\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 222}

\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=222}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0222}
\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=0.6055, line join=round, line cap=round]
\draw[line width=0.5pt] (6.04,7.028) -- (6.202,7.0) -- (6.366,6.976) -- (6.53,6.956) -- (6.694,6.94) -- (6.859,6.928) -- (7.024,6.919) -- (7.189,6.915) -- (7.354,6.914) -- (7.519,6.917) -- (7.684,6.924) -- (7.849,6.935) -- (8.014,6.95) -- (8.178,6.969) -- (8.341,6.991);
\draw[line width=0.5pt] (6.04,7.028) -- (6.076,6.845) -- (6.108,6.66) -- (6.135,6.476) -- (6.158,6.29) -- (6.177,6.104) -- (6.192,5.918) -- (6.202,5.731) -- (6.207,5.544) -- (6.209,5.357) -- (6.206,5.17) -- (6.198,4.983) -- (6.186,4.797) -- (6.17,4.61) -- (6.149,4.425);
\draw[line width=0.5pt] (8.341,6.991) -- (8.52,6.963) -- (8.699,6.939) -- (8.878,6.919) -- (9.058,6.903) -- (9.238,6.892) -- (9.419,6.884) -- (9.6,6.881) -- (9.781,6.882) -- (9.961,6.888) -- (10.142,6.898) -- (10.322,6.912) -- (10.502,6.93) -- (10.681,6.952) -- (10.859,6.979);
\draw[line width=0.5pt] (8.341,6.991) -- (8.403,6.791) -- (8.461,6.589) -- (8.514,6.386) -- (8.562,6.181) -- (8.605,5.976) -- (8.643,5.769) -- (8.676,5.561) -- (8.705,5.353) -- (8.729,5.144) -- (8.747,4.935) -- (8.761,4.726) -- (8.77,4.516) -- (8.774,4.306) -- (8.773,4.096);
\draw[line width=0.5pt] (10.859,6.979) -- (10.866,6.792) -- (10.876,6.606) -- (10.891,6.42) -- (10.91,6.234) -- (10.934,6.049) -- (10.962,5.864) -- (10.994,5.68) -- (11.031,5.497) -- (11.072,5.315) -- (11.117,5.133) -- (11.167,4.953) -- (11.22,4.775) -- (11.278,4.597) -- (11.34,4.421);
\draw[line width=0.5pt] (1.34,6.699) -- (1.47,6.688) -- (1.599,6.674) -- (1.728,6.657) -- (1.857,6.636) -- (1.985,6.613) -- (2.113,6.586) -- (2.239,6.557) -- (2.366,6.525) -- (2.491,6.489) -- (2.616,6.451) -- (2.739,6.41) -- (2.862,6.366) -- (2.983,6.319) -- (3.103,6.269);
\draw[line width=0.5pt] (3.103,6.269) -- (3.32,6.291) -- (3.536,6.317) -- (3.751,6.349) -- (3.965,6.386) -- (4.179,6.428) -- (4.391,6.475) -- (4.603,6.527) -- (4.813,6.584) -- (5.022,6.646) -- (5.229,6.713) -- (5.434,6.784) -- (5.638,6.861) -- (5.84,6.942) -- (6.04,7.028);
\draw[line width=0.5pt] (3.103,6.269) -- (3.133,6.113) -- (3.159,5.956) -- (3.181,5.798) -- (3.199,5.64) -- (3.213,5.482) -- (3.224,5.323) -- (3.231,5.164) -- (3.235,5.005) -- (3.235,4.846) -- (3.23,4.687) -- (3.223,4.528) -- (3.211,4.369) -- (3.196,4.211) -- (3.177,4.053);
\draw[line width=0.5pt] (6.149,4.425) -- (6.331,4.373) -- (6.515,4.325) -- (6.7,4.282) -- (6.885,4.243) -- (7.072,4.208) -- (7.259,4.178) -- (7.447,4.152) -- (7.636,4.131) -- (7.825,4.114) -- (8.014,4.101) -- (8.204,4.093) -- (8.394,4.09) -- (8.583,4.091) -- (8.773,4.096);
\draw[line width=0.5pt] (6.149,4.425) -- (6.148,4.216) -- (6.142,4.008) -- (6.131,3.8) -- (6.115,3.592) -- (6.094,3.384) -- (6.068,3.177) -- (6.037,2.97) -- (6.002,2.765) -- (5.962,2.56) -- (5.917,2.356) -- (5.867,2.154) -- (5.812,1.952) -- (5.753,1.753) -- (5.689,1.554);
\draw[line width=0.5pt] (8.773,4.096) -- (8.958,4.091) -- (9.144,4.09) -- (9.329,4.094) -- (9.515,4.102) -- (9.7,4.115) -- (9.885,4.131) -- (10.07,4.153) -- (10.254,4.178) -- (10.437,4.208) -- (10.62,4.242) -- (10.801,4.28) -- (10.982,4.323) -- (11.162,4.37) -- (11.34,4.421);
\draw[line width=0.5pt] (8.773,4.096) -- (8.8,3.919) -- (8.824,3.742) -- (8.843,3.564) -- (8.858,3.385) -- (8.869,3.207) -- (8.876,3.028) -- (8.878,2.848) -- (8.876,2.669) -- (8.87,2.49) -- (8.86,2.311) -- (8.846,2.133) -- (8.827,1.955) -- (8.804,1.777) -- (8.777,1.6);
\draw[line width=0.5pt] (3.177,4.053) -- (3.391,4.047) -- (3.606,4.046) -- (3.821,4.05) -- (4.036,4.059) -- (4.25,4.073) -- (4.465,4.092) -- (4.678,4.116) -- (4.891,4.146) -- (5.104,4.18) -- (5.315,4.219) -- (5.525,4.263) -- (5.735,4.312) -- (5.942,4.366) -- (6.149,4.425);
\draw[line width=0.5pt] (3.177,4.053) -- (3.208,3.895) -- (3.236,3.736) -- (3.26,3.577) -- (3.281,3.416) -- (3.297,3.256) -- (3.31,3.095) -- (3.319,2.934) -- (3.324,2.772) -- (3.326,2.611) -- (3.324,2.449) -- (3.317,2.288) -- (3.308,2.127) -- (3.294,1.966) -- (3.276,1.806);
\draw[line width=0.5pt] (3.276,1.806) -- (3.444,1.762) -- (3.614,1.721) -- (3.784,1.685) -- (3.955,1.653) -- (4.127,1.625) -- (4.299,1.601) -- (4.472,1.581) -- (4.646,1.564) -- (4.819,1.553) -- (4.993,1.545) -- (5.167,1.541) -- (5.341,1.541) -- (5.515,1.546) -- (5.689,1.554);
\draw[line width=0.5pt] (5.689,1.554) -- (5.907,1.591) -- (6.127,1.623) -- (6.347,1.65) -- (6.567,1.671) -- (6.788,1.688) -- (7.01,1.699) -- (7.231,1.705) -- (7.453,1.705) -- (7.675,1.701) -- (7.896,1.691) -- (8.117,1.676) -- (8.338,1.656) -- (8.558,1.631) -- (8.777,1.6);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.2,7.028) -- (6.178,7.108) -- (6.12,7.166) -- (6.04,7.188) -- (5.96,7.166) -- (5.901,7.108) -- (5.88,7.028) -- (5.901,6.948) -- (5.96,6.889) -- (6.04,6.868) -- (6.12,6.889) -- (6.178,6.948) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.501,6.991) -- (8.48,7.071) -- (8.421,7.13) -- (8.341,7.151) -- (8.261,7.13) -- (8.203,7.071) -- (8.181,6.991) -- (8.203,6.911) -- (8.261,6.853) -- (8.341,6.831) -- (8.421,6.853) -- (8.48,6.911) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.019,6.979) -- (10.998,7.059) -- (10.939,7.117) -- (10.859,7.139) -- (10.779,7.117) -- (10.721,7.059) -- (10.699,6.979) -- (10.721,6.899) -- (10.779,6.84) -- (10.859,6.819) -- (10.939,6.84) -- (10.998,6.899) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.5,6.699) -- (1.479,6.779) -- (1.42,6.838) -- (1.34,6.859) -- (1.26,6.838) -- (1.202,6.779) -- (1.18,6.699) -- (1.202,6.619) -- (1.26,6.561) -- (1.34,6.539) -- (1.42,6.561) -- (1.479,6.619) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.263,6.269) -- (3.242,6.349) -- (3.183,6.408) -- (3.103,6.429) -- (3.023,6.408) -- (2.965,6.349) -- (2.943,6.269) -- (2.965,6.189) -- (3.023,6.131) -- (3.103,6.109) -- (3.183,6.131) -- (3.242,6.189) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.309,4.425) -- (6.288,4.505) -- (6.229,4.563) -- (6.149,4.585) -- (6.069,4.563) -- (6.01,4.505) -- (5.989,4.425) -- (6.01,4.345) -- (6.069,4.286) -- (6.149,4.265) -- (6.229,4.286) -- (6.288,4.345) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.5,4.421) -- (11.479,4.501) -- (11.42,4.56) -- (11.34,4.581) -- (11.26,4.56) -- (11.202,4.501) -- (11.18,4.421) -- (11.202,4.341) -- (11.26,4.283) -- (11.34,4.261) -- (11.42,4.283) -- (11.479,4.341) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.933,4.096) -- (8.911,4.176) -- (8.853,4.235) -- (8.773,4.256) -- (8.693,4.235) -- (8.634,4.176) -- (8.613,4.096) -- (8.634,4.016) -- (8.693,3.957) -- (8.773,3.936) -- (8.853,3.957) -- (8.911,4.016) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.337,4.053) -- (3.315,4.133) -- (3.257,4.192) -- (3.177,4.213) -- (3.097,4.192) -- (3.038,4.133) -- (3.017,4.053) -- (3.038,3.973) -- (3.097,3.915) -- (3.177,3.893) -- (3.257,3.915) -- (3.315,3.973) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.436,1.806) -- (3.415,1.886) -- (3.356,1.945) -- (3.276,1.966) -- (3.196,1.945) -- (3.138,1.886) -- (3.116,1.806) -- (3.138,1.726) -- (3.196,1.667) -- (3.276,1.646) -- (3.356,1.667) -- (3.415,1.726) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.937,1.6) -- (8.916,1.68) -- (8.857,1.739) -- (8.777,1.76) -- (8.697,1.739) -- (8.639,1.68) -- (8.617,1.6) -- (8.639,1.52) -- (8.697,1.462) -- (8.777,1.44) -- (8.857,1.462) -- (8.916,1.52) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.849,1.554) -- (5.828,1.634) -- (5.769,1.693) -- (5.689,1.714) -- (5.609,1.693) -- (5.55,1.634) -- (5.529,1.554) -- (5.55,1.474) -- (5.609,1.416) -- (5.689,1.394) -- (5.769,1.416) -- (5.828,1.474) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.195,6.915) -- (7.317,6.934) -- (7.427,6.99) -- (7.514,7.077) -- (7.57,7.187) -- (7.589,7.309) -- (7.57,7.43) -- (7.514,7.54) -- (7.427,7.627) -- (7.317,7.683) -- (7.195,7.702) -- (7.195,7.702) -- (7.073,7.683) -- (6.964,7.627) -- (6.876,7.54) -- (6.82,7.43) -- (6.801,7.309) -- (6.82,7.187) -- (6.876,7.077) -- (6.964,6.99) -- (7.073,6.934) -- (7.195,6.915) -- cycle;
\node at (7.195,7.309) {\scalebox{1.652}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.595,5.353) -- (6.717,5.373) -- (6.827,5.429) -- (6.914,5.516) -- (6.97,5.626) -- (6.989,5.747) -- (6.97,5.869) -- (6.914,5.979) -- (6.827,6.066) -- (6.717,6.122) -- (6.595,6.141) -- (6.595,6.141) -- (6.473,6.122) -- (6.364,6.066) -- (6.277,5.979) -- (6.221,5.869) -- (6.201,5.747) -- (6.221,5.626) -- (6.277,5.516) -- (6.364,5.429) -- (6.473,5.373) -- (6.595,5.353) -- cycle;
\node at (6.595,5.747) {\scalebox{1.652}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.602,6.881) -- (9.723,6.901) -- (9.833,6.956) -- (9.92,7.044) -- (9.976,7.153) -- (9.996,7.275) -- (9.976,7.397) -- (9.92,7.507) -- (9.833,7.594) -- (9.723,7.65) -- (9.602,7.669) -- (9.602,7.669) -- (9.48,7.65) -- (9.37,7.594) -- (9.283,7.507) -- (9.227,7.397) -- (9.208,7.275) -- (9.227,7.153) -- (9.283,7.044) -- (9.37,6.956) -- (9.48,6.901) -- (9.602,6.881) -- cycle;
\node at (9.602,7.275) {\scalebox{1.652}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.066,5.226) -- (9.188,5.245) -- (9.298,5.301) -- (9.385,5.388) -- (9.441,5.498) -- (9.46,5.62) -- (9.441,5.741) -- (9.385,5.851) -- (9.298,5.938) -- (9.188,5.994) -- (9.066,6.013) -- (9.066,6.013) -- (8.944,5.994) -- (8.834,5.938) -- (8.747,5.851) -- (8.691,5.741) -- (8.672,5.62) -- (8.691,5.498) -- (8.747,5.388) -- (8.834,5.301) -- (8.944,5.245) -- (9.066,5.226) -- cycle;
\node at (9.066,5.62) {\scalebox{1.652}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.381,5.359) -- (11.503,5.378) -- (11.613,5.434) -- (11.7,5.521) -- (11.756,5.631) -- (11.775,5.753) -- (11.756,5.875) -- (11.7,5.984) -- (11.613,6.072) -- (11.503,6.128) -- (11.381,6.147) -- (11.381,6.147) -- (11.26,6.128) -- (11.15,6.072) -- (11.063,5.984) -- (11.007,5.875) -- (10.988,5.753) -- (11.007,5.631) -- (11.063,5.521) -- (11.15,5.434) -- (11.26,5.378) -- (11.381,5.359) -- cycle;
\node at (11.381,5.753) {\scalebox{1.652}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.954,5.468) -- (2.075,5.488) -- (2.185,5.543) -- (2.272,5.631) -- (2.328,5.74) -- (2.348,5.862) -- (2.328,5.984) -- (2.272,6.094) -- (2.185,6.181) -- (2.075,6.237) -- (1.954,6.256) -- (1.954,6.256) -- (1.832,6.237) -- (1.722,6.181) -- (1.635,6.094) -- (1.579,5.984) -- (1.56,5.862) -- (1.579,5.74) -- (1.635,5.631) -- (1.722,5.543) -- (1.832,5.488) -- (1.954,5.468) -- cycle;
\node at (1.954,5.862) {\scalebox{1.652}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.504,6.515) -- (4.626,6.534) -- (4.736,6.59) -- (4.823,6.677) -- (4.879,6.787) -- (4.898,6.909) -- (4.879,7.03) -- (4.823,7.14) -- (4.736,7.227) -- (4.626,7.283) -- (4.504,7.303) -- (4.504,7.303) -- (4.382,7.283) -- (4.273,7.227) -- (4.186,7.14) -- (4.13,7.03) -- (4.11,6.909) -- (4.13,6.787) -- (4.186,6.677) -- (4.273,6.59) -- (4.382,6.534) -- (4.504,6.515) -- cycle;
\node at (4.504,6.909) {\scalebox{1.652}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.625,4.783) -- (3.747,4.803) -- (3.857,4.859) -- (3.944,4.946) -- (4.0,5.055) -- (4.019,5.177) -- (4.0,5.299) -- (3.944,5.409) -- (3.857,5.496) -- (3.747,5.552) -- (3.625,5.571) -- (3.625,5.571) -- (3.503,5.552) -- (3.394,5.496) -- (3.306,5.409) -- (3.251,5.299) -- (3.231,5.177) -- (3.251,5.055) -- (3.306,4.946) -- (3.394,4.859) -- (3.503,4.803) -- (3.625,4.783) -- cycle;
\node at (3.625,5.177) {\scalebox{1.652}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.496,4.149) -- (7.618,4.168) -- (7.728,4.224) -- (7.815,4.311) -- (7.871,4.421) -- (7.89,4.543) -- (7.871,4.665) -- (7.815,4.774) -- (7.728,4.862) -- (7.618,4.918) -- (7.496,4.937) -- (7.496,4.937) -- (7.375,4.918) -- (7.265,4.862) -- (7.178,4.774) -- (7.122,4.665) -- (7.102,4.543) -- (7.122,4.421) -- (7.178,4.311) -- (7.265,4.224) -- (7.375,4.168) -- (7.496,4.149) -- cycle;
\node at (7.496,4.543) {\scalebox{1.652}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.762,2.247) -- (5.883,2.266) -- (5.993,2.322) -- (6.08,2.409) -- (6.136,2.519) -- (6.155,2.641) -- (6.136,2.762) -- (6.08,2.872) -- (5.993,2.959) -- (5.883,3.015) -- (5.762,3.034) -- (5.391,3.034) -- (5.269,3.015) -- (5.159,2.959) -- (5.072,2.872) -- (5.016,2.762) -- (4.997,2.641) -- (5.016,2.519) -- (5.072,2.409) -- (5.159,2.322) -- (5.269,2.266) -- (5.391,2.247) -- cycle;
\node at (5.576,2.641) {\scalebox{1.652}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.215,4.076) -- (10.337,4.095) -- (10.447,4.151) -- (10.534,4.239) -- (10.59,4.348) -- (10.609,4.47) -- (10.59,4.592) -- (10.534,4.702) -- (10.447,4.789) -- (10.337,4.845) -- (10.215,4.864) -- (9.844,4.864) -- (9.723,4.845) -- (9.613,4.789) -- (9.526,4.702) -- (9.47,4.592) -- (9.45,4.47) -- (9.47,4.348) -- (9.526,4.239) -- (9.613,4.151) -- (9.723,4.095) -- (9.844,4.076) -- cycle;
\node at (10.03,4.47) {\scalebox{1.652}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.457,2.455) -- (9.579,2.475) -- (9.689,2.53) -- (9.776,2.618) -- (9.832,2.727) -- (9.851,2.849) -- (9.832,2.971) -- (9.776,3.081) -- (9.689,3.168) -- (9.579,3.224) -- (9.457,3.243) -- (9.087,3.243) -- (8.965,3.224) -- (8.855,3.168) -- (8.768,3.081) -- (8.712,2.971) -- (8.693,2.849) -- (8.712,2.727) -- (8.768,2.618) -- (8.855,2.53) -- (8.965,2.475) -- (9.087,2.455) -- cycle;
\node at (9.272,2.849) {\scalebox{1.652}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.815,4.113) -- (4.936,4.133) -- (5.046,4.189) -- (5.133,4.276) -- (5.189,4.385) -- (5.209,4.507) -- (5.189,4.629) -- (5.133,4.739) -- (5.046,4.826) -- (4.936,4.882) -- (4.815,4.901) -- (4.444,4.901) -- (4.322,4.882) -- (4.212,4.826) -- (4.125,4.739) -- (4.069,4.629) -- (4.05,4.507) -- (4.069,4.385) -- (4.125,4.276) -- (4.212,4.189) -- (4.322,4.133) -- (4.444,4.113) -- cycle;
\node at (4.629,4.507) {\scalebox{1.652}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.898,2.557) -- (4.02,2.576) -- (4.13,2.632) -- (4.217,2.72) -- (4.273,2.829) -- (4.292,2.951) -- (4.273,3.073) -- (4.217,3.183) -- (4.13,3.27) -- (4.02,3.326) -- (3.898,3.345) -- (3.527,3.345) -- (3.406,3.326) -- (3.296,3.27) -- (3.209,3.183) -- (3.153,3.073) -- (3.133,2.951) -- (3.153,2.829) -- (3.209,2.72) -- (3.296,2.632) -- (3.406,2.576) -- (3.527,2.557) -- cycle;
\node at (3.713,2.951) {\scalebox{1.652}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.617,0.795) -- (4.738,0.814) -- (4.848,0.87) -- (4.935,0.957) -- (4.991,1.067) -- (5.011,1.189) -- (4.991,1.31) -- (4.935,1.42) -- (4.848,1.507) -- (4.738,1.563) -- (4.617,1.583) -- (4.246,1.583) -- (4.124,1.563) -- (4.014,1.507) -- (3.927,1.42) -- (3.871,1.31) -- (3.852,1.189) -- (3.871,1.067) -- (3.927,0.957) -- (4.014,0.87) -- (4.124,0.814) -- (4.246,0.795) -- cycle;
\node at (4.431,1.189) {\scalebox{1.652}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.411,1.705) -- (7.532,1.724) -- (7.642,1.78) -- (7.729,1.867) -- (7.785,1.977) -- (7.805,2.099) -- (7.785,2.22) -- (7.729,2.33) -- (7.642,2.417) -- (7.532,2.473) -- (7.411,2.492) -- (7.04,2.492) -- (6.918,2.473) -- (6.808,2.417) -- (6.721,2.33) -- (6.665,2.22) -- (6.646,2.099) -- (6.665,1.977) -- (6.721,1.867) -- (6.808,1.78) -- (6.918,1.724) -- (7.04,1.705) -- cycle;
\node at (7.225,2.099) {\scalebox{1.652}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (11.04,4.121) -- (11.64,4.121) -- (11.64,4.721) -- (11.04,4.721) -- cycle;
\node at (11.34,4.421) {\scalebox{1.652}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.357,4.363) -- (9.957,4.363) -- (9.957,4.963) -- (9.357,4.963) -- cycle;
\node at (9.657,4.663) {\scalebox{1.652}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.34,3.023) -- (6.94,3.023) -- (6.94,3.623) -- (6.34,3.623) -- cycle;
\node at (6.64,3.323) {\scalebox{1.652}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (10.112,2.463) -- (10.712,2.463) -- (10.712,3.063) -- (10.112,3.063) -- cycle;
\node at (10.412,2.763) {\scalebox{1.652}{\small d}};
\node at (5.945,7.641) {\scalebox{1.652}{\small A}};
\node at (8.737,7.468) {\scalebox{1.652}{\small B}};
\node at (11.409,7.265) {\scalebox{1.652}{\small C}};
\node at (0.765,6.931) {\scalebox{1.652}{\small D}};
\node at (2.546,6.541) {\scalebox{1.652}{\small E}};
\node at (5.692,4.005) {\scalebox{1.652}{\small F}};
\node at (11.959,4.391) {\scalebox{1.652}{\small G}};
\node at (9.376,3.952) {\scalebox{1.652}{\small H}};
\node at (2.567,3.939) {\scalebox{1.652}{\small I}};
\node at (2.818,1.388) {\scalebox{1.652}{\small J}};
\node at (9.157,1.11) {\scalebox{1.652}{\small K}};
\node at (5.55,0.95) {\scalebox{1.652}{\small L}};
\draw[line width=0.5pt] (12.64,6.078) -- (12.64,7.178);
\draw[line width=0.5pt] (12.409,6.793) -- (12.64,7.178) -- (12.871,6.793);
\draw[line width=0.5pt] (12.338,6.628) -- (12.943,6.628);
\node at (12.64,7.563) {\scalebox{1.652}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour G, 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 boulevard du Parc & Ⓐ Rond-point de la Poste & Ⓙ Placette des Ormes \\ \hline
① rue du Four & ⑩ chemin de Ronde & Ⓑ Carrefour de la Mairie & Ⓚ Place de la Fontaine \\ \hline
② rue des Jardins & ⑪ rue des Lilas & Ⓒ Coin du Lavoir & Ⓛ Carrefour Saint-Roch \\ \hline
③ rue Neuve & ⑫ passage du Marché & Ⓓ Place du Vieux-Puits & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ allée des Peupliers & ⑬ rue du Pont & Ⓔ Carrefour du Lavoir & \textbf{a} l'école \\ \hline
⑤ rue du Puits & ⑭ allée des Tilleuls & Ⓕ Carrefour des Tilleuls & \textbf{b} la bibliothèque \\ \hline
⑥ impasse des Cerisiers & ⑮ rue du Stade & Ⓖ Place Sainte-Croix & \textbf{c} le marché couvert \\ \hline
⑦ chemin des Vignes & ⑯ sentier des Prés & Ⓗ Carrefour du Moulin & \textbf{d} la gendarmerie \\ \hline
⑧ rue de la Mairie & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour de l'Hôpital &  \\ \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=0.6203, line join=round, line cap=round]
\draw[line width=0.5pt] (6.04,7.028) -- (6.202,7.0) -- (6.366,6.976) -- (6.53,6.956) -- (6.694,6.94) -- (6.859,6.928) -- (7.024,6.919) -- (7.189,6.915) -- (7.354,6.914) -- (7.519,6.917) -- (7.684,6.924) -- (7.849,6.935) -- (8.014,6.95) -- (8.178,6.969) -- (8.341,6.991);
\draw[line width=0.5pt] (6.04,7.028) -- (6.076,6.845) -- (6.108,6.66) -- (6.135,6.476) -- (6.158,6.29) -- (6.177,6.104) -- (6.192,5.918) -- (6.202,5.731) -- (6.207,5.544) -- (6.209,5.357) -- (6.206,5.17) -- (6.198,4.983) -- (6.186,4.797) -- (6.17,4.61) -- (6.149,4.425);
\draw[line width=0.5pt] (8.341,6.991) -- (8.52,6.963) -- (8.699,6.939) -- (8.878,6.919) -- (9.058,6.903) -- (9.238,6.892) -- (9.419,6.884) -- (9.6,6.881) -- (9.781,6.882) -- (9.961,6.888) -- (10.142,6.898) -- (10.322,6.912) -- (10.502,6.93) -- (10.681,6.952) -- (10.859,6.979);
\draw[line width=0.5pt] (8.341,6.991) -- (8.403,6.791) -- (8.461,6.589) -- (8.514,6.386) -- (8.562,6.181) -- (8.605,5.976) -- (8.643,5.769) -- (8.676,5.561) -- (8.705,5.353) -- (8.729,5.144) -- (8.747,4.935) -- (8.761,4.726) -- (8.77,4.516) -- (8.774,4.306) -- (8.773,4.096);
\draw[line width=0.5pt] (10.859,6.979) -- (10.866,6.792) -- (10.876,6.606) -- (10.891,6.42) -- (10.91,6.234) -- (10.934,6.049) -- (10.962,5.864) -- (10.994,5.68) -- (11.031,5.497) -- (11.072,5.315) -- (11.117,5.133) -- (11.167,4.953) -- (11.22,4.775) -- (11.278,4.597) -- (11.34,4.421);
\draw[line width=0.5pt] (1.34,6.699) -- (1.47,6.688) -- (1.599,6.674) -- (1.728,6.657) -- (1.857,6.636) -- (1.985,6.613) -- (2.113,6.586) -- (2.239,6.557) -- (2.366,6.525) -- (2.491,6.489) -- (2.616,6.451) -- (2.739,6.41) -- (2.862,6.366) -- (2.983,6.319) -- (3.103,6.269);
\draw[line width=0.5pt] (3.103,6.269) -- (3.32,6.291) -- (3.536,6.317) -- (3.751,6.349) -- (3.965,6.386) -- (4.179,6.428) -- (4.391,6.475) -- (4.603,6.527) -- (4.813,6.584) -- (5.022,6.646) -- (5.229,6.713) -- (5.434,6.784) -- (5.638,6.861) -- (5.84,6.942) -- (6.04,7.028);
\draw[line width=0.5pt] (3.103,6.269) -- (3.133,6.113) -- (3.159,5.956) -- (3.181,5.798) -- (3.199,5.64) -- (3.213,5.482) -- (3.224,5.323) -- (3.231,5.164) -- (3.235,5.005) -- (3.235,4.846) -- (3.23,4.687) -- (3.223,4.528) -- (3.211,4.369) -- (3.196,4.211) -- (3.177,4.053);
\draw[line width=0.5pt] (6.149,4.425) -- (6.331,4.373) -- (6.515,4.325) -- (6.7,4.282) -- (6.885,4.243) -- (7.072,4.208) -- (7.259,4.178) -- (7.447,4.152) -- (7.636,4.131) -- (7.825,4.114) -- (8.014,4.101) -- (8.204,4.093) -- (8.394,4.09) -- (8.583,4.091) -- (8.773,4.096);
\draw[line width=0.5pt] (6.149,4.425) -- (6.148,4.216) -- (6.142,4.008) -- (6.131,3.8) -- (6.115,3.592) -- (6.094,3.384) -- (6.068,3.177) -- (6.037,2.97) -- (6.002,2.765) -- (5.962,2.56) -- (5.917,2.356) -- (5.867,2.154) -- (5.812,1.952) -- (5.753,1.753) -- (5.689,1.554);
\draw[line width=0.5pt] (8.773,4.096) -- (8.958,4.091) -- (9.144,4.09) -- (9.329,4.094) -- (9.515,4.102) -- (9.7,4.115) -- (9.885,4.131) -- (10.07,4.153) -- (10.254,4.178) -- (10.437,4.208) -- (10.62,4.242) -- (10.801,4.28) -- (10.982,4.323) -- (11.162,4.37) -- (11.34,4.421);
\draw[line width=0.5pt] (8.773,4.096) -- (8.8,3.919) -- (8.824,3.742) -- (8.843,3.564) -- (8.858,3.385) -- (8.869,3.207) -- (8.876,3.028) -- (8.878,2.848) -- (8.876,2.669) -- (8.87,2.49) -- (8.86,2.311) -- (8.846,2.133) -- (8.827,1.955) -- (8.804,1.777) -- (8.777,1.6);
\draw[line width=0.5pt] (3.177,4.053) -- (3.391,4.047) -- (3.606,4.046) -- (3.821,4.05) -- (4.036,4.059) -- (4.25,4.073) -- (4.465,4.092) -- (4.678,4.116) -- (4.891,4.146) -- (5.104,4.18) -- (5.315,4.219) -- (5.525,4.263) -- (5.735,4.312) -- (5.942,4.366) -- (6.149,4.425);
\draw[line width=0.5pt] (3.177,4.053) -- (3.208,3.895) -- (3.236,3.736) -- (3.26,3.577) -- (3.281,3.416) -- (3.297,3.256) -- (3.31,3.095) -- (3.319,2.934) -- (3.324,2.772) -- (3.326,2.611) -- (3.324,2.449) -- (3.317,2.288) -- (3.308,2.127) -- (3.294,1.966) -- (3.276,1.806);
\draw[line width=0.5pt] (3.276,1.806) -- (3.444,1.762) -- (3.614,1.721) -- (3.784,1.685) -- (3.955,1.653) -- (4.127,1.625) -- (4.299,1.601) -- (4.472,1.581) -- (4.646,1.564) -- (4.819,1.553) -- (4.993,1.545) -- (5.167,1.541) -- (5.341,1.541) -- (5.515,1.546) -- (5.689,1.554);
\draw[line width=0.5pt] (5.689,1.554) -- (5.907,1.591) -- (6.127,1.623) -- (6.347,1.65) -- (6.567,1.671) -- (6.788,1.688) -- (7.01,1.699) -- (7.231,1.705) -- (7.453,1.705) -- (7.675,1.701) -- (7.896,1.691) -- (8.117,1.676) -- (8.338,1.656) -- (8.558,1.631) -- (8.777,1.6);
\filldraw[fill=white, draw=black, line width=0.5pt] (6.413,7.028) -- (6.384,7.171) -- (6.303,7.292) -- (6.182,7.373) -- (6.04,7.401) -- (5.897,7.373) -- (5.776,7.292) -- (5.695,7.171) -- (5.666,7.028) -- (5.695,6.885) -- (5.776,6.764) -- (5.897,6.683) -- (6.04,6.655) -- (6.182,6.683) -- (6.303,6.764) -- (6.384,6.885) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.714,6.991) -- (8.686,7.134) -- (8.605,7.255) -- (8.484,7.336) -- (8.341,7.365) -- (8.198,7.336) -- (8.077,7.255) -- (7.996,7.134) -- (7.968,6.991) -- (7.996,6.849) -- (8.077,6.727) -- (8.198,6.647) -- (8.341,6.618) -- (8.484,6.647) -- (8.605,6.727) -- (8.686,6.849) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.233,6.979) -- (11.204,7.122) -- (11.123,7.243) -- (11.002,7.324) -- (10.859,7.352) -- (10.717,7.324) -- (10.596,7.243) -- (10.515,7.122) -- (10.486,6.979) -- (10.515,6.836) -- (10.596,6.715) -- (10.717,6.634) -- (10.859,6.606) -- (11.002,6.634) -- (11.123,6.715) -- (11.204,6.836) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.713,6.699) -- (1.685,6.842) -- (1.604,6.963) -- (1.483,7.044) -- (1.34,7.073) -- (1.197,7.044) -- (1.076,6.963) -- (0.995,6.842) -- (0.967,6.699) -- (0.995,6.557) -- (1.076,6.435) -- (1.197,6.355) -- (1.34,6.326) -- (1.483,6.355) -- (1.604,6.435) -- (1.685,6.557) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (3.477,6.269) -- (3.448,6.412) -- (3.367,6.533) -- (3.246,6.614) -- (3.103,6.642) -- (2.961,6.614) -- (2.839,6.533) -- (2.759,6.412) -- (2.73,6.269) -- (2.759,6.126) -- (2.839,6.005) -- (2.961,5.924) -- (3.103,5.896) -- (3.246,5.924) -- (3.367,6.005) -- (3.448,6.126) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.522,4.425) -- (6.494,4.568) -- (6.413,4.689) -- (6.292,4.77) -- (6.149,4.798) -- (6.006,4.77) -- (5.885,4.689) -- (5.804,4.568) -- (5.776,4.425) -- (5.804,4.282) -- (5.885,4.161) -- (6.006,4.08) -- (6.149,4.052) -- (6.292,4.08) -- (6.413,4.161) -- (6.494,4.282) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.713,4.421) -- (11.685,4.564) -- (11.604,4.685) -- (11.483,4.766) -- (11.34,4.794) -- (11.197,4.766) -- (11.076,4.685) -- (10.995,4.564) -- (10.967,4.421) -- (10.995,4.278) -- (11.076,4.157) -- (11.197,4.076) -- (11.34,4.048) -- (11.483,4.076) -- (11.604,4.157) -- (11.685,4.278) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.146,4.096) -- (9.118,4.239) -- (9.037,4.36) -- (8.916,4.441) -- (8.773,4.469) -- (8.63,4.441) -- (8.509,4.36) -- (8.428,4.239) -- (8.4,4.096) -- (8.428,3.953) -- (8.509,3.832) -- (8.63,3.751) -- (8.773,3.723) -- (8.916,3.751) -- (9.037,3.832) -- (9.118,3.953) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.55,4.053) -- (3.522,4.196) -- (3.441,4.317) -- (3.32,4.398) -- (3.177,4.426) -- (3.034,4.398) -- (2.913,4.317) -- (2.832,4.196) -- (2.804,4.053) -- (2.832,3.91) -- (2.913,3.789) -- (3.034,3.708) -- (3.177,3.68) -- (3.32,3.708) -- (3.441,3.789) -- (3.522,3.91) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (3.649,1.806) -- (3.621,1.949) -- (3.54,2.07) -- (3.419,2.151) -- (3.276,2.179) -- (3.133,2.151) -- (3.012,2.07) -- (2.931,1.949) -- (2.903,1.806) -- (2.931,1.663) -- (3.012,1.542) -- (3.133,1.461) -- (3.276,1.433) -- (3.419,1.461) -- (3.54,1.542) -- (3.621,1.663) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.151,1.6) -- (9.122,1.743) -- (9.041,1.864) -- (8.92,1.945) -- (8.777,1.974) -- (8.635,1.945) -- (8.514,1.864) -- (8.433,1.743) -- (8.404,1.6) -- (8.433,1.458) -- (8.514,1.337) -- (8.635,1.256) -- (8.777,1.227) -- (8.92,1.256) -- (9.041,1.337) -- (9.122,1.458) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.062,1.554) -- (6.034,1.697) -- (5.953,1.818) -- (5.832,1.899) -- (5.689,1.927) -- (5.546,1.899) -- (5.425,1.818) -- (5.344,1.697) -- (5.316,1.554) -- (5.344,1.411) -- (5.425,1.29) -- (5.546,1.209) -- (5.689,1.181) -- (5.832,1.209) -- (5.953,1.29) -- (6.034,1.411) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.195,6.915) -- (7.314,6.933) -- (7.421,6.988) -- (7.506,7.073) -- (7.561,7.18) -- (7.579,7.299) -- (7.561,7.418) -- (7.506,7.525) -- (7.421,7.61) -- (7.314,7.665) -- (7.195,7.684) -- (7.195,7.684) -- (7.076,7.665) -- (6.969,7.61) -- (6.884,7.525) -- (6.829,7.418) -- (6.81,7.299) -- (6.829,7.18) -- (6.884,7.073) -- (6.969,6.988) -- (7.076,6.933) -- (7.195,6.915) -- cycle;
\node at (7.195,7.299) {\scalebox{1.612}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.586,5.362) -- (6.705,5.381) -- (6.812,5.436) -- (6.897,5.521) -- (6.952,5.628) -- (6.97,5.747) -- (6.952,5.866) -- (6.897,5.973) -- (6.812,6.058) -- (6.705,6.113) -- (6.586,6.131) -- (6.586,6.131) -- (6.467,6.113) -- (6.36,6.058) -- (6.275,5.973) -- (6.22,5.866) -- (6.201,5.747) -- (6.22,5.628) -- (6.275,5.521) -- (6.36,5.436) -- (6.467,5.381) -- (6.586,5.362) -- cycle;
\node at (6.586,5.747) {\scalebox{1.612}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.602,6.881) -- (9.721,6.9) -- (9.828,6.955) -- (9.913,7.04) -- (9.967,7.147) -- (9.986,7.266) -- (9.967,7.385) -- (9.913,7.492) -- (9.828,7.577) -- (9.721,7.631) -- (9.602,7.65) -- (9.602,7.65) -- (9.483,7.631) -- (9.376,7.577) -- (9.291,7.492) -- (9.236,7.385) -- (9.217,7.266) -- (9.236,7.147) -- (9.291,7.04) -- (9.376,6.955) -- (9.483,6.9) -- (9.602,6.881) -- cycle;
\node at (9.602,7.266) {\scalebox{1.612}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.057,5.234) -- (9.176,5.252) -- (9.283,5.307) -- (9.368,5.392) -- (9.422,5.499) -- (9.441,5.618) -- (9.422,5.737) -- (9.368,5.844) -- (9.283,5.929) -- (9.176,5.984) -- (9.057,6.003) -- (9.057,6.003) -- (8.938,5.984) -- (8.831,5.929) -- (8.746,5.844) -- (8.691,5.737) -- (8.672,5.618) -- (8.691,5.499) -- (8.746,5.392) -- (8.831,5.307) -- (8.938,5.252) -- (9.057,5.234) -- cycle;
\node at (9.057,5.618) {\scalebox{1.612}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.372,5.367) -- (11.491,5.385) -- (11.598,5.44) -- (11.683,5.525) -- (11.738,5.632) -- (11.757,5.751) -- (11.738,5.87) -- (11.683,5.977) -- (11.598,6.062) -- (11.491,6.117) -- (11.372,6.136) -- (11.372,6.136) -- (11.253,6.117) -- (11.146,6.062) -- (11.061,5.977) -- (11.006,5.87) -- (10.988,5.751) -- (11.006,5.632) -- (11.061,5.525) -- (11.146,5.44) -- (11.253,5.385) -- (11.372,5.367) -- cycle;
\node at (11.372,5.751) {\scalebox{1.612}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.958,5.495) -- (2.076,5.514) -- (2.184,5.568) -- (2.269,5.653) -- (2.323,5.761) -- (2.342,5.879) -- (2.323,5.998) -- (2.269,6.105) -- (2.184,6.19) -- (2.076,6.245) -- (1.958,6.264) -- (1.958,6.264) -- (1.839,6.245) -- (1.731,6.19) -- (1.646,6.105) -- (1.592,5.998) -- (1.573,5.879) -- (1.592,5.761) -- (1.646,5.653) -- (1.731,5.568) -- (1.839,5.514) -- (1.958,5.495) -- cycle;
\node at (1.958,5.879) {\scalebox{1.612}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.507,6.515) -- (4.625,6.534) -- (4.733,6.589) -- (4.818,6.674) -- (4.872,6.781) -- (4.891,6.9) -- (4.872,7.018) -- (4.818,7.126) -- (4.733,7.211) -- (4.625,7.265) -- (4.507,7.284) -- (4.507,7.284) -- (4.388,7.265) -- (4.281,7.211) -- (4.195,7.126) -- (4.141,7.018) -- (4.122,6.9) -- (4.141,6.781) -- (4.195,6.674) -- (4.281,6.589) -- (4.388,6.534) -- (4.507,6.515) -- cycle;
\node at (4.507,6.9) {\scalebox{1.612}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.616,4.792) -- (3.735,4.811) -- (3.842,4.866) -- (3.927,4.951) -- (3.981,5.058) -- (4.0,5.177) -- (3.981,5.296) -- (3.927,5.403) -- (3.842,5.488) -- (3.735,5.543) -- (3.616,5.561) -- (3.616,5.561) -- (3.497,5.543) -- (3.39,5.488) -- (3.305,5.403) -- (3.25,5.296) -- (3.231,5.177) -- (3.25,5.058) -- (3.305,4.951) -- (3.39,4.866) -- (3.497,4.811) -- (3.616,4.792) -- cycle;
\node at (3.616,5.177) {\scalebox{1.612}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.495,4.149) -- (7.614,4.168) -- (7.721,4.223) -- (7.806,4.308) -- (7.861,4.415) -- (7.88,4.534) -- (7.861,4.652) -- (7.806,4.76) -- (7.721,4.845) -- (7.614,4.899) -- (7.495,4.918) -- (7.495,4.918) -- (7.376,4.899) -- (7.269,4.845) -- (7.184,4.76) -- (7.129,4.652) -- (7.111,4.534) -- (7.129,4.415) -- (7.184,4.308) -- (7.269,4.223) -- (7.376,4.168) -- (7.495,4.149) -- cycle;
\node at (7.495,4.534) {\scalebox{1.612}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.766,2.254) -- (5.885,2.273) -- (5.992,2.328) -- (6.077,2.413) -- (6.132,2.52) -- (6.151,2.639) -- (6.132,2.757) -- (6.077,2.865) -- (5.992,2.95) -- (5.885,3.004) -- (5.766,3.023) -- (5.404,3.023) -- (5.286,3.004) -- (5.178,2.95) -- (5.093,2.865) -- (5.039,2.757) -- (5.02,2.639) -- (5.039,2.52) -- (5.093,2.413) -- (5.178,2.328) -- (5.286,2.273) -- (5.404,2.254) -- cycle;
\node at (5.585,2.639) {\scalebox{1.612}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.211,4.086) -- (10.329,4.104) -- (10.437,4.159) -- (10.522,4.244) -- (10.576,4.351) -- (10.595,4.47) -- (10.576,4.589) -- (10.522,4.696) -- (10.437,4.781) -- (10.329,4.836) -- (10.211,4.855) -- (9.849,4.855) -- (9.73,4.836) -- (9.623,4.781) -- (9.538,4.696) -- (9.483,4.589) -- (9.464,4.47) -- (9.483,4.351) -- (9.538,4.244) -- (9.623,4.159) -- (9.73,4.104) -- (9.849,4.086) -- cycle;
\node at (10.03,4.47) {\scalebox{1.612}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.443,2.465) -- (9.562,2.483) -- (9.67,2.538) -- (9.755,2.623) -- (9.809,2.73) -- (9.828,2.849) -- (9.809,2.968) -- (9.755,3.075) -- (9.67,3.16) -- (9.562,3.215) -- (9.443,3.234) -- (9.082,3.234) -- (8.963,3.215) -- (8.856,3.16) -- (8.77,3.075) -- (8.716,2.968) -- (8.697,2.849) -- (8.716,2.73) -- (8.77,2.623) -- (8.856,2.538) -- (8.963,2.483) -- (9.082,2.465) -- cycle;
\node at (9.263,2.849) {\scalebox{1.612}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.811,4.113) -- (4.93,4.132) -- (5.038,4.187) -- (5.123,4.272) -- (5.177,4.379) -- (5.196,4.498) -- (5.177,4.617) -- (5.123,4.724) -- (5.038,4.809) -- (4.93,4.864) -- (4.811,4.882) -- (4.45,4.882) -- (4.331,4.864) -- (4.224,4.809) -- (4.139,4.724) -- (4.084,4.617) -- (4.065,4.498) -- (4.084,4.379) -- (4.139,4.272) -- (4.224,4.187) -- (4.331,4.132) -- (4.45,4.113) -- cycle;
\node at (4.631,4.498) {\scalebox{1.612}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.884,2.566) -- (4.003,2.585) -- (4.11,2.64) -- (4.195,2.725) -- (4.25,2.832) -- (4.269,2.951) -- (4.25,3.07) -- (4.195,3.177) -- (4.11,3.262) -- (4.003,3.316) -- (3.884,3.335) -- (3.522,3.335) -- (3.404,3.316) -- (3.296,3.262) -- (3.211,3.177) -- (3.157,3.07) -- (3.138,2.951) -- (3.157,2.832) -- (3.211,2.725) -- (3.296,2.64) -- (3.404,2.585) -- (3.522,2.566) -- cycle;
\node at (3.703,2.951) {\scalebox{1.612}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.693,1.578) -- (4.812,1.597) -- (4.919,1.652) -- (5.004,1.737) -- (5.059,1.844) -- (5.078,1.963) -- (5.059,2.082) -- (5.004,2.189) -- (4.919,2.274) -- (4.812,2.329) -- (4.693,2.348) -- (4.331,2.348) -- (4.212,2.329) -- (4.105,2.274) -- (4.02,2.189) -- (3.966,2.082) -- (3.947,1.963) -- (3.966,1.844) -- (4.02,1.737) -- (4.105,1.652) -- (4.212,1.597) -- (4.331,1.578) -- cycle;
\node at (4.512,1.963) {\scalebox{1.612}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.406,1.705) -- (7.525,1.723) -- (7.632,1.778) -- (7.718,1.863) -- (7.772,1.97) -- (7.791,2.089) -- (7.772,2.208) -- (7.718,2.315) -- (7.632,2.4) -- (7.525,2.455) -- (7.406,2.474) -- (7.045,2.474) -- (6.926,2.455) -- (6.819,2.4) -- (6.733,2.315) -- (6.679,2.208) -- (6.66,2.089) -- (6.679,1.97) -- (6.733,1.863) -- (6.819,1.778) -- (6.926,1.723) -- (7.045,1.705) -- cycle;
\node at (7.226,2.089) {\scalebox{1.612}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (11.04,4.121) -- (11.64,4.121) -- (11.64,4.721) -- (11.04,4.721) -- cycle;
\node at (11.34,4.421) {\scalebox{1.612}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.357,4.363) -- (9.957,4.363) -- (9.957,4.963) -- (9.357,4.963) -- cycle;
\node at (9.657,4.663) {\scalebox{1.612}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.34,3.023) -- (6.94,3.023) -- (6.94,3.623) -- (6.34,3.623) -- cycle;
\node at (6.64,3.323) {\scalebox{1.612}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (10.112,2.463) -- (10.712,2.463) -- (10.712,3.063) -- (10.112,3.063) -- cycle;
\node at (10.412,2.763) {\scalebox{1.612}{\small d}};
\node at (5.945,7.641) {\scalebox{1.612}{\small A}};
\node at (8.737,7.468) {\scalebox{1.612}{\small B}};
\node at (11.409,7.265) {\scalebox{1.612}{\small C}};
\node at (0.765,6.931) {\scalebox{1.612}{\small D}};
\node at (2.546,6.541) {\scalebox{1.612}{\small E}};
\node at (5.692,4.005) {\scalebox{1.612}{\small F}};
\node at (11.959,4.391) {\scalebox{1.612}{\small G}};
\node at (9.376,3.952) {\scalebox{1.612}{\small H}};
\node at (2.567,3.939) {\scalebox{1.612}{\small I}};
\node at (2.818,1.388) {\scalebox{1.612}{\small J}};
\node at (9.157,1.11) {\scalebox{1.612}{\small K}};
\node at (5.55,0.95) {\scalebox{1.612}{\small L}};
\draw[line width=0.5pt] (12.64,6.078) -- (12.64,7.178);
\draw[line width=0.5pt] (12.409,6.793) -- (12.64,7.178) -- (12.871,6.793);
\draw[line width=0.5pt] (12.338,6.628) -- (12.943,6.628);
\node at (12.64,7.563) {\scalebox{1.612}{\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 & B, E, F, G, J, K & 6 \\ \hline
clairs & A, C, D, H, I, 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{Non.} Si l'on retire le carrefour \textbf{E}, le quartier tombe en 2 morceaux : toute route de l'un à l'autre passe par lui. Un circuit qui reviendrait à l'école devrait donc s'y arrêter au moins \textbf{deux fois} — c'est interdit.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{G} :}\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 du Puits ⑤} vers le nord jusqu'à Coin du Lavoir \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Neuve ③}, vers l'ouest, jusqu'à Carrefour de la Mairie \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Four ①}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Jardins ②}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{boulevard du Parc ⑨}, vers l'est, jusqu'à Carrefour du Moulin \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{passage du Marché ⑫}, vers le sud, jusqu'à Place de la Fontaine \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ⑯}, vers l'ouest, jusqu'à Carrefour Saint-Roch \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Stade ⑮}, vers l'ouest, jusqu'à Placette des Ormes \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Tilleuls ⑭}, vers le nord, jusqu'à Carrefour de l'Hôpital \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de la Mairie ⑧}, vers le nord, jusqu'à Carrefour du Lavoir \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse des Cerisiers ⑥}, vers l'ouest, jusqu'à Place du Vieux-Puits \textbf{(D)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 12 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
