\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 345}

\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=345}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0345}
\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=0.6037, line join=round, line cap=round]
\draw[line width=0.5pt] (11.38,7.326) -- (11.317,7.125) -- (11.259,6.922) -- (11.206,6.718) -- (11.158,6.513) -- (11.115,6.307) -- (11.077,6.1) -- (11.043,5.891) -- (11.014,5.682) -- (10.991,5.473) -- (10.972,5.263) -- (10.958,5.053) -- (10.949,4.842) -- (10.945,4.631) -- (10.946,4.421);
\draw[line width=0.5pt] (6.089,7.18) -- (6.27,7.193) -- (6.451,7.202) -- (6.633,7.206) -- (6.815,7.206) -- (6.997,7.202) -- (7.179,7.193) -- (7.36,7.181) -- (7.542,7.164) -- (7.722,7.142) -- (7.903,7.117) -- (8.082,7.087) -- (8.261,7.053) -- (8.439,7.015) -- (8.615,6.973);
\draw[line width=0.5pt] (6.089,7.18) -- (6.043,6.989) -- (6.002,6.797) -- (5.966,6.604) -- (5.934,6.41) -- (5.906,6.215) -- (5.884,6.02) -- (5.866,5.824) -- (5.852,5.628) -- (5.843,5.431) -- (5.839,5.235) -- (5.839,5.038) -- (5.844,4.842) -- (5.854,4.646) -- (5.868,4.45);
\draw[line width=0.5pt] (8.615,6.973) -- (8.807,7.028) -- (9.001,7.079) -- (9.195,7.125) -- (9.391,7.167) -- (9.587,7.204) -- (9.785,7.236) -- (9.983,7.264) -- (10.182,7.287) -- (10.381,7.305) -- (10.58,7.319) -- (10.78,7.328) -- (10.98,7.332) -- (11.18,7.331) -- (11.38,7.326);
\draw[line width=0.5pt] (3.567,6.878) -- (3.749,6.871) -- (3.931,6.87) -- (4.113,6.872) -- (4.295,6.879) -- (4.477,6.89) -- (4.659,6.905) -- (4.84,6.925) -- (5.021,6.949) -- (5.201,6.977) -- (5.38,7.009) -- (5.559,7.045) -- (5.737,7.086) -- (5.913,7.131) -- (6.089,7.18);
\draw[line width=0.5pt] (3.567,6.878) -- (3.579,6.702) -- (3.586,6.526) -- (3.59,6.35) -- (3.589,6.173) -- (3.585,5.997) -- (3.576,5.821) -- (3.563,5.645) -- (3.546,5.47) -- (3.525,5.295) -- (3.499,5.12) -- (3.47,4.947) -- (3.437,4.774) -- (3.399,4.602) -- (3.357,4.43);
\draw[line width=0.5pt] (1.384,5.129) -- (1.516,5.058) -- (1.65,4.99) -- (1.785,4.925) -- (1.922,4.864) -- (2.061,4.805) -- (2.201,4.75) -- (2.342,4.698) -- (2.484,4.65) -- (2.627,4.605) -- (2.772,4.563) -- (2.917,4.525) -- (3.063,4.49) -- (3.21,4.458) -- (3.357,4.43);
\draw[line width=0.5pt] (1.384,5.129) -- (1.418,4.901) -- (1.448,4.672) -- (1.472,4.443) -- (1.491,4.213) -- (1.504,3.983) -- (1.512,3.752) -- (1.514,3.521) -- (1.511,3.29) -- (1.503,3.06) -- (1.489,2.829) -- (1.47,2.599) -- (1.445,2.37) -- (1.415,2.141) -- (1.38,1.913);
\draw[line width=0.5pt] (8.658,4.499) -- (8.82,4.468) -- (8.982,4.442) -- (9.144,4.419) -- (9.307,4.4) -- (9.471,4.384) -- (9.635,4.373) -- (9.799,4.365) -- (9.963,4.362) -- (10.128,4.362) -- (10.292,4.366) -- (10.456,4.374) -- (10.62,4.386) -- (10.783,4.401) -- (10.946,4.421);
\draw[line width=0.5pt] (5.868,4.45) -- (6.065,4.484) -- (6.263,4.513) -- (6.462,4.538) -- (6.661,4.558) -- (6.861,4.573) -- (7.061,4.584) -- (7.261,4.589) -- (7.461,4.591) -- (7.662,4.587) -- (7.862,4.579) -- (8.062,4.566) -- (8.261,4.548) -- (8.46,4.526) -- (8.658,4.499);
\draw[line width=0.5pt] (5.868,4.45) -- (5.882,4.3) -- (5.9,4.151) -- (5.921,4.002) -- (5.946,3.854) -- (5.975,3.706) -- (6.006,3.559) -- (6.042,3.413) -- (6.08,3.267) -- (6.123,3.123) -- (6.168,2.979) -- (6.217,2.837) -- (6.269,2.696) -- (6.325,2.556) -- (6.383,2.418);
\draw[line width=0.5pt] (3.357,4.43) -- (3.535,4.459) -- (3.714,4.484) -- (3.892,4.504) -- (4.072,4.52) -- (4.252,4.532) -- (4.432,4.54) -- (4.612,4.544) -- (4.792,4.543) -- (4.972,4.538) -- (5.152,4.529) -- (5.332,4.515) -- (5.511,4.498) -- (5.69,4.476) -- (5.868,4.45);
\draw[line width=0.5pt] (3.357,4.43) -- (3.324,4.275) -- (3.294,4.12) -- (3.268,3.963) -- (3.246,3.806) -- (3.227,3.649) -- (3.212,3.491) -- (3.201,3.332) -- (3.193,3.174) -- (3.189,3.015) -- (3.189,2.856) -- (3.192,2.698) -- (3.2,2.539) -- (3.211,2.381) -- (3.226,2.223);
\draw[line width=0.5pt] (3.226,2.223) -- (3.447,2.272) -- (3.67,2.315) -- (3.894,2.352) -- (4.119,2.385) -- (4.344,2.412) -- (4.57,2.434) -- (4.796,2.451) -- (5.023,2.462) -- (5.25,2.468) -- (5.477,2.469) -- (5.704,2.464) -- (5.931,2.454) -- (6.157,2.439) -- (6.383,2.418);
\draw[line width=0.5pt] (1.38,1.913) -- (1.514,1.915) -- (1.648,1.92) -- (1.782,1.928) -- (1.916,1.94) -- (2.049,1.954) -- (2.183,1.972) -- (2.315,1.992) -- (2.448,2.016) -- (2.579,2.043) -- (2.71,2.073) -- (2.84,2.106) -- (2.97,2.142) -- (3.098,2.181) -- (3.226,2.223);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.54,7.326) -- (11.518,7.406) -- (11.46,7.465) -- (11.38,7.486) -- (11.3,7.465) -- (11.241,7.406) -- (11.22,7.326) -- (11.241,7.246) -- (11.3,7.188) -- (11.38,7.166) -- (11.46,7.188) -- (11.518,7.246) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.249,7.18) -- (6.227,7.26) -- (6.169,7.319) -- (6.089,7.34) -- (6.009,7.319) -- (5.95,7.26) -- (5.929,7.18) -- (5.95,7.1) -- (6.009,7.041) -- (6.089,7.02) -- (6.169,7.041) -- (6.227,7.1) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.775,6.973) -- (8.754,7.053) -- (8.695,7.111) -- (8.615,7.133) -- (8.535,7.111) -- (8.477,7.053) -- (8.455,6.973) -- (8.477,6.893) -- (8.535,6.834) -- (8.615,6.813) -- (8.695,6.834) -- (8.754,6.893) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.727,6.878) -- (3.705,6.958) -- (3.647,7.016) -- (3.567,7.038) -- (3.487,7.016) -- (3.428,6.958) -- (3.407,6.878) -- (3.428,6.798) -- (3.487,6.739) -- (3.567,6.718) -- (3.647,6.739) -- (3.705,6.798) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.544,5.129) -- (1.522,5.209) -- (1.464,5.268) -- (1.384,5.289) -- (1.304,5.268) -- (1.245,5.209) -- (1.224,5.129) -- (1.245,5.049) -- (1.304,4.99) -- (1.384,4.969) -- (1.464,4.99) -- (1.522,5.049) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.818,4.499) -- (8.797,4.579) -- (8.738,4.637) -- (8.658,4.659) -- (8.578,4.637) -- (8.52,4.579) -- (8.498,4.499) -- (8.52,4.419) -- (8.578,4.36) -- (8.658,4.339) -- (8.738,4.36) -- (8.797,4.419) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.028,4.45) -- (6.006,4.53) -- (5.948,4.588) -- (5.868,4.61) -- (5.788,4.588) -- (5.729,4.53) -- (5.708,4.45) -- (5.729,4.37) -- (5.788,4.311) -- (5.868,4.29) -- (5.948,4.311) -- (6.006,4.37) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.517,4.43) -- (3.496,4.51) -- (3.437,4.569) -- (3.357,4.59) -- (3.277,4.569) -- (3.219,4.51) -- (3.197,4.43) -- (3.219,4.35) -- (3.277,4.292) -- (3.357,4.27) -- (3.437,4.292) -- (3.496,4.35) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.106,4.421) -- (11.085,4.501) -- (11.026,4.559) -- (10.946,4.581) -- (10.866,4.559) -- (10.808,4.501) -- (10.786,4.421) -- (10.808,4.341) -- (10.866,4.282) -- (10.946,4.261) -- (11.026,4.282) -- (11.085,4.341) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.543,2.418) -- (6.522,2.498) -- (6.463,2.556) -- (6.383,2.578) -- (6.303,2.556) -- (6.245,2.498) -- (6.223,2.418) -- (6.245,2.338) -- (6.303,2.279) -- (6.383,2.258) -- (6.463,2.279) -- (6.522,2.338) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.386,2.223) -- (3.364,2.303) -- (3.306,2.362) -- (3.226,2.383) -- (3.146,2.362) -- (3.087,2.303) -- (3.066,2.223) -- (3.087,2.143) -- (3.146,2.085) -- (3.226,2.063) -- (3.306,2.085) -- (3.364,2.143) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.54,1.913) -- (1.518,1.993) -- (1.46,2.052) -- (1.38,2.073) -- (1.3,2.052) -- (1.241,1.993) -- (1.22,1.913) -- (1.241,1.833) -- (1.3,1.775) -- (1.38,1.753) -- (1.46,1.775) -- (1.518,1.833) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.433,5.439) -- (11.554,5.459) -- (11.664,5.515) -- (11.751,5.602) -- (11.807,5.712) -- (11.827,5.833) -- (11.807,5.955) -- (11.751,6.065) -- (11.664,6.152) -- (11.554,6.208) -- (11.433,6.227) -- (11.433,6.227) -- (11.311,6.208) -- (11.201,6.152) -- (11.114,6.065) -- (11.058,5.955) -- (11.039,5.833) -- (11.058,5.712) -- (11.114,5.602) -- (11.201,5.515) -- (11.311,5.459) -- (11.433,5.439) -- cycle;
\node at (11.433,5.833) {\scalebox{1.656}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.393,7.179) -- (7.514,7.199) -- (7.624,7.255) -- (7.711,7.342) -- (7.767,7.452) -- (7.787,7.573) -- (7.767,7.695) -- (7.711,7.805) -- (7.624,7.892) -- (7.514,7.948) -- (7.393,7.967) -- (7.393,7.967) -- (7.271,7.948) -- (7.161,7.892) -- (7.074,7.805) -- (7.018,7.695) -- (6.999,7.573) -- (7.018,7.452) -- (7.074,7.342) -- (7.161,7.255) -- (7.271,7.199) -- (7.393,7.179) -- cycle;
\node at (7.393,7.573) {\scalebox{1.656}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.258,5.398) -- (6.38,5.418) -- (6.49,5.474) -- (6.577,5.561) -- (6.633,5.671) -- (6.652,5.792) -- (6.633,5.914) -- (6.577,6.024) -- (6.49,6.111) -- (6.38,6.167) -- (6.258,6.186) -- (6.258,6.186) -- (6.136,6.167) -- (6.027,6.111) -- (5.939,6.024) -- (5.884,5.914) -- (5.864,5.792) -- (5.884,5.671) -- (5.939,5.561) -- (6.027,5.474) -- (6.136,5.418) -- (6.258,5.398) -- cycle;
\node at (6.258,5.792) {\scalebox{1.656}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.933,7.26) -- (10.055,7.28) -- (10.164,7.336) -- (10.252,7.423) -- (10.308,7.533) -- (10.327,7.654) -- (10.308,7.776) -- (10.252,7.886) -- (10.164,7.973) -- (10.055,8.029) -- (9.933,8.048) -- (9.933,8.048) -- (9.811,8.029) -- (9.701,7.973) -- (9.614,7.886) -- (9.558,7.776) -- (9.539,7.654) -- (9.558,7.533) -- (9.614,7.423) -- (9.701,7.336) -- (9.811,7.28) -- (9.933,7.26) -- cycle;
\node at (9.933,7.654) {\scalebox{1.656}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.793,6.922) -- (4.915,6.941) -- (5.025,6.997) -- (5.112,7.084) -- (5.168,7.194) -- (5.187,7.316) -- (5.168,7.438) -- (5.112,7.547) -- (5.025,7.635) -- (4.915,7.69) -- (4.793,7.71) -- (4.793,7.71) -- (4.671,7.69) -- (4.562,7.635) -- (4.474,7.547) -- (4.418,7.438) -- (4.399,7.316) -- (4.418,7.194) -- (4.474,7.084) -- (4.562,6.997) -- (4.671,6.941) -- (4.793,6.922) -- cycle;
\node at (4.793,7.316) {\scalebox{1.656}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.17,5.285) -- (3.292,5.304) -- (3.402,5.36) -- (3.489,5.447) -- (3.545,5.557) -- (3.564,5.679) -- (3.545,5.801) -- (3.489,5.91) -- (3.402,5.998) -- (3.292,6.054) -- (3.17,6.073) -- (3.17,6.073) -- (3.049,6.054) -- (2.939,5.998) -- (2.852,5.91) -- (2.796,5.801) -- (2.777,5.679) -- (2.796,5.557) -- (2.852,5.447) -- (2.939,5.36) -- (3.049,5.304) -- (3.17,5.285) -- cycle;
\node at (3.17,5.679) {\scalebox{1.656}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.218,5.149) -- (2.34,5.169) -- (2.45,5.225) -- (2.537,5.312) -- (2.593,5.422) -- (2.612,5.543) -- (2.593,5.665) -- (2.537,5.775) -- (2.45,5.862) -- (2.34,5.918) -- (2.218,5.937) -- (2.218,5.937) -- (2.097,5.918) -- (1.987,5.862) -- (1.9,5.775) -- (1.844,5.665) -- (1.824,5.543) -- (1.844,5.422) -- (1.9,5.312) -- (1.987,5.225) -- (2.097,5.169) -- (2.218,5.149) -- cycle;
\node at (2.218,5.543) {\scalebox{1.656}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.908,3.127) -- (2.03,3.146) -- (2.14,3.202) -- (2.227,3.289) -- (2.283,3.399) -- (2.302,3.521) -- (2.283,3.642) -- (2.227,3.752) -- (2.14,3.839) -- (2.03,3.895) -- (1.908,3.914) -- (1.908,3.914) -- (1.786,3.895) -- (1.677,3.839) -- (1.589,3.752) -- (1.534,3.642) -- (1.514,3.521) -- (1.534,3.399) -- (1.589,3.289) -- (1.677,3.202) -- (1.786,3.146) -- (1.908,3.127) -- cycle;
\node at (1.908,3.521) {\scalebox{1.656}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.123,4.362) -- (10.244,4.381) -- (10.354,4.437) -- (10.441,4.524) -- (10.497,4.634) -- (10.516,4.756) -- (10.497,4.878) -- (10.441,4.987) -- (10.354,5.074) -- (10.244,5.13) -- (10.123,5.15) -- (10.123,5.15) -- (10.001,5.13) -- (9.891,5.074) -- (9.804,4.987) -- (9.748,4.878) -- (9.729,4.756) -- (9.748,4.634) -- (9.804,4.524) -- (9.891,4.437) -- (10.001,4.381) -- (10.123,4.362) -- cycle;
\node at (10.123,4.756) {\scalebox{1.656}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.439,4.589) -- (7.561,4.609) -- (7.671,4.665) -- (7.758,4.752) -- (7.814,4.862) -- (7.833,4.983) -- (7.814,5.105) -- (7.758,5.215) -- (7.671,5.302) -- (7.561,5.358) -- (7.439,5.377) -- (7.069,5.377) -- (6.947,5.358) -- (6.837,5.302) -- (6.75,5.215) -- (6.694,5.105) -- (6.675,4.983) -- (6.694,4.862) -- (6.75,4.752) -- (6.837,4.665) -- (6.947,4.609) -- (7.069,4.589) -- cycle;
\node at (7.254,4.983) {\scalebox{1.656}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.148,3.252) -- (7.27,3.271) -- (7.379,3.327) -- (7.467,3.415) -- (7.523,3.524) -- (7.542,3.646) -- (7.523,3.768) -- (7.467,3.878) -- (7.379,3.965) -- (7.27,4.021) -- (7.148,4.04) -- (6.777,4.04) -- (6.655,4.021) -- (6.546,3.965) -- (6.459,3.878) -- (6.403,3.768) -- (6.383,3.646) -- (6.403,3.524) -- (6.459,3.415) -- (6.546,3.327) -- (6.655,3.271) -- (6.777,3.252) -- cycle;
\node at (6.963,3.646) {\scalebox{1.656}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.794,4.544) -- (4.916,4.563) -- (5.026,4.619) -- (5.113,4.706) -- (5.169,4.816) -- (5.188,4.938) -- (5.169,5.059) -- (5.113,5.169) -- (5.026,5.256) -- (4.916,5.312) -- (4.794,5.331) -- (4.423,5.331) -- (4.302,5.312) -- (4.192,5.256) -- (4.105,5.169) -- (4.049,5.059) -- (4.03,4.938) -- (4.049,4.816) -- (4.105,4.706) -- (4.192,4.619) -- (4.302,4.563) -- (4.423,4.544) -- cycle;
\node at (4.609,4.938) {\scalebox{1.656}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.779,2.915) -- (3.901,2.934) -- (4.011,2.99) -- (4.098,3.077) -- (4.154,3.187) -- (4.173,3.309) -- (4.154,3.431) -- (4.098,3.54) -- (4.011,3.627) -- (3.901,3.683) -- (3.779,3.703) -- (3.408,3.703) -- (3.287,3.683) -- (3.177,3.627) -- (3.09,3.54) -- (3.034,3.431) -- (3.014,3.309) -- (3.034,3.187) -- (3.09,3.077) -- (3.177,2.99) -- (3.287,2.934) -- (3.408,2.915) -- cycle;
\node at (3.594,3.309) {\scalebox{1.656}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.958,2.45) -- (5.079,2.469) -- (5.189,2.525) -- (5.276,2.612) -- (5.332,2.722) -- (5.352,2.844) -- (5.332,2.966) -- (5.276,3.075) -- (5.189,3.163) -- (5.079,3.219) -- (4.958,3.238) -- (4.587,3.238) -- (4.465,3.219) -- (4.355,3.163) -- (4.268,3.075) -- (4.212,2.966) -- (4.193,2.844) -- (4.212,2.722) -- (4.268,2.612) -- (4.355,2.525) -- (4.465,2.469) -- (4.587,2.45) -- cycle;
\node at (4.772,2.844) {\scalebox{1.656}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.192,0.6) -- (2.314,0.619) -- (2.424,0.675) -- (2.511,0.762) -- (2.567,0.872) -- (2.586,0.994) -- (2.567,1.116) -- (2.511,1.225) -- (2.424,1.313) -- (2.314,1.369) -- (2.192,1.388) -- (1.821,1.388) -- (1.7,1.369) -- (1.59,1.313) -- (1.503,1.225) -- (1.447,1.116) -- (1.427,0.994) -- (1.447,0.872) -- (1.503,0.762) -- (1.59,0.675) -- (1.7,0.619) -- (1.821,0.6) -- cycle;
\node at (2.007,0.994) {\scalebox{1.656}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (3.267,6.578) -- (3.867,6.578) -- (3.867,7.178) -- (3.267,7.178) -- cycle;
\node at (3.567,6.878) {\scalebox{1.656}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (0.737,1.64) -- (1.337,1.64) -- (1.337,2.24) -- (0.737,2.24) -- cycle;
\node at (1.037,1.94) {\scalebox{1.656}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.677,3.481) -- (1.277,3.481) -- (1.277,4.081) -- (0.677,4.081) -- cycle;
\node at (0.977,3.781) {\scalebox{1.656}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (3.834,5.676) -- (4.434,5.676) -- (4.434,6.276) -- (3.834,6.276) -- cycle;
\node at (4.134,5.976) {\scalebox{1.656}{\small d}};
\node at (11.943,7.584) {\scalebox{1.656}{\small A}};
\node at (6.137,7.798) {\scalebox{1.656}{\small B}};
\node at (9.101,7.358) {\scalebox{1.656}{\small C}};
\node at (3.101,7.287) {\scalebox{1.656}{\small D}};
\node at (0.765,5.171) {\scalebox{1.656}{\small E}};
\node at (9.274,4.427) {\scalebox{1.656}{\small F}};
\node at (5.807,3.833) {\scalebox{1.656}{\small G}};
\node at (2.745,4.337) {\scalebox{1.656}{\small H}};
\node at (11.564,4.372) {\scalebox{1.656}{\small I}};
\node at (6.505,1.81) {\scalebox{1.656}{\small J}};
\node at (2.781,1.792) {\scalebox{1.656}{\small K}};
\node at (0.858,1.578) {\scalebox{1.656}{\small L}};
\draw[line width=0.5pt] (12.68,6.376) -- (12.68,7.476);
\draw[line width=0.5pt] (12.449,7.091) -- (12.68,7.476) -- (12.911,7.091);
\draw[line width=0.5pt] (12.377,6.926) -- (12.982,6.926);
\node at (12.68,7.861) {\scalebox{1.656}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour D, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ chemin du Lavoir & Ⓑ Carrefour de la Mairie & Ⓚ Carrefour des Tilleuls \\ \hline
① sentier des Prés & ⑩ rue Haute & Ⓒ Place Sainte-Croix & Ⓛ Fontaine Notre-Dame \\ \hline
② rue des Lilas & ⑪ rue du Pont & Ⓓ Place du Vieux-Puits & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ rue des Jardins & ⑫ rue Basse & Ⓔ Coin du Lavoir & \textbf{a} l'école \\ \hline
④ avenue de la Gare & ⑬ boulevard du Parc & Ⓕ Placette des Ormes & \textbf{b} la boulangerie \\ \hline
⑤ venelle du Chat & ⑭ rue de la Mairie & Ⓖ Rond-point des Vaches & \textbf{c} le cimetière \\ \hline
⑥ passage du Marché & ⑮ rue de la Poste & Ⓗ Rond-point du Pont & \textbf{d} la mairie \\ \hline
⑦ rue du Moulin & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Place des Tanneurs &  \\ \hline
⑧ rue de l'Église & Ⓐ Place de la Fontaine & Ⓙ Place de l'Église &  \\ \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=0.6185, line join=round, line cap=round]
\draw[line width=0.5pt] (11.38,7.294) -- (11.317,7.093) -- (11.259,6.891) -- (11.206,6.687) -- (11.158,6.481) -- (11.115,6.275) -- (11.077,6.068) -- (11.043,5.859) -- (11.014,5.651) -- (10.991,5.441) -- (10.972,5.231) -- (10.958,5.021) -- (10.949,4.81) -- (10.945,4.599) -- (10.946,4.389);
\draw[line width=0.5pt] (6.089,7.148) -- (6.27,7.161) -- (6.451,7.17) -- (6.633,7.174) -- (6.815,7.174) -- (6.997,7.17) -- (7.179,7.162) -- (7.36,7.149) -- (7.542,7.132) -- (7.722,7.111) -- (7.903,7.085) -- (8.082,7.055) -- (8.261,7.021) -- (8.439,6.983) -- (8.615,6.941);
\draw[line width=0.5pt] (6.089,7.148) -- (6.043,6.957) -- (6.002,6.765) -- (5.966,6.572) -- (5.934,6.378) -- (5.906,6.183) -- (5.884,5.988) -- (5.866,5.792) -- (5.852,5.596) -- (5.843,5.4) -- (5.839,5.203) -- (5.839,5.006) -- (5.844,4.81) -- (5.854,4.614) -- (5.868,4.418);
\draw[line width=0.5pt] (8.615,6.941) -- (8.807,6.996) -- (9.001,7.047) -- (9.195,7.093) -- (9.391,7.135) -- (9.587,7.172) -- (9.785,7.204) -- (9.983,7.232) -- (10.182,7.255) -- (10.381,7.273) -- (10.58,7.287) -- (10.78,7.296) -- (10.98,7.3) -- (11.18,7.3) -- (11.38,7.294);
\draw[line width=0.5pt] (3.567,6.846) -- (3.749,6.839) -- (3.931,6.838) -- (4.113,6.84) -- (4.295,6.847) -- (4.477,6.858) -- (4.659,6.873) -- (4.84,6.893) -- (5.021,6.917) -- (5.201,6.945) -- (5.38,6.977) -- (5.559,7.014) -- (5.737,7.054) -- (5.913,7.099) -- (6.089,7.148);
\draw[line width=0.5pt] (3.567,6.846) -- (3.579,6.67) -- (3.586,6.494) -- (3.59,6.318) -- (3.589,6.142) -- (3.585,5.965) -- (3.576,5.789) -- (3.563,5.613) -- (3.546,5.438) -- (3.525,5.263) -- (3.499,5.089) -- (3.47,4.915) -- (3.437,4.742) -- (3.399,4.57) -- (3.357,4.399);
\draw[line width=0.5pt] (1.384,5.097) -- (1.516,5.026) -- (1.65,4.958) -- (1.785,4.893) -- (1.922,4.832) -- (2.061,4.773) -- (2.201,4.718) -- (2.342,4.666) -- (2.484,4.618) -- (2.627,4.573) -- (2.772,4.531) -- (2.917,4.493) -- (3.063,4.458) -- (3.21,4.426) -- (3.357,4.399);
\draw[line width=0.5pt] (1.384,5.097) -- (1.418,4.869) -- (1.448,4.64) -- (1.472,4.411) -- (1.491,4.181) -- (1.504,3.951) -- (1.512,3.72) -- (1.514,3.489) -- (1.511,3.258) -- (1.503,3.028) -- (1.489,2.797) -- (1.47,2.567) -- (1.445,2.338) -- (1.415,2.109) -- (1.38,1.882);
\draw[line width=0.5pt] (8.658,4.467) -- (8.82,4.436) -- (8.982,4.41) -- (9.144,4.387) -- (9.307,4.368) -- (9.471,4.352) -- (9.635,4.341) -- (9.799,4.334) -- (9.963,4.33) -- (10.128,4.33) -- (10.292,4.334) -- (10.456,4.342) -- (10.62,4.354) -- (10.783,4.369) -- (10.946,4.389);
\draw[line width=0.5pt] (5.868,4.418) -- (6.065,4.452) -- (6.263,4.481) -- (6.462,4.506) -- (6.661,4.526) -- (6.861,4.541) -- (7.061,4.552) -- (7.261,4.558) -- (7.461,4.559) -- (7.662,4.555) -- (7.862,4.547) -- (8.062,4.534) -- (8.261,4.516) -- (8.46,4.494) -- (8.658,4.467);
\draw[line width=0.5pt] (5.868,4.418) -- (5.882,4.268) -- (5.9,4.119) -- (5.921,3.97) -- (5.946,3.822) -- (5.975,3.674) -- (6.006,3.527) -- (6.042,3.381) -- (6.08,3.235) -- (6.123,3.091) -- (6.168,2.947) -- (6.217,2.805) -- (6.269,2.664) -- (6.325,2.524) -- (6.383,2.386);
\draw[line width=0.5pt] (3.357,4.399) -- (3.535,4.427) -- (3.714,4.452) -- (3.892,4.472) -- (4.072,4.489) -- (4.252,4.501) -- (4.432,4.508) -- (4.612,4.512) -- (4.792,4.511) -- (4.972,4.506) -- (5.152,4.497) -- (5.332,4.483) -- (5.511,4.466) -- (5.69,4.444) -- (5.868,4.418);
\draw[line width=0.5pt] (3.357,4.399) -- (3.324,4.244) -- (3.294,4.088) -- (3.268,3.931) -- (3.246,3.774) -- (3.227,3.617) -- (3.212,3.459) -- (3.201,3.3) -- (3.193,3.142) -- (3.189,2.983) -- (3.189,2.824) -- (3.192,2.666) -- (3.2,2.507) -- (3.211,2.349) -- (3.226,2.191);
\draw[line width=0.5pt] (3.226,2.191) -- (3.447,2.24) -- (3.67,2.283) -- (3.894,2.321) -- (4.119,2.353) -- (4.344,2.38) -- (4.57,2.402) -- (4.796,2.419) -- (5.023,2.43) -- (5.25,2.436) -- (5.477,2.437) -- (5.704,2.432) -- (5.931,2.422) -- (6.157,2.407) -- (6.383,2.386);
\draw[line width=0.5pt] (1.38,1.882) -- (1.514,1.883) -- (1.648,1.888) -- (1.782,1.896) -- (1.916,1.908) -- (2.049,1.922) -- (2.183,1.94) -- (2.315,1.96) -- (2.448,1.984) -- (2.579,2.011) -- (2.71,2.041) -- (2.84,2.074) -- (2.97,2.11) -- (3.098,2.149) -- (3.226,2.191);
\filldraw[fill=white, draw=black, line width=0.5pt] (11.753,7.294) -- (11.724,7.437) -- (11.644,7.558) -- (11.522,7.639) -- (11.38,7.668) -- (11.237,7.639) -- (11.116,7.558) -- (11.035,7.437) -- (11.006,7.294) -- (11.035,7.152) -- (11.116,7.03) -- (11.237,6.95) -- (11.38,6.921) -- (11.522,6.95) -- (11.644,7.03) -- (11.724,7.152) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.462,7.148) -- (6.433,7.291) -- (6.352,7.412) -- (6.231,7.493) -- (6.089,7.521) -- (5.946,7.493) -- (5.825,7.412) -- (5.744,7.291) -- (5.715,7.148) -- (5.744,7.005) -- (5.825,6.884) -- (5.946,6.803) -- (6.089,6.775) -- (6.231,6.803) -- (6.352,6.884) -- (6.433,7.005) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.989,6.941) -- (8.96,7.084) -- (8.879,7.205) -- (8.758,7.286) -- (8.615,7.314) -- (8.472,7.286) -- (8.351,7.205) -- (8.271,7.084) -- (8.242,6.941) -- (8.271,6.798) -- (8.351,6.677) -- (8.472,6.596) -- (8.615,6.568) -- (8.758,6.596) -- (8.879,6.677) -- (8.96,6.798) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (3.94,6.846) -- (3.911,6.988) -- (3.83,7.11) -- (3.709,7.19) -- (3.567,7.219) -- (3.424,7.19) -- (3.303,7.11) -- (3.222,6.988) -- (3.193,6.846) -- (3.222,6.703) -- (3.303,6.582) -- (3.424,6.501) -- (3.567,6.472) -- (3.709,6.501) -- (3.83,6.582) -- (3.911,6.703) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.757,5.097) -- (1.728,5.24) -- (1.647,5.361) -- (1.526,5.442) -- (1.384,5.47) -- (1.241,5.442) -- (1.12,5.361) -- (1.039,5.24) -- (1.01,5.097) -- (1.039,4.954) -- (1.12,4.833) -- (1.241,4.752) -- (1.384,4.724) -- (1.526,4.752) -- (1.647,4.833) -- (1.728,4.954) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.032,4.467) -- (9.003,4.61) -- (8.922,4.731) -- (8.801,4.812) -- (8.658,4.84) -- (8.516,4.812) -- (8.394,4.731) -- (8.314,4.61) -- (8.285,4.467) -- (8.314,4.324) -- (8.394,4.203) -- (8.516,4.122) -- (8.658,4.094) -- (8.801,4.122) -- (8.922,4.203) -- (9.003,4.324) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.241,4.418) -- (6.213,4.561) -- (6.132,4.682) -- (6.011,4.763) -- (5.868,4.791) -- (5.725,4.763) -- (5.604,4.682) -- (5.523,4.561) -- (5.495,4.418) -- (5.523,4.275) -- (5.604,4.154) -- (5.725,4.073) -- (5.868,4.045) -- (6.011,4.073) -- (6.132,4.154) -- (6.213,4.275) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.731,4.399) -- (3.702,4.541) -- (3.621,4.662) -- (3.5,4.743) -- (3.357,4.772) -- (3.215,4.743) -- (3.094,4.662) -- (3.013,4.541) -- (2.984,4.399) -- (3.013,4.256) -- (3.094,4.135) -- (3.215,4.054) -- (3.357,4.025) -- (3.5,4.054) -- (3.621,4.135) -- (3.702,4.256) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.319,4.389) -- (11.291,4.532) -- (11.21,4.653) -- (11.089,4.734) -- (10.946,4.762) -- (10.803,4.734) -- (10.682,4.653) -- (10.601,4.532) -- (10.573,4.389) -- (10.601,4.246) -- (10.682,4.125) -- (10.803,4.044) -- (10.946,4.016) -- (11.089,4.044) -- (11.21,4.125) -- (11.291,4.246) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.757,2.386) -- (6.728,2.529) -- (6.647,2.65) -- (6.526,2.731) -- (6.383,2.759) -- (6.24,2.731) -- (6.119,2.65) -- (6.039,2.529) -- (6.01,2.386) -- (6.039,2.243) -- (6.119,2.122) -- (6.24,2.041) -- (6.383,2.013) -- (6.526,2.041) -- (6.647,2.122) -- (6.728,2.243) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (3.599,2.191) -- (3.57,2.334) -- (3.49,2.455) -- (3.369,2.536) -- (3.226,2.565) -- (3.083,2.536) -- (2.962,2.455) -- (2.881,2.334) -- (2.852,2.191) -- (2.881,2.049) -- (2.962,1.927) -- (3.083,1.847) -- (3.226,1.818) -- (3.369,1.847) -- (3.49,1.927) -- (3.57,2.049) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.753,1.882) -- (1.724,2.024) -- (1.644,2.145) -- (1.522,2.226) -- (1.38,2.255) -- (1.237,2.226) -- (1.116,2.145) -- (1.035,2.024) -- (1.006,1.882) -- (1.035,1.739) -- (1.116,1.618) -- (1.237,1.537) -- (1.38,1.508) -- (1.522,1.537) -- (1.644,1.618) -- (1.724,1.739) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.423,5.418) -- (11.542,5.437) -- (11.649,5.492) -- (11.734,5.577) -- (11.789,5.684) -- (11.808,5.803) -- (11.789,5.922) -- (11.734,6.029) -- (11.649,6.114) -- (11.542,6.168) -- (11.423,6.187) -- (11.423,6.187) -- (11.305,6.168) -- (11.197,6.114) -- (11.112,6.029) -- (11.058,5.922) -- (11.039,5.803) -- (11.058,5.684) -- (11.112,5.577) -- (11.197,5.492) -- (11.305,5.437) -- (11.423,5.418) -- cycle;
\node at (11.423,5.803) {\scalebox{1.617}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.392,7.147) -- (7.511,7.166) -- (7.618,7.221) -- (7.703,7.306) -- (7.758,7.413) -- (7.776,7.532) -- (7.758,7.651) -- (7.703,7.758) -- (7.618,7.843) -- (7.511,7.898) -- (7.392,7.917) -- (7.392,7.917) -- (7.273,7.898) -- (7.166,7.843) -- (7.081,7.758) -- (7.026,7.651) -- (7.007,7.532) -- (7.026,7.413) -- (7.081,7.306) -- (7.166,7.221) -- (7.273,7.166) -- (7.392,7.147) -- cycle;
\node at (7.392,7.532) {\scalebox{1.617}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.249,5.377) -- (6.368,5.395) -- (6.475,5.45) -- (6.56,5.535) -- (6.615,5.642) -- (6.633,5.761) -- (6.615,5.88) -- (6.56,5.987) -- (6.475,6.072) -- (6.368,6.127) -- (6.249,6.146) -- (6.249,6.146) -- (6.13,6.127) -- (6.023,6.072) -- (5.938,5.987) -- (5.883,5.88) -- (5.864,5.761) -- (5.883,5.642) -- (5.938,5.535) -- (6.023,5.45) -- (6.13,5.395) -- (6.249,5.377) -- cycle;
\node at (6.249,5.761) {\scalebox{1.617}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.934,7.229) -- (10.053,7.247) -- (10.16,7.302) -- (10.245,7.387) -- (10.3,7.494) -- (10.319,7.613) -- (10.3,7.732) -- (10.245,7.839) -- (10.16,7.924) -- (10.053,7.979) -- (9.934,7.998) -- (9.934,7.998) -- (9.815,7.979) -- (9.708,7.924) -- (9.623,7.839) -- (9.568,7.732) -- (9.55,7.613) -- (9.568,7.494) -- (9.623,7.387) -- (9.708,7.302) -- (9.815,7.247) -- (9.934,7.229) -- cycle;
\node at (9.934,7.613) {\scalebox{1.617}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.794,6.89) -- (4.913,6.909) -- (5.02,6.964) -- (5.105,7.049) -- (5.16,7.156) -- (5.179,7.275) -- (5.16,7.393) -- (5.105,7.501) -- (5.02,7.586) -- (4.913,7.64) -- (4.794,7.659) -- (4.794,7.659) -- (4.675,7.64) -- (4.568,7.586) -- (4.483,7.501) -- (4.429,7.393) -- (4.41,7.275) -- (4.429,7.156) -- (4.483,7.049) -- (4.568,6.964) -- (4.675,6.909) -- (4.794,6.89) -- cycle;
\node at (4.794,7.275) {\scalebox{1.617}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.18,5.262) -- (3.299,5.281) -- (3.406,5.335) -- (3.491,5.42) -- (3.546,5.527) -- (3.564,5.646) -- (3.546,5.765) -- (3.491,5.872) -- (3.406,5.957) -- (3.299,6.012) -- (3.18,6.031) -- (3.18,6.031) -- (3.061,6.012) -- (2.954,5.957) -- (2.869,5.872) -- (2.814,5.765) -- (2.795,5.646) -- (2.814,5.527) -- (2.869,5.42) -- (2.954,5.335) -- (3.061,5.281) -- (3.18,5.262) -- cycle;
\node at (3.18,5.646) {\scalebox{1.617}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.337,4.693) -- (2.456,4.712) -- (2.563,4.766) -- (2.648,4.852) -- (2.703,4.959) -- (2.722,5.078) -- (2.703,5.196) -- (2.648,5.304) -- (2.563,5.389) -- (2.456,5.443) -- (2.337,5.462) -- (2.337,5.462) -- (2.219,5.443) -- (2.111,5.389) -- (2.026,5.304) -- (1.972,5.196) -- (1.953,5.078) -- (1.972,4.959) -- (2.026,4.852) -- (2.111,4.766) -- (2.219,4.712) -- (2.337,4.693) -- cycle;
\node at (2.337,5.078) {\scalebox{1.617}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.899,3.104) -- (2.018,3.123) -- (2.125,3.178) -- (2.21,3.263) -- (2.264,3.37) -- (2.283,3.489) -- (2.264,3.608) -- (2.21,3.715) -- (2.125,3.8) -- (2.018,3.854) -- (1.899,3.873) -- (1.899,3.873) -- (1.78,3.854) -- (1.673,3.8) -- (1.588,3.715) -- (1.533,3.608) -- (1.514,3.489) -- (1.533,3.37) -- (1.588,3.263) -- (1.673,3.178) -- (1.78,3.123) -- (1.899,3.104) -- cycle;
\node at (1.899,3.489) {\scalebox{1.617}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.123,4.33) -- (10.242,4.349) -- (10.349,4.403) -- (10.434,4.489) -- (10.488,4.596) -- (10.507,4.715) -- (10.488,4.833) -- (10.434,4.941) -- (10.349,5.026) -- (10.242,5.08) -- (10.123,5.099) -- (10.123,5.099) -- (10.004,5.08) -- (9.897,5.026) -- (9.812,4.941) -- (9.757,4.833) -- (9.738,4.715) -- (9.757,4.596) -- (9.812,4.489) -- (9.897,4.403) -- (10.004,4.349) -- (10.123,4.33) -- cycle;
\node at (10.123,4.715) {\scalebox{1.617}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.435,4.557) -- (7.554,4.576) -- (7.661,4.631) -- (7.746,4.716) -- (7.801,4.823) -- (7.82,4.942) -- (7.801,5.061) -- (7.746,5.168) -- (7.661,5.253) -- (7.554,5.308) -- (7.435,5.327) -- (7.073,5.327) -- (6.955,5.308) -- (6.847,5.253) -- (6.762,5.168) -- (6.708,5.061) -- (6.689,4.942) -- (6.708,4.823) -- (6.762,4.716) -- (6.847,4.631) -- (6.955,4.576) -- (7.073,4.557) -- cycle;
\node at (7.254,4.942) {\scalebox{1.617}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.122,3.224) -- (7.24,3.243) -- (7.348,3.298) -- (7.433,3.383) -- (7.487,3.49) -- (7.506,3.609) -- (7.487,3.727) -- (7.433,3.835) -- (7.348,3.92) -- (7.24,3.974) -- (7.122,3.993) -- (6.76,3.993) -- (6.641,3.974) -- (6.534,3.92) -- (6.449,3.835) -- (6.394,3.727) -- (6.375,3.609) -- (6.394,3.49) -- (6.449,3.383) -- (6.534,3.298) -- (6.641,3.243) -- (6.76,3.224) -- cycle;
\node at (6.941,3.609) {\scalebox{1.617}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.79,4.512) -- (4.909,4.531) -- (5.016,4.585) -- (5.101,4.67) -- (5.156,4.777) -- (5.174,4.896) -- (5.156,5.015) -- (5.101,5.122) -- (5.016,5.207) -- (4.909,5.262) -- (4.79,5.281) -- (4.428,5.281) -- (4.309,5.262) -- (4.202,5.207) -- (4.117,5.122) -- (4.062,5.015) -- (4.043,4.896) -- (4.062,4.777) -- (4.117,4.67) -- (4.202,4.585) -- (4.309,4.531) -- (4.428,4.512) -- cycle;
\node at (4.609,4.896) {\scalebox{1.617}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.765,2.893) -- (3.884,2.912) -- (3.991,2.966) -- (4.076,3.051) -- (4.131,3.159) -- (4.15,3.277) -- (4.131,3.396) -- (4.076,3.503) -- (3.991,3.589) -- (3.884,3.643) -- (3.765,3.662) -- (3.403,3.662) -- (3.285,3.643) -- (3.177,3.589) -- (3.092,3.503) -- (3.038,3.396) -- (3.019,3.277) -- (3.038,3.159) -- (3.092,3.051) -- (3.177,2.966) -- (3.285,2.912) -- (3.403,2.893) -- cycle;
\node at (3.584,3.277) {\scalebox{1.617}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.954,2.418) -- (5.073,2.437) -- (5.18,2.492) -- (5.265,2.577) -- (5.319,2.684) -- (5.338,2.803) -- (5.319,2.921) -- (5.265,3.029) -- (5.18,3.114) -- (5.073,3.168) -- (4.954,3.187) -- (4.592,3.187) -- (4.473,3.168) -- (4.366,3.114) -- (4.281,3.029) -- (4.226,2.921) -- (4.207,2.803) -- (4.226,2.684) -- (4.281,2.577) -- (4.366,2.492) -- (4.473,2.437) -- (4.592,2.418) -- cycle;
\node at (4.773,2.803) {\scalebox{1.617}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.186,0.6) -- (2.304,0.619) -- (2.412,0.673) -- (2.497,0.759) -- (2.551,0.866) -- (2.57,0.985) -- (2.551,1.103) -- (2.497,1.211) -- (2.412,1.296) -- (2.304,1.35) -- (2.186,1.369) -- (1.824,1.369) -- (1.705,1.35) -- (1.598,1.296) -- (1.513,1.211) -- (1.458,1.103) -- (1.439,0.985) -- (1.458,0.866) -- (1.513,0.759) -- (1.598,0.673) -- (1.705,0.619) -- (1.824,0.6) -- cycle;
\node at (2.005,0.985) {\scalebox{1.617}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (3.267,6.546) -- (3.867,6.546) -- (3.867,7.146) -- (3.267,7.146) -- cycle;
\node at (3.567,6.846) {\scalebox{1.617}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (0.737,1.608) -- (1.337,1.608) -- (1.337,2.208) -- (0.737,2.208) -- cycle;
\node at (1.037,1.908) {\scalebox{1.617}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.677,3.449) -- (1.277,3.449) -- (1.277,4.049) -- (0.677,4.049) -- cycle;
\node at (0.977,3.749) {\scalebox{1.617}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (3.834,5.644) -- (4.434,5.644) -- (4.434,6.244) -- (3.834,6.244) -- cycle;
\node at (4.134,5.944) {\scalebox{1.617}{\small d}};
\node at (11.943,7.552) {\scalebox{1.617}{\small A}};
\node at (6.137,7.766) {\scalebox{1.617}{\small B}};
\node at (9.101,7.327) {\scalebox{1.617}{\small C}};
\node at (3.101,7.255) {\scalebox{1.617}{\small D}};
\node at (0.765,5.139) {\scalebox{1.617}{\small E}};
\node at (9.274,4.395) {\scalebox{1.617}{\small F}};
\node at (5.807,3.801) {\scalebox{1.617}{\small G}};
\node at (2.745,4.305) {\scalebox{1.617}{\small H}};
\node at (11.564,4.34) {\scalebox{1.617}{\small I}};
\node at (6.505,1.778) {\scalebox{1.617}{\small J}};
\node at (2.781,1.76) {\scalebox{1.617}{\small K}};
\node at (0.858,1.546) {\scalebox{1.617}{\small L}};
\draw[line width=0.5pt] (12.68,6.344) -- (12.68,7.444);
\draw[line width=0.5pt] (12.449,7.059) -- (12.68,7.444) -- (12.911,7.059);
\draw[line width=0.5pt] (12.377,6.894) -- (12.982,6.894);
\node at (12.68,7.829) {\scalebox{1.617}{\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 & C, D, E, G, I, K & 6 \\ \hline
clairs & A, B, F, H, J, L & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{6} clairs. Il y en a \textbf{autant} de chaque côté.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 12 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui.} Un circuit existe :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{venelle du Chat ⑤} vers l'est jusqu'à Carrefour de la Mairie \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Lilas ②}, vers l'est, jusqu'à Place Sainte-Croix \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{avenue de la Gare ④}, vers l'est, jusqu'à Place de la Fontaine \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ①}, vers le sud, jusqu'à Place des Tanneurs \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin du Lavoir ⑨}, vers l'ouest, jusqu'à Placette des Ormes \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Haute ⑩}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Pont ⑪}, vers le sud, jusqu'à Place de l'Église \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Mairie ⑭}, vers l'ouest, jusqu'à Carrefour des Tilleuls \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de la Poste ⑮}, vers l'ouest, jusqu'à Fontaine Notre-Dame \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de l'Église ⑧}, vers le nord, jusqu'à Coin du Lavoir \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Moulin ⑦}, vers l'est, jusqu'à Rond-point du Pont \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{passage du Marché ⑥}, vers le nord, jusqu'à Place du Vieux-Puits \textbf{(D)}.\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}
