\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 156}

\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=156}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0156}
\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=0.8177, line join=round, line cap=round]
\draw[line width=0.5pt] (5.166,11.504) -- (5.353,11.463) -- (5.54,11.428) -- (5.729,11.396) -- (5.918,11.37) -- (6.107,11.347) -- (6.298,11.329) -- (6.488,11.316) -- (6.679,11.307) -- (6.87,11.302) -- (7.061,11.302) -- (7.252,11.307) -- (7.443,11.316) -- (7.634,11.329) -- (7.824,11.347);
\draw[line width=0.5pt] (7.824,11.347) -- (7.855,11.147) -- (7.881,10.946) -- (7.902,10.745) -- (7.919,10.542) -- (7.931,10.34) -- (7.939,10.137) -- (7.941,9.935) -- (7.939,9.732) -- (7.932,9.529) -- (7.92,9.327) -- (7.903,9.125) -- (7.882,8.923) -- (7.856,8.722) -- (7.825,8.522);
\draw[line width=0.5pt] (1.585,10.775) -- (1.847,10.788) -- (2.108,10.806) -- (2.369,10.831) -- (2.63,10.862) -- (2.889,10.899) -- (3.148,10.942) -- (3.406,10.992) -- (3.662,11.047) -- (3.917,11.108) -- (4.171,11.176) -- (4.422,11.249) -- (4.672,11.328) -- (4.92,11.413) -- (5.166,11.504);
\draw[line width=0.5pt] (1.585,10.775) -- (1.606,10.597) -- (1.623,10.418) -- (1.636,10.239) -- (1.645,10.06) -- (1.649,9.88) -- (1.649,9.7) -- (1.645,9.521) -- (1.636,9.341) -- (1.624,9.162) -- (1.607,8.983) -- (1.586,8.805) -- (1.561,8.627) -- (1.531,8.45) -- (1.498,8.274);
\draw[line width=0.5pt] (7.825,8.522) -- (7.759,8.303) -- (7.697,8.083) -- (7.641,7.862) -- (7.59,7.639) -- (7.545,7.414) -- (7.504,7.189) -- (7.469,6.963) -- (7.44,6.736) -- (7.415,6.509) -- (7.396,6.281) -- (7.382,6.053) -- (7.374,5.824) -- (7.371,5.596) -- (7.374,5.367);
\draw[line width=0.5pt] (1.498,8.274) -- (1.698,8.273) -- (1.898,8.268) -- (2.099,8.258) -- (2.299,8.243) -- (2.498,8.224) -- (2.698,8.2) -- (2.896,8.171) -- (3.094,8.138) -- (3.291,8.1) -- (3.487,8.058) -- (3.682,8.01) -- (3.876,7.959) -- (4.068,7.902) -- (4.259,7.842);
\draw[line width=0.5pt] (1.498,8.274) -- (1.474,8.054) -- (1.455,7.835) -- (1.442,7.614) -- (1.434,7.394) -- (1.431,7.173) -- (1.433,6.953) -- (1.441,6.732) -- (1.453,6.512) -- (1.471,6.292) -- (1.494,6.072) -- (1.522,5.854) -- (1.555,5.636) -- (1.594,5.418) -- (1.637,5.202);
\draw[line width=0.5pt] (4.259,7.842) -- (4.504,7.929) -- (4.752,8.01) -- (5.001,8.086) -- (5.252,8.155) -- (5.505,8.219) -- (5.759,8.277) -- (6.014,8.329) -- (6.271,8.375) -- (6.528,8.415) -- (6.787,8.448) -- (7.046,8.476) -- (7.305,8.497) -- (7.565,8.513) -- (7.825,8.522);
\draw[line width=0.5pt] (4.259,7.842) -- (4.263,7.671) -- (4.271,7.5) -- (4.282,7.33) -- (4.298,7.16) -- (4.318,6.99) -- (4.342,6.821) -- (4.369,6.652) -- (4.401,6.484) -- (4.437,6.317) -- (4.476,6.151) -- (4.52,5.985) -- (4.567,5.821) -- (4.618,5.658) -- (4.673,5.497);
\draw[line width=0.5pt] (4.673,5.497) -- (4.866,5.517) -- (5.059,5.533) -- (5.253,5.544) -- (5.447,5.551) -- (5.641,5.553) -- (5.835,5.55) -- (6.029,5.543) -- (6.223,5.532) -- (6.416,5.516) -- (6.609,5.495) -- (6.801,5.47) -- (6.993,5.44) -- (7.184,5.406) -- (7.374,5.367);
\draw[line width=0.5pt] (4.673,5.497) -- (4.64,5.266) -- (4.612,5.034) -- (4.589,4.802) -- (4.572,4.569) -- (4.56,4.336) -- (4.554,4.102) -- (4.553,3.869) -- (4.558,3.635) -- (4.568,3.402) -- (4.584,3.169) -- (4.605,2.936) -- (4.632,2.705) -- (4.664,2.473) -- (4.701,2.243);
\draw[line width=0.5pt] (7.374,5.367) -- (7.418,5.123) -- (7.457,4.879) -- (7.49,4.633) -- (7.518,4.386) -- (7.539,4.139) -- (7.555,3.892) -- (7.565,3.644) -- (7.569,3.396) -- (7.567,3.148) -- (7.559,2.9) -- (7.546,2.652) -- (7.527,2.405) -- (7.502,2.158) -- (7.471,1.912);
\draw[line width=0.5pt] (1.637,5.202) -- (1.855,5.19) -- (2.074,5.183) -- (2.293,5.181) -- (2.512,5.184) -- (2.731,5.192) -- (2.949,5.206) -- (3.167,5.224) -- (3.385,5.248) -- (3.602,5.277) -- (3.818,5.311) -- (4.034,5.35) -- (4.248,5.394) -- (4.461,5.443) -- (4.673,5.497);
\draw[line width=0.5pt] (1.637,5.202) -- (1.708,4.945) -- (1.773,4.686) -- (1.833,4.425) -- (1.885,4.163) -- (1.932,3.9) -- (1.973,3.636) -- (2.007,3.371) -- (2.035,3.105) -- (2.057,2.839) -- (2.072,2.572) -- (2.081,2.305) -- (2.084,2.038) -- (2.081,1.77) -- (2.071,1.504);
\draw[line width=0.5pt] (4.701,2.243) -- (4.894,2.189) -- (5.088,2.14) -- (5.283,2.096) -- (5.479,2.056) -- (5.676,2.02) -- (5.874,1.99) -- (6.072,1.964) -- (6.271,1.942) -- (6.471,1.925) -- (6.671,1.913) -- (6.871,1.906) -- (7.071,1.903) -- (7.271,1.906) -- (7.471,1.912);
\draw[line width=0.5pt] (2.071,1.504) -- (2.25,1.585) -- (2.43,1.662) -- (2.612,1.734) -- (2.796,1.803) -- (2.981,1.867) -- (3.168,1.927) -- (3.356,1.982) -- (3.545,2.033) -- (3.736,2.079) -- (3.927,2.121) -- (4.12,2.158) -- (4.313,2.191) -- (4.507,2.22) -- (4.701,2.243);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.326,11.504) -- (5.305,11.584) -- (5.246,11.642) -- (5.166,11.664) -- (5.086,11.642) -- (5.027,11.584) -- (5.006,11.504) -- (5.027,11.424) -- (5.086,11.365) -- (5.166,11.344) -- (5.246,11.365) -- (5.305,11.424) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.984,11.347) -- (7.962,11.427) -- (7.904,11.486) -- (7.824,11.507) -- (7.744,11.486) -- (7.685,11.427) -- (7.664,11.347) -- (7.685,11.267) -- (7.744,11.209) -- (7.824,11.187) -- (7.904,11.209) -- (7.962,11.267) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.745,10.775) -- (1.724,10.855) -- (1.665,10.914) -- (1.585,10.935) -- (1.505,10.914) -- (1.447,10.855) -- (1.425,10.775) -- (1.447,10.695) -- (1.505,10.637) -- (1.585,10.615) -- (1.665,10.637) -- (1.724,10.695) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.985,8.522) -- (7.964,8.602) -- (7.905,8.661) -- (7.825,8.682) -- (7.745,8.661) -- (7.687,8.602) -- (7.665,8.522) -- (7.687,8.442) -- (7.745,8.383) -- (7.825,8.362) -- (7.905,8.383) -- (7.964,8.442) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.658,8.274) -- (1.636,8.354) -- (1.578,8.412) -- (1.498,8.434) -- (1.418,8.412) -- (1.359,8.354) -- (1.338,8.274) -- (1.359,8.194) -- (1.418,8.135) -- (1.498,8.114) -- (1.578,8.135) -- (1.636,8.194) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.419,7.842) -- (4.398,7.922) -- (4.339,7.98) -- (4.259,8.002) -- (4.179,7.98) -- (4.121,7.922) -- (4.099,7.842) -- (4.121,7.762) -- (4.179,7.703) -- (4.259,7.682) -- (4.339,7.703) -- (4.398,7.762) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.833,5.497) -- (4.812,5.577) -- (4.753,5.635) -- (4.673,5.657) -- (4.593,5.635) -- (4.535,5.577) -- (4.513,5.497) -- (4.535,5.417) -- (4.593,5.358) -- (4.673,5.337) -- (4.753,5.358) -- (4.812,5.417) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.534,5.367) -- (7.512,5.447) -- (7.454,5.506) -- (7.374,5.527) -- (7.294,5.506) -- (7.235,5.447) -- (7.214,5.367) -- (7.235,5.287) -- (7.294,5.229) -- (7.374,5.207) -- (7.454,5.229) -- (7.512,5.287) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.797,5.202) -- (1.776,5.282) -- (1.717,5.341) -- (1.637,5.362) -- (1.557,5.341) -- (1.499,5.282) -- (1.477,5.202) -- (1.499,5.122) -- (1.557,5.064) -- (1.637,5.042) -- (1.717,5.064) -- (1.776,5.122) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.861,2.243) -- (4.84,2.323) -- (4.781,2.382) -- (4.701,2.403) -- (4.621,2.382) -- (4.563,2.323) -- (4.541,2.243) -- (4.563,2.163) -- (4.621,2.105) -- (4.701,2.083) -- (4.781,2.105) -- (4.84,2.163) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.631,1.912) -- (7.609,1.992) -- (7.551,2.051) -- (7.471,2.072) -- (7.391,2.051) -- (7.332,1.992) -- (7.311,1.912) -- (7.332,1.832) -- (7.391,1.774) -- (7.471,1.752) -- (7.551,1.774) -- (7.609,1.832) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.231,1.504) -- (2.21,1.584) -- (2.151,1.642) -- (2.071,1.664) -- (1.991,1.642) -- (1.933,1.584) -- (1.911,1.504) -- (1.933,1.424) -- (1.991,1.365) -- (2.071,1.344) -- (2.151,1.365) -- (2.21,1.424) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.512,11.315) -- (6.633,11.334) -- (6.743,11.39) -- (6.83,11.477) -- (6.886,11.587) -- (6.905,11.709) -- (6.886,11.831) -- (6.83,11.941) -- (6.743,12.028) -- (6.633,12.084) -- (6.512,12.103) -- (6.512,12.103) -- (6.39,12.084) -- (6.28,12.028) -- (6.193,11.941) -- (6.137,11.831) -- (6.118,11.709) -- (6.137,11.587) -- (6.193,11.477) -- (6.28,11.39) -- (6.39,11.334) -- (6.512,11.315) -- cycle;
\node at (6.512,11.709) {\scalebox{1.223}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.335,9.541) -- (8.457,9.56) -- (8.566,9.616) -- (8.654,9.703) -- (8.71,9.813) -- (8.729,9.935) -- (8.71,10.057) -- (8.654,10.166) -- (8.566,10.254) -- (8.457,10.31) -- (8.335,10.329) -- (8.335,10.329) -- (8.213,10.31) -- (8.103,10.254) -- (8.016,10.166) -- (7.96,10.057) -- (7.941,9.935) -- (7.96,9.813) -- (8.016,9.703) -- (8.103,9.616) -- (8.213,9.56) -- (8.335,9.541) -- cycle;
\node at (8.335,9.935) {\scalebox{1.223}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.327,10.984) -- (3.449,11.003) -- (3.559,11.059) -- (3.646,11.146) -- (3.702,11.256) -- (3.721,11.378) -- (3.702,11.499) -- (3.646,11.609) -- (3.559,11.696) -- (3.449,11.752) -- (3.327,11.772) -- (3.327,11.772) -- (3.205,11.752) -- (3.096,11.696) -- (3.009,11.609) -- (2.953,11.499) -- (2.933,11.378) -- (2.953,11.256) -- (3.009,11.146) -- (3.096,11.059) -- (3.205,11.003) -- (3.327,10.984) -- cycle;
\node at (3.327,11.378) {\scalebox{1.223}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.038,9.113) -- (2.16,9.132) -- (2.27,9.188) -- (2.357,9.275) -- (2.413,9.385) -- (2.432,9.507) -- (2.413,9.629) -- (2.357,9.739) -- (2.27,9.826) -- (2.16,9.882) -- (2.038,9.901) -- (2.038,9.901) -- (1.917,9.882) -- (1.807,9.826) -- (1.72,9.739) -- (1.664,9.629) -- (1.644,9.507) -- (1.664,9.385) -- (1.72,9.275) -- (1.807,9.188) -- (1.917,9.132) -- (2.038,9.113) -- cycle;
\node at (2.038,9.507) {\scalebox{1.223}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.859,6.513) -- (7.981,6.533) -- (8.091,6.589) -- (8.178,6.676) -- (8.234,6.786) -- (8.253,6.907) -- (8.234,7.029) -- (8.178,7.139) -- (8.091,7.226) -- (7.981,7.282) -- (7.859,7.301) -- (7.859,7.301) -- (7.738,7.282) -- (7.628,7.226) -- (7.541,7.139) -- (7.485,7.029) -- (7.465,6.907) -- (7.485,6.786) -- (7.541,6.676) -- (7.628,6.589) -- (7.738,6.533) -- (7.859,6.513) -- cycle;
\node at (7.859,6.907) {\scalebox{1.223}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.957,8.167) -- (3.079,8.186) -- (3.189,8.242) -- (3.276,8.329) -- (3.332,8.439) -- (3.351,8.561) -- (3.332,8.682) -- (3.276,8.792) -- (3.189,8.879) -- (3.079,8.935) -- (2.957,8.955) -- (2.957,8.955) -- (2.835,8.935) -- (2.726,8.879) -- (2.638,8.792) -- (2.582,8.682) -- (2.563,8.561) -- (2.582,8.439) -- (2.638,8.329) -- (2.726,8.242) -- (2.835,8.186) -- (2.957,8.167) -- cycle;
\node at (2.957,8.561) {\scalebox{1.223}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.834,6.356) -- (1.956,6.375) -- (2.066,6.431) -- (2.153,6.518) -- (2.209,6.628) -- (2.228,6.75) -- (2.209,6.872) -- (2.153,6.982) -- (2.066,7.069) -- (1.956,7.125) -- (1.834,7.144) -- (1.834,7.144) -- (1.712,7.125) -- (1.603,7.069) -- (1.516,6.982) -- (1.46,6.872) -- (1.44,6.75) -- (1.46,6.628) -- (1.516,6.518) -- (1.603,6.431) -- (1.712,6.375) -- (1.834,6.356) -- cycle;
\node at (1.834,6.75) {\scalebox{1.223}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.94,8.322) -- (6.062,8.341) -- (6.172,8.397) -- (6.259,8.484) -- (6.315,8.594) -- (6.334,8.716) -- (6.315,8.838) -- (6.259,8.947) -- (6.172,9.034) -- (6.062,9.09) -- (5.94,9.11) -- (5.94,9.11) -- (5.819,9.09) -- (5.709,9.034) -- (5.622,8.947) -- (5.566,8.838) -- (5.546,8.716) -- (5.566,8.594) -- (5.622,8.484) -- (5.709,8.397) -- (5.819,8.341) -- (5.94,8.322) -- cycle;
\node at (5.94,8.716) {\scalebox{1.223}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.757,6.327) -- (4.879,6.346) -- (4.989,6.402) -- (5.076,6.489) -- (5.132,6.599) -- (5.151,6.721) -- (5.132,6.842) -- (5.076,6.952) -- (4.989,7.039) -- (4.879,7.095) -- (4.757,7.115) -- (4.757,7.115) -- (4.636,7.095) -- (4.526,7.039) -- (4.439,6.952) -- (4.383,6.842) -- (4.363,6.721) -- (4.383,6.599) -- (4.439,6.489) -- (4.526,6.402) -- (4.636,6.346) -- (4.757,6.327) -- cycle;
\node at (4.757,6.721) {\scalebox{1.223}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.233,5.543) -- (6.355,5.562) -- (6.465,5.618) -- (6.552,5.705) -- (6.608,5.815) -- (6.627,5.937) -- (6.608,6.058) -- (6.552,6.168) -- (6.465,6.255) -- (6.355,6.311) -- (6.233,6.331) -- (5.862,6.331) -- (5.741,6.311) -- (5.631,6.255) -- (5.544,6.168) -- (5.488,6.058) -- (5.468,5.937) -- (5.488,5.815) -- (5.544,5.705) -- (5.631,5.618) -- (5.741,5.562) -- (5.862,5.543) -- cycle;
\node at (6.048,5.937) {\scalebox{1.223}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.345,3.472) -- (4.466,3.491) -- (4.576,3.547) -- (4.663,3.634) -- (4.719,3.744) -- (4.738,3.865) -- (4.719,3.987) -- (4.663,4.097) -- (4.576,4.184) -- (4.466,4.24) -- (4.345,4.259) -- (3.974,4.259) -- (3.852,4.24) -- (3.742,4.184) -- (3.655,4.097) -- (3.599,3.987) -- (3.58,3.865) -- (3.599,3.744) -- (3.655,3.634) -- (3.742,3.547) -- (3.852,3.491) -- (3.974,3.472) -- cycle;
\node at (4.159,3.865) {\scalebox{1.223}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.144,3.261) -- (8.265,3.28) -- (8.375,3.336) -- (8.462,3.423) -- (8.518,3.533) -- (8.538,3.655) -- (8.518,3.777) -- (8.462,3.886) -- (8.375,3.973) -- (8.265,4.029) -- (8.144,4.049) -- (7.773,4.049) -- (7.651,4.029) -- (7.541,3.973) -- (7.454,3.886) -- (7.398,3.777) -- (7.379,3.655) -- (7.398,3.533) -- (7.454,3.423) -- (7.541,3.336) -- (7.651,3.28) -- (7.773,3.261) -- cycle;
\node at (7.958,3.655) {\scalebox{1.223}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.315,5.222) -- (3.436,5.242) -- (3.546,5.298) -- (3.633,5.385) -- (3.689,5.495) -- (3.709,5.616) -- (3.689,5.738) -- (3.633,5.848) -- (3.546,5.935) -- (3.436,5.991) -- (3.315,6.01) -- (2.944,6.01) -- (2.822,5.991) -- (2.712,5.935) -- (2.625,5.848) -- (2.569,5.738) -- (2.55,5.616) -- (2.569,5.495) -- (2.625,5.385) -- (2.712,5.298) -- (2.822,5.242) -- (2.944,5.222) -- cycle;
\node at (3.129,5.616) {\scalebox{1.223}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.583,3.023) -- (2.705,3.042) -- (2.815,3.098) -- (2.902,3.185) -- (2.958,3.295) -- (2.977,3.417) -- (2.958,3.539) -- (2.902,3.648) -- (2.815,3.735) -- (2.705,3.791) -- (2.583,3.811) -- (2.213,3.811) -- (2.091,3.791) -- (1.981,3.735) -- (1.894,3.648) -- (1.838,3.539) -- (1.819,3.417) -- (1.838,3.295) -- (1.894,3.185) -- (1.981,3.098) -- (2.091,3.042) -- (2.213,3.023) -- cycle;
\node at (2.398,3.417) {\scalebox{1.223}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.304,1.961) -- (6.426,1.98) -- (6.536,2.036) -- (6.623,2.123) -- (6.679,2.233) -- (6.698,2.355) -- (6.679,2.476) -- (6.623,2.586) -- (6.536,2.673) -- (6.426,2.729) -- (6.304,2.749) -- (5.934,2.749) -- (5.812,2.729) -- (5.702,2.673) -- (5.615,2.586) -- (5.559,2.476) -- (5.54,2.355) -- (5.559,2.233) -- (5.615,2.123) -- (5.702,2.036) -- (5.812,1.98) -- (5.934,1.961) -- cycle;
\node at (6.119,2.355) {\scalebox{1.223}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.435,1.967) -- (3.556,1.986) -- (3.666,2.042) -- (3.753,2.13) -- (3.809,2.239) -- (3.828,2.361) -- (3.809,2.483) -- (3.753,2.593) -- (3.666,2.68) -- (3.556,2.736) -- (3.435,2.755) -- (3.064,2.755) -- (2.942,2.736) -- (2.832,2.68) -- (2.745,2.593) -- (2.689,2.483) -- (2.67,2.361) -- (2.689,2.239) -- (2.745,2.13) -- (2.832,2.042) -- (2.942,1.986) -- (3.064,1.967) -- cycle;
\node at (3.249,2.361) {\scalebox{1.223}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.373,5.197) -- (4.973,5.197) -- (4.973,5.797) -- (4.373,5.797) -- cycle;
\node at (4.673,5.497) {\scalebox{1.223}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (2.067,1.081) -- (2.667,1.081) -- (2.667,1.681) -- (2.067,1.681) -- cycle;
\node at (2.367,1.381) {\scalebox{1.223}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.575,10.45) -- (9.175,10.45) -- (9.175,11.05) -- (8.575,11.05) -- cycle;
\node at (8.875,10.75) {\scalebox{1.223}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.6,7.076) -- (1.2,7.076) -- (1.2,7.676) -- (0.6,7.676) -- cycle;
\node at (0.9,7.376) {\scalebox{1.223}{\small d}};
\node at (5.229,12.12) {\scalebox{1.223}{\small A}};
\node at (8.17,11.862) {\scalebox{1.223}{\small B}};
\node at (1.213,11.271) {\scalebox{1.223}{\small C}};
\node at (8.36,8.837) {\scalebox{1.223}{\small D}};
\node at (0.944,8.554) {\scalebox{1.223}{\small E}};
\node at (4.053,8.426) {\scalebox{1.223}{\small F}};
\node at (4.673,4.877) {\scalebox{1.223}{\small G}};
\node at (7.932,5.098) {\scalebox{1.223}{\small H}};
\node at (1.079,4.933) {\scalebox{1.223}{\small I}};
\node at (4.705,1.623) {\scalebox{1.223}{\small J}};
\node at (7.785,1.378) {\scalebox{1.223}{\small K}};
\node at (1.792,0.95) {\scalebox{1.223}{\small L}};
\draw[line width=0.5pt] (9.125,10.554) -- (9.125,11.654);
\draw[line width=0.5pt] (8.894,11.269) -- (9.125,11.654) -- (9.356,11.269);
\draw[line width=0.5pt] (8.823,11.104) -- (9.428,11.104);
\node at (9.125,12.039) {\scalebox{1.223}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour G, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ rue Verte & Ⓐ Coin du Lavoir & Ⓙ Carrefour du Moulin \\ \hline
① rue des Lilas & ⑩ quai aux Fleurs & Ⓑ Place aux Herbes & Ⓚ Fontaine Notre-Dame \\ \hline
② rue du Moulin & ⑪ rue de l'Église & Ⓒ Place Sainte-Croix & Ⓛ Place des Tanneurs \\ \hline
③ avenue de la Gare & ⑫ venelle du Chat & Ⓓ Rond-point de la Gare & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ rue du Puits & ⑬ chemin Creux & Ⓔ Carrefour de la Mairie & \textbf{a} l'école \\ \hline
⑤ boulevard du Parc & ⑭ allée des Tilleuls & Ⓕ Carrefour du Lavoir & \textbf{b} la mairie \\ \hline
⑥ rue des Écoles & ⑮ rue du Stade & Ⓖ Rond-point des Vaches & \textbf{c} la bibliothèque \\ \hline
⑦ chemin des Vignes & ⑯ rue Neuve & Ⓗ Carrefour de l'Hôpital & \textbf{d} le stade \\ \hline
⑧ rue Haute & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Rond-point de la Poste &  \\ \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=0.8377, line join=round, line cap=round]
\draw[line width=0.5pt] (5.166,11.504) -- (5.353,11.463) -- (5.54,11.428) -- (5.729,11.396) -- (5.918,11.37) -- (6.107,11.347) -- (6.298,11.329) -- (6.488,11.316) -- (6.679,11.307) -- (6.87,11.302) -- (7.061,11.302) -- (7.252,11.307) -- (7.443,11.316) -- (7.634,11.329) -- (7.824,11.347);
\draw[line width=0.5pt] (7.824,11.347) -- (7.855,11.147) -- (7.881,10.946) -- (7.902,10.745) -- (7.919,10.542) -- (7.931,10.34) -- (7.939,10.137) -- (7.941,9.935) -- (7.939,9.732) -- (7.932,9.529) -- (7.92,9.327) -- (7.903,9.125) -- (7.882,8.923) -- (7.856,8.722) -- (7.825,8.522);
\draw[line width=0.5pt] (1.585,10.775) -- (1.847,10.788) -- (2.108,10.806) -- (2.369,10.831) -- (2.63,10.862) -- (2.889,10.899) -- (3.148,10.942) -- (3.406,10.992) -- (3.662,11.047) -- (3.917,11.108) -- (4.171,11.176) -- (4.422,11.249) -- (4.672,11.328) -- (4.92,11.413) -- (5.166,11.504);
\draw[line width=0.5pt] (1.585,10.775) -- (1.606,10.597) -- (1.623,10.418) -- (1.636,10.239) -- (1.645,10.06) -- (1.649,9.88) -- (1.649,9.7) -- (1.645,9.521) -- (1.636,9.341) -- (1.624,9.162) -- (1.607,8.983) -- (1.586,8.805) -- (1.561,8.627) -- (1.531,8.45) -- (1.498,8.274);
\draw[line width=0.5pt] (7.825,8.522) -- (7.759,8.303) -- (7.697,8.083) -- (7.641,7.862) -- (7.59,7.639) -- (7.545,7.414) -- (7.504,7.189) -- (7.469,6.963) -- (7.44,6.736) -- (7.415,6.509) -- (7.396,6.281) -- (7.382,6.053) -- (7.374,5.824) -- (7.371,5.596) -- (7.374,5.367);
\draw[line width=0.5pt] (1.498,8.274) -- (1.698,8.273) -- (1.898,8.268) -- (2.099,8.258) -- (2.299,8.243) -- (2.498,8.224) -- (2.698,8.2) -- (2.896,8.171) -- (3.094,8.138) -- (3.291,8.1) -- (3.487,8.058) -- (3.682,8.01) -- (3.876,7.959) -- (4.068,7.902) -- (4.259,7.842);
\draw[line width=0.5pt] (1.498,8.274) -- (1.474,8.054) -- (1.455,7.835) -- (1.442,7.614) -- (1.434,7.394) -- (1.431,7.173) -- (1.433,6.953) -- (1.441,6.732) -- (1.453,6.512) -- (1.471,6.292) -- (1.494,6.072) -- (1.522,5.854) -- (1.555,5.636) -- (1.594,5.418) -- (1.637,5.202);
\draw[line width=0.5pt] (4.259,7.842) -- (4.504,7.929) -- (4.752,8.01) -- (5.001,8.086) -- (5.252,8.155) -- (5.505,8.219) -- (5.759,8.277) -- (6.014,8.329) -- (6.271,8.375) -- (6.528,8.415) -- (6.787,8.448) -- (7.046,8.476) -- (7.305,8.497) -- (7.565,8.513) -- (7.825,8.522);
\draw[line width=0.5pt] (4.259,7.842) -- (4.263,7.671) -- (4.271,7.5) -- (4.282,7.33) -- (4.298,7.16) -- (4.318,6.99) -- (4.342,6.821) -- (4.369,6.652) -- (4.401,6.484) -- (4.437,6.317) -- (4.476,6.151) -- (4.52,5.985) -- (4.567,5.821) -- (4.618,5.658) -- (4.673,5.497);
\draw[line width=0.5pt] (4.673,5.497) -- (4.866,5.517) -- (5.059,5.533) -- (5.253,5.544) -- (5.447,5.551) -- (5.641,5.553) -- (5.835,5.55) -- (6.029,5.543) -- (6.223,5.532) -- (6.416,5.516) -- (6.609,5.495) -- (6.801,5.47) -- (6.993,5.44) -- (7.184,5.406) -- (7.374,5.367);
\draw[line width=0.5pt] (4.673,5.497) -- (4.64,5.266) -- (4.612,5.034) -- (4.589,4.802) -- (4.572,4.569) -- (4.56,4.336) -- (4.554,4.102) -- (4.553,3.869) -- (4.558,3.635) -- (4.568,3.402) -- (4.584,3.169) -- (4.605,2.936) -- (4.632,2.705) -- (4.664,2.473) -- (4.701,2.243);
\draw[line width=0.5pt] (7.374,5.367) -- (7.418,5.123) -- (7.457,4.879) -- (7.49,4.633) -- (7.518,4.386) -- (7.539,4.139) -- (7.555,3.892) -- (7.565,3.644) -- (7.569,3.396) -- (7.567,3.148) -- (7.559,2.9) -- (7.546,2.652) -- (7.527,2.405) -- (7.502,2.158) -- (7.471,1.912);
\draw[line width=0.5pt] (1.637,5.202) -- (1.855,5.19) -- (2.074,5.183) -- (2.293,5.181) -- (2.512,5.184) -- (2.731,5.192) -- (2.949,5.206) -- (3.167,5.224) -- (3.385,5.248) -- (3.602,5.277) -- (3.818,5.311) -- (4.034,5.35) -- (4.248,5.394) -- (4.461,5.443) -- (4.673,5.497);
\draw[line width=0.5pt] (1.637,5.202) -- (1.708,4.945) -- (1.773,4.686) -- (1.833,4.425) -- (1.885,4.163) -- (1.932,3.9) -- (1.973,3.636) -- (2.007,3.371) -- (2.035,3.105) -- (2.057,2.839) -- (2.072,2.572) -- (2.081,2.305) -- (2.084,2.038) -- (2.081,1.77) -- (2.071,1.504);
\draw[line width=0.5pt] (4.701,2.243) -- (4.894,2.189) -- (5.088,2.14) -- (5.283,2.096) -- (5.479,2.056) -- (5.676,2.02) -- (5.874,1.99) -- (6.072,1.964) -- (6.271,1.942) -- (6.471,1.925) -- (6.671,1.913) -- (6.871,1.906) -- (7.071,1.903) -- (7.271,1.906) -- (7.471,1.912);
\draw[line width=0.5pt] (2.071,1.504) -- (2.25,1.585) -- (2.43,1.662) -- (2.612,1.734) -- (2.796,1.803) -- (2.981,1.867) -- (3.168,1.927) -- (3.356,1.982) -- (3.545,2.033) -- (3.736,2.079) -- (3.927,2.121) -- (4.12,2.158) -- (4.313,2.191) -- (4.507,2.22) -- (4.701,2.243);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.539,11.504) -- (5.511,11.646) -- (5.43,11.768) -- (5.309,11.848) -- (5.166,11.877) -- (5.023,11.848) -- (4.902,11.768) -- (4.821,11.646) -- (4.793,11.504) -- (4.821,11.361) -- (4.902,11.24) -- (5.023,11.159) -- (5.166,11.13) -- (5.309,11.159) -- (5.43,11.24) -- (5.511,11.361) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.197,11.347) -- (8.168,11.49) -- (8.088,11.611) -- (7.966,11.692) -- (7.824,11.72) -- (7.681,11.692) -- (7.56,11.611) -- (7.479,11.49) -- (7.45,11.347) -- (7.479,11.204) -- (7.56,11.083) -- (7.681,11.002) -- (7.824,10.974) -- (7.966,11.002) -- (8.088,11.083) -- (8.168,11.204) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.959,10.775) -- (1.93,10.918) -- (1.849,11.039) -- (1.728,11.12) -- (1.585,11.149) -- (1.442,11.12) -- (1.321,11.039) -- (1.241,10.918) -- (1.212,10.775) -- (1.241,10.632) -- (1.321,10.511) -- (1.442,10.43) -- (1.585,10.402) -- (1.728,10.43) -- (1.849,10.511) -- (1.93,10.632) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.199,8.522) -- (8.17,8.665) -- (8.089,8.786) -- (7.968,8.867) -- (7.825,8.895) -- (7.682,8.867) -- (7.561,8.786) -- (7.48,8.665) -- (7.452,8.522) -- (7.48,8.379) -- (7.561,8.258) -- (7.682,8.177) -- (7.825,8.149) -- (7.968,8.177) -- (8.089,8.258) -- (8.17,8.379) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.871,8.274) -- (1.842,8.416) -- (1.762,8.537) -- (1.64,8.618) -- (1.498,8.647) -- (1.355,8.618) -- (1.234,8.537) -- (1.153,8.416) -- (1.124,8.274) -- (1.153,8.131) -- (1.234,8.01) -- (1.355,7.929) -- (1.498,7.9) -- (1.64,7.929) -- (1.762,8.01) -- (1.842,8.131) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.632,7.842) -- (4.604,7.984) -- (4.523,8.105) -- (4.402,8.186) -- (4.259,8.215) -- (4.116,8.186) -- (3.995,8.105) -- (3.914,7.984) -- (3.886,7.842) -- (3.914,7.699) -- (3.995,7.578) -- (4.116,7.497) -- (4.259,7.468) -- (4.402,7.497) -- (4.523,7.578) -- (4.604,7.699) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.047,5.497) -- (5.018,5.64) -- (4.937,5.761) -- (4.816,5.842) -- (4.673,5.87) -- (4.53,5.842) -- (4.409,5.761) -- (4.328,5.64) -- (4.3,5.497) -- (4.328,5.354) -- (4.409,5.233) -- (4.53,5.152) -- (4.673,5.124) -- (4.816,5.152) -- (4.937,5.233) -- (5.018,5.354) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.747,5.367) -- (7.718,5.51) -- (7.638,5.631) -- (7.516,5.712) -- (7.374,5.74) -- (7.231,5.712) -- (7.11,5.631) -- (7.029,5.51) -- (7.0,5.367) -- (7.029,5.224) -- (7.11,5.103) -- (7.231,5.022) -- (7.374,4.994) -- (7.516,5.022) -- (7.638,5.103) -- (7.718,5.224) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.01,5.202) -- (1.982,5.345) -- (1.901,5.466) -- (1.78,5.547) -- (1.637,5.575) -- (1.494,5.547) -- (1.373,5.466) -- (1.292,5.345) -- (1.264,5.202) -- (1.292,5.059) -- (1.373,4.938) -- (1.494,4.857) -- (1.637,4.829) -- (1.78,4.857) -- (1.901,4.938) -- (1.982,5.059) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.075,2.243) -- (5.046,2.386) -- (4.965,2.507) -- (4.844,2.588) -- (4.701,2.616) -- (4.558,2.588) -- (4.437,2.507) -- (4.357,2.386) -- (4.328,2.243) -- (4.357,2.1) -- (4.437,1.979) -- (4.558,1.898) -- (4.701,1.87) -- (4.844,1.898) -- (4.965,1.979) -- (5.046,2.1) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.844,1.912) -- (7.815,2.055) -- (7.735,2.176) -- (7.613,2.257) -- (7.471,2.286) -- (7.328,2.257) -- (7.207,2.176) -- (7.126,2.055) -- (7.097,1.912) -- (7.126,1.77) -- (7.207,1.649) -- (7.328,1.568) -- (7.471,1.539) -- (7.613,1.568) -- (7.735,1.649) -- (7.815,1.77) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.445,1.504) -- (2.416,1.646) -- (2.335,1.768) -- (2.214,1.848) -- (2.071,1.877) -- (1.929,1.848) -- (1.807,1.768) -- (1.727,1.646) -- (1.698,1.504) -- (1.727,1.361) -- (1.807,1.24) -- (1.929,1.159) -- (2.071,1.13) -- (2.214,1.159) -- (2.335,1.24) -- (2.416,1.361) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.511,11.315) -- (6.63,11.334) -- (6.737,11.389) -- (6.822,11.474) -- (6.877,11.581) -- (6.895,11.7) -- (6.877,11.818) -- (6.822,11.926) -- (6.737,12.011) -- (6.63,12.065) -- (6.511,12.084) -- (6.511,12.084) -- (6.392,12.065) -- (6.285,12.011) -- (6.2,11.926) -- (6.145,11.818) -- (6.126,11.7) -- (6.145,11.581) -- (6.2,11.474) -- (6.285,11.389) -- (6.392,11.334) -- (6.511,11.315) -- cycle;
\node at (6.511,11.7) {\scalebox{1.194}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.326,9.55) -- (8.444,9.569) -- (8.552,9.624) -- (8.637,9.709) -- (8.691,9.816) -- (8.71,9.935) -- (8.691,10.054) -- (8.637,10.161) -- (8.552,10.246) -- (8.444,10.301) -- (8.326,10.319) -- (8.326,10.319) -- (8.207,10.301) -- (8.1,10.246) -- (8.014,10.161) -- (7.96,10.054) -- (7.941,9.935) -- (7.96,9.816) -- (8.014,9.709) -- (8.1,9.624) -- (8.207,9.569) -- (8.326,9.55) -- cycle;
\node at (8.326,9.935) {\scalebox{1.194}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.329,10.984) -- (3.448,11.003) -- (3.555,11.057) -- (3.64,11.143) -- (3.695,11.25) -- (3.714,11.369) -- (3.695,11.487) -- (3.64,11.595) -- (3.555,11.68) -- (3.448,11.734) -- (3.329,11.753) -- (3.329,11.753) -- (3.21,11.734) -- (3.103,11.68) -- (3.018,11.595) -- (2.963,11.487) -- (2.945,11.369) -- (2.963,11.25) -- (3.018,11.143) -- (3.103,11.057) -- (3.21,11.003) -- (3.329,10.984) -- cycle;
\node at (3.329,11.369) {\scalebox{1.194}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.029,9.123) -- (2.148,9.142) -- (2.255,9.196) -- (2.34,9.281) -- (2.395,9.389) -- (2.413,9.507) -- (2.395,9.626) -- (2.34,9.733) -- (2.255,9.818) -- (2.148,9.873) -- (2.029,9.892) -- (2.029,9.892) -- (1.91,9.873) -- (1.803,9.818) -- (1.718,9.733) -- (1.663,9.626) -- (1.644,9.507) -- (1.663,9.389) -- (1.718,9.281) -- (1.803,9.196) -- (1.91,9.142) -- (2.029,9.123) -- cycle;
\node at (2.029,9.507) {\scalebox{1.194}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.85,6.524) -- (7.969,6.543) -- (8.076,6.598) -- (8.161,6.683) -- (8.216,6.79) -- (8.234,6.909) -- (8.216,7.028) -- (8.161,7.135) -- (8.076,7.22) -- (7.969,7.274) -- (7.85,7.293) -- (7.85,7.293) -- (7.731,7.274) -- (7.624,7.22) -- (7.539,7.135) -- (7.484,7.028) -- (7.465,6.909) -- (7.484,6.79) -- (7.539,6.683) -- (7.624,6.598) -- (7.731,6.543) -- (7.85,6.524) -- cycle;
\node at (7.85,6.909) {\scalebox{1.194}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.956,8.167) -- (3.074,8.186) -- (3.182,8.24) -- (3.267,8.325) -- (3.321,8.433) -- (3.34,8.551) -- (3.321,8.67) -- (3.267,8.777) -- (3.182,8.862) -- (3.074,8.917) -- (2.956,8.936) -- (2.956,8.936) -- (2.837,8.917) -- (2.73,8.862) -- (2.645,8.777) -- (2.59,8.67) -- (2.571,8.551) -- (2.59,8.433) -- (2.645,8.325) -- (2.73,8.24) -- (2.837,8.186) -- (2.956,8.167) -- cycle;
\node at (2.956,8.551) {\scalebox{1.194}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.825,6.365) -- (1.944,6.384) -- (2.051,6.439) -- (2.136,6.524) -- (2.191,6.631) -- (2.209,6.75) -- (2.191,6.868) -- (2.136,6.976) -- (2.051,7.061) -- (1.944,7.115) -- (1.825,7.134) -- (1.825,7.134) -- (1.706,7.115) -- (1.599,7.061) -- (1.514,6.976) -- (1.459,6.868) -- (1.44,6.75) -- (1.459,6.631) -- (1.514,6.524) -- (1.599,6.439) -- (1.706,6.384) -- (1.825,6.365) -- cycle;
\node at (1.825,6.75) {\scalebox{1.194}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.942,8.322) -- (6.061,8.341) -- (6.168,8.396) -- (6.253,8.481) -- (6.308,8.588) -- (6.327,8.707) -- (6.308,8.825) -- (6.253,8.933) -- (6.168,9.018) -- (6.061,9.072) -- (5.942,9.091) -- (5.942,9.091) -- (5.823,9.072) -- (5.716,9.018) -- (5.631,8.933) -- (5.576,8.825) -- (5.558,8.707) -- (5.576,8.588) -- (5.631,8.481) -- (5.716,8.396) -- (5.823,8.341) -- (5.942,8.322) -- cycle;
\node at (5.942,8.707) {\scalebox{1.194}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.748,6.334) -- (4.867,6.353) -- (4.974,6.408) -- (5.059,6.493) -- (5.114,6.6) -- (5.133,6.719) -- (5.114,6.838) -- (5.059,6.945) -- (4.974,7.03) -- (4.867,7.085) -- (4.748,7.103) -- (4.748,7.103) -- (4.629,7.085) -- (4.522,7.03) -- (4.437,6.945) -- (4.382,6.838) -- (4.364,6.719) -- (4.382,6.6) -- (4.437,6.493) -- (4.522,6.408) -- (4.629,6.353) -- (4.748,6.334) -- cycle;
\node at (4.748,6.719) {\scalebox{1.194}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.228,5.543) -- (6.347,5.562) -- (6.454,5.616) -- (6.539,5.701) -- (6.594,5.809) -- (6.613,5.927) -- (6.594,6.046) -- (6.539,6.153) -- (6.454,6.238) -- (6.347,6.293) -- (6.228,6.312) -- (5.866,6.312) -- (5.747,6.293) -- (5.64,6.238) -- (5.555,6.153) -- (5.501,6.046) -- (5.482,5.927) -- (5.501,5.809) -- (5.555,5.701) -- (5.64,5.616) -- (5.747,5.562) -- (5.866,5.543) -- cycle;
\node at (6.047,5.927) {\scalebox{1.194}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.35,3.481) -- (4.468,3.5) -- (4.576,3.554) -- (4.661,3.639) -- (4.715,3.747) -- (4.734,3.866) -- (4.715,3.984) -- (4.661,4.092) -- (4.576,4.177) -- (4.468,4.231) -- (4.35,4.25) -- (3.988,4.25) -- (3.869,4.231) -- (3.762,4.177) -- (3.677,4.092) -- (3.622,3.984) -- (3.603,3.866) -- (3.622,3.747) -- (3.677,3.639) -- (3.762,3.554) -- (3.869,3.5) -- (3.988,3.481) -- cycle;
\node at (4.169,3.866) {\scalebox{1.194}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.13,3.27) -- (8.249,3.289) -- (8.356,3.343) -- (8.441,3.429) -- (8.496,3.536) -- (8.514,3.655) -- (8.496,3.773) -- (8.441,3.881) -- (8.356,3.966) -- (8.249,4.02) -- (8.13,4.039) -- (7.768,4.039) -- (7.649,4.02) -- (7.542,3.966) -- (7.457,3.881) -- (7.402,3.773) -- (7.384,3.655) -- (7.402,3.536) -- (7.457,3.429) -- (7.542,3.343) -- (7.649,3.289) -- (7.768,3.27) -- cycle;
\node at (7.949,3.655) {\scalebox{1.194}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.311,5.222) -- (3.43,5.241) -- (3.537,5.296) -- (3.622,5.381) -- (3.677,5.488) -- (3.696,5.607) -- (3.677,5.726) -- (3.622,5.833) -- (3.537,5.918) -- (3.43,5.973) -- (3.311,5.992) -- (2.949,5.992) -- (2.83,5.973) -- (2.723,5.918) -- (2.638,5.833) -- (2.584,5.726) -- (2.565,5.607) -- (2.584,5.488) -- (2.638,5.381) -- (2.723,5.296) -- (2.83,5.241) -- (2.949,5.222) -- cycle;
\node at (3.13,5.607) {\scalebox{1.194}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.57,3.031) -- (2.688,3.05) -- (2.796,3.105) -- (2.881,3.19) -- (2.935,3.297) -- (2.954,3.416) -- (2.935,3.535) -- (2.881,3.642) -- (2.796,3.727) -- (2.688,3.781) -- (2.57,3.8) -- (2.208,3.8) -- (2.089,3.781) -- (1.982,3.727) -- (1.897,3.642) -- (1.842,3.535) -- (1.823,3.416) -- (1.842,3.297) -- (1.897,3.19) -- (1.982,3.105) -- (2.089,3.05) -- (2.208,3.031) -- cycle;
\node at (2.389,3.416) {\scalebox{1.194}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.299,1.961) -- (6.418,1.98) -- (6.525,2.034) -- (6.61,2.119) -- (6.665,2.227) -- (6.683,2.345) -- (6.665,2.464) -- (6.61,2.571) -- (6.525,2.656) -- (6.418,2.711) -- (6.299,2.73) -- (5.937,2.73) -- (5.818,2.711) -- (5.711,2.656) -- (5.626,2.571) -- (5.571,2.464) -- (5.552,2.345) -- (5.571,2.227) -- (5.626,2.119) -- (5.711,2.034) -- (5.818,1.98) -- (5.937,1.961) -- cycle;
\node at (6.118,2.345) {\scalebox{1.194}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.433,1.968) -- (3.552,1.986) -- (3.659,2.041) -- (3.744,2.126) -- (3.798,2.233) -- (3.817,2.352) -- (3.798,2.471) -- (3.744,2.578) -- (3.659,2.663) -- (3.552,2.718) -- (3.433,2.737) -- (3.071,2.737) -- (2.952,2.718) -- (2.845,2.663) -- (2.76,2.578) -- (2.705,2.471) -- (2.686,2.352) -- (2.705,2.233) -- (2.76,2.126) -- (2.845,2.041) -- (2.952,1.986) -- (3.071,1.968) -- cycle;
\node at (3.252,2.352) {\scalebox{1.194}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.373,5.197) -- (4.973,5.197) -- (4.973,5.797) -- (4.373,5.797) -- cycle;
\node at (4.673,5.497) {\scalebox{1.194}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (2.067,1.081) -- (2.667,1.081) -- (2.667,1.681) -- (2.067,1.681) -- cycle;
\node at (2.367,1.381) {\scalebox{1.194}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.575,10.45) -- (9.175,10.45) -- (9.175,11.05) -- (8.575,11.05) -- cycle;
\node at (8.875,10.75) {\scalebox{1.194}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.6,7.076) -- (1.2,7.076) -- (1.2,7.676) -- (0.6,7.676) -- cycle;
\node at (0.9,7.376) {\scalebox{1.194}{\small d}};
\node at (5.229,12.12) {\scalebox{1.194}{\small A}};
\node at (8.17,11.862) {\scalebox{1.194}{\small B}};
\node at (1.213,11.271) {\scalebox{1.194}{\small C}};
\node at (8.36,8.837) {\scalebox{1.194}{\small D}};
\node at (0.944,8.554) {\scalebox{1.194}{\small E}};
\node at (4.053,8.426) {\scalebox{1.194}{\small F}};
\node at (4.673,4.877) {\scalebox{1.194}{\small G}};
\node at (7.932,5.098) {\scalebox{1.194}{\small H}};
\node at (1.079,4.933) {\scalebox{1.194}{\small I}};
\node at (4.705,1.623) {\scalebox{1.194}{\small J}};
\node at (7.785,1.378) {\scalebox{1.194}{\small K}};
\node at (1.792,0.95) {\scalebox{1.194}{\small L}};
\draw[line width=0.5pt] (9.125,10.554) -- (9.125,11.654);
\draw[line width=0.5pt] (8.894,11.269) -- (9.125,11.654) -- (9.356,11.269);
\draw[line width=0.5pt] (8.823,11.104) -- (9.428,11.104);
\node at (9.125,12.039) {\scalebox{1.194}{\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, D, E, G, K, L & 6 \\ \hline
clairs & B, C, F, H, I, J & 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 Verte ⑨} vers le nord jusqu'à Carrefour du Lavoir \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Haute ⑧}, vers l'est, jusqu'à Rond-point de la Gare \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Moulin ②}, vers le nord, jusqu'à Place aux Herbes \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ①}, vers l'ouest, jusqu'à Coin du Lavoir \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{avenue de la Gare ③}, vers l'ouest, jusqu'à Place Sainte-Croix \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ④}, vers le sud, jusqu'à Carrefour de la Mairie \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin des Vignes ⑦}, vers le sud, jusqu'à Rond-point de la Poste \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Tilleuls ⑭}, vers le sud, jusqu'à Place des Tanneurs \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Neuve ⑯}, vers l'est, jusqu'à Carrefour du Moulin \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Stade ⑮}, vers l'est, jusqu'à Fontaine Notre-Dame \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{venelle du Chat ⑫}, vers le nord, jusqu'à Carrefour de l'Hôpital \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{quai aux Fleurs ⑩}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(G)}.\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}
