.265) -- (12.905,6.88);
\draw[line width=0.5pt] (12.372,6.715) -- (12.977,6.715);
\node at (12.674,7.65) {\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, C, F, H, K & 5 \\ \hline
clairs & B, D, E, G, I, J & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{6} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 11 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 5 carrefours sombres et 6 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{D} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend l'\textbf{avenue de la Gare ①} vers le nord jusqu'à Coin du Lavoir \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Écoles ②}, vers l'ouest, jusqu'à Place de la Fontaine \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Moulin ③}, vers le sud, jusqu'à Carrefour du Lavoir \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin Creux ⑦}, vers l'ouest, jusqu'à Carrefour de la Mairie \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin des Vignes ④}, vers l'ouest, jusqu'à Place du Vieux-Puits \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ⑤}, vers le sud, jusqu'à Place Sainte-Croix \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de l'Église ⑪}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Stade ⑫}, vers l'est, jusqu'à Rond-point du Pont \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de la Poste ⑬}, vers l'est, jusqu'à Rond-point du Stade \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{impasse Sainte-Anne ⑭}, vers l'est, jusqu'à Fontaine Notre-Dame \textbf{(I)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 11 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
