\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 48}

\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=48}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0048}
\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=0.6068, line join=round, line cap=round]
\draw[line width=0.5pt] (7.708,11.557) -- (7.96,11.497) -- (8.214,11.444) -- (8.469,11.396) -- (8.725,11.355) -- (8.982,11.32) -- (9.24,11.291) -- (9.498,11.267) -- (9.757,11.25) -- (10.016,11.239) -- (10.276,11.234) -- (10.535,11.235) -- (10.794,11.242) -- (11.053,11.256) -- (11.312,11.275);
\draw[line width=0.5pt] (7.708,11.557) -- (7.71,11.339) -- (7.706,11.121) -- (7.698,10.903) -- (7.684,10.686) -- (7.665,10.469) -- (7.641,10.252) -- (7.612,10.036) -- (7.578,9.821) -- (7.539,9.606) -- (7.494,9.393) -- (7.445,9.181) -- (7.391,8.97) -- (7.332,8.76) -- (7.268,8.552);
\draw[line width=0.5pt] (11.312,11.275) -- (11.246,11.071) -- (11.185,10.864) -- (11.129,10.657) -- (11.079,10.448) -- (11.033,10.238) -- (10.992,10.026) -- (10.955,9.814) -- (10.924,9.601) -- (10.898,9.388) -- (10.877,9.174) -- (10.861,8.959) -- (10.85,8.744) -- (10.844,8.529) -- (10.844,8.315);
\draw[line width=0.5pt] (4.795,11.251) -- (4.998,11.305) -- (5.203,11.353) -- (5.408,11.397) -- (5.615,11.436) -- (5.822,11.47) -- (6.03,11.5) -- (6.239,11.524) -- (6.448,11.543) -- (6.658,11.558) -- (6.868,11.568) -- (7.078,11.572) -- (7.288,11.572) -- (7.498,11.567) -- (7.708,11.557);
\draw[line width=0.5pt] (4.795,11.251) -- (4.769,11.018) -- (4.748,10.785) -- (4.733,10.551) -- (4.723,10.317) -- (4.719,10.083) -- (4.72,9.849) -- (4.727,9.614) -- (4.739,9.38) -- (4.757,9.147) -- (4.78,8.914) -- (4.808,8.681) -- (4.842,8.449) -- (4.882,8.219) -- (4.927,7.989);
\draw[line width=0.5pt] (1.87,11.196) -- (2.076,11.232) -- (2.284,11.263) -- (2.492,11.289) -- (2.701,11.311) -- (2.911,11.327) -- (3.12,11.338) -- (3.33,11.344) -- (3.54,11.346) -- (3.75,11.342) -- (3.96,11.334) -- (4.17,11.321) -- (4.379,11.302) -- (4.587,11.279) -- (4.795,11.251);
\draw[line width=0.5pt] (1.87,11.196) -- (1.801,10.982) -- (1.738,10.766) -- (1.679,10.548) -- (1.626,10.329) -- (1.578,10.108) -- (1.535,9.887) -- (1.498,9.665) -- (1.465,9.442) -- (1.438,9.218) -- (1.416,8.993) -- (1.4,8.768) -- (1.389,8.543) -- (1.383,8.318) -- (1.382,8.093);
\draw[line width=0.5pt] (7.268,8.552) -- (7.524,8.574) -- (7.78,8.59) -- (8.037,8.601) -- (8.294,8.605) -- (8.551,8.603) -- (8.809,8.595) -- (9.065,8.581) -- (9.322,8.561) -- (9.578,8.535) -- (9.833,8.503) -- (10.087,8.464) -- (10.341,8.42) -- (10.593,8.37) -- (10.844,8.315);
\draw[line width=0.5pt] (7.268,8.552) -- (7.27,8.329) -- (7.279,8.106) -- (7.292,7.884) -- (7.311,7.661) -- (7.334,7.44) -- (7.363,7.218) -- (7.398,6.998) -- (7.437,6.778) -- (7.482,6.56) -- (7.531,6.342) -- (7.586,6.126) -- (7.646,5.911) -- (7.711,5.698) -- (7.78,5.486);
\draw[line width=0.5pt] (10.844,8.315) -- (10.86,8.096) -- (10.871,7.877) -- (10.877,7.657) -- (10.878,7.438) -- (10.873,7.218) -- (10.864,6.999) -- (10.849,6.78) -- (10.829,6.561) -- (10.804,6.343) -- (10.774,6.125) -- (10.739,5.909) -- (10.699,5.693) -- (10.654,5.478) -- (10.603,5.265);
\draw[line width=0.5pt] (1.382,8.093) -- (1.632,8.047) -- (1.883,8.006) -- (2.135,7.972) -- (2.388,7.944) -- (2.642,7.921) -- (2.896,7.905) -- (3.15,7.895) -- (3.405,7.89) -- (3.659,7.892) -- (3.914,7.899) -- (4.168,7.912) -- (4.422,7.932) -- (4.675,7.957) -- (4.927,7.989);
\draw[line width=0.5pt] (1.382,8.093) -- (1.423,7.913) -- (1.46,7.731) -- (1.493,7.549) -- (1.521,7.366) -- (1.545,7.183) -- (1.565,6.999) -- (1.58,6.814) -- (1.591,6.63) -- (1.598,6.445) -- (1.6,6.26) -- (1.598,6.075) -- (1.592,5.89) -- (1.581,5.705) -- (1.566,5.521);
\draw[line width=0.5pt] (4.927,7.989) -- (5.099,8.003) -- (5.271,8.021) -- (5.442,8.044) -- (5.613,8.07) -- (5.783,8.101) -- (5.952,8.135) -- (6.121,8.174) -- (6.288,8.216) -- (6.455,8.262) -- (6.62,8.313) -- (6.784,8.367) -- (6.947,8.425) -- (7.108,8.486) -- (7.268,8.552);
\draw[line width=0.5pt] (7.78,5.486) -- (7.845,5.211) -- (7.902,4.934) -- (7.953,4.656) -- (7.998,4.377) -- (8.035,4.097) -- (8.067,3.815) -- (8.091,3.534) -- (8.11,3.251) -- (8.121,2.969) -- (8.126,2.686) -- (8.124,2.403) -- (8.115,2.121) -- (8.1,1.838) -- (8.078,1.557);
\draw[line width=0.5pt] (10.603,5.265) -- (10.554,5.052) -- (10.51,4.839) -- (10.471,4.624) -- (10.437,4.409) -- (10.408,4.192) -- (10.385,3.975) -- (10.366,3.758) -- (10.353,3.54) -- (10.344,3.322) -- (10.341,3.104) -- (10.343,2.886) -- (10.35,2.668) -- (10.362,2.45) -- (10.379,2.233);
\draw[line width=0.5pt] (8.078,1.557) -- (8.234,1.63) -- (8.391,1.699) -- (8.55,1.765) -- (8.711,1.827) -- (8.873,1.885) -- (9.036,1.939) -- (9.201,1.99) -- (9.367,2.036) -- (9.533,2.079) -- (9.701,2.118) -- (9.87,2.152) -- (10.039,2.183) -- (10.209,2.21) -- (10.379,2.233);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.868,11.557) -- (7.847,11.637) -- (7.788,11.695) -- (7.708,11.717) -- (7.628,11.695) -- (7.57,11.637) -- (7.548,11.557) -- (7.57,11.477) -- (7.628,11.418) -- (7.708,11.397) -- (7.788,11.418) -- (7.847,11.477) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.472,11.275) -- (11.45,11.355) -- (11.392,11.414) -- (11.312,11.435) -- (11.232,11.414) -- (11.173,11.355) -- (11.152,11.275) -- (11.173,11.195) -- (11.232,11.137) -- (11.312,11.115) -- (11.392,11.137) -- (11.45,11.195) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.955,11.251) -- (4.934,11.331) -- (4.875,11.39) -- (4.795,11.411) -- (4.715,11.39) -- (4.657,11.331) -- (4.635,11.251) -- (4.657,11.171) -- (4.715,11.113) -- (4.795,11.091) -- (4.875,11.113) -- (4.934,11.171) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.03,11.196) -- (2.008,11.276) -- (1.95,11.335) -- (1.87,11.356) -- (1.79,11.335) -- (1.731,11.276) -- (1.71,11.196) -- (1.731,11.116) -- (1.79,11.058) -- (1.87,11.036) -- (1.95,11.058) -- (2.008,11.116) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.428,8.552) -- (7.406,8.632) -- (7.348,8.69) -- (7.268,8.712) -- (7.188,8.69) -- (7.129,8.632) -- (7.108,8.552) -- (7.129,8.472) -- (7.188,8.413) -- (7.268,8.392) -- (7.348,8.413) -- (7.406,8.472) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.004,8.315) -- (10.982,8.395) -- (10.924,8.453) -- (10.844,8.475) -- (10.764,8.453) -- (10.705,8.395) -- (10.684,8.315) -- (10.705,8.235) -- (10.764,8.176) -- (10.844,8.155) -- (10.924,8.176) -- (10.982,8.235) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.542,8.093) -- (1.521,8.173) -- (1.462,8.231) -- (1.382,8.253) -- (1.302,8.231) -- (1.243,8.173) -- (1.222,8.093) -- (1.243,8.013) -- (1.302,7.954) -- (1.382,7.933) -- (1.462,7.954) -- (1.521,8.013) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.087,7.989) -- (5.066,8.069) -- (5.007,8.127) -- (4.927,8.149) -- (4.847,8.127) -- (4.789,8.069) -- (4.767,7.989) -- (4.789,7.909) -- (4.847,7.85) -- (4.927,7.829) -- (5.007,7.85) -- (5.066,7.909) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.726,5.521) -- (1.705,5.601) -- (1.646,5.659) -- (1.566,5.681) -- (1.486,5.659) -- (1.428,5.601) -- (1.406,5.521) -- (1.428,5.441) -- (1.486,5.382) -- (1.566,5.361) -- (1.646,5.382) -- (1.705,5.441) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.94,5.486) -- (7.919,5.566) -- (7.86,5.625) -- (7.78,5.646) -- (7.7,5.625) -- (7.642,5.566) -- (7.62,5.486) -- (7.642,5.406) -- (7.7,5.348) -- (7.78,5.326) -- (7.86,5.348) -- (7.919,5.406) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.763,5.265) -- (10.742,5.345) -- (10.683,5.403) -- (10.603,5.425) -- (10.523,5.403) -- (10.465,5.345) -- (10.443,5.265) -- (10.465,5.185) -- (10.523,5.126) -- (10.603,5.105) -- (10.683,5.126) -- (10.742,5.185) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.539,2.233) -- (10.518,2.313) -- (10.459,2.371) -- (10.379,2.393) -- (10.299,2.371) -- (10.241,2.313) -- (10.219,2.233) -- (10.241,2.153) -- (10.299,2.094) -- (10.379,2.073) -- (10.459,2.094) -- (10.518,2.153) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.238,1.557) -- (8.217,1.637) -- (8.158,1.695) -- (8.078,1.717) -- (7.998,1.695) -- (7.94,1.637) -- (7.918,1.557) -- (7.94,1.477) -- (7.998,1.418) -- (8.078,1.397) -- (8.158,1.418) -- (8.217,1.477) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.529,11.266) -- (9.651,11.285) -- (9.76,11.341) -- (9.848,11.429) -- (9.904,11.538) -- (9.923,11.66) -- (9.904,11.782) -- (9.848,11.892) -- (9.76,11.979) -- (9.651,12.035) -- (9.529,12.054) -- (9.529,12.054) -- (9.407,12.035) -- (9.297,11.979) -- (9.21,11.892) -- (9.154,11.782) -- (9.135,11.66) -- (9.154,11.538) -- (9.21,11.429) -- (9.297,11.341) -- (9.407,11.285) -- (9.529,11.266) -- cycle;
\node at (9.529,11.66) {\scalebox{1.648}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.002,9.585) -- (8.123,9.604) -- (8.233,9.66) -- (8.32,9.747) -- (8.376,9.857) -- (8.396,9.979) -- (8.376,10.101) -- (8.32,10.21) -- (8.233,10.298) -- (8.123,10.354) -- (8.002,10.373) -- (8.002,10.373) -- (7.88,10.354) -- (7.77,10.298) -- (7.683,10.21) -- (7.627,10.101) -- (7.608,9.979) -- (7.627,9.857) -- (7.683,9.747) -- (7.77,9.66) -- (7.88,9.604) -- (8.002,9.585) -- cycle;
\node at (8.002,9.979) {\scalebox{1.648}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.345,9.359) -- (11.466,9.378) -- (11.576,9.434) -- (11.663,9.521) -- (11.719,9.631) -- (11.738,9.753) -- (11.719,9.874) -- (11.663,9.984) -- (11.576,10.071) -- (11.466,10.127) -- (11.345,10.147) -- (11.345,10.147) -- (11.223,10.127) -- (11.113,10.071) -- (11.026,9.984) -- (10.97,9.874) -- (10.951,9.753) -- (10.97,9.631) -- (11.026,9.521) -- (11.113,9.434) -- (11.223,9.378) -- (11.345,9.359) -- cycle;
\node at (11.345,9.753) {\scalebox{1.648}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.28,10.738) -- (6.402,10.758) -- (6.512,10.814) -- (6.599,10.901) -- (6.655,11.011) -- (6.674,11.132) -- (6.655,11.254) -- (6.599,11.364) -- (6.512,11.451) -- (6.402,11.507) -- (6.28,11.526) -- (6.28,11.526) -- (6.159,11.507) -- (6.049,11.451) -- (5.962,11.364) -- (5.906,11.254) -- (5.886,11.132) -- (5.906,11.011) -- (5.962,10.901) -- (6.049,10.814) -- (6.159,10.758) -- (6.28,10.738) -- cycle;
\node at (6.28,11.132) {\scalebox{1.648}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.12,9.236) -- (5.242,9.256) -- (5.352,9.312) -- (5.439,9.399) -- (5.495,9.509) -- (5.514,9.63) -- (5.495,9.752) -- (5.439,9.862) -- (5.352,9.949) -- (5.242,10.005) -- (5.12,10.024) -- (5.12,10.024) -- (4.999,10.005) -- (4.889,9.949) -- (4.802,9.862) -- (4.746,9.752) -- (4.726,9.63) -- (4.746,9.509) -- (4.802,9.399) -- (4.889,9.312) -- (4.999,9.256) -- (5.12,9.236) -- cycle;
\node at (5.12,9.63) {\scalebox{1.648}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.323,11.344) -- (3.445,11.364) -- (3.554,11.42) -- (3.642,11.507) -- (3.698,11.617) -- (3.717,11.738) -- (3.698,11.86) -- (3.642,11.97) -- (3.554,12.057) -- (3.445,12.113) -- (3.323,12.132) -- (3.323,12.132) -- (3.201,12.113) -- (3.091,12.057) -- (3.004,11.97) -- (2.948,11.86) -- (2.929,11.738) -- (2.948,11.617) -- (3.004,11.507) -- (3.091,11.42) -- (3.201,11.364) -- (3.323,11.344) -- cycle;
\node at (3.323,11.738) {\scalebox{1.648}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.887,9.21) -- (2.009,9.229) -- (2.119,9.285) -- (2.206,9.372) -- (2.262,9.482) -- (2.281,9.604) -- (2.262,9.725) -- (2.206,9.835) -- (2.119,9.922) -- (2.009,9.978) -- (1.887,9.997) -- (1.887,9.997) -- (1.765,9.978) -- (1.655,9.922) -- (1.568,9.835) -- (1.512,9.725) -- (1.493,9.604) -- (1.512,9.482) -- (1.568,9.372) -- (1.655,9.285) -- (1.765,9.229) -- (1.887,9.21) -- cycle;
\node at (1.887,9.604) {\scalebox{1.648}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.092,8.58) -- (9.213,8.599) -- (9.323,8.655) -- (9.41,8.742) -- (9.466,8.852) -- (9.485,8.974) -- (9.466,9.095) -- (9.41,9.205) -- (9.323,9.292) -- (9.213,9.348) -- (9.092,9.368) -- (9.092,9.368) -- (8.97,9.348) -- (8.86,9.292) -- (8.773,9.205) -- (8.717,9.095) -- (8.698,8.974) -- (8.717,8.852) -- (8.773,8.742) -- (8.86,8.655) -- (8.97,8.599) -- (9.092,8.58) -- cycle;
\node at (9.092,8.974) {\scalebox{1.648}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.786,6.669) -- (7.908,6.688) -- (8.018,6.744) -- (8.105,6.831) -- (8.161,6.941) -- (8.18,7.063) -- (8.161,7.185) -- (8.105,7.294) -- (8.018,7.381) -- (7.908,7.437) -- (7.786,7.457) -- (7.786,7.457) -- (7.664,7.437) -- (7.555,7.381) -- (7.467,7.294) -- (7.411,7.185) -- (7.392,7.063) -- (7.411,6.941) -- (7.467,6.831) -- (7.555,6.744) -- (7.664,6.688) -- (7.786,6.669) -- cycle;
\node at (7.786,7.063) {\scalebox{1.648}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.427,6.355) -- (11.549,6.374) -- (11.659,6.43) -- (11.746,6.517) -- (11.802,6.627) -- (11.821,6.749) -- (11.802,6.871) -- (11.746,6.98) -- (11.659,7.067) -- (11.549,7.123) -- (11.427,7.143) -- (11.057,7.143) -- (10.935,7.123) -- (10.825,7.067) -- (10.738,6.98) -- (10.682,6.871) -- (10.663,6.749) -- (10.682,6.627) -- (10.738,6.517) -- (10.825,6.43) -- (10.935,6.374) -- (11.057,6.355) -- cycle;
\node at (11.242,6.749) {\scalebox{1.648}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.347,7.894) -- (3.469,7.914) -- (3.579,7.97) -- (3.666,8.057) -- (3.722,8.167) -- (3.741,8.288) -- (3.722,8.41) -- (3.666,8.52) -- (3.579,8.607) -- (3.469,8.663) -- (3.347,8.682) -- (2.976,8.682) -- (2.855,8.663) -- (2.745,8.607) -- (2.658,8.52) -- (2.602,8.41) -- (2.583,8.288) -- (2.602,8.167) -- (2.658,8.057) -- (2.745,7.97) -- (2.855,7.914) -- (2.976,7.894) -- cycle;
\node at (3.162,8.288) {\scalebox{1.648}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.158,6.449) -- (2.28,6.468) -- (2.39,6.524) -- (2.477,6.611) -- (2.533,6.721) -- (2.552,6.843) -- (2.533,6.964) -- (2.477,7.074) -- (2.39,7.161) -- (2.28,7.217) -- (2.158,7.236) -- (1.788,7.236) -- (1.666,7.217) -- (1.556,7.161) -- (1.469,7.074) -- (1.413,6.964) -- (1.394,6.843) -- (1.413,6.721) -- (1.469,6.611) -- (1.556,6.524) -- (1.666,6.468) -- (1.788,6.449) -- cycle;
\node at (1.973,6.843) {\scalebox{1.648}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.436,7.239) -- (6.558,7.258) -- (6.668,7.314) -- (6.755,7.402) -- (6.811,7.511) -- (6.83,7.633) -- (6.811,7.755) -- (6.755,7.865) -- (6.668,7.952) -- (6.558,8.008) -- (6.436,8.027) -- (6.065,8.027) -- (5.944,8.008) -- (5.834,7.952) -- (5.747,7.865) -- (5.691,7.755) -- (5.671,7.633) -- (5.691,7.511) -- (5.747,7.402) -- (5.834,7.314) -- (5.944,7.258) -- (6.065,7.239) -- cycle;
\node at (6.251,7.633) {\scalebox{1.648}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.67,3.17) -- (8.791,3.189) -- (8.901,3.245) -- (8.988,3.332) -- (9.044,3.442) -- (9.063,3.563) -- (9.044,3.685) -- (8.988,3.795) -- (8.901,3.882) -- (8.791,3.938) -- (8.67,3.957) -- (8.299,3.957) -- (8.177,3.938) -- (8.067,3.882) -- (7.98,3.795) -- (7.924,3.685) -- (7.905,3.563) -- (7.924,3.442) -- (7.98,3.332) -- (8.067,3.245) -- (8.177,3.189) -- (8.299,3.17) -- cycle;
\node at (8.484,3.563) {\scalebox{1.648}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.944,3.335) -- (11.066,3.354) -- (11.176,3.41) -- (11.263,3.497) -- (11.319,3.607) -- (11.338,3.729) -- (11.319,3.851) -- (11.263,3.961) -- (11.176,4.048) -- (11.066,4.104) -- (10.944,4.123) -- (10.574,4.123) -- (10.452,4.104) -- (10.342,4.048) -- (10.255,3.961) -- (10.199,3.851) -- (10.18,3.729) -- (10.199,3.607) -- (10.255,3.497) -- (10.342,3.41) -- (10.452,3.354) -- (10.574,3.335) -- cycle;
\node at (10.759,3.729) {\scalebox{1.648}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.296,1.903) -- (9.418,1.922) -- (9.528,1.978) -- (9.615,2.065) -- (9.671,2.175) -- (9.69,2.297) -- (9.671,2.418) -- (9.615,2.528) -- (9.528,2.615) -- (9.418,2.671) -- (9.296,2.691) -- (8.925,2.691) -- (8.804,2.671) -- (8.694,2.615) -- (8.607,2.528) -- (8.551,2.418) -- (8.531,2.297) -- (8.551,2.175) -- (8.607,2.065) -- (8.694,1.978) -- (8.804,1.922) -- (8.925,1.903) -- cycle;
\node at (9.111,2.297) {\scalebox{1.648}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.57,10.896) -- (2.17,10.896) -- (2.17,11.496) -- (1.57,11.496) -- cycle;
\node at (1.87,11.196) {\scalebox{1.648}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.771,1.071) -- (9.371,1.071) -- (9.371,1.671) -- (8.771,1.671) -- cycle;
\node at (9.071,1.371) {\scalebox{1.648}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.311,12.249) -- (5.911,12.249) -- (5.911,12.849) -- (5.311,12.849) -- cycle;
\node at (5.611,12.549) {\scalebox{1.648}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.379,8.341) -- (5.979,8.341) -- (5.979,8.941) -- (5.379,8.941) -- cycle;
\node at (5.679,8.641) {\scalebox{1.648}{\small d}};
\node at (7.844,12.161) {\scalebox{1.648}{\small A}};
\node at (11.79,11.67) {\scalebox{1.648}{\small B}};
\node at (4.498,11.795) {\scalebox{1.648}{\small C}};
\node at (1.37,11.564) {\scalebox{1.648}{\small D}};
\node at (7.528,9.114) {\scalebox{1.648}{\small E}};
\node at (11.453,8.428) {\scalebox{1.648}{\small F}};
\node at (0.765,8.153) {\scalebox{1.648}{\small G}};
\node at (4.323,8.126) {\scalebox{1.648}{\small H}};
\node at (0.988,5.296) {\scalebox{1.648}{\small I}};
\node at (8.043,4.925) {\scalebox{1.648}{\small J}};
\node at (11.134,4.944) {\scalebox{1.648}{\small K}};
\node at (10.724,1.718) {\scalebox{1.648}{\small L}};
\node at (8.207,0.95) {\scalebox{1.648}{\small M}};
\draw[line width=0.5pt] (12.612,10.607) -- (12.612,11.707);
\draw[line width=0.5pt] (12.381,11.322) -- (12.612,11.707) -- (12.843,11.322);
\draw[line width=0.5pt] (12.309,11.157) -- (12.914,11.157);
\node at (12.612,12.092) {\scalebox{1.648}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour D, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ passage du Marché & Ⓐ Carrefour du Moulin & Ⓙ Placette des Ormes \\ \hline
① chemin des Vignes & ⑩ avenue de la Gare & Ⓑ Coin du Lavoir & Ⓚ Place du Marché \\ \hline
② allée des Peupliers & ⑪ rue Verte & Ⓒ Carrefour des Tilleuls & Ⓛ Place de la Fontaine \\ \hline
③ sentier des Prés & ⑫ impasse du Vieux-Four & Ⓓ Place des Tanneurs & Ⓜ Place aux Herbes \\ \hline
④ rue du Puits & ⑬ rue du Moulin & Ⓔ Rond-point de la Poste & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ rue de la Poste & ⑭ rue Pasteur & Ⓕ Carrefour Saint-Roch & \textbf{a} l'école \\ \hline
⑥ rue Haute & ⑮ rue Basse & Ⓖ Rond-point du Stade & \textbf{b} le stade \\ \hline
⑦ chemin de Ronde & ⑯ rue des Écoles & Ⓗ Carrefour du Lavoir & \textbf{c} la boulangerie \\ \hline
⑧ chemin Creux & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Place de l'Église & \textbf{d} le marché couvert \\ \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=0.6216, line join=round, line cap=round]
\draw[line width=0.5pt] (7.708,11.557) -- (7.96,11.497) -- (8.214,11.444) -- (8.469,11.396) -- (8.725,11.355) -- (8.982,11.32) -- (9.24,11.291) -- (9.498,11.267) -- (9.757,11.25) -- (10.016,11.239) -- (10.276,11.234) -- (10.535,11.235) -- (10.794,11.242) -- (11.053,11.256) -- (11.312,11.275);
\draw[line width=0.5pt] (7.708,11.557) -- (7.71,11.339) -- (7.706,11.121) -- (7.698,10.903) -- (7.684,10.686) -- (7.665,10.469) -- (7.641,10.252) -- (7.612,10.036) -- (7.578,9.821) -- (7.539,9.606) -- (7.494,9.393) -- (7.445,9.181) -- (7.391,8.97) -- (7.332,8.76) -- (7.268,8.552);
\draw[line width=0.5pt] (11.312,11.275) -- (11.246,11.071) -- (11.185,10.864) -- (11.129,10.657) -- (11.079,10.448) -- (11.033,10.238) -- (10.992,10.026) -- (10.955,9.814) -- (10.924,9.601) -- (10.898,9.388) -- (10.877,9.174) -- (10.861,8.959) -- (10.85,8.744) -- (10.844,8.529) -- (10.844,8.315);
\draw[line width=0.5pt] (4.795,11.251) -- (4.998,11.305) -- (5.203,11.353) -- (5.408,11.397) -- (5.615,11.436) -- (5.822,11.47) -- (6.03,11.5) -- (6.239,11.524) -- (6.448,11.543) -- (6.658,11.558) -- (6.868,11.568) -- (7.078,11.572) -- (7.288,11.572) -- (7.498,11.567) -- (7.708,11.557);
\draw[line width=0.5pt] (4.795,11.251) -- (4.769,11.018) -- (4.748,10.785) -- (4.733,10.551) -- (4.723,10.317) -- (4.719,10.083) -- (4.72,9.849) -- (4.727,9.614) -- (4.739,9.38) -- (4.757,9.147) -- (4.78,8.914) -- (4.808,8.681) -- (4.842,8.449) -- (4.882,8.219) -- (4.927,7.989);
\draw[line width=0.5pt] (1.87,11.196) -- (2.076,11.232) -- (2.284,11.263) -- (2.492,11.289) -- (2.701,11.311) -- (2.911,11.327) -- (3.12,11.338) -- (3.33,11.344) -- (3.54,11.346) -- (3.75,11.342) -- (3.96,11.334) -- (4.17,11.321) -- (4.379,11.302) -- (4.587,11.279) -- (4.795,11.251);
\draw[line width=0.5pt] (1.87,11.196) -- (1.801,10.982) -- (1.738,10.766) -- (1.679,10.548) -- (1.626,10.329) -- (1.578,10.108) -- (1.535,9.887) -- (1.498,9.665) -- (1.465,9.442) -- (1.438,9.218) -- (1.416,8.993) -- (1.4,8.768) -- (1.389,8.543) -- (1.383,8.318) -- (1.382,8.093);
\draw[line width=0.5pt] (7.268,8.552) -- (7.524,8.574) -- (7.78,8.59) -- (8.037,8.601) -- (8.294,8.605) -- (8.551,8.603) -- (8.809,8.595) -- (9.065,8.581) -- (9.322,8.561) -- (9.578,8.535) -- (9.833,8.503) -- (10.087,8.464) -- (10.341,8.42) -- (10.593,8.37) -- (10.844,8.315);
\draw[line width=0.5pt] (7.268,8.552) -- (7.27,8.329) -- (7.279,8.106) -- (7.292,7.884) -- (7.311,7.661) -- (7.334,7.44) -- (7.363,7.218) -- (7.398,6.998) -- (7.437,6.778) -- (7.482,6.56) -- (7.531,6.342) -- (7.586,6.126) -- (7.646,5.911) -- (7.711,5.698) -- (7.78,5.486);
\draw[line width=0.5pt] (10.844,8.315) -- (10.86,8.096) -- (10.871,7.877) -- (10.877,7.657) -- (10.878,7.438) -- (10.873,7.218) -- (10.864,6.999) -- (10.849,6.78) -- (10.829,6.561) -- (10.804,6.343) -- (10.774,6.125) -- (10.739,5.909) -- (10.699,5.693) -- (10.654,5.478) -- (10.603,5.265);
\draw[line width=0.5pt] (1.382,8.093) -- (1.632,8.047) -- (1.883,8.006) -- (2.135,7.972) -- (2.388,7.944) -- (2.642,7.921) -- (2.896,7.905) -- (3.15,7.895) -- (3.405,7.89) -- (3.659,7.892) -- (3.914,7.899) -- (4.168,7.912) -- (4.422,7.932) -- (4.675,7.957) -- (4.927,7.989);
\draw[line width=0.5pt] (1.382,8.093) -- (1.423,7.913) -- (1.46,7.731) -- (1.493,7.549) -- (1.521,7.366) -- (1.545,7.183) -- (1.565,6.999) -- (1.58,6.814) -- (1.591,6.63) -- (1.598,6.445) -- (1.6,6.26) -- (1.598,6.075) -- (1.592,5.89) -- (1.581,5.705) -- (1.566,5.521);
\draw[line width=0.5pt] (4.927,7.989) -- (5.099,8.003) -- (5.271,8.021) -- (5.442,8.044) -- (5.613,8.07) -- (5.783,8.101) -- (5.952,8.135) -- (6.121,8.174) -- (6.288,8.216) -- (6.455,8.262) -- (6.62,8.313) -- (6.784,8.367) -- (6.947,8.425) -- (7.108,8.486) -- (7.268,8.552);
\draw[line width=0.5pt] (7.78,5.486) -- (7.845,5.211) -- (7.902,4.934) -- (7.953,4.656) -- (7.998,4.377) -- (8.035,4.097) -- (8.067,3.815) -- (8.091,3.534) -- (8.11,3.251) -- (8.121,2.969) -- (8.126,2.686) -- (8.124,2.403) -- (8.115,2.121) -- (8.1,1.838) -- (8.078,1.557);
\draw[line width=0.5pt] (10.603,5.265) -- (10.554,5.052) -- (10.51,4.839) -- (10.471,4.624) -- (10.437,4.409) -- (10.408,4.192) -- (10.385,3.975) -- (10.366,3.758) -- (10.353,3.54) -- (10.344,3.322) -- (10.341,3.104) -- (10.343,2.886) -- (10.35,2.668) -- (10.362,2.45) -- (10.379,2.233);
\draw[line width=0.5pt] (8.078,1.557) -- (8.234,1.63) -- (8.391,1.699) -- (8.55,1.765) -- (8.711,1.827) -- (8.873,1.885) -- (9.036,1.939) -- (9.201,1.99) -- (9.367,2.036) -- (9.533,2.079) -- (9.701,2.118) -- (9.87,2.152) -- (10.039,2.183) -- (10.209,2.21) -- (10.379,2.233);
\filldraw[fill=white, draw=black, line width=0.5pt] (8.081,11.557) -- (8.053,11.699) -- (7.972,11.82) -- (7.851,11.901) -- (7.708,11.93) -- (7.565,11.901) -- (7.444,11.82) -- (7.363,11.699) -- (7.335,11.557) -- (7.363,11.414) -- (7.444,11.293) -- (7.565,11.212) -- (7.708,11.183) -- (7.851,11.212) -- (7.972,11.293) -- (8.053,11.414) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.685,11.275) -- (11.656,11.418) -- (11.575,11.539) -- (11.454,11.62) -- (11.312,11.649) -- (11.169,11.62) -- (11.048,11.539) -- (10.967,11.418) -- (10.938,11.275) -- (10.967,11.133) -- (11.048,11.011) -- (11.169,10.931) -- (11.312,10.902) -- (11.454,10.931) -- (11.575,11.011) -- (11.656,11.133) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.169,11.251) -- (5.14,11.394) -- (5.059,11.515) -- (4.938,11.596) -- (4.795,11.624) -- (4.653,11.596) -- (4.531,11.515) -- (4.451,11.394) -- (4.422,11.251) -- (4.451,11.108) -- (4.531,10.987) -- (4.653,10.906) -- (4.795,10.878) -- (4.938,10.906) -- (5.059,10.987) -- (5.14,11.108) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.243,11.196) -- (2.215,11.339) -- (2.134,11.46) -- (2.013,11.541) -- (1.87,11.57) -- (1.727,11.541) -- (1.606,11.46) -- (1.525,11.339) -- (1.496,11.196) -- (1.525,11.054) -- (1.606,10.933) -- (1.727,10.852) -- (1.87,10.823) -- (2.013,10.852) -- (2.134,10.933) -- (2.215,11.054) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.641,8.552) -- (7.613,8.695) -- (7.532,8.816) -- (7.411,8.897) -- (7.268,8.925) -- (7.125,8.897) -- (7.004,8.816) -- (6.923,8.695) -- (6.895,8.552) -- (6.923,8.409) -- (7.004,8.288) -- (7.125,8.207) -- (7.268,8.179) -- (7.411,8.207) -- (7.532,8.288) -- (7.613,8.409) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.217,8.315) -- (11.188,8.457) -- (11.108,8.578) -- (10.986,8.659) -- (10.844,8.688) -- (10.701,8.659) -- (10.58,8.578) -- (10.499,8.457) -- (10.47,8.315) -- (10.499,8.172) -- (10.58,8.051) -- (10.701,7.97) -- (10.844,7.941) -- (10.986,7.97) -- (11.108,8.051) -- (11.188,8.172) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.755,8.093) -- (1.727,8.236) -- (1.646,8.357) -- (1.525,8.438) -- (1.382,8.466) -- (1.239,8.438) -- (1.118,8.357) -- (1.037,8.236) -- (1.009,8.093) -- (1.037,7.95) -- (1.118,7.829) -- (1.239,7.748) -- (1.382,7.72) -- (1.525,7.748) -- (1.646,7.829) -- (1.727,7.95) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.3,7.989) -- (5.272,8.132) -- (5.191,8.253) -- (5.07,8.334) -- (4.927,8.362) -- (4.784,8.334) -- (4.663,8.253) -- (4.582,8.132) -- (4.554,7.989) -- (4.582,7.846) -- (4.663,7.725) -- (4.784,7.644) -- (4.927,7.616) -- (5.07,7.644) -- (5.191,7.725) -- (5.272,7.846) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.939,5.521) -- (1.911,5.664) -- (1.83,5.785) -- (1.709,5.866) -- (1.566,5.894) -- (1.423,5.866) -- (1.302,5.785) -- (1.221,5.664) -- (1.193,5.521) -- (1.221,5.378) -- (1.302,5.257) -- (1.423,5.176) -- (1.566,5.148) -- (1.709,5.176) -- (1.83,5.257) -- (1.911,5.378) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.154,5.486) -- (8.125,5.629) -- (8.044,5.75) -- (7.923,5.831) -- (7.78,5.859) -- (7.638,5.831) -- (7.517,5.75) -- (7.436,5.629) -- (7.407,5.486) -- (7.436,5.343) -- (7.517,5.222) -- (7.638,5.141) -- (7.78,5.113) -- (7.923,5.141) -- (8.044,5.222) -- (8.125,5.343) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (10.976,5.265) -- (10.948,5.408) -- (10.867,5.529) -- (10.746,5.61) -- (10.603,5.638) -- (10.46,5.61) -- (10.339,5.529) -- (10.258,5.408) -- (10.23,5.265) -- (10.258,5.122) -- (10.339,5.001) -- (10.46,4.92) -- (10.603,4.892) -- (10.746,4.92) -- (10.867,5.001) -- (10.948,5.122) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.752,2.233) -- (10.724,2.376) -- (10.643,2.497) -- (10.522,2.578) -- (10.379,2.606) -- (10.236,2.578) -- (10.115,2.497) -- (10.034,2.376) -- (10.006,2.233) -- (10.034,2.09) -- (10.115,1.969) -- (10.236,1.888) -- (10.379,1.86) -- (10.522,1.888) -- (10.643,1.969) -- (10.724,2.09) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.451,1.557) -- (8.423,1.699) -- (8.342,1.82) -- (8.221,1.901) -- (8.078,1.93) -- (7.935,1.901) -- (7.814,1.82) -- (7.733,1.699) -- (7.705,1.557) -- (7.733,1.414) -- (7.814,1.293) -- (7.935,1.212) -- (8.078,1.183) -- (8.221,1.212) -- (8.342,1.293) -- (8.423,1.414) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.528,11.266) -- (9.647,11.285) -- (9.754,11.34) -- (9.839,11.425) -- (9.894,11.532) -- (9.913,11.651) -- (9.894,11.77) -- (9.839,11.877) -- (9.754,11.962) -- (9.647,12.016) -- (9.528,12.035) -- (9.528,12.035) -- (9.409,12.016) -- (9.302,11.962) -- (9.217,11.877) -- (9.162,11.77) -- (9.144,11.651) -- (9.162,11.532) -- (9.217,11.425) -- (9.302,11.34) -- (9.409,11.285) -- (9.528,11.266) -- cycle;
\node at (9.528,11.651) {\scalebox{1.609}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.992,9.596) -- (8.111,9.615) -- (8.218,9.669) -- (8.303,9.754) -- (8.358,9.861) -- (8.377,9.98) -- (8.358,10.099) -- (8.303,10.206) -- (8.218,10.291) -- (8.111,10.346) -- (7.992,10.365) -- (7.992,10.365) -- (7.874,10.346) -- (7.766,10.291) -- (7.681,10.206) -- (7.627,10.099) -- (7.608,9.98) -- (7.627,9.861) -- (7.681,9.754) -- (7.766,9.669) -- (7.874,9.615) -- (7.992,9.596) -- cycle;
\node at (7.992,9.98) {\scalebox{1.609}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.335,9.37) -- (11.454,9.389) -- (11.561,9.443) -- (11.646,9.528) -- (11.701,9.635) -- (11.72,9.754) -- (11.701,9.873) -- (11.646,9.98) -- (11.561,10.065) -- (11.454,10.12) -- (11.335,10.139) -- (11.335,10.139) -- (11.216,10.12) -- (11.109,10.065) -- (11.024,9.98) -- (10.97,9.873) -- (10.951,9.754) -- (10.97,9.635) -- (11.024,9.528) -- (11.109,9.443) -- (11.216,9.389) -- (11.335,9.37) -- cycle;
\node at (11.335,9.754) {\scalebox{1.609}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.199,11.522) -- (6.318,11.541) -- (6.425,11.595) -- (6.51,11.68) -- (6.565,11.788) -- (6.584,11.906) -- (6.565,12.025) -- (6.51,12.132) -- (6.425,12.218) -- (6.318,12.272) -- (6.199,12.291) -- (6.199,12.291) -- (6.08,12.272) -- (5.973,12.218) -- (5.888,12.132) -- (5.833,12.025) -- (5.815,11.906) -- (5.833,11.788) -- (5.888,11.68) -- (5.973,11.595) -- (6.08,11.541) -- (6.199,11.522) -- cycle;
\node at (6.199,11.906) {\scalebox{1.609}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.111,9.245) -- (5.23,9.264) -- (5.337,9.319) -- (5.422,9.404) -- (5.477,9.511) -- (5.495,9.63) -- (5.477,9.749) -- (5.422,9.856) -- (5.337,9.941) -- (5.23,9.996) -- (5.111,10.015) -- (5.111,10.015) -- (4.992,9.996) -- (4.885,9.941) -- (4.8,9.856) -- (4.745,9.749) -- (4.726,9.63) -- (4.745,9.511) -- (4.8,9.404) -- (4.885,9.319) -- (4.992,9.264) -- (5.111,9.245) -- cycle;
\node at (5.111,9.63) {\scalebox{1.609}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.323,11.344) -- (3.442,11.363) -- (3.549,11.418) -- (3.634,11.503) -- (3.689,11.61) -- (3.708,11.729) -- (3.689,11.848) -- (3.634,11.955) -- (3.549,12.04) -- (3.442,12.095) -- (3.323,12.113) -- (3.323,12.113) -- (3.204,12.095) -- (3.097,12.04) -- (3.012,11.955) -- (2.957,11.848) -- (2.939,11.729) -- (2.957,11.61) -- (3.012,11.503) -- (3.097,11.418) -- (3.204,11.363) -- (3.323,11.344) -- cycle;
\node at (3.323,11.729) {\scalebox{1.609}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.878,9.221) -- (1.997,9.239) -- (2.104,9.294) -- (2.189,9.379) -- (2.243,9.486) -- (2.262,9.605) -- (2.243,9.724) -- (2.189,9.831) -- (2.104,9.916) -- (1.997,9.971) -- (1.878,9.99) -- (1.878,9.99) -- (1.759,9.971) -- (1.652,9.916) -- (1.567,9.831) -- (1.512,9.724) -- (1.493,9.605) -- (1.512,9.486) -- (1.567,9.379) -- (1.652,9.294) -- (1.759,9.239) -- (1.878,9.221) -- cycle;
\node at (1.878,9.605) {\scalebox{1.609}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.091,8.58) -- (9.21,8.599) -- (9.317,8.653) -- (9.402,8.738) -- (9.457,8.846) -- (9.475,8.964) -- (9.457,9.083) -- (9.402,9.19) -- (9.317,9.275) -- (9.21,9.33) -- (9.091,9.349) -- (9.091,9.349) -- (8.972,9.33) -- (8.865,9.275) -- (8.78,9.19) -- (8.725,9.083) -- (8.706,8.964) -- (8.725,8.846) -- (8.78,8.738) -- (8.865,8.653) -- (8.972,8.599) -- (9.091,8.58) -- cycle;
\node at (9.091,8.964) {\scalebox{1.609}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.777,6.677) -- (7.896,6.696) -- (8.003,6.75) -- (8.088,6.835) -- (8.143,6.942) -- (8.161,7.061) -- (8.143,7.18) -- (8.088,7.287) -- (8.003,7.372) -- (7.896,7.427) -- (7.777,7.446) -- (7.777,7.446) -- (7.658,7.427) -- (7.551,7.372) -- (7.466,7.287) -- (7.411,7.18) -- (7.392,7.061) -- (7.411,6.942) -- (7.466,6.835) -- (7.551,6.75) -- (7.658,6.696) -- (7.777,6.677) -- cycle;
\node at (7.777,7.061) {\scalebox{1.609}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.413,6.365) -- (11.532,6.384) -- (11.64,6.438) -- (11.725,6.524) -- (11.779,6.631) -- (11.798,6.75) -- (11.779,6.868) -- (11.725,6.976) -- (11.64,7.061) -- (11.532,7.115) -- (11.413,7.134) -- (11.052,7.134) -- (10.933,7.115) -- (10.826,7.061) -- (10.741,6.976) -- (10.686,6.868) -- (10.667,6.75) -- (10.686,6.631) -- (10.741,6.524) -- (10.826,6.438) -- (10.933,6.384) -- (11.052,6.365) -- cycle;
\node at (11.233,6.75) {\scalebox{1.609}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.342,7.894) -- (3.461,7.913) -- (3.569,7.968) -- (3.654,8.053) -- (3.708,8.16) -- (3.727,8.279) -- (3.708,8.398) -- (3.654,8.505) -- (3.569,8.59) -- (3.461,8.645) -- (3.342,8.663) -- (2.981,8.663) -- (2.862,8.645) -- (2.755,8.59) -- (2.669,8.505) -- (2.615,8.398) -- (2.596,8.279) -- (2.615,8.16) -- (2.669,8.053) -- (2.755,7.968) -- (2.862,7.913) -- (2.981,7.894) -- cycle;
\node at (3.162,8.279) {\scalebox{1.609}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.145,6.457) -- (2.263,6.476) -- (2.371,6.531) -- (2.456,6.616) -- (2.51,6.723) -- (2.529,6.842) -- (2.51,6.961) -- (2.456,7.068) -- (2.371,7.153) -- (2.263,7.208) -- (2.145,7.226) -- (1.783,7.226) -- (1.664,7.208) -- (1.557,7.153) -- (1.472,7.068) -- (1.417,6.961) -- (1.398,6.842) -- (1.417,6.723) -- (1.472,6.616) -- (1.557,6.531) -- (1.664,6.476) -- (1.783,6.457) -- cycle;
\node at (1.964,6.842) {\scalebox{1.609}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.429,7.261) -- (6.547,7.28) -- (6.655,7.335) -- (6.74,7.42) -- (6.794,7.527) -- (6.813,7.646) -- (6.794,7.765) -- (6.74,7.872) -- (6.655,7.957) -- (6.547,8.012) -- (6.429,8.03) -- (6.067,8.03) -- (5.948,8.012) -- (5.841,7.957) -- (5.756,7.872) -- (5.701,7.765) -- (5.682,7.646) -- (5.701,7.527) -- (5.756,7.42) -- (5.841,7.335) -- (5.948,7.28) -- (6.067,7.261) -- cycle;
\node at (6.248,7.646) {\scalebox{1.609}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.656,3.178) -- (8.775,3.197) -- (8.882,3.252) -- (8.967,3.337) -- (9.022,3.444) -- (9.04,3.563) -- (9.022,3.682) -- (8.967,3.789) -- (8.882,3.874) -- (8.775,3.928) -- (8.656,3.947) -- (8.294,3.947) -- (8.175,3.928) -- (8.068,3.874) -- (7.983,3.789) -- (7.928,3.682) -- (7.909,3.563) -- (7.928,3.444) -- (7.983,3.337) -- (8.068,3.252) -- (8.175,3.197) -- (8.294,3.178) -- cycle;
\node at (8.475,3.563) {\scalebox{1.609}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.931,3.345) -- (11.049,3.364) -- (11.157,3.419) -- (11.242,3.504) -- (11.296,3.611) -- (11.315,3.73) -- (11.296,3.849) -- (11.242,3.956) -- (11.157,4.041) -- (11.049,4.095) -- (10.931,4.114) -- (10.569,4.114) -- (10.45,4.095) -- (10.343,4.041) -- (10.258,3.956) -- (10.203,3.849) -- (10.184,3.73) -- (10.203,3.611) -- (10.258,3.504) -- (10.343,3.419) -- (10.45,3.364) -- (10.569,3.345) -- cycle;
\node at (10.75,3.73) {\scalebox{1.609}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.586,2.222) -- (9.705,2.241) -- (9.812,2.295) -- (9.897,2.38) -- (9.952,2.487) -- (9.97,2.606) -- (9.952,2.725) -- (9.897,2.832) -- (9.812,2.917) -- (9.705,2.972) -- (9.586,2.991) -- (9.224,2.991) -- (9.105,2.972) -- (8.998,2.917) -- (8.913,2.832) -- (8.858,2.725) -- (8.839,2.606) -- (8.858,2.487) -- (8.913,2.38) -- (8.998,2.295) -- (9.105,2.241) -- (9.224,2.222) -- cycle;
\node at (9.405,2.606) {\scalebox{1.609}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.57,10.896) -- (2.17,10.896) -- (2.17,11.496) -- (1.57,11.496) -- cycle;
\node at (1.87,11.196) {\scalebox{1.609}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.771,1.071) -- (9.371,1.071) -- (9.371,1.671) -- (8.771,1.671) -- cycle;
\node at (9.071,1.371) {\scalebox{1.609}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.311,12.249) -- (5.911,12.249) -- (5.911,12.849) -- (5.311,12.849) -- cycle;
\node at (5.611,12.549) {\scalebox{1.609}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.379,8.341) -- (5.979,8.341) -- (5.979,8.941) -- (5.379,8.941) -- cycle;
\node at (5.679,8.641) {\scalebox{1.609}{\small d}};
\node at (7.844,12.161) {\scalebox{1.609}{\small A}};
\node at (11.79,11.67) {\scalebox{1.609}{\small B}};
\node at (4.498,11.795) {\scalebox{1.609}{\small C}};
\node at (1.37,11.564) {\scalebox{1.609}{\small D}};
\node at (7.528,9.114) {\scalebox{1.609}{\small E}};
\node at (11.453,8.428) {\scalebox{1.609}{\small F}};
\node at (0.765,8.153) {\scalebox{1.609}{\small G}};
\node at (4.323,8.126) {\scalebox{1.609}{\small H}};
\node at (0.988,5.296) {\scalebox{1.609}{\small I}};
\node at (8.043,4.925) {\scalebox{1.609}{\small J}};
\node at (11.134,4.944) {\scalebox{1.609}{\small K}};
\node at (10.724,1.718) {\scalebox{1.609}{\small L}};
\node at (8.207,0.95) {\scalebox{1.609}{\small M}};
\draw[line width=0.5pt] (12.612,10.607) -- (12.612,11.707);
\draw[line width=0.5pt] (12.381,11.322) -- (12.612,11.707) -- (12.843,11.322);
\draw[line width=0.5pt] (12.309,11.157) -- (12.914,11.157);
\node at (12.612,12.092) {\scalebox{1.609}{\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, C, E, G, K, M & 6 \\ \hline
clairs & A, D, F, H, I, J, L & 7 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{7} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 13 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 6 carrefours sombres et 7 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{D} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{rue Haute ⑥} vers l'est jusqu'à Carrefour des Tilleuls \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Puits ④}, vers l'est, jusqu'à Carrefour du Moulin \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin des Vignes ①}, vers l'est, jusqu'à Coin du Lavoir \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ③}, vers le sud, jusqu'à Carrefour Saint-Roch \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{avenue de la Gare ⑩}, vers le sud, jusqu'à Place du Marché \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Basse ⑮}, vers le sud, jusqu'à Place de la Fontaine \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Écoles ⑯}, vers l'ouest, jusqu'à Place aux Herbes \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Pasteur ⑭}, vers le nord, jusqu'à Placette des Ormes \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{passage du Marché ⑨}, vers le nord, jusqu'à Rond-point de la Poste \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Moulin ⑬}, vers l'ouest, jusqu'à Carrefour du Lavoir \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Verte ⑪}, vers l'ouest, jusqu'à Rond-point du Stade \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse du Vieux-Four ⑫}, vers le sud, jusqu'à Place de l'Église \textbf{(I)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 13 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
