\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 271}

\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=271}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0271}
\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=0.6075, line join=round, line cap=round]
\draw[line width=0.5pt] (4.234,7.522) -- (4.494,7.481) -- (4.756,7.447) -- (5.018,7.418) -- (5.281,7.396) -- (5.545,7.38) -- (5.808,7.37) -- (6.072,7.366) -- (6.336,7.369) -- (6.6,7.377) -- (6.864,7.392) -- (7.127,7.413) -- (7.389,7.44) -- (7.651,7.474) -- (7.912,7.514);
\draw[line width=0.5pt] (4.234,7.522) -- (4.328,7.303) -- (4.417,7.082) -- (4.5,6.858) -- (4.579,6.633) -- (4.652,6.406) -- (4.72,6.177) -- (4.782,5.946) -- (4.839,5.715) -- (4.89,5.481) -- (4.936,5.247) -- (4.977,5.012) -- (5.012,4.776) -- (5.041,4.539) -- (5.065,4.302);
\draw[line width=0.5pt] (7.912,7.514) -- (8.146,7.443) -- (8.383,7.378) -- (8.62,7.319) -- (8.859,7.265) -- (9.1,7.217) -- (9.341,7.175) -- (9.584,7.138) -- (9.827,7.107) -- (10.071,7.082) -- (10.315,7.063) -- (10.56,7.049) -- (10.805,7.041) -- (11.05,7.039) -- (11.295,7.042);
\draw[line width=0.5pt] (7.912,7.514) -- (7.927,7.286) -- (7.937,7.058) -- (7.941,6.83) -- (7.94,6.602) -- (7.934,6.374) -- (7.922,6.146) -- (7.905,5.919) -- (7.883,5.692) -- (7.855,5.465) -- (7.822,5.24) -- (7.783,5.015) -- (7.74,4.791) -- (7.691,4.568) -- (7.637,4.347);
\draw[line width=0.5pt] (11.295,7.042) -- (11.249,6.862) -- (11.207,6.681) -- (11.17,6.499) -- (11.136,6.317) -- (11.108,6.133) -- (11.083,5.949) -- (11.063,5.764) -- (11.047,5.579) -- (11.036,5.393) -- (11.028,5.207) -- (11.026,5.021) -- (11.027,4.836) -- (11.033,4.65) -- (11.044,4.465);
\draw[line width=0.5pt] (7.637,4.347) -- (7.877,4.392) -- (8.118,4.432) -- (8.36,4.467) -- (8.603,4.495) -- (8.847,4.518) -- (9.091,4.535) -- (9.335,4.546) -- (9.58,4.552) -- (9.825,4.552) -- (10.069,4.546) -- (10.314,4.534) -- (10.558,4.517) -- (10.801,4.494) -- (11.044,4.465);
\draw[line width=0.5pt] (7.637,4.347) -- (7.599,4.155) -- (7.567,3.962) -- (7.538,3.769) -- (7.515,3.575) -- (7.495,3.38) -- (7.481,3.185) -- (7.471,2.989) -- (7.465,2.794) -- (7.465,2.598) -- (7.468,2.402) -- (7.477,2.207) -- (7.49,2.012) -- (7.507,1.817) -- (7.53,1.623);
\draw[line width=0.5pt] (5.065,4.302) -- (5.246,4.333) -- (5.429,4.36) -- (5.612,4.383) -- (5.796,4.402) -- (5.98,4.416) -- (6.164,4.425) -- (6.349,4.431) -- (6.534,4.432) -- (6.718,4.428) -- (6.903,4.421) -- (7.087,4.409) -- (7.271,4.392) -- (7.454,4.372) -- (7.637,4.347);
\draw[line width=0.5pt] (5.065,4.302) -- (5.009,4.118) -- (4.958,3.933) -- (4.912,3.746) -- (4.87,3.558) -- (4.832,3.369) -- (4.799,3.18) -- (4.77,2.99) -- (4.745,2.799) -- (4.726,2.607) -- (4.71,2.415) -- (4.699,2.223) -- (4.693,2.031) -- (4.691,1.839) -- (4.694,1.646);
\draw[line width=0.5pt] (1.518,4.156) -- (1.768,4.206) -- (2.019,4.249) -- (2.271,4.286) -- (2.523,4.317) -- (2.777,4.343) -- (3.031,4.362) -- (3.285,4.376) -- (3.54,4.383) -- (3.795,4.385) -- (4.049,4.38) -- (4.304,4.37) -- (4.558,4.353) -- (4.812,4.331) -- (5.065,4.302);
\draw[line width=0.5pt] (1.518,4.156) -- (1.472,3.964) -- (1.431,3.771) -- (1.394,3.576) -- (1.362,3.381) -- (1.334,3.185) -- (1.311,2.988) -- (1.293,2.791) -- (1.279,2.594) -- (1.27,2.396) -- (1.266,2.198) -- (1.266,2.0) -- (1.271,1.802) -- (1.281,1.605) -- (1.295,1.408);
\draw[line width=0.5pt] (4.694,1.646) -- (4.895,1.676) -- (5.097,1.7) -- (5.3,1.72) -- (5.503,1.735) -- (5.706,1.746) -- (5.909,1.751) -- (6.113,1.752) -- (6.316,1.748) -- (6.52,1.739) -- (6.723,1.725) -- (6.925,1.707) -- (7.128,1.684) -- (7.329,1.656) -- (7.53,1.623);
\draw[line width=0.5pt] (1.295,1.408) -- (1.538,1.387) -- (1.782,1.373) -- (2.027,1.364) -- (2.271,1.361) -- (2.516,1.364) -- (2.76,1.373) -- (3.004,1.387) -- (3.248,1.407) -- (3.491,1.433) -- (3.734,1.464) -- (3.976,1.501) -- (4.216,1.544) -- (4.456,1.592) -- (4.694,1.646);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.394,7.522) -- (4.372,7.602) -- (4.314,7.661) -- (4.234,7.682) -- (4.154,7.661) -- (4.095,7.602) -- (4.074,7.522) -- (4.095,7.442) -- (4.154,7.384) -- (4.234,7.362) -- (4.314,7.384) -- (4.372,7.442) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.072,7.514) -- (8.05,7.594) -- (7.992,7.652) -- (7.912,7.674) -- (7.832,7.652) -- (7.773,7.594) -- (7.752,7.514) -- (7.773,7.434) -- (7.832,7.375) -- (7.912,7.354) -- (7.992,7.375) -- (8.05,7.434) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.455,7.042) -- (11.433,7.122) -- (11.375,7.181) -- (11.295,7.202) -- (11.215,7.181) -- (11.156,7.122) -- (11.135,7.042) -- (11.156,6.962) -- (11.215,6.904) -- (11.295,6.882) -- (11.375,6.904) -- (11.433,6.962) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.204,4.465) -- (11.182,4.545) -- (11.124,4.603) -- (11.044,4.625) -- (10.964,4.603) -- (10.905,4.545) -- (10.884,4.465) -- (10.905,4.385) -- (10.964,4.326) -- (11.044,4.305) -- (11.124,4.326) -- (11.182,4.385) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.797,4.347) -- (7.775,4.427) -- (7.717,4.485) -- (7.637,4.507) -- (7.557,4.485) -- (7.498,4.427) -- (7.477,4.347) -- (7.498,4.267) -- (7.557,4.208) -- (7.637,4.187) -- (7.717,4.208) -- (7.775,4.267) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.225,4.302) -- (5.203,4.382) -- (5.145,4.441) -- (5.065,4.462) -- (4.985,4.441) -- (4.926,4.382) -- (4.905,4.302) -- (4.926,4.222) -- (4.985,4.164) -- (5.065,4.142) -- (5.145,4.164) -- (5.203,4.222) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.678,4.156) -- (1.657,4.236) -- (1.598,4.295) -- (1.518,4.316) -- (1.438,4.295) -- (1.379,4.236) -- (1.358,4.156) -- (1.379,4.076) -- (1.438,4.018) -- (1.518,3.996) -- (1.598,4.018) -- (1.657,4.076) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.854,1.646) -- (4.833,1.726) -- (4.774,1.785) -- (4.694,1.806) -- (4.614,1.785) -- (4.555,1.726) -- (4.534,1.646) -- (4.555,1.566) -- (4.614,1.508) -- (4.694,1.486) -- (4.774,1.508) -- (4.833,1.566) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.69,1.623) -- (7.668,1.703) -- (7.61,1.762) -- (7.53,1.783) -- (7.45,1.762) -- (7.391,1.703) -- (7.37,1.623) -- (7.391,1.543) -- (7.45,1.484) -- (7.53,1.463) -- (7.61,1.484) -- (7.668,1.543) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.455,1.408) -- (1.433,1.488) -- (1.375,1.546) -- (1.295,1.568) -- (1.215,1.546) -- (1.156,1.488) -- (1.135,1.408) -- (1.156,1.328) -- (1.215,1.269) -- (1.295,1.248) -- (1.375,1.269) -- (1.433,1.328) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.073,7.366) -- (6.195,7.385) -- (6.305,7.441) -- (6.392,7.528) -- (6.448,7.638) -- (6.467,7.76) -- (6.448,7.882) -- (6.392,7.992) -- (6.305,8.079) -- (6.195,8.135) -- (6.073,8.154) -- (6.073,8.154) -- (5.952,8.135) -- (5.842,8.079) -- (5.755,7.992) -- (5.699,7.882) -- (5.679,7.76) -- (5.699,7.638) -- (5.755,7.528) -- (5.842,7.441) -- (5.952,7.385) -- (6.073,7.366) -- cycle;
\node at (6.073,7.76) {\scalebox{1.646}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.163,5.651) -- (5.285,5.67) -- (5.395,5.726) -- (5.482,5.813) -- (5.538,5.923) -- (5.557,6.045) -- (5.538,6.167) -- (5.482,6.276) -- (5.395,6.363) -- (5.285,6.419) -- (5.163,6.439) -- (5.163,6.439) -- (5.042,6.419) -- (4.932,6.363) -- (4.845,6.276) -- (4.789,6.167) -- (4.769,6.045) -- (4.789,5.923) -- (4.845,5.813) -- (4.932,5.726) -- (5.042,5.67) -- (5.163,5.651) -- cycle;
\node at (5.163,6.045) {\scalebox{1.646}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.638,7.135) -- (9.76,7.154) -- (9.87,7.21) -- (9.957,7.297) -- (10.013,7.407) -- (10.032,7.529) -- (10.013,7.65) -- (9.957,7.76) -- (9.87,7.847) -- (9.76,7.903) -- (9.638,7.922) -- (9.638,7.922) -- (9.516,7.903) -- (9.407,7.847) -- (9.319,7.76) -- (9.264,7.65) -- (9.244,7.529) -- (9.264,7.407) -- (9.319,7.297) -- (9.407,7.21) -- (9.516,7.154) -- (9.638,7.135) -- cycle;
\node at (9.638,7.529) {\scalebox{1.646}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.297,5.491) -- (8.419,5.51) -- (8.529,5.566) -- (8.616,5.653) -- (8.672,5.763) -- (8.691,5.885) -- (8.672,6.006) -- (8.616,6.116) -- (8.529,6.203) -- (8.419,6.259) -- (8.297,6.279) -- (8.297,6.279) -- (8.176,6.259) -- (8.066,6.203) -- (7.979,6.116) -- (7.923,6.006) -- (7.903,5.885) -- (7.923,5.763) -- (7.979,5.653) -- (8.066,5.566) -- (8.176,5.51) -- (8.297,5.491) -- cycle;
\node at (8.297,5.885) {\scalebox{1.646}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.455,5.332) -- (11.577,5.351) -- (11.687,5.407) -- (11.774,5.494) -- (11.83,5.604) -- (11.849,5.726) -- (11.83,5.847) -- (11.774,5.957) -- (11.687,6.044) -- (11.577,6.1) -- (11.455,6.12) -- (11.455,6.12) -- (11.333,6.1) -- (11.223,6.044) -- (11.136,5.957) -- (11.08,5.847) -- (11.061,5.726) -- (11.08,5.604) -- (11.136,5.494) -- (11.223,5.407) -- (11.333,5.351) -- (11.455,5.332) -- cycle;
\node at (11.455,5.726) {\scalebox{1.646}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.322,4.546) -- (9.444,4.565) -- (9.553,4.621) -- (9.64,4.708) -- (9.696,4.818) -- (9.716,4.94) -- (9.696,5.062) -- (9.64,5.171) -- (9.553,5.259) -- (9.444,5.315) -- (9.322,5.334) -- (9.322,5.334) -- (9.2,5.315) -- (9.09,5.259) -- (9.003,5.171) -- (8.947,5.062) -- (8.928,4.94) -- (8.947,4.818) -- (9.003,4.708) -- (9.09,4.621) -- (9.2,4.565) -- (9.322,4.546) -- cycle;
\node at (9.322,4.94) {\scalebox{1.646}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.864,2.58) -- (7.986,2.599) -- (8.096,2.655) -- (8.183,2.742) -- (8.239,2.852) -- (8.258,2.974) -- (8.239,3.096) -- (8.183,3.205) -- (8.096,3.292) -- (7.986,3.348) -- (7.864,3.368) -- (7.864,3.368) -- (7.743,3.348) -- (7.633,3.292) -- (7.546,3.205) -- (7.49,3.096) -- (7.471,2.974) -- (7.49,2.852) -- (7.546,2.742) -- (7.633,2.655) -- (7.743,2.599) -- (7.864,2.58) -- cycle;
\node at (7.864,2.974) {\scalebox{1.646}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.342,4.431) -- (6.464,4.45) -- (6.574,4.506) -- (6.661,4.593) -- (6.717,4.703) -- (6.736,4.824) -- (6.717,4.946) -- (6.661,5.056) -- (6.574,5.143) -- (6.464,5.199) -- (6.342,5.218) -- (6.342,5.218) -- (6.22,5.199) -- (6.11,5.143) -- (6.023,5.056) -- (5.967,4.946) -- (5.948,4.824) -- (5.967,4.703) -- (6.023,4.593) -- (6.11,4.506) -- (6.22,4.45) -- (6.342,4.431) -- cycle;
\node at (6.342,4.824) {\scalebox{1.646}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.16,2.541) -- (5.282,2.561) -- (5.391,2.617) -- (5.479,2.704) -- (5.534,2.813) -- (5.554,2.935) -- (5.534,3.057) -- (5.479,3.167) -- (5.391,3.254) -- (5.282,3.31) -- (5.16,3.329) -- (5.16,3.329) -- (5.038,3.31) -- (4.928,3.254) -- (4.841,3.167) -- (4.785,3.057) -- (4.766,2.935) -- (4.785,2.813) -- (4.841,2.704) -- (4.928,2.617) -- (5.038,2.561) -- (5.16,2.541) -- cycle;
\node at (5.16,2.935) {\scalebox{1.646}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.454,4.375) -- (3.576,4.395) -- (3.686,4.45) -- (3.773,4.538) -- (3.829,4.647) -- (3.848,4.769) -- (3.829,4.891) -- (3.773,5.001) -- (3.686,5.088) -- (3.576,5.144) -- (3.454,5.163) -- (3.084,5.163) -- (2.962,5.144) -- (2.852,5.088) -- (2.765,5.001) -- (2.709,4.891) -- (2.69,4.769) -- (2.709,4.647) -- (2.765,4.538) -- (2.852,4.45) -- (2.962,4.395) -- (3.084,4.375) -- cycle;
\node at (3.269,4.769) {\scalebox{1.646}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.871,2.365) -- (1.993,2.385) -- (2.103,2.441) -- (2.19,2.528) -- (2.246,2.638) -- (2.265,2.759) -- (2.246,2.881) -- (2.19,2.991) -- (2.103,3.078) -- (1.993,3.134) -- (1.871,3.153) -- (1.5,3.153) -- (1.379,3.134) -- (1.269,3.078) -- (1.182,2.991) -- (1.126,2.881) -- (1.106,2.759) -- (1.126,2.638) -- (1.182,2.528) -- (1.269,2.441) -- (1.379,2.385) -- (1.5,2.365) -- cycle;
\node at (1.686,2.759) {\scalebox{1.646}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.301,1.752) -- (6.423,1.771) -- (6.533,1.827) -- (6.62,1.914) -- (6.676,2.024) -- (6.695,2.146) -- (6.676,2.267) -- (6.62,2.377) -- (6.533,2.464) -- (6.423,2.52) -- (6.301,2.54) -- (5.931,2.54) -- (5.809,2.52) -- (5.699,2.464) -- (5.612,2.377) -- (5.556,2.267) -- (5.537,2.146) -- (5.556,2.024) -- (5.612,1.914) -- (5.699,1.827) -- (5.809,1.771) -- (5.931,1.752) -- cycle;
\node at (6.116,2.146) {\scalebox{1.646}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.217,0.6) -- (3.339,0.619) -- (3.449,0.675) -- (3.536,0.762) -- (3.592,0.872) -- (3.611,0.994) -- (3.592,1.116) -- (3.536,1.225) -- (3.449,1.313) -- (3.339,1.369) -- (3.217,1.388) -- (2.847,1.388) -- (2.725,1.369) -- (2.615,1.313) -- (2.528,1.225) -- (2.472,1.116) -- (2.453,0.994) -- (2.472,0.872) -- (2.528,0.762) -- (2.615,0.675) -- (2.725,0.619) -- (2.847,0.6) -- cycle;
\node at (3.032,0.994) {\scalebox{1.646}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.765,4.002) -- (5.365,4.002) -- (5.365,4.602) -- (4.765,4.602) -- cycle;
\node at (5.065,4.302) {\scalebox{1.646}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.746,1.308) -- (2.346,1.308) -- (2.346,1.908) -- (1.746,1.908) -- cycle;
\node at (2.046,1.608) {\scalebox{1.646}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.113,0.929) -- (5.713,0.929) -- (5.713,1.529) -- (5.113,1.529) -- cycle;
\node at (5.413,1.229) {\scalebox{1.646}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.884,4.667) -- (2.484,4.667) -- (2.484,5.267) -- (1.884,5.267) -- cycle;
\node at (2.184,4.967) {\scalebox{1.646}{\small d}};
\node at (3.901,8.045) {\scalebox{1.646}{\small A}};
\node at (8.208,8.058) {\scalebox{1.646}{\small B}};
\node at (11.845,7.329) {\scalebox{1.646}{\small C}};
\node at (11.664,4.473) {\scalebox{1.646}{\small D}};
\node at (8.256,4.322) {\scalebox{1.646}{\small E}};
\node at (4.447,4.249) {\scalebox{1.646}{\small F}};
\node at (0.899,4.124) {\scalebox{1.646}{\small G}};
\node at (4.393,1.104) {\scalebox{1.646}{\small H}};
\node at (7.794,1.062) {\scalebox{1.646}{\small I}};
\node at (0.765,1.086) {\scalebox{1.646}{\small J}};
\draw[line width=0.5pt] (12.595,6.572) -- (12.595,7.672);
\draw[line width=0.5pt] (12.364,7.287) -- (12.595,7.672) -- (12.826,7.287);
\draw[line width=0.5pt] (12.292,7.122) -- (12.897,7.122);
\node at (12.595,8.057) {\scalebox{1.646}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour F, 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 & ⑧ rue Pasteur & Ⓑ Carrefour Saint-Roch & Ⓙ Coin du Lavoir \\ \hline
① passage du Marché & ⑨ sentier des Prés & Ⓒ Place aux Herbes & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
② rue Verte & ⑩ rue de la Mairie & Ⓓ Place de l'Église & \textbf{a} l'école \\ \hline
③ quai aux Fleurs & ⑪ rue du Moulin & Ⓔ Rond-point du Pont & \textbf{b} le lavoir \\ \hline
④ rue Basse & ⑫ rue des Lilas & Ⓕ Carrefour de l'Hôpital & \textbf{c} la boulangerie \\ \hline
⑤ rue du Four & ⑬ rue de l'Église & Ⓖ Place du Vieux-Puits & \textbf{d} le stade \\ \hline
⑥ chemin Creux & \cellcolor{black!32}\bfseries Les carrefours & Ⓗ Rond-point de la Poste &  \\ \hline
⑦ chemin des Vignes & Ⓐ Placette des Ormes & Ⓘ Fontaine Notre-Dame &  \\ \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=0.6223, line join=round, line cap=round]
\draw[line width=0.5pt] (4.234,7.503) -- (4.494,7.462) -- (4.756,7.428) -- (5.018,7.399) -- (5.281,7.377) -- (5.545,7.361) -- (5.808,7.351) -- (6.072,7.347) -- (6.336,7.35) -- (6.6,7.359) -- (6.864,7.373) -- (7.127,7.394) -- (7.389,7.422) -- (7.651,7.455) -- (7.912,7.495);
\draw[line width=0.5pt] (4.234,7.503) -- (4.328,7.284) -- (4.417,7.063) -- (4.5,6.84) -- (4.579,6.614) -- (4.652,6.387) -- (4.72,6.158) -- (4.782,5.928) -- (4.839,5.696) -- (4.89,5.463) -- (4.936,5.229) -- (4.977,4.993) -- (5.012,4.757) -- (5.041,4.521) -- (5.065,4.283);
\draw[line width=0.5pt] (7.912,7.495) -- (8.146,7.424) -- (8.383,7.36) -- (8.62,7.3) -- (8.859,7.247) -- (9.1,7.199) -- (9.341,7.156) -- (9.584,7.12) -- (9.827,7.089) -- (10.071,7.063) -- (10.315,7.044) -- (10.56,7.03) -- (10.805,7.022) -- (11.05,7.02) -- (11.295,7.024);
\draw[line width=0.5pt] (7.912,7.495) -- (7.927,7.267) -- (7.937,7.04) -- (7.941,6.812) -- (7.94,6.584) -- (7.934,6.355) -- (7.922,6.128) -- (7.905,5.9) -- (7.883,5.673) -- (7.855,5.447) -- (7.822,5.221) -- (7.783,4.996) -- (7.74,4.772) -- (7.691,4.549) -- (7.637,4.328);
\draw[line width=0.5pt] (11.295,7.024) -- (11.249,6.844) -- (11.207,6.663) -- (11.17,6.481) -- (11.136,6.298) -- (11.108,6.114) -- (11.083,5.93) -- (11.063,5.745) -- (11.047,5.56) -- (11.036,5.374) -- (11.028,5.189) -- (11.026,5.003) -- (11.027,4.817) -- (11.033,4.631) -- (11.044,4.446);
\draw[line width=0.5pt] (7.637,4.328) -- (7.877,4.374) -- (8.118,4.414) -- (8.36,4.448) -- (8.603,4.476) -- (8.847,4.499) -- (9.091,4.516) -- (9.335,4.527) -- (9.58,4.533) -- (9.825,4.533) -- (10.069,4.527) -- (10.314,4.515) -- (10.558,4.498) -- (10.801,4.475) -- (11.044,4.446);
\draw[line width=0.5pt] (7.637,4.328) -- (7.599,4.136) -- (7.567,3.943) -- (7.538,3.75) -- (7.515,3.556) -- (7.495,3.361) -- (7.481,3.166) -- (7.471,2.971) -- (7.465,2.775) -- (7.465,2.579) -- (7.468,2.384) -- (7.477,2.188) -- (7.49,1.993) -- (7.507,1.798) -- (7.53,1.604);
\draw[line width=0.5pt] (5.065,4.283) -- (5.246,4.315) -- (5.429,4.342) -- (5.612,4.364) -- (5.796,4.383) -- (5.98,4.397) -- (6.164,4.406) -- (6.349,4.412) -- (6.534,4.413) -- (6.718,4.41) -- (6.903,4.402) -- (7.087,4.39) -- (7.271,4.374) -- (7.454,4.353) -- (7.637,4.328);
\draw[line width=0.5pt] (5.065,4.283) -- (5.009,4.099) -- (4.958,3.914) -- (4.912,3.727) -- (4.87,3.539) -- (4.832,3.351) -- (4.799,3.161) -- (4.77,2.971) -- (4.745,2.78) -- (4.726,2.589) -- (4.71,2.397) -- (4.699,2.205) -- (4.693,2.012) -- (4.691,1.82) -- (4.694,1.628);
\draw[line width=0.5pt] (1.518,4.138) -- (1.768,4.187) -- (2.019,4.23) -- (2.271,4.267) -- (2.523,4.299) -- (2.777,4.324) -- (3.031,4.343) -- (3.285,4.357) -- (3.54,4.364) -- (3.795,4.366) -- (4.049,4.361) -- (4.304,4.351) -- (4.558,4.334) -- (4.812,4.312) -- (5.065,4.283);
\draw[line width=0.5pt] (1.518,4.138) -- (1.472,3.945) -- (1.431,3.752) -- (1.394,3.557) -- (1.362,3.362) -- (1.334,3.166) -- (1.311,2.97) -- (1.293,2.772) -- (1.279,2.575) -- (1.27,2.377) -- (1.266,2.179) -- (1.266,1.981) -- (1.271,1.784) -- (1.281,1.586) -- (1.295,1.389);
\draw[line width=0.5pt] (4.694,1.628) -- (4.895,1.657) -- (5.097,1.682) -- (5.3,1.702) -- (5.503,1.717) -- (5.706,1.727) -- (5.909,1.732) -- (6.113,1.733) -- (6.316,1.729) -- (6.52,1.72) -- (6.723,1.706) -- (6.925,1.688) -- (7.128,1.665) -- (7.329,1.637) -- (7.53,1.604);
\draw[line width=0.5pt] (1.295,1.389) -- (1.538,1.369) -- (1.782,1.354) -- (2.027,1.345) -- (2.271,1.342) -- (2.516,1.345) -- (2.76,1.354) -- (3.004,1.368) -- (3.248,1.388) -- (3.491,1.414) -- (3.734,1.445) -- (3.976,1.482) -- (4.216,1.525) -- (4.456,1.574) -- (4.694,1.628);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.607,7.503) -- (4.579,7.646) -- (4.498,7.767) -- (4.377,7.848) -- (4.234,7.877) -- (4.091,7.848) -- (3.97,7.767) -- (3.889,7.646) -- (3.861,7.503) -- (3.889,7.361) -- (3.97,7.239) -- (4.091,7.159) -- (4.234,7.13) -- (4.377,7.159) -- (4.498,7.239) -- (4.579,7.361) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.285,7.495) -- (8.257,7.638) -- (8.176,7.759) -- (8.055,7.84) -- (7.912,7.868) -- (7.769,7.84) -- (7.648,7.759) -- (7.567,7.638) -- (7.539,7.495) -- (7.567,7.352) -- (7.648,7.231) -- (7.769,7.15) -- (7.912,7.122) -- (8.055,7.15) -- (8.176,7.231) -- (8.257,7.352) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.668,7.024) -- (11.64,7.166) -- (11.559,7.288) -- (11.438,7.368) -- (11.295,7.397) -- (11.152,7.368) -- (11.031,7.288) -- (10.95,7.166) -- (10.922,7.024) -- (10.95,6.881) -- (11.031,6.76) -- (11.152,6.679) -- (11.295,6.65) -- (11.438,6.679) -- (11.559,6.76) -- (11.64,6.881) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.417,4.446) -- (11.389,4.589) -- (11.308,4.71) -- (11.187,4.791) -- (11.044,4.819) -- (10.901,4.791) -- (10.78,4.71) -- (10.699,4.589) -- (10.671,4.446) -- (10.699,4.303) -- (10.78,4.182) -- (10.901,4.101) -- (11.044,4.073) -- (11.187,4.101) -- (11.308,4.182) -- (11.389,4.303) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.01,4.328) -- (7.982,4.471) -- (7.901,4.592) -- (7.78,4.673) -- (7.637,4.701) -- (7.494,4.673) -- (7.373,4.592) -- (7.292,4.471) -- (7.264,4.328) -- (7.292,4.185) -- (7.373,4.064) -- (7.494,3.983) -- (7.637,3.955) -- (7.78,3.983) -- (7.901,4.064) -- (7.982,4.185) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.438,4.283) -- (5.409,4.426) -- (5.328,4.547) -- (5.207,4.628) -- (5.065,4.657) -- (4.922,4.628) -- (4.801,4.547) -- (4.72,4.426) -- (4.691,4.283) -- (4.72,4.141) -- (4.801,4.02) -- (4.922,3.939) -- (5.065,3.91) -- (5.207,3.939) -- (5.328,4.02) -- (5.409,4.141) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.891,4.138) -- (1.863,4.28) -- (1.782,4.402) -- (1.661,4.482) -- (1.518,4.511) -- (1.375,4.482) -- (1.254,4.402) -- (1.173,4.28) -- (1.145,4.138) -- (1.173,3.995) -- (1.254,3.874) -- (1.375,3.793) -- (1.518,3.764) -- (1.661,3.793) -- (1.782,3.874) -- (1.863,3.995) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.067,1.628) -- (5.039,1.771) -- (4.958,1.892) -- (4.837,1.973) -- (4.694,2.001) -- (4.551,1.973) -- (4.43,1.892) -- (4.349,1.771) -- (4.321,1.628) -- (4.349,1.485) -- (4.43,1.364) -- (4.551,1.283) -- (4.694,1.255) -- (4.837,1.283) -- (4.958,1.364) -- (5.039,1.485) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.903,1.604) -- (7.874,1.747) -- (7.794,1.868) -- (7.672,1.949) -- (7.53,1.977) -- (7.387,1.949) -- (7.266,1.868) -- (7.185,1.747) -- (7.156,1.604) -- (7.185,1.461) -- (7.266,1.34) -- (7.387,1.259) -- (7.53,1.231) -- (7.672,1.259) -- (7.794,1.34) -- (7.874,1.461) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.668,1.389) -- (1.64,1.532) -- (1.559,1.653) -- (1.438,1.734) -- (1.295,1.762) -- (1.152,1.734) -- (1.031,1.653) -- (0.95,1.532) -- (0.922,1.389) -- (0.95,1.246) -- (1.031,1.125) -- (1.152,1.044) -- (1.295,1.016) -- (1.438,1.044) -- (1.559,1.125) -- (1.64,1.246) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.073,7.347) -- (6.192,7.366) -- (6.299,7.421) -- (6.384,7.506) -- (6.439,7.613) -- (6.458,7.732) -- (6.439,7.851) -- (6.384,7.958) -- (6.299,8.043) -- (6.192,8.098) -- (6.073,8.116) -- (6.073,8.116) -- (5.954,8.098) -- (5.847,8.043) -- (5.762,7.958) -- (5.708,7.851) -- (5.689,7.732) -- (5.708,7.613) -- (5.762,7.506) -- (5.847,7.421) -- (5.954,7.366) -- (6.073,7.347) -- cycle;
\node at (6.073,7.732) {\scalebox{1.607}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.154,5.639) -- (5.273,5.658) -- (5.38,5.713) -- (5.465,5.798) -- (5.52,5.905) -- (5.539,6.024) -- (5.52,6.143) -- (5.465,6.25) -- (5.38,6.335) -- (5.273,6.389) -- (5.154,6.408) -- (5.154,6.408) -- (5.035,6.389) -- (4.928,6.335) -- (4.843,6.25) -- (4.789,6.143) -- (4.77,6.024) -- (4.789,5.905) -- (4.843,5.798) -- (4.928,5.713) -- (5.035,5.658) -- (5.154,5.639) -- cycle;
\node at (5.154,6.024) {\scalebox{1.607}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.637,7.116) -- (9.756,7.135) -- (9.863,7.189) -- (9.948,7.274) -- (10.003,7.382) -- (10.021,7.501) -- (10.003,7.619) -- (9.948,7.727) -- (9.863,7.812) -- (9.756,7.866) -- (9.637,7.885) -- (9.637,7.885) -- (9.518,7.866) -- (9.411,7.812) -- (9.326,7.727) -- (9.271,7.619) -- (9.252,7.501) -- (9.271,7.382) -- (9.326,7.274) -- (9.411,7.189) -- (9.518,7.135) -- (9.637,7.116) -- cycle;
\node at (9.637,7.501) {\scalebox{1.607}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.288,5.482) -- (8.407,5.501) -- (8.514,5.556) -- (8.599,5.641) -- (8.654,5.748) -- (8.673,5.867) -- (8.654,5.986) -- (8.599,6.093) -- (8.514,6.178) -- (8.407,6.233) -- (8.288,6.251) -- (8.288,6.251) -- (8.169,6.233) -- (8.062,6.178) -- (7.977,6.093) -- (7.922,5.986) -- (7.903,5.867) -- (7.922,5.748) -- (7.977,5.641) -- (8.062,5.556) -- (8.169,5.501) -- (8.288,5.482) -- cycle;
\node at (8.288,5.867) {\scalebox{1.607}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.446,5.323) -- (11.564,5.342) -- (11.672,5.397) -- (11.757,5.482) -- (11.811,5.589) -- (11.83,5.708) -- (11.811,5.827) -- (11.757,5.934) -- (11.672,6.019) -- (11.564,6.074) -- (11.446,6.092) -- (11.446,6.092) -- (11.327,6.074) -- (11.22,6.019) -- (11.135,5.934) -- (11.08,5.827) -- (11.061,5.708) -- (11.08,5.589) -- (11.135,5.482) -- (11.22,5.397) -- (11.327,5.342) -- (11.446,5.323) -- cycle;
\node at (11.446,5.708) {\scalebox{1.607}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.322,4.527) -- (9.441,4.546) -- (9.548,4.601) -- (9.633,4.686) -- (9.688,4.793) -- (9.707,4.912) -- (9.688,5.031) -- (9.633,5.138) -- (9.548,5.223) -- (9.441,5.277) -- (9.322,5.296) -- (9.322,5.296) -- (9.203,5.277) -- (9.096,5.223) -- (9.011,5.138) -- (8.956,5.031) -- (8.938,4.912) -- (8.956,4.793) -- (9.011,4.686) -- (9.096,4.601) -- (9.203,4.546) -- (9.322,4.527) -- cycle;
\node at (9.322,4.912) {\scalebox{1.607}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.855,2.571) -- (7.974,2.59) -- (8.081,2.644) -- (8.166,2.729) -- (8.221,2.837) -- (8.24,2.955) -- (8.221,3.074) -- (8.166,3.181) -- (8.081,3.267) -- (7.974,3.321) -- (7.855,3.34) -- (7.855,3.34) -- (7.736,3.321) -- (7.629,3.267) -- (7.544,3.181) -- (7.489,3.074) -- (7.471,2.955) -- (7.489,2.837) -- (7.544,2.729) -- (7.629,2.644) -- (7.736,2.59) -- (7.855,2.571) -- cycle;
\node at (7.855,2.955) {\scalebox{1.607}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.342,4.412) -- (6.461,4.431) -- (6.568,4.485) -- (6.653,4.57) -- (6.708,4.677) -- (6.727,4.796) -- (6.708,4.915) -- (6.653,5.022) -- (6.568,5.107) -- (6.461,5.162) -- (6.342,5.181) -- (6.342,5.181) -- (6.223,5.162) -- (6.116,5.107) -- (6.031,5.022) -- (5.976,4.915) -- (5.958,4.796) -- (5.976,4.677) -- (6.031,4.57) -- (6.116,4.485) -- (6.223,4.431) -- (6.342,4.412) -- cycle;
\node at (6.342,4.796) {\scalebox{1.607}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.151,2.533) -- (5.269,2.552) -- (5.377,2.607) -- (5.462,2.692) -- (5.516,2.799) -- (5.535,2.918) -- (5.516,3.037) -- (5.462,3.144) -- (5.377,3.229) -- (5.269,3.283) -- (5.151,3.302) -- (5.151,3.302) -- (5.032,3.283) -- (4.925,3.229) -- (4.839,3.144) -- (4.785,3.037) -- (4.766,2.918) -- (4.785,2.799) -- (4.839,2.692) -- (4.925,2.607) -- (5.032,2.552) -- (5.151,2.533) -- cycle;
\node at (5.151,2.918) {\scalebox{1.607}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.45,4.357) -- (3.569,4.375) -- (3.676,4.43) -- (3.762,4.515) -- (3.816,4.622) -- (3.835,4.741) -- (3.816,4.86) -- (3.762,4.967) -- (3.676,5.052) -- (3.569,5.107) -- (3.45,5.126) -- (3.089,5.126) -- (2.97,5.107) -- (2.862,5.052) -- (2.777,4.967) -- (2.723,4.86) -- (2.704,4.741) -- (2.723,4.622) -- (2.777,4.515) -- (2.862,4.43) -- (2.97,4.375) -- (3.089,4.357) -- cycle;
\node at (3.269,4.741) {\scalebox{1.607}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.857,2.357) -- (1.976,2.376) -- (2.083,2.43) -- (2.168,2.515) -- (2.223,2.623) -- (2.242,2.741) -- (2.223,2.86) -- (2.168,2.967) -- (2.083,3.052) -- (1.976,3.107) -- (1.857,3.126) -- (1.495,3.126) -- (1.377,3.107) -- (1.269,3.052) -- (1.184,2.967) -- (1.13,2.86) -- (1.111,2.741) -- (1.13,2.623) -- (1.184,2.515) -- (1.269,2.43) -- (1.377,2.376) -- (1.495,2.357) -- cycle;
\node at (1.676,2.741) {\scalebox{1.607}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.297,1.733) -- (6.416,1.752) -- (6.523,1.806) -- (6.608,1.891) -- (6.663,1.999) -- (6.681,2.117) -- (6.663,2.236) -- (6.608,2.344) -- (6.523,2.429) -- (6.416,2.483) -- (6.297,2.502) -- (5.935,2.502) -- (5.816,2.483) -- (5.709,2.429) -- (5.624,2.344) -- (5.569,2.236) -- (5.551,2.117) -- (5.569,1.999) -- (5.624,1.891) -- (5.709,1.806) -- (5.816,1.752) -- (5.935,1.733) -- cycle;
\node at (6.116,2.117) {\scalebox{1.607}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.212,0.6) -- (3.331,0.619) -- (3.438,0.673) -- (3.523,0.759) -- (3.578,0.866) -- (3.597,0.985) -- (3.578,1.103) -- (3.523,1.211) -- (3.438,1.296) -- (3.331,1.35) -- (3.212,1.369) -- (2.85,1.369) -- (2.731,1.35) -- (2.624,1.296) -- (2.539,1.211) -- (2.485,1.103) -- (2.466,0.985) -- (2.485,0.866) -- (2.539,0.759) -- (2.624,0.673) -- (2.731,0.619) -- (2.85,0.6) -- cycle;
\node at (3.031,0.985) {\scalebox{1.607}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.765,3.983) -- (5.365,3.983) -- (5.365,4.583) -- (4.765,4.583) -- cycle;
\node at (5.065,4.283) {\scalebox{1.607}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.746,1.289) -- (2.346,1.289) -- (2.346,1.889) -- (1.746,1.889) -- cycle;
\node at (2.046,1.589) {\scalebox{1.607}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.113,0.911) -- (5.713,0.911) -- (5.713,1.511) -- (5.113,1.511) -- cycle;
\node at (5.413,1.211) {\scalebox{1.607}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (1.884,4.648) -- (2.484,4.648) -- (2.484,5.248) -- (1.884,5.248) -- cycle;
\node at (2.184,4.948) {\scalebox{1.607}{\small d}};
\node at (3.901,8.026) {\scalebox{1.607}{\small A}};
\node at (8.208,8.04) {\scalebox{1.607}{\small B}};
\node at (11.845,7.31) {\scalebox{1.607}{\small C}};
\node at (11.664,4.454) {\scalebox{1.607}{\small D}};
\node at (8.256,4.304) {\scalebox{1.607}{\small E}};
\node at (4.447,4.23) {\scalebox{1.607}{\small F}};
\node at (0.899,4.105) {\scalebox{1.607}{\small G}};
\node at (4.393,1.086) {\scalebox{1.607}{\small H}};
\node at (7.794,1.043) {\scalebox{1.607}{\small I}};
\node at (0.765,1.067) {\scalebox{1.607}{\small J}};
\draw[line width=0.5pt] (12.595,6.553) -- (12.595,7.653);
\draw[line width=0.5pt] (12.364,7.268) -- (12.595,7.653) -- (12.826,7.268);
\draw[line width=0.5pt] (12.292,7.103) -- (12.897,7.103);
\node at (12.595,8.038) {\scalebox{1.607}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, C, E, G, H & 5 \\ \hline
clairs & B, D, F, I, J & 5 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{5} 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 10 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 ②} vers le nord jusqu'à Placette des Ormes \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{passage du Marché ①}, vers l'est, jusqu'à Carrefour Saint-Roch \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{quai aux Fleurs ③}, vers l'est, jusqu'à Place aux Herbes \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Four ⑤}, vers le sud, jusqu'à Place de l'Église \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin Creux ⑥}, vers l'ouest, jusqu'à Rond-point du Pont \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin des Vignes ⑦}, vers le sud, jusqu'à Fontaine Notre-Dame \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Lilas ⑫}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de l'Église ⑬}, vers l'ouest, jusqu'à Coin du Lavoir \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Moulin ⑪}, vers le nord, jusqu'à Place du Vieux-Puits \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Mairie ⑩}, vers l'est, jusqu'à Carrefour de l'Hôpital \textbf{(F)}.\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}
