\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 221}

\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=221}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0221}
\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=0.5916, line join=round, line cap=round]
\draw[line width=0.5pt] (9.224,6.997) -- (9.158,6.786) -- (9.097,6.574) -- (9.04,6.36) -- (8.989,6.145) -- (8.943,5.929) -- (8.902,5.712) -- (8.866,5.494) -- (8.835,5.275) -- (8.809,5.055) -- (8.789,4.835) -- (8.773,4.615) -- (8.763,4.394) -- (8.758,4.173) -- (8.758,3.952);
\draw[line width=0.5pt] (6.578,6.972) -- (6.765,7.003) -- (6.953,7.029) -- (7.142,7.051) -- (7.331,7.068) -- (7.521,7.081) -- (7.71,7.09) -- (7.9,7.094) -- (8.09,7.093) -- (8.28,7.088) -- (8.47,7.079) -- (8.659,7.065) -- (8.848,7.047) -- (9.037,7.024) -- (9.224,6.997);
\draw[line width=0.5pt] (6.578,6.972) -- (6.547,6.805) -- (6.519,6.637) -- (6.495,6.468) -- (6.476,6.299) -- (6.46,6.129) -- (6.449,5.959) -- (6.441,5.788) -- (6.437,5.618) -- (6.438,5.448) -- (6.442,5.277) -- (6.45,5.107) -- (6.463,4.937) -- (6.479,4.767) -- (6.499,4.598);
\draw[line width=0.5pt] (6.499,4.598) -- (6.666,4.577) -- (6.833,4.552) -- (6.999,4.523) -- (7.165,4.491) -- (7.329,4.454) -- (7.493,4.413) -- (7.655,4.369) -- (7.817,4.32) -- (7.978,4.268) -- (8.137,4.213) -- (8.294,4.153) -- (8.451,4.09) -- (8.605,4.023) -- (8.758,3.952);
\draw[line width=0.5pt] (6.499,4.598) -- (6.496,4.408) -- (6.498,4.218) -- (6.504,4.028) -- (6.514,3.838) -- (6.529,3.649) -- (6.549,3.459) -- (6.573,3.271) -- (6.601,3.082) -- (6.634,2.895) -- (6.671,2.708) -- (6.712,2.523) -- (6.758,2.338) -- (6.808,2.155) -- (6.863,1.973);
\draw[line width=0.5pt] (1.658,4.554) -- (1.802,4.538) -- (1.946,4.519) -- (2.089,4.497) -- (2.232,4.471) -- (2.375,4.442) -- (2.516,4.41) -- (2.657,4.374) -- (2.797,4.335) -- (2.935,4.293) -- (3.073,4.247) -- (3.21,4.198) -- (3.345,4.146) -- (3.479,4.091) -- (3.612,4.033);
\draw[line width=0.5pt] (1.658,4.554) -- (1.637,4.36) -- (1.621,4.165) -- (1.609,3.971) -- (1.602,3.775) -- (1.599,3.58) -- (1.602,3.385) -- (1.608,3.189) -- (1.619,2.994) -- (1.635,2.8) -- (1.656,2.605) -- (1.68,2.412) -- (1.71,2.219) -- (1.744,2.026) -- (1.783,1.835);
\draw[line width=0.5pt] (11.511,4.333) -- (11.551,4.142) -- (11.586,3.95) -- (11.618,3.757) -- (11.644,3.563) -- (11.666,3.369) -- (11.684,3.174) -- (11.697,2.979) -- (11.705,2.783) -- (11.709,2.588) -- (11.708,2.392) -- (11.702,2.197) -- (11.692,2.002) -- (11.677,1.807) -- (11.658,1.612);
\draw[line width=0.5pt] (3.612,4.033) -- (3.823,4.041) -- (4.034,4.055) -- (4.244,4.073) -- (4.454,4.097) -- (4.663,4.125) -- (4.871,4.158) -- (5.079,4.196) -- (5.286,4.24) -- (5.492,4.287) -- (5.696,4.34) -- (5.899,4.398) -- (6.101,4.46) -- (6.301,4.527) -- (6.499,4.598);
\draw[line width=0.5pt] (3.612,4.033) -- (3.671,3.899) -- (3.727,3.764) -- (3.78,3.628) -- (3.829,3.491) -- (3.875,3.352) -- (3.918,3.212) -- (3.958,3.072) -- (3.994,2.93) -- (4.027,2.788) -- (4.057,2.645) -- (4.083,2.501) -- (4.106,2.357) -- (4.125,2.212) -- (4.141,2.067);
\draw[line width=0.5pt] (8.758,3.952) -- (8.958,3.949) -- (9.157,3.951) -- (9.356,3.957) -- (9.555,3.968) -- (9.754,3.984) -- (9.952,4.004) -- (10.15,4.029) -- (10.347,4.059) -- (10.544,4.093) -- (10.739,4.132) -- (10.934,4.175) -- (11.128,4.223) -- (11.32,4.276) -- (11.511,4.333);
\draw[line width=0.5pt] (8.758,3.952) -- (8.768,3.814) -- (8.775,3.676) -- (8.778,3.537) -- (8.778,3.398) -- (8.774,3.26) -- (8.768,3.121) -- (8.758,2.983) -- (8.745,2.844) -- (8.729,2.707) -- (8.709,2.569) -- (8.686,2.432) -- (8.66,2.296) -- (8.631,2.161) -- (8.599,2.026);
\draw[line width=0.5pt] (4.141,2.067) -- (4.333,2.03) -- (4.526,1.998) -- (4.719,1.971) -- (4.913,1.948) -- (5.108,1.93) -- (5.303,1.916) -- (5.498,1.907) -- (5.694,1.903) -- (5.889,1.903) -- (6.085,1.908) -- (6.28,1.917) -- (6.475,1.931) -- (6.669,1.95) -- (6.863,1.973);
\draw[line width=0.5pt] (8.599,2.026) -- (8.82,2.03) -- (9.041,2.029) -- (9.263,2.023) -- (9.484,2.011) -- (9.705,1.994) -- (9.926,1.972) -- (10.146,1.945) -- (10.365,1.913) -- (10.583,1.876) -- (10.801,1.833) -- (11.017,1.786) -- (11.232,1.733) -- (11.446,1.675) -- (11.658,1.612);
\draw[line width=0.5pt] (6.863,1.973) -- (6.987,1.957) -- (7.11,1.945) -- (7.235,1.936) -- (7.359,1.929) -- (7.484,1.926) -- (7.608,1.925) -- (7.733,1.928) -- (7.858,1.933) -- (7.982,1.941) -- (8.106,1.952) -- (8.23,1.966) -- (8.353,1.983) -- (8.476,2.003) -- (8.599,2.026);
\draw[line width=0.5pt] (1.783,1.835) -- (1.947,1.877) -- (2.113,1.916) -- (2.279,1.95) -- (2.447,1.981) -- (2.615,2.007) -- (2.783,2.03) -- (2.952,2.048) -- (3.122,2.063) -- (3.292,2.074) -- (3.462,2.08) -- (3.632,2.083) -- (3.802,2.082) -- (3.972,2.076) -- (4.141,2.067);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.384,6.997) -- (9.363,7.077) -- (9.304,7.135) -- (9.224,7.157) -- (9.144,7.135) -- (9.086,7.077) -- (9.064,6.997) -- (9.086,6.917) -- (9.144,6.858) -- (9.224,6.837) -- (9.304,6.858) -- (9.363,6.917) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.738,6.972) -- (6.717,7.052) -- (6.658,7.111) -- (6.578,7.132) -- (6.498,7.111) -- (6.44,7.052) -- (6.418,6.972) -- (6.44,6.892) -- (6.498,6.833) -- (6.578,6.812) -- (6.658,6.833) -- (6.717,6.892) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.659,4.598) -- (6.638,4.678) -- (6.579,4.737) -- (6.499,4.758) -- (6.419,4.737) -- (6.361,4.678) -- (6.339,4.598) -- (6.361,4.518) -- (6.419,4.46) -- (6.499,4.438) -- (6.579,4.46) -- (6.638,4.518) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.818,4.554) -- (1.797,4.634) -- (1.738,4.693) -- (1.658,4.714) -- (1.578,4.693) -- (1.52,4.634) -- (1.498,4.554) -- (1.52,4.474) -- (1.578,4.415) -- (1.658,4.394) -- (1.738,4.415) -- (1.797,4.474) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.671,4.333) -- (11.649,4.413) -- (11.591,4.471) -- (11.511,4.493) -- (11.431,4.471) -- (11.372,4.413) -- (11.351,4.333) -- (11.372,4.253) -- (11.431,4.194) -- (11.511,4.173) -- (11.591,4.194) -- (11.649,4.253) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.772,4.033) -- (3.751,4.113) -- (3.692,4.171) -- (3.612,4.193) -- (3.532,4.171) -- (3.474,4.113) -- (3.452,4.033) -- (3.474,3.953) -- (3.532,3.894) -- (3.612,3.873) -- (3.692,3.894) -- (3.751,3.953) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.918,3.952) -- (8.897,4.032) -- (8.838,4.091) -- (8.758,4.112) -- (8.678,4.091) -- (8.62,4.032) -- (8.598,3.952) -- (8.62,3.872) -- (8.678,3.814) -- (8.758,3.792) -- (8.838,3.814) -- (8.897,3.872) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.301,2.067) -- (4.28,2.147) -- (4.221,2.205) -- (4.141,2.227) -- (4.061,2.205) -- (4.003,2.147) -- (3.981,2.067) -- (4.003,1.987) -- (4.061,1.928) -- (4.141,1.907) -- (4.221,1.928) -- (4.28,1.987) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.759,2.026) -- (8.737,2.106) -- (8.679,2.164) -- (8.599,2.186) -- (8.519,2.164) -- (8.46,2.106) -- (8.439,2.026) -- (8.46,1.946) -- (8.519,1.887) -- (8.599,1.866) -- (8.679,1.887) -- (8.737,1.946) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.023,1.973) -- (7.001,2.053) -- (6.943,2.111) -- (6.863,2.133) -- (6.783,2.111) -- (6.724,2.053) -- (6.703,1.973) -- (6.724,1.893) -- (6.783,1.834) -- (6.863,1.813) -- (6.943,1.834) -- (7.001,1.893) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.943,1.835) -- (1.921,1.915) -- (1.863,1.974) -- (1.783,1.995) -- (1.703,1.974) -- (1.644,1.915) -- (1.623,1.835) -- (1.644,1.755) -- (1.703,1.697) -- (1.783,1.675) -- (1.863,1.697) -- (1.921,1.755) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.818,1.612) -- (11.797,1.692) -- (11.738,1.751) -- (11.658,1.772) -- (11.578,1.751) -- (11.52,1.692) -- (11.498,1.612) -- (11.52,1.532) -- (11.578,1.474) -- (11.658,1.452) -- (11.738,1.474) -- (11.797,1.532) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.255,5.04) -- (9.377,5.06) -- (9.487,5.116) -- (9.574,5.203) -- (9.63,5.313) -- (9.649,5.434) -- (9.63,5.556) -- (9.574,5.666) -- (9.487,5.753) -- (9.377,5.809) -- (9.255,5.828) -- (9.255,5.828) -- (9.133,5.809) -- (9.024,5.753) -- (8.937,5.666) -- (8.881,5.556) -- (8.861,5.434) -- (8.881,5.313) -- (8.937,5.203) -- (9.024,5.116) -- (9.133,5.06) -- (9.255,5.04) -- cycle;
\node at (9.255,5.434) {\scalebox{1.69}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.897,7.094) -- (8.018,7.113) -- (8.128,7.169) -- (8.215,7.256) -- (8.271,7.366) -- (8.291,7.487) -- (8.271,7.609) -- (8.215,7.719) -- (8.128,7.806) -- (8.018,7.862) -- (7.897,7.881) -- (7.897,7.881) -- (7.775,7.862) -- (7.665,7.806) -- (7.578,7.719) -- (7.522,7.609) -- (7.503,7.487) -- (7.522,7.366) -- (7.578,7.256) -- (7.665,7.169) -- (7.775,7.113) -- (7.897,7.094) -- cycle;
\node at (7.897,7.487) {\scalebox{1.69}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.842,5.543) -- (6.964,5.562) -- (7.073,5.618) -- (7.16,5.705) -- (7.216,5.815) -- (7.236,5.936) -- (7.216,6.058) -- (7.16,6.168) -- (7.073,6.255) -- (6.964,6.311) -- (6.842,6.33) -- (6.842,6.33) -- (6.72,6.311) -- (6.61,6.255) -- (6.523,6.168) -- (6.467,6.058) -- (6.448,5.936) -- (6.467,5.815) -- (6.523,5.705) -- (6.61,5.618) -- (6.72,5.562) -- (6.842,5.543) -- cycle;
\node at (6.842,5.936) {\scalebox{1.69}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.764,4.353) -- (7.886,4.373) -- (7.995,4.429) -- (8.082,4.516) -- (8.138,4.626) -- (8.158,4.747) -- (8.138,4.869) -- (8.082,4.979) -- (7.995,5.066) -- (7.886,5.122) -- (7.764,5.141) -- (7.764,5.141) -- (7.642,5.122) -- (7.532,5.066) -- (7.445,4.979) -- (7.389,4.869) -- (7.37,4.747) -- (7.389,4.626) -- (7.445,4.516) -- (7.532,4.429) -- (7.642,4.373) -- (7.764,4.353) -- cycle;
\node at (7.764,4.747) {\scalebox{1.69}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.963,2.931) -- (7.085,2.95) -- (7.194,3.006) -- (7.282,3.093) -- (7.338,3.203) -- (7.357,3.325) -- (7.338,3.446) -- (7.282,3.556) -- (7.194,3.643) -- (7.085,3.699) -- (6.963,3.718) -- (6.963,3.718) -- (6.841,3.699) -- (6.731,3.643) -- (6.644,3.556) -- (6.588,3.446) -- (6.569,3.325) -- (6.588,3.203) -- (6.644,3.093) -- (6.731,3.006) -- (6.841,2.95) -- (6.963,2.931) -- cycle;
\node at (6.963,3.325) {\scalebox{1.69}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.647,4.556) -- (2.768,4.576) -- (2.878,4.632) -- (2.965,4.719) -- (3.021,4.829) -- (3.041,4.95) -- (3.021,5.072) -- (2.965,5.182) -- (2.878,5.269) -- (2.768,5.325) -- (2.647,5.344) -- (2.647,5.344) -- (2.525,5.325) -- (2.415,5.269) -- (2.328,5.182) -- (2.272,5.072) -- (2.253,4.95) -- (2.272,4.829) -- (2.328,4.719) -- (2.415,4.632) -- (2.525,4.576) -- (2.647,4.556) -- cycle;
\node at (2.647,4.95) {\scalebox{1.69}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.002,2.814) -- (2.123,2.833) -- (2.233,2.889) -- (2.32,2.976) -- (2.376,3.086) -- (2.396,3.207) -- (2.376,3.329) -- (2.32,3.439) -- (2.233,3.526) -- (2.123,3.582) -- (2.002,3.601) -- (2.002,3.601) -- (1.88,3.582) -- (1.77,3.526) -- (1.683,3.439) -- (1.627,3.329) -- (1.608,3.207) -- (1.627,3.086) -- (1.683,2.976) -- (1.77,2.889) -- (1.88,2.833) -- (2.002,2.814) -- cycle;
\node at (2.002,3.207) {\scalebox{1.69}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (12.09,2.606) -- (12.212,2.625) -- (12.321,2.681) -- (12.409,2.769) -- (12.465,2.878) -- (12.484,3.0) -- (12.465,3.122) -- (12.409,3.232) -- (12.321,3.319) -- (12.212,3.375) -- (12.09,3.394) -- (12.09,3.394) -- (11.968,3.375) -- (11.858,3.319) -- (11.771,3.232) -- (11.715,3.122) -- (11.696,3.0) -- (11.715,2.878) -- (11.771,2.769) -- (11.858,2.681) -- (11.968,2.625) -- (12.09,2.606) -- cycle;
\node at (12.09,3.0) {\scalebox{1.69}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.003,4.189) -- (5.125,4.208) -- (5.235,4.264) -- (5.322,4.352) -- (5.378,4.461) -- (5.397,4.583) -- (5.378,4.705) -- (5.322,4.815) -- (5.235,4.902) -- (5.125,4.958) -- (5.003,4.977) -- (5.003,4.977) -- (4.882,4.958) -- (4.772,4.902) -- (4.685,4.815) -- (4.629,4.705) -- (4.609,4.583) -- (4.629,4.461) -- (4.685,4.352) -- (4.772,4.264) -- (4.882,4.208) -- (5.003,4.189) -- cycle;
\node at (5.003,4.583) {\scalebox{1.69}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.524,2.78) -- (4.645,2.799) -- (4.755,2.855) -- (4.842,2.942) -- (4.898,3.052) -- (4.918,3.174) -- (4.898,3.296) -- (4.842,3.405) -- (4.755,3.493) -- (4.645,3.549) -- (4.524,3.568) -- (4.153,3.568) -- (4.031,3.549) -- (3.921,3.493) -- (3.834,3.405) -- (3.778,3.296) -- (3.759,3.174) -- (3.778,3.052) -- (3.834,2.942) -- (3.921,2.855) -- (4.031,2.799) -- (4.153,2.78) -- cycle;
\node at (4.338,3.174) {\scalebox{1.69}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.801,3.312) -- (10.923,3.331) -- (11.033,3.387) -- (11.12,3.474) -- (11.176,3.584) -- (11.195,3.706) -- (11.176,3.827) -- (11.12,3.937) -- (11.033,4.024) -- (10.923,4.08) -- (10.801,4.1) -- (10.431,4.1) -- (10.309,4.08) -- (10.199,4.024) -- (10.112,3.937) -- (10.056,3.827) -- (10.037,3.706) -- (10.056,3.584) -- (10.112,3.474) -- (10.199,3.387) -- (10.309,3.331) -- (10.431,3.312) -- cycle;
\node at (10.616,3.706) {\scalebox{1.69}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.336,2.556) -- (9.458,2.575) -- (9.568,2.631) -- (9.655,2.718) -- (9.711,2.828) -- (9.73,2.95) -- (9.711,3.072) -- (9.655,3.182) -- (9.568,3.269) -- (9.458,3.325) -- (9.336,3.344) -- (8.965,3.344) -- (8.844,3.325) -- (8.734,3.269) -- (8.647,3.182) -- (8.591,3.072) -- (8.571,2.95) -- (8.591,2.828) -- (8.647,2.718) -- (8.734,2.631) -- (8.844,2.575) -- (8.965,2.556) -- cycle;
\node at (9.151,2.95) {\scalebox{1.69}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.697,1.907) -- (5.819,1.926) -- (5.929,1.982) -- (6.016,2.07) -- (6.072,2.179) -- (6.091,2.301) -- (6.072,2.423) -- (6.016,2.533) -- (5.929,2.62) -- (5.819,2.676) -- (5.697,2.695) -- (5.327,2.695) -- (5.205,2.676) -- (5.095,2.62) -- (5.008,2.533) -- (4.952,2.423) -- (4.933,2.301) -- (4.952,2.179) -- (5.008,2.07) -- (5.095,1.982) -- (5.205,1.926) -- (5.327,1.907) -- cycle;
\node at (5.512,2.301) {\scalebox{1.69}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.84,1.869) -- (10.961,1.889) -- (11.071,1.944) -- (11.158,2.032) -- (11.214,2.141) -- (11.233,2.263) -- (11.214,2.385) -- (11.158,2.495) -- (11.071,2.582) -- (10.961,2.638) -- (10.84,2.657) -- (10.469,2.657) -- (10.347,2.638) -- (10.237,2.582) -- (10.15,2.495) -- (10.094,2.385) -- (10.075,2.263) -- (10.094,2.141) -- (10.15,2.032) -- (10.237,1.944) -- (10.347,1.889) -- (10.469,1.869) -- cycle;
\node at (10.654,2.263) {\scalebox{1.69}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.241,0.6) -- (8.363,0.619) -- (8.472,0.675) -- (8.56,0.762) -- (8.616,0.872) -- (8.635,0.994) -- (8.616,1.116) -- (8.56,1.225) -- (8.472,1.313) -- (8.363,1.369) -- (8.241,1.388) -- (7.87,1.388) -- (7.748,1.369) -- (7.639,1.313) -- (7.551,1.225) -- (7.496,1.116) -- (7.476,0.994) -- (7.496,0.872) -- (7.551,0.762) -- (7.639,0.675) -- (7.748,0.619) -- (7.87,0.6) -- cycle;
\node at (8.056,0.994) {\scalebox{1.69}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.921,2.027) -- (3.043,2.046) -- (3.153,2.102) -- (3.24,2.189) -- (3.296,2.299) -- (3.315,2.421) -- (3.296,2.542) -- (3.24,2.652) -- (3.153,2.739) -- (3.043,2.795) -- (2.921,2.815) -- (2.55,2.815) -- (2.429,2.795) -- (2.319,2.739) -- (2.232,2.652) -- (2.176,2.542) -- (2.156,2.421) -- (2.176,2.299) -- (2.232,2.189) -- (2.319,2.102) -- (2.429,2.046) -- (2.55,2.027) -- cycle;
\node at (2.736,2.421) {\scalebox{1.69}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.483,1.535) -- (2.083,1.535) -- (2.083,2.135) -- (1.483,2.135) -- cycle;
\node at (1.783,1.835) {\scalebox{1.69}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.777,4.585) -- (4.377,4.585) -- (4.377,5.185) -- (3.777,5.185) -- cycle;
\node at (4.077,4.885) {\scalebox{1.69}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.6,1.787) -- (1.2,1.787) -- (1.2,2.387) -- (0.6,2.387) -- cycle;
\node at (0.9,2.087) {\scalebox{1.69}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.621,5.852) -- (10.221,5.852) -- (10.221,6.452) -- (9.621,6.452) -- cycle;
\node at (9.921,6.152) {\scalebox{1.69}{\small d}};
\node at (9.601,7.489) {\scalebox{1.69}{\small A}};
\node at (6.547,7.591) {\scalebox{1.69}{\small B}};
\node at (6.331,5.195) {\scalebox{1.69}{\small C}};
\node at (1.046,4.651) {\scalebox{1.69}{\small D}};
\node at (12.126,4.409) {\scalebox{1.69}{\small E}};
\node at (2.995,4.089) {\scalebox{1.69}{\small F}};
\node at (9.375,4.016) {\scalebox{1.69}{\small G}};
\node at (3.621,1.73) {\scalebox{1.69}{\small H}};
\node at (9.054,1.605) {\scalebox{1.69}{\small I}};
\node at (6.906,1.354) {\scalebox{1.69}{\small J}};
\node at (1.204,1.612) {\scalebox{1.69}{\small K}};
\node at (12.227,1.366) {\scalebox{1.69}{\small L}};
\draw[line width=0.5pt] (12.958,6.047) -- (12.958,7.147);
\draw[line width=0.5pt] (12.727,6.762) -- (12.958,7.147) -- (13.189,6.762);
\draw[line width=0.5pt] (12.656,6.597) -- (13.261,6.597);
\node at (12.958,7.532) {\scalebox{1.69}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour K, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ rue de l'Église & Ⓐ Place du Vieux-Puits & Ⓙ Carrefour Saint-Roch \\ \hline
① rue du Puits & ⑩ passage du Marché & Ⓑ Place du Marché & Ⓚ Placette des Ormes \\ \hline
② chemin des Vignes & ⑪ rue des Lilas & Ⓒ Carrefour du Lavoir & Ⓛ Coin du Lavoir \\ \hline
③ rue Basse & ⑫ quai aux Fleurs & Ⓓ Place de l'Église & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ rue des Jardins & ⑬ rue Pasteur & Ⓔ Place de la Fontaine & \textbf{a} l'école \\ \hline
⑤ rue Neuve & ⑭ rue du Pont & Ⓕ Carrefour de l'Hôpital & \textbf{b} le lavoir \\ \hline
⑥ chemin Creux & ⑮ rue de la Poste & Ⓖ Carrefour des Tilleuls & \textbf{c} l'église \\ \hline
⑦ rue du Four & ⑯ venelle du Chat & Ⓗ Rond-point du Stade & \textbf{d} la pharmacie \\ \hline
⑧ allée des Peupliers & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Rond-point de la Gare &  \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\begin{enumerate}
  \item Colorier les carrefours du plan avec \textbf{deux couleurs}, de sorte que deux carrefours \textbf{reliés par une rue} ne soient jamais de la même couleur. Combien y en a-t-il de chaque couleur ?
  \item Le car roule d'un carrefour au suivant. Que peut-on dire de la \textbf{couleur} des arrêts successifs ? Qu'en déduire sur les couleurs des arrêts d'une tournée qui les dessert tous ?
  \item \textbf{Le car peut-il desservir tous les carrefours, une fois chacun, et revenir à l'école ?} Si oui, décrire un trajet ; sinon, démontrer que c'est impossible.
  \item \textbf{Et s'il n'est pas obligé de revenir à l'école ?} Même consigne.
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage — Correction}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.606, line join=round, line cap=round]
\draw[line width=0.5pt] (9.224,6.593) -- (9.158,6.382) -- (9.097,6.17) -- (9.04,5.956) -- (8.989,5.741) -- (8.943,5.525) -- (8.902,5.308) -- (8.866,5.09) -- (8.835,4.871) -- (8.809,4.651) -- (8.789,4.431) -- (8.773,4.211) -- (8.763,3.99) -- (8.758,3.769) -- (8.758,3.548);
\draw[line width=0.5pt] (6.578,6.568) -- (6.765,6.599) -- (6.953,6.625) -- (7.142,6.647) -- (7.331,6.664) -- (7.521,6.677) -- (7.71,6.685) -- (7.9,6.689) -- (8.09,6.689) -- (8.28,6.684) -- (8.47,6.675) -- (8.659,6.661) -- (8.848,6.643) -- (9.037,6.62) -- (9.224,6.593);
\draw[line width=0.5pt] (6.578,6.568) -- (6.547,6.401) -- (6.519,6.232) -- (6.495,6.064) -- (6.476,5.894) -- (6.46,5.725) -- (6.449,5.555) -- (6.441,5.384) -- (6.437,5.214) -- (6.438,5.043) -- (6.442,4.873) -- (6.45,4.703) -- (6.463,4.533) -- (6.479,4.363) -- (6.499,4.194);
\draw[line width=0.5pt] (6.499,4.194) -- (6.666,4.173) -- (6.833,4.148) -- (6.999,4.119) -- (7.165,4.086) -- (7.329,4.05) -- (7.493,4.009) -- (7.655,3.965) -- (7.817,3.916) -- (7.978,3.864) -- (8.137,3.808) -- (8.294,3.749) -- (8.451,3.686) -- (8.605,3.619) -- (8.758,3.548);
\draw[line width=0.5pt] (6.499,4.194) -- (6.496,4.004) -- (6.498,3.814) -- (6.504,3.624) -- (6.514,3.434) -- (6.529,3.244) -- (6.549,3.055) -- (6.573,2.866) -- (6.601,2.678) -- (6.634,2.491) -- (6.671,2.304) -- (6.712,2.119) -- (6.758,1.934) -- (6.808,1.751) -- (6.863,1.569);
\draw[line width=0.5pt] (1.658,4.15) -- (1.802,4.134) -- (1.946,4.115) -- (2.089,4.093) -- (2.232,4.067) -- (2.375,4.038) -- (2.516,4.006) -- (2.657,3.97) -- (2.797,3.931) -- (2.935,3.888) -- (3.073,3.843) -- (3.21,3.794) -- (3.345,3.742) -- (3.479,3.687) -- (3.612,3.629);
\draw[line width=0.5pt] (1.658,4.15) -- (1.637,3.956) -- (1.621,3.761) -- (1.609,3.566) -- (1.602,3.371) -- (1.599,3.176) -- (1.602,2.981) -- (1.608,2.785) -- (1.619,2.59) -- (1.635,2.396) -- (1.656,2.201) -- (1.68,2.008) -- (1.71,1.815) -- (1.744,1.622) -- (1.783,1.431);
\draw[line width=0.5pt] (11.511,3.929) -- (11.551,3.738) -- (11.586,3.545) -- (11.618,3.352) -- (11.644,3.159) -- (11.666,2.965) -- (11.684,2.77) -- (11.697,2.575) -- (11.705,2.379) -- (11.709,2.184) -- (11.708,1.988) -- (11.702,1.793) -- (11.692,1.598) -- (11.677,1.403) -- (11.658,1.208);
\draw[line width=0.5pt] (3.612,3.629) -- (3.823,3.637) -- (4.034,3.651) -- (4.244,3.669) -- (4.454,3.693) -- (4.663,3.721) -- (4.871,3.754) -- (5.079,3.792) -- (5.286,3.835) -- (5.492,3.883) -- (5.696,3.936) -- (5.899,3.993) -- (6.101,4.056) -- (6.301,4.123) -- (6.499,4.194);
\draw[line width=0.5pt] (3.612,3.629) -- (3.671,3.495) -- (3.727,3.36) -- (3.78,3.224) -- (3.829,3.086) -- (3.875,2.948) -- (3.918,2.808) -- (3.958,2.667) -- (3.994,2.526) -- (4.027,2.384) -- (4.057,2.24) -- (4.083,2.097) -- (4.106,1.952) -- (4.125,1.808) -- (4.141,1.663);
\draw[line width=0.5pt] (8.758,3.548) -- (8.958,3.545) -- (9.157,3.547) -- (9.356,3.553) -- (9.555,3.564) -- (9.754,3.58) -- (9.952,3.6) -- (10.15,3.625) -- (10.347,3.655) -- (10.544,3.689) -- (10.739,3.728) -- (10.934,3.771) -- (11.128,3.819) -- (11.32,3.872) -- (11.511,3.929);
\draw[line width=0.5pt] (8.758,3.548) -- (8.768,3.41) -- (8.775,3.272) -- (8.778,3.133) -- (8.778,2.994) -- (8.774,2.855) -- (8.768,2.717) -- (8.758,2.578) -- (8.745,2.44) -- (8.729,2.302) -- (8.709,2.165) -- (8.686,2.028) -- (8.66,1.892) -- (8.631,1.756) -- (8.599,1.622);
\draw[line width=0.5pt] (4.141,1.663) -- (4.333,1.626) -- (4.526,1.594) -- (4.719,1.567) -- (4.913,1.544) -- (5.108,1.526) -- (5.303,1.512) -- (5.498,1.503) -- (5.694,1.499) -- (5.889,1.499) -- (6.085,1.504) -- (6.28,1.513) -- (6.475,1.527) -- (6.669,1.545) -- (6.863,1.569);
\draw[line width=0.5pt] (8.599,1.622) -- (8.82,1.626) -- (9.041,1.625) -- (9.263,1.618) -- (9.484,1.607) -- (9.705,1.59) -- (9.926,1.568) -- (10.146,1.541) -- (10.365,1.509) -- (10.583,1.472) -- (10.801,1.429) -- (11.017,1.382) -- (11.232,1.329) -- (11.446,1.271) -- (11.658,1.208);
\draw[line width=0.5pt] (6.863,1.569) -- (6.987,1.553) -- (7.11,1.541) -- (7.235,1.532) -- (7.359,1.525) -- (7.484,1.522) -- (7.608,1.521) -- (7.733,1.523) -- (7.858,1.529) -- (7.982,1.537) -- (8.106,1.548) -- (8.23,1.562) -- (8.353,1.579) -- (8.476,1.599) -- (8.599,1.622);
\draw[line width=0.5pt] (1.783,1.431) -- (1.947,1.473) -- (2.113,1.512) -- (2.279,1.546) -- (2.447,1.576) -- (2.615,1.603) -- (2.783,1.626) -- (2.952,1.644) -- (3.122,1.659) -- (3.292,1.669) -- (3.462,1.676) -- (3.632,1.679) -- (3.802,1.677) -- (3.972,1.672) -- (4.141,1.663);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.598,6.593) -- (9.569,6.735) -- (9.488,6.857) -- (9.367,6.937) -- (9.224,6.966) -- (9.082,6.937) -- (8.96,6.857) -- (8.88,6.735) -- (8.851,6.593) -- (8.88,6.45) -- (8.96,6.329) -- (9.082,6.248) -- (9.224,6.219) -- (9.367,6.248) -- (9.488,6.329) -- (9.569,6.45) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.951,6.568) -- (6.923,6.711) -- (6.842,6.832) -- (6.721,6.913) -- (6.578,6.941) -- (6.435,6.913) -- (6.314,6.832) -- (6.233,6.711) -- (6.205,6.568) -- (6.233,6.425) -- (6.314,6.304) -- (6.435,6.223) -- (6.578,6.195) -- (6.721,6.223) -- (6.842,6.304) -- (6.923,6.425) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.872,4.194) -- (6.844,4.337) -- (6.763,4.458) -- (6.642,4.539) -- (6.499,4.568) -- (6.356,4.539) -- (6.235,4.458) -- (6.154,4.337) -- (6.126,4.194) -- (6.154,4.052) -- (6.235,3.93) -- (6.356,3.85) -- (6.499,3.821) -- (6.642,3.85) -- (6.763,3.93) -- (6.844,4.052) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.031,4.15) -- (2.003,4.293) -- (1.922,4.414) -- (1.801,4.495) -- (1.658,4.523) -- (1.515,4.495) -- (1.394,4.414) -- (1.313,4.293) -- (1.285,4.15) -- (1.313,4.007) -- (1.394,3.886) -- (1.515,3.805) -- (1.658,3.777) -- (1.801,3.805) -- (1.922,3.886) -- (2.003,4.007) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.884,3.929) -- (11.855,4.072) -- (11.774,4.193) -- (11.653,4.273) -- (11.511,4.302) -- (11.368,4.273) -- (11.247,4.193) -- (11.166,4.072) -- (11.137,3.929) -- (11.166,3.786) -- (11.247,3.665) -- (11.368,3.584) -- (11.511,3.555) -- (11.653,3.584) -- (11.774,3.665) -- (11.855,3.786) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.985,3.629) -- (3.957,3.771) -- (3.876,3.893) -- (3.755,3.973) -- (3.612,4.002) -- (3.469,3.973) -- (3.348,3.893) -- (3.267,3.771) -- (3.239,3.629) -- (3.267,3.486) -- (3.348,3.365) -- (3.469,3.284) -- (3.612,3.255) -- (3.755,3.284) -- (3.876,3.365) -- (3.957,3.486) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.132,3.548) -- (9.103,3.691) -- (9.022,3.812) -- (8.901,3.893) -- (8.758,3.922) -- (8.616,3.893) -- (8.495,3.812) -- (8.414,3.691) -- (8.385,3.548) -- (8.414,3.406) -- (8.495,3.284) -- (8.616,3.204) -- (8.758,3.175) -- (8.901,3.204) -- (9.022,3.284) -- (9.103,3.406) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.515,1.663) -- (4.486,1.805) -- (4.405,1.927) -- (4.284,2.007) -- (4.141,2.036) -- (3.999,2.007) -- (3.878,1.927) -- (3.797,1.805) -- (3.768,1.663) -- (3.797,1.52) -- (3.878,1.399) -- (3.999,1.318) -- (4.141,1.289) -- (4.284,1.318) -- (4.405,1.399) -- (4.486,1.52) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.972,1.622) -- (8.944,1.764) -- (8.863,1.886) -- (8.742,1.966) -- (8.599,1.995) -- (8.456,1.966) -- (8.335,1.886) -- (8.254,1.764) -- (8.226,1.622) -- (8.254,1.479) -- (8.335,1.358) -- (8.456,1.277) -- (8.599,1.248) -- (8.742,1.277) -- (8.863,1.358) -- (8.944,1.479) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.236,1.569) -- (7.208,1.711) -- (7.127,1.832) -- (7.006,1.913) -- (6.863,1.942) -- (6.72,1.913) -- (6.599,1.832) -- (6.518,1.711) -- (6.49,1.569) -- (6.518,1.426) -- (6.599,1.305) -- (6.72,1.224) -- (6.863,1.195) -- (7.006,1.224) -- (7.127,1.305) -- (7.208,1.426) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.156,1.431) -- (2.127,1.574) -- (2.046,1.695) -- (1.925,1.776) -- (1.783,1.804) -- (1.64,1.776) -- (1.519,1.695) -- (1.438,1.574) -- (1.409,1.431) -- (1.438,1.288) -- (1.519,1.167) -- (1.64,1.086) -- (1.783,1.058) -- (1.925,1.086) -- (2.046,1.167) -- (2.127,1.288) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (12.031,1.208) -- (12.003,1.351) -- (11.922,1.472) -- (11.801,1.553) -- (11.658,1.582) -- (11.515,1.553) -- (11.394,1.472) -- (11.313,1.351) -- (11.285,1.208) -- (11.313,1.066) -- (11.394,0.944) -- (11.515,0.864) -- (11.658,0.835) -- (11.801,0.864) -- (11.922,0.944) -- (12.003,1.066) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.246,4.647) -- (9.365,4.666) -- (9.472,4.72) -- (9.557,4.806) -- (9.612,4.913) -- (9.63,5.032) -- (9.612,5.15) -- (9.557,5.258) -- (9.472,5.343) -- (9.365,5.397) -- (9.246,5.416) -- (9.246,5.416) -- (9.127,5.397) -- (9.02,5.343) -- (8.935,5.258) -- (8.88,5.15) -- (8.861,5.032) -- (8.88,4.913) -- (8.935,4.806) -- (9.02,4.72) -- (9.127,4.666) -- (9.246,4.647) -- cycle;
\node at (9.246,5.032) {\scalebox{1.65}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.897,6.689) -- (8.016,6.708) -- (8.123,6.763) -- (8.208,6.848) -- (8.262,6.955) -- (8.281,7.074) -- (8.262,7.193) -- (8.208,7.3) -- (8.123,7.385) -- (8.016,7.44) -- (7.897,7.458) -- (7.897,7.458) -- (7.778,7.44) -- (7.671,7.385) -- (7.586,7.3) -- (7.531,7.193) -- (7.512,7.074) -- (7.531,6.955) -- (7.586,6.848) -- (7.671,6.763) -- (7.778,6.708) -- (7.897,6.689) -- cycle;
\node at (7.897,7.074) {\scalebox{1.65}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.832,5.148) -- (6.951,5.167) -- (7.058,5.222) -- (7.144,5.307) -- (7.198,5.414) -- (7.217,5.533) -- (7.198,5.652) -- (7.144,5.759) -- (7.058,5.844) -- (6.951,5.899) -- (6.832,5.917) -- (6.832,5.917) -- (6.714,5.899) -- (6.606,5.844) -- (6.521,5.759) -- (6.467,5.652) -- (6.448,5.533) -- (6.467,5.414) -- (6.521,5.307) -- (6.606,5.222) -- (6.714,5.167) -- (6.832,5.148) -- cycle;
\node at (6.832,5.533) {\scalebox{1.65}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.761,3.95) -- (7.88,3.969) -- (7.987,4.023) -- (8.072,4.108) -- (8.127,4.215) -- (8.146,4.334) -- (8.127,4.453) -- (8.072,4.56) -- (7.987,4.645) -- (7.88,4.7) -- (7.761,4.719) -- (7.761,4.719) -- (7.642,4.7) -- (7.535,4.645) -- (7.45,4.56) -- (7.396,4.453) -- (7.377,4.334) -- (7.396,4.215) -- (7.45,4.108) -- (7.535,4.023) -- (7.642,3.969) -- (7.761,3.95) -- cycle;
\node at (7.761,4.334) {\scalebox{1.65}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.954,2.535) -- (7.072,2.553) -- (7.18,2.608) -- (7.265,2.693) -- (7.319,2.8) -- (7.338,2.919) -- (7.319,3.038) -- (7.265,3.145) -- (7.18,3.23) -- (7.072,3.285) -- (6.954,3.304) -- (6.954,3.304) -- (6.835,3.285) -- (6.728,3.23) -- (6.643,3.145) -- (6.588,3.038) -- (6.569,2.919) -- (6.588,2.8) -- (6.643,2.693) -- (6.728,2.608) -- (6.835,2.553) -- (6.954,2.535) -- cycle;
\node at (6.954,2.919) {\scalebox{1.65}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.644,4.149) -- (2.762,4.168) -- (2.87,4.222) -- (2.955,4.307) -- (3.009,4.414) -- (3.028,4.533) -- (3.009,4.652) -- (2.955,4.759) -- (2.87,4.844) -- (2.762,4.899) -- (2.644,4.918) -- (2.644,4.918) -- (2.525,4.899) -- (2.418,4.844) -- (2.333,4.759) -- (2.278,4.652) -- (2.259,4.533) -- (2.278,4.414) -- (2.333,4.307) -- (2.418,4.222) -- (2.525,4.168) -- (2.644,4.149) -- cycle;
\node at (2.644,4.533) {\scalebox{1.65}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.992,2.418) -- (2.111,2.437) -- (2.218,2.492) -- (2.303,2.577) -- (2.358,2.684) -- (2.377,2.803) -- (2.358,2.922) -- (2.303,3.029) -- (2.218,3.114) -- (2.111,3.169) -- (1.992,3.187) -- (1.992,3.187) -- (1.873,3.169) -- (1.766,3.114) -- (1.681,3.029) -- (1.627,2.922) -- (1.608,2.803) -- (1.627,2.684) -- (1.681,2.577) -- (1.766,2.492) -- (1.873,2.437) -- (1.992,2.418) -- cycle;
\node at (1.992,2.803) {\scalebox{1.65}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (12.081,2.211) -- (12.199,2.23) -- (12.307,2.284) -- (12.392,2.369) -- (12.446,2.477) -- (12.465,2.595) -- (12.446,2.714) -- (12.392,2.821) -- (12.307,2.907) -- (12.199,2.961) -- (12.081,2.98) -- (12.081,2.98) -- (11.962,2.961) -- (11.855,2.907) -- (11.769,2.821) -- (11.715,2.714) -- (11.696,2.595) -- (11.715,2.477) -- (11.769,2.369) -- (11.855,2.284) -- (11.962,2.23) -- (12.081,2.211) -- cycle;
\node at (12.081,2.595) {\scalebox{1.65}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.005,3.785) -- (5.124,3.804) -- (5.231,3.859) -- (5.316,3.944) -- (5.371,4.051) -- (5.39,4.17) -- (5.371,4.289) -- (5.316,4.396) -- (5.231,4.481) -- (5.124,4.535) -- (5.005,4.554) -- (5.005,4.554) -- (4.886,4.535) -- (4.779,4.481) -- (4.694,4.396) -- (4.639,4.289) -- (4.621,4.17) -- (4.639,4.051) -- (4.694,3.944) -- (4.779,3.859) -- (4.886,3.804) -- (5.005,3.785) -- cycle;
\node at (5.005,4.17) {\scalebox{1.65}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.51,2.383) -- (4.629,2.402) -- (4.736,2.456) -- (4.821,2.541) -- (4.876,2.649) -- (4.895,2.767) -- (4.876,2.886) -- (4.821,2.993) -- (4.736,3.078) -- (4.629,3.133) -- (4.51,3.152) -- (4.148,3.152) -- (4.029,3.133) -- (3.922,3.078) -- (3.837,2.993) -- (3.783,2.886) -- (3.764,2.767) -- (3.783,2.649) -- (3.837,2.541) -- (3.922,2.456) -- (4.029,2.402) -- (4.148,2.383) -- cycle;
\node at (4.329,2.767) {\scalebox{1.65}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.654,3.682) -- (10.773,3.701) -- (10.88,3.756) -- (10.965,3.841) -- (11.02,3.948) -- (11.039,4.067) -- (11.02,4.186) -- (10.965,4.293) -- (10.88,4.378) -- (10.773,4.433) -- (10.654,4.451) -- (10.292,4.451) -- (10.174,4.433) -- (10.066,4.378) -- (9.981,4.293) -- (9.927,4.186) -- (9.908,4.067) -- (9.927,3.948) -- (9.981,3.841) -- (10.066,3.756) -- (10.174,3.701) -- (10.292,3.682) -- cycle;
\node at (10.473,4.067) {\scalebox{1.65}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.322,2.162) -- (9.441,2.181) -- (9.548,2.236) -- (9.633,2.321) -- (9.688,2.428) -- (9.707,2.547) -- (9.688,2.665) -- (9.633,2.773) -- (9.548,2.858) -- (9.441,2.912) -- (9.322,2.931) -- (8.96,2.931) -- (8.841,2.912) -- (8.734,2.858) -- (8.649,2.773) -- (8.595,2.665) -- (8.576,2.547) -- (8.595,2.428) -- (8.649,2.321) -- (8.734,2.236) -- (8.841,2.181) -- (8.96,2.162) -- cycle;
\node at (9.141,2.547) {\scalebox{1.65}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.693,1.503) -- (5.811,1.522) -- (5.919,1.577) -- (6.004,1.662) -- (6.058,1.769) -- (6.077,1.888) -- (6.058,2.006) -- (6.004,2.114) -- (5.919,2.199) -- (5.811,2.253) -- (5.693,2.272) -- (5.331,2.272) -- (5.212,2.253) -- (5.105,2.199) -- (5.02,2.114) -- (4.965,2.006) -- (4.946,1.888) -- (4.965,1.769) -- (5.02,1.662) -- (5.105,1.577) -- (5.212,1.522) -- (5.331,1.503) -- cycle;
\node at (5.512,1.888) {\scalebox{1.65}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.275,0.776) -- (10.394,0.794) -- (10.501,0.849) -- (10.586,0.934) -- (10.641,1.041) -- (10.66,1.16) -- (10.641,1.279) -- (10.586,1.386) -- (10.501,1.471) -- (10.394,1.526) -- (10.275,1.545) -- (9.913,1.545) -- (9.794,1.526) -- (9.687,1.471) -- (9.602,1.386) -- (9.547,1.279) -- (9.529,1.16) -- (9.547,1.041) -- (9.602,0.934) -- (9.687,0.849) -- (9.794,0.794) -- (9.913,0.776) -- cycle;
\node at (10.094,1.16) {\scalebox{1.65}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.091,2.077) -- (8.21,2.096) -- (8.317,2.15) -- (8.402,2.236) -- (8.457,2.343) -- (8.476,2.462) -- (8.457,2.58) -- (8.402,2.688) -- (8.317,2.773) -- (8.21,2.827) -- (8.091,2.846) -- (7.729,2.846) -- (7.61,2.827) -- (7.503,2.773) -- (7.418,2.688) -- (7.363,2.58) -- (7.345,2.462) -- (7.363,2.343) -- (7.418,2.236) -- (7.503,2.15) -- (7.61,2.096) -- (7.729,2.077) -- cycle;
\node at (7.91,2.462) {\scalebox{1.65}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.918,1.623) -- (3.037,1.642) -- (3.144,1.696) -- (3.229,1.781) -- (3.283,1.888) -- (3.302,2.007) -- (3.283,2.126) -- (3.229,2.233) -- (3.144,2.318) -- (3.037,2.373) -- (2.918,2.392) -- (2.556,2.392) -- (2.437,2.373) -- (2.33,2.318) -- (2.245,2.233) -- (2.19,2.126) -- (2.171,2.007) -- (2.19,1.888) -- (2.245,1.781) -- (2.33,1.696) -- (2.437,1.642) -- (2.556,1.623) -- cycle;
\node at (2.737,2.007) {\scalebox{1.65}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.483,1.131) -- (2.083,1.131) -- (2.083,1.731) -- (1.483,1.731) -- cycle;
\node at (1.783,1.431) {\scalebox{1.65}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.777,4.181) -- (4.377,4.181) -- (4.377,4.781) -- (3.777,4.781) -- cycle;
\node at (4.077,4.481) {\scalebox{1.65}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.6,1.383) -- (1.2,1.383) -- (1.2,1.983) -- (0.6,1.983) -- cycle;
\node at (0.9,1.683) {\scalebox{1.65}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.621,5.448) -- (10.221,5.448) -- (10.221,6.048) -- (9.621,6.048) -- cycle;
\node at (9.921,5.748) {\scalebox{1.65}{\small d}};
\node at (9.601,7.085) {\scalebox{1.65}{\small A}};
\node at (6.547,7.187) {\scalebox{1.65}{\small B}};
\node at (6.331,4.791) {\scalebox{1.65}{\small C}};
\node at (1.046,4.247) {\scalebox{1.65}{\small D}};
\node at (12.126,4.004) {\scalebox{1.65}{\small E}};
\node at (2.995,3.685) {\scalebox{1.65}{\small F}};
\node at (9.375,3.611) {\scalebox{1.65}{\small G}};
\node at (3.621,1.326) {\scalebox{1.65}{\small H}};
\node at (9.054,1.2) {\scalebox{1.65}{\small I}};
\node at (6.906,0.95) {\scalebox{1.65}{\small J}};
\node at (1.204,1.208) {\scalebox{1.65}{\small K}};
\node at (12.227,0.962) {\scalebox{1.65}{\small L}};
\draw[line width=0.5pt] (12.958,5.643) -- (12.958,6.743);
\draw[line width=0.5pt] (12.727,6.358) -- (12.958,6.743) -- (13.189,6.358);
\draw[line width=0.5pt] (12.656,6.193) -- (13.261,6.193);
\node at (12.958,7.128) {\scalebox{1.65}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, C, D, E, H, I & 6 \\ \hline
clairs & B, F, G, J, K, L & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{6} clairs. Il y en a \textbf{autant} de chaque côté.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 12 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui.} Un circuit existe :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{rue du Four ⑦} vers le nord jusqu'à Place de l'Église \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin Creux ⑥}, vers l'est, jusqu'à Carrefour de l'Hôpital \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de l'Église ⑨}, vers l'est, jusqu'à Carrefour du Lavoir \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ③}, vers le nord, jusqu'à Place du Marché \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin des Vignes ②}, vers l'est, jusqu'à Place du Vieux-Puits \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Puits ①}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ⑪}, vers l'est, jusqu'à Place de la Fontaine \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Peupliers ⑧}, vers le sud, jusqu'à Coin du Lavoir \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Pont ⑭}, vers l'ouest, jusqu'à Rond-point de la Gare \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de la Poste ⑮}, vers l'ouest, jusqu'à Carrefour Saint-Roch \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Pasteur ⑬}, vers l'ouest, jusqu'à Rond-point du Stade \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{venelle du Chat ⑯}, vers l'ouest, jusqu'à Placette des Ormes \textbf{(K)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Le car est passé une fois par chaque carrefour et se retrouve à l'école.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, et c'est déjà fait : le trajet ci-dessus dessert tous les carrefours. Il suffit au car de s'arrêter avant le dernier trajet de retour.}\par\addvspace{0.4em}
\end{document}
