\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 265}

\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=265}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0265}
\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=0.5982, line join=round, line cap=round]
\draw[line width=0.5pt] (5.038,9.088) -- (5.211,9.035) -- (5.386,8.986) -- (5.562,8.942) -- (5.739,8.902) -- (5.917,8.866) -- (6.096,8.834) -- (6.275,8.806) -- (6.456,8.783) -- (6.636,8.764) -- (6.817,8.749) -- (6.998,8.738) -- (7.18,8.732) -- (7.361,8.729) -- (7.543,8.732);
\draw[line width=0.5pt] (5.038,9.088) -- (5.049,8.831) -- (5.054,8.575) -- (5.054,8.318) -- (5.047,8.061) -- (5.034,7.805) -- (5.015,7.548) -- (4.99,7.293) -- (4.959,7.038) -- (4.922,6.783) -- (4.879,6.53) -- (4.83,6.278) -- (4.776,6.027) -- (4.715,5.778) -- (4.648,5.53);
\draw[line width=0.5pt] (11.305,8.839) -- (11.271,8.594) -- (11.243,8.348) -- (11.221,8.101) -- (11.204,7.854) -- (11.194,7.606) -- (11.189,7.358) -- (11.19,7.11) -- (11.197,6.862) -- (11.21,6.615) -- (11.228,6.367) -- (11.252,6.121) -- (11.282,5.875) -- (11.318,5.63) -- (11.36,5.385);
\draw[line width=0.5pt] (7.543,8.732) -- (7.808,8.78) -- (8.075,8.823) -- (8.342,8.859) -- (8.611,8.889) -- (8.88,8.913) -- (9.149,8.93) -- (9.419,8.941) -- (9.69,8.945) -- (9.96,8.943) -- (10.23,8.935) -- (10.499,8.921) -- (10.769,8.9) -- (11.037,8.873) -- (11.305,8.839);
\draw[line width=0.5pt] (7.543,8.732) -- (7.495,8.488) -- (7.453,8.244) -- (7.417,7.999) -- (7.387,7.753) -- (7.362,7.506) -- (7.344,7.259) -- (7.331,7.011) -- (7.324,6.763) -- (7.322,6.515) -- (7.327,6.267) -- (7.337,6.019) -- (7.354,5.772) -- (7.376,5.525) -- (7.404,5.279);
\draw[line width=0.5pt] (1.36,8.136) -- (1.61,8.244) -- (1.863,8.345) -- (2.118,8.441) -- (2.375,8.531) -- (2.635,8.615) -- (2.896,8.692) -- (3.16,8.764) -- (3.424,8.829) -- (3.691,8.888) -- (3.958,8.94) -- (4.227,8.987) -- (4.496,9.027) -- (4.767,9.06) -- (5.038,9.088);
\draw[line width=0.5pt] (1.36,8.136) -- (1.327,7.91) -- (1.299,7.684) -- (1.277,7.457) -- (1.26,7.229) -- (1.248,7.001) -- (1.242,6.773) -- (1.241,6.545) -- (1.245,6.317) -- (1.255,6.089) -- (1.27,5.861) -- (1.29,5.633) -- (1.316,5.407) -- (1.347,5.181) -- (1.383,4.956);
\draw[line width=0.5pt] (4.648,5.53) -- (4.846,5.542) -- (5.045,5.55) -- (5.243,5.553) -- (5.442,5.551) -- (5.64,5.545) -- (5.838,5.534) -- (6.036,5.518) -- (6.234,5.498) -- (6.431,5.473) -- (6.627,5.443) -- (6.823,5.409) -- (7.018,5.37) -- (7.211,5.327) -- (7.404,5.279);
\draw[line width=0.5pt] (11.36,5.385) -- (11.313,5.15) -- (11.272,4.914) -- (11.237,4.676) -- (11.207,4.438) -- (11.183,4.199) -- (11.164,3.96) -- (11.151,3.72) -- (11.144,3.48) -- (11.142,3.24) -- (11.146,3.0) -- (11.156,2.76) -- (11.171,2.521) -- (11.192,2.282) -- (11.218,2.043);
\draw[line width=0.5pt] (7.404,5.279) -- (7.683,5.33) -- (7.963,5.374) -- (8.245,5.411) -- (8.527,5.442) -- (8.81,5.467) -- (9.094,5.484) -- (9.377,5.495) -- (9.661,5.5) -- (9.946,5.497) -- (10.229,5.488) -- (10.513,5.473) -- (10.796,5.45) -- (11.078,5.421) -- (11.36,5.385);
\draw[line width=0.5pt] (7.404,5.279) -- (7.41,5.005) -- (7.422,4.73) -- (7.441,4.457) -- (7.466,4.183) -- (7.498,3.91) -- (7.537,3.638) -- (7.581,3.367) -- (7.632,3.098) -- (7.689,2.829) -- (7.753,2.562) -- (7.823,2.296) -- (7.899,2.032) -- (7.981,1.771) -- (8.069,1.511);
\draw[line width=0.5pt] (1.383,4.956) -- (1.609,5.032) -- (1.835,5.103) -- (2.064,5.169) -- (2.294,5.229) -- (2.526,5.284) -- (2.758,5.333) -- (2.992,5.377) -- (3.227,5.416) -- (3.463,5.449) -- (3.699,5.476) -- (3.936,5.498) -- (4.173,5.514) -- (4.411,5.525) -- (4.648,5.53);
\draw[line width=0.5pt] (1.383,4.956) -- (1.37,4.763) -- (1.36,4.569) -- (1.355,4.376) -- (1.355,4.182) -- (1.359,3.988) -- (1.368,3.795) -- (1.382,3.601) -- (1.4,3.408) -- (1.422,3.216) -- (1.449,3.024) -- (1.481,2.833) -- (1.517,2.642) -- (1.557,2.453) -- (1.602,2.265);
\draw[line width=0.5pt] (1.602,2.265) -- (1.811,2.266) -- (2.019,2.263) -- (2.228,2.254) -- (2.436,2.241) -- (2.644,2.222) -- (2.852,2.199) -- (3.059,2.171) -- (3.265,2.138) -- (3.471,2.1) -- (3.675,2.058) -- (3.878,2.01) -- (4.081,1.958) -- (4.281,1.901) -- (4.481,1.84);
\draw[line width=0.5pt] (4.481,1.84) -- (4.731,1.777) -- (4.984,1.721) -- (5.237,1.67) -- (5.492,1.625) -- (5.748,1.587) -- (6.004,1.554) -- (6.262,1.527) -- (6.519,1.507) -- (6.778,1.492) -- (7.036,1.484) -- (7.295,1.481) -- (7.553,1.485) -- (7.812,1.495) -- (8.069,1.511);
\draw[line width=0.5pt] (8.069,1.511) -- (8.287,1.583) -- (8.506,1.65) -- (8.726,1.712) -- (8.948,1.769) -- (9.172,1.82) -- (9.396,1.866) -- (9.622,1.907) -- (9.848,1.943) -- (10.076,1.973) -- (10.304,1.998) -- (10.532,2.017) -- (10.761,2.031) -- (10.989,2.04) -- (11.218,2.043);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.198,9.088) -- (5.176,9.168) -- (5.118,9.226) -- (5.038,9.248) -- (4.958,9.226) -- (4.899,9.168) -- (4.878,9.088) -- (4.899,9.008) -- (4.958,8.949) -- (5.038,8.928) -- (5.118,8.949) -- (5.176,9.008) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.465,8.839) -- (11.444,8.919) -- (11.385,8.978) -- (11.305,8.999) -- (11.225,8.978) -- (11.166,8.919) -- (11.145,8.839) -- (11.166,8.759) -- (11.225,8.701) -- (11.305,8.679) -- (11.385,8.701) -- (11.444,8.759) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.703,8.732) -- (7.681,8.812) -- (7.623,8.87) -- (7.543,8.892) -- (7.463,8.87) -- (7.404,8.812) -- (7.383,8.732) -- (7.404,8.652) -- (7.463,8.593) -- (7.543,8.572) -- (7.623,8.593) -- (7.681,8.652) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.52,8.136) -- (1.499,8.216) -- (1.44,8.275) -- (1.36,8.296) -- (1.28,8.275) -- (1.221,8.216) -- (1.2,8.136) -- (1.221,8.056) -- (1.28,7.998) -- (1.36,7.976) -- (1.44,7.998) -- (1.499,8.056) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.808,5.53) -- (4.787,5.61) -- (4.728,5.668) -- (4.648,5.69) -- (4.568,5.668) -- (4.51,5.61) -- (4.488,5.53) -- (4.51,5.45) -- (4.568,5.391) -- (4.648,5.37) -- (4.728,5.391) -- (4.787,5.45) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.52,5.385) -- (11.499,5.465) -- (11.44,5.524) -- (11.36,5.545) -- (11.28,5.524) -- (11.221,5.465) -- (11.2,5.385) -- (11.221,5.305) -- (11.28,5.247) -- (11.36,5.225) -- (11.44,5.247) -- (11.499,5.305) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.564,5.279) -- (7.542,5.359) -- (7.484,5.417) -- (7.404,5.439) -- (7.324,5.417) -- (7.265,5.359) -- (7.244,5.279) -- (7.265,5.199) -- (7.324,5.14) -- (7.404,5.119) -- (7.484,5.14) -- (7.542,5.199) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.543,4.956) -- (1.522,5.036) -- (1.463,5.094) -- (1.383,5.116) -- (1.303,5.094) -- (1.245,5.036) -- (1.223,4.956) -- (1.245,4.876) -- (1.303,4.817) -- (1.383,4.796) -- (1.463,4.817) -- (1.522,4.876) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.762,2.265) -- (1.741,2.345) -- (1.682,2.403) -- (1.602,2.425) -- (1.522,2.403) -- (1.464,2.345) -- (1.442,2.265) -- (1.464,2.185) -- (1.522,2.126) -- (1.602,2.105) -- (1.682,2.126) -- (1.741,2.185) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.378,2.043) -- (11.357,2.123) -- (11.298,2.182) -- (11.218,2.203) -- (11.138,2.182) -- (11.08,2.123) -- (11.058,2.043) -- (11.08,1.963) -- (11.138,1.905) -- (11.218,1.883) -- (11.298,1.905) -- (11.357,1.963) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.641,1.84) -- (4.619,1.92) -- (4.561,1.978) -- (4.481,2.0) -- (4.401,1.978) -- (4.342,1.92) -- (4.321,1.84) -- (4.342,1.76) -- (4.401,1.701) -- (4.481,1.68) -- (4.561,1.701) -- (4.619,1.76) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.229,1.511) -- (8.208,1.591) -- (8.149,1.649) -- (8.069,1.671) -- (7.989,1.649) -- (7.931,1.591) -- (7.909,1.511) -- (7.931,1.431) -- (7.989,1.372) -- (8.069,1.351) -- (8.149,1.372) -- (8.208,1.431) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.331,8.802) -- (6.453,8.822) -- (6.562,8.878) -- (6.65,8.965) -- (6.705,9.075) -- (6.725,9.196) -- (6.705,9.318) -- (6.65,9.428) -- (6.562,9.515) -- (6.453,9.571) -- (6.331,9.59) -- (6.331,9.59) -- (6.209,9.571) -- (6.099,9.515) -- (6.012,9.428) -- (5.956,9.318) -- (5.937,9.196) -- (5.956,9.075) -- (6.012,8.965) -- (6.099,8.878) -- (6.209,8.822) -- (6.331,8.802) -- cycle;
\node at (6.331,9.196) {\scalebox{1.672}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.381,6.856) -- (5.503,6.875) -- (5.613,6.931) -- (5.7,7.018) -- (5.756,7.128) -- (5.775,7.25) -- (5.756,7.372) -- (5.7,7.481) -- (5.613,7.569) -- (5.503,7.624) -- (5.381,7.644) -- (5.381,7.644) -- (5.26,7.624) -- (5.15,7.569) -- (5.063,7.481) -- (5.007,7.372) -- (4.988,7.25) -- (5.007,7.128) -- (5.063,7.018) -- (5.15,6.931) -- (5.26,6.875) -- (5.381,6.856) -- cycle;
\node at (5.381,7.25) {\scalebox{1.672}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.584,6.722) -- (11.706,6.742) -- (11.815,6.798) -- (11.903,6.885) -- (11.959,6.995) -- (11.978,7.116) -- (11.959,7.238) -- (11.903,7.348) -- (11.815,7.435) -- (11.706,7.491) -- (11.584,7.51) -- (11.584,7.51) -- (11.462,7.491) -- (11.352,7.435) -- (11.265,7.348) -- (11.209,7.238) -- (11.19,7.116) -- (11.209,6.995) -- (11.265,6.885) -- (11.352,6.798) -- (11.462,6.742) -- (11.584,6.722) -- cycle;
\node at (11.584,7.116) {\scalebox{1.672}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.408,8.94) -- (9.53,8.96) -- (9.64,9.016) -- (9.727,9.103) -- (9.783,9.213) -- (9.802,9.334) -- (9.783,9.456) -- (9.727,9.566) -- (9.64,9.653) -- (9.53,9.709) -- (9.408,9.728) -- (9.408,9.728) -- (9.286,9.709) -- (9.177,9.653) -- (9.089,9.566) -- (9.033,9.456) -- (9.014,9.334) -- (9.033,9.213) -- (9.089,9.103) -- (9.177,9.016) -- (9.286,8.96) -- (9.408,8.94) -- cycle;
\node at (9.408,9.334) {\scalebox{1.672}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.724,6.601) -- (7.846,6.621) -- (7.956,6.676) -- (8.043,6.764) -- (8.099,6.873) -- (8.118,6.995) -- (8.099,7.117) -- (8.043,7.227) -- (7.956,7.314) -- (7.846,7.37) -- (7.724,7.389) -- (7.724,7.389) -- (7.603,7.37) -- (7.493,7.314) -- (7.406,7.227) -- (7.35,7.117) -- (7.33,6.995) -- (7.35,6.873) -- (7.406,6.764) -- (7.493,6.676) -- (7.603,6.621) -- (7.724,6.601) -- cycle;
\node at (7.724,6.995) {\scalebox{1.672}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.061,8.751) -- (3.183,8.77) -- (3.292,8.826) -- (3.38,8.913) -- (3.436,9.023) -- (3.455,9.145) -- (3.436,9.267) -- (3.38,9.376) -- (3.292,9.464) -- (3.183,9.52) -- (3.061,9.539) -- (3.061,9.539) -- (2.939,9.52) -- (2.829,9.464) -- (2.742,9.376) -- (2.686,9.267) -- (2.667,9.145) -- (2.686,9.023) -- (2.742,8.913) -- (2.829,8.826) -- (2.939,8.77) -- (3.061,8.751) -- cycle;
\node at (3.061,9.145) {\scalebox{1.672}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.634,6.154) -- (1.756,6.173) -- (1.866,6.229) -- (1.953,6.316) -- (2.009,6.426) -- (2.028,6.548) -- (2.009,6.67) -- (1.953,6.779) -- (1.866,6.866) -- (1.756,6.922) -- (1.634,6.942) -- (1.634,6.942) -- (1.513,6.922) -- (1.403,6.866) -- (1.316,6.779) -- (1.26,6.67) -- (1.241,6.548) -- (1.26,6.426) -- (1.316,6.316) -- (1.403,6.229) -- (1.513,6.173) -- (1.634,6.154) -- cycle;
\node at (1.634,6.548) {\scalebox{1.672}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.072,5.516) -- (6.194,5.536) -- (6.304,5.592) -- (6.391,5.679) -- (6.447,5.789) -- (6.466,5.91) -- (6.447,6.032) -- (6.391,6.142) -- (6.304,6.229) -- (6.194,6.285) -- (6.072,6.304) -- (6.072,6.304) -- (5.95,6.285) -- (5.841,6.229) -- (5.754,6.142) -- (5.698,6.032) -- (5.678,5.91) -- (5.698,5.789) -- (5.754,5.679) -- (5.841,5.592) -- (5.95,5.536) -- (6.072,5.516) -- cycle;
\node at (6.072,5.91) {\scalebox{1.672}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.545,3.31) -- (11.667,3.329) -- (11.776,3.385) -- (11.864,3.472) -- (11.92,3.582) -- (11.939,3.704) -- (11.92,3.825) -- (11.864,3.935) -- (11.776,4.022) -- (11.667,4.078) -- (11.545,4.098) -- (11.545,4.098) -- (11.423,4.078) -- (11.313,4.022) -- (11.226,3.935) -- (11.17,3.825) -- (11.151,3.704) -- (11.17,3.582) -- (11.226,3.472) -- (11.313,3.385) -- (11.423,3.329) -- (11.545,3.31) -- cycle;
\node at (11.545,3.704) {\scalebox{1.672}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.552,5.495) -- (9.674,5.514) -- (9.784,5.57) -- (9.871,5.658) -- (9.927,5.767) -- (9.946,5.889) -- (9.927,6.011) -- (9.871,6.121) -- (9.784,6.208) -- (9.674,6.264) -- (9.552,6.283) -- (9.181,6.283) -- (9.06,6.264) -- (8.95,6.208) -- (8.863,6.121) -- (8.807,6.011) -- (8.788,5.889) -- (8.807,5.767) -- (8.863,5.658) -- (8.95,5.57) -- (9.06,5.514) -- (9.181,5.495) -- cycle;
\node at (9.367,5.889) {\scalebox{1.672}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.154,3.042) -- (8.276,3.061) -- (8.386,3.117) -- (8.473,3.204) -- (8.529,3.314) -- (8.548,3.436) -- (8.529,3.558) -- (8.473,3.667) -- (8.386,3.755) -- (8.276,3.811) -- (8.154,3.83) -- (7.784,3.83) -- (7.662,3.811) -- (7.552,3.755) -- (7.465,3.667) -- (7.409,3.558) -- (7.39,3.436) -- (7.409,3.314) -- (7.465,3.204) -- (7.552,3.117) -- (7.662,3.061) -- (7.784,3.042) -- cycle;
\node at (7.969,3.436) {\scalebox{1.672}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.109,5.371) -- (3.231,5.391) -- (3.341,5.447) -- (3.428,5.534) -- (3.484,5.644) -- (3.503,5.765) -- (3.484,5.887) -- (3.428,5.997) -- (3.341,6.084) -- (3.231,6.14) -- (3.109,6.159) -- (2.739,6.159) -- (2.617,6.14) -- (2.507,6.084) -- (2.42,5.997) -- (2.364,5.887) -- (2.345,5.765) -- (2.364,5.644) -- (2.42,5.534) -- (2.507,5.447) -- (2.617,5.391) -- (2.739,5.371) -- cycle;
\node at (2.924,5.765) {\scalebox{1.672}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.96,3.239) -- (2.081,3.258) -- (2.191,3.314) -- (2.278,3.402) -- (2.334,3.511) -- (2.354,3.633) -- (2.334,3.755) -- (2.278,3.865) -- (2.191,3.952) -- (2.081,4.008) -- (1.96,4.027) -- (1.589,4.027) -- (1.467,4.008) -- (1.358,3.952) -- (1.27,3.865) -- (1.214,3.755) -- (1.195,3.633) -- (1.214,3.511) -- (1.27,3.402) -- (1.358,3.314) -- (1.467,3.258) -- (1.589,3.239) -- cycle;
\node at (1.774,3.633) {\scalebox{1.672}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.302,2.167) -- (3.424,2.186) -- (3.533,2.242) -- (3.621,2.329) -- (3.676,2.439) -- (3.696,2.561) -- (3.676,2.682) -- (3.621,2.792) -- (3.533,2.879) -- (3.424,2.935) -- (3.302,2.955) -- (2.931,2.955) -- (2.809,2.935) -- (2.7,2.879) -- (2.612,2.792) -- (2.557,2.682) -- (2.537,2.561) -- (2.557,2.439) -- (2.612,2.329) -- (2.7,2.242) -- (2.809,2.186) -- (2.931,2.167) -- cycle;
\node at (3.117,2.561) {\scalebox{1.672}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.483,1.526) -- (6.605,1.545) -- (6.714,1.601) -- (6.802,1.688) -- (6.857,1.798) -- (6.877,1.92) -- (6.857,2.041) -- (6.802,2.151) -- (6.714,2.238) -- (6.605,2.294) -- (6.483,2.313) -- (6.112,2.313) -- (5.99,2.294) -- (5.881,2.238) -- (5.793,2.151) -- (5.738,2.041) -- (5.718,1.92) -- (5.738,1.798) -- (5.793,1.688) -- (5.881,1.601) -- (5.99,1.545) -- (6.112,1.526) -- cycle;
\node at (6.297,1.92) {\scalebox{1.672}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.742,1.902) -- (9.863,1.921) -- (9.973,1.977) -- (10.06,2.064) -- (10.116,2.174) -- (10.136,2.295) -- (10.116,2.417) -- (10.06,2.527) -- (9.973,2.614) -- (9.863,2.67) -- (9.742,2.689) -- (9.371,2.689) -- (9.249,2.67) -- (9.139,2.614) -- (9.052,2.527) -- (8.996,2.417) -- (8.977,2.295) -- (8.996,2.174) -- (9.052,2.064) -- (9.139,1.977) -- (9.249,1.921) -- (9.371,1.902) -- cycle;
\node at (9.556,2.295) {\scalebox{1.672}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.348,5.23) -- (4.948,5.23) -- (4.948,5.83) -- (4.348,5.83) -- cycle;
\node at (4.648,5.53) {\scalebox{1.672}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (12.507,3.623) -- (13.107,3.623) -- (13.107,4.223) -- (12.507,4.223) -- cycle;
\node at (12.807,3.923) {\scalebox{1.672}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.114,10.098) -- (9.714,10.098) -- (9.714,10.698) -- (9.114,10.698) -- cycle;
\node at (9.414,10.398) {\scalebox{1.672}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (7.941,7.597) -- (8.541,7.597) -- (8.541,8.197) -- (7.941,8.197) -- cycle;
\node at (8.241,7.897) {\scalebox{1.672}{\small d}};
\node at (4.844,9.677) {\scalebox{1.672}{\small A}};
\node at (11.812,9.196) {\scalebox{1.672}{\small B}};
\node at (7.756,9.314) {\scalebox{1.672}{\small C}};
\node at (0.823,8.445) {\scalebox{1.672}{\small D}};
\node at (4.035,5.616) {\scalebox{1.672}{\small E}};
\node at (11.98,5.396) {\scalebox{1.672}{\small F}};
\node at (8.024,5.267) {\scalebox{1.672}{\small G}};
\node at (0.765,4.912) {\scalebox{1.672}{\small H}};
\node at (1.082,1.928) {\scalebox{1.672}{\small I}};
\node at (11.736,1.702) {\scalebox{1.672}{\small J}};
\node at (4.194,1.29) {\scalebox{1.672}{\small K}};
\node at (8.334,0.95) {\scalebox{1.672}{\small L}};
\draw[line width=0.5pt] (12.66,8.138) -- (12.66,9.238);
\draw[line width=0.5pt] (12.429,8.853) -- (12.66,9.238) -- (12.891,8.853);
\draw[line width=0.5pt] (12.357,8.688) -- (12.962,8.688);
\node at (12.66,9.623) {\scalebox{1.672}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour E, 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 du Stade & Ⓐ Carrefour Saint-Roch & Ⓙ Placette des Ormes \\ \hline
① rue Verte & ⑩ rue des Écoles & Ⓑ Place de la Fontaine & Ⓚ Coin du Lavoir \\ \hline
② venelle du Chat & ⑪ passage du Marché & Ⓒ Rond-point des Vaches & Ⓛ Carrefour de l'Hôpital \\ \hline
③ rue Haute & ⑫ rue du Pont & Ⓓ Place Sainte-Croix & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ chemin de Ronde & ⑬ quai aux Fleurs & Ⓔ Rond-point du Pont & \textbf{a} l'école \\ \hline
⑤ allée des Peupliers & ⑭ rue de la Mairie & Ⓕ Rond-point de la Gare & \textbf{b} le marché couvert \\ \hline
⑥ rue des Jardins & ⑮ rue de la Poste & Ⓖ Carrefour de la Mairie & \textbf{c} le lavoir \\ \hline
⑦ rue de l'Église & ⑯ rue Basse & Ⓗ Rond-point du Stade & \textbf{d} la mairie \\ \hline
⑧ rue des Lilas & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ 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=0.6128, line join=round, line cap=round]
\draw[line width=0.5pt] (5.038,9.088) -- (5.211,9.035) -- (5.386,8.986) -- (5.562,8.942) -- (5.739,8.902) -- (5.917,8.866) -- (6.096,8.834) -- (6.275,8.806) -- (6.456,8.783) -- (6.636,8.764) -- (6.817,8.749) -- (6.998,8.738) -- (7.18,8.732) -- (7.361,8.729) -- (7.543,8.732);
\draw[line width=0.5pt] (5.038,9.088) -- (5.049,8.831) -- (5.054,8.575) -- (5.054,8.318) -- (5.047,8.061) -- (5.034,7.805) -- (5.015,7.548) -- (4.99,7.293) -- (4.959,7.038) -- (4.922,6.783) -- (4.879,6.53) -- (4.83,6.278) -- (4.776,6.027) -- (4.715,5.778) -- (4.648,5.53);
\draw[line width=0.5pt] (11.305,8.839) -- (11.271,8.594) -- (11.243,8.348) -- (11.221,8.101) -- (11.204,7.854) -- (11.194,7.606) -- (11.189,7.358) -- (11.19,7.11) -- (11.197,6.862) -- (11.21,6.615) -- (11.228,6.367) -- (11.252,6.121) -- (11.282,5.875) -- (11.318,5.63) -- (11.36,5.385);
\draw[line width=0.5pt] (7.543,8.732) -- (7.808,8.78) -- (8.075,8.823) -- (8.342,8.859) -- (8.611,8.889) -- (8.88,8.913) -- (9.149,8.93) -- (9.419,8.941) -- (9.69,8.945) -- (9.96,8.943) -- (10.23,8.935) -- (10.499,8.921) -- (10.769,8.9) -- (11.037,8.873) -- (11.305,8.839);
\draw[line width=0.5pt] (7.543,8.732) -- (7.495,8.488) -- (7.453,8.244) -- (7.417,7.999) -- (7.387,7.753) -- (7.362,7.506) -- (7.344,7.259) -- (7.331,7.011) -- (7.324,6.763) -- (7.322,6.515) -- (7.327,6.267) -- (7.337,6.019) -- (7.354,5.772) -- (7.376,5.525) -- (7.404,5.279);
\draw[line width=0.5pt] (1.36,8.136) -- (1.61,8.244) -- (1.863,8.345) -- (2.118,8.441) -- (2.375,8.531) -- (2.635,8.615) -- (2.896,8.692) -- (3.16,8.764) -- (3.424,8.829) -- (3.691,8.888) -- (3.958,8.94) -- (4.227,8.987) -- (4.496,9.027) -- (4.767,9.06) -- (5.038,9.088);
\draw[line width=0.5pt] (1.36,8.136) -- (1.327,7.91) -- (1.299,7.684) -- (1.277,7.457) -- (1.26,7.229) -- (1.248,7.001) -- (1.242,6.773) -- (1.241,6.545) -- (1.245,6.317) -- (1.255,6.089) -- (1.27,5.861) -- (1.29,5.633) -- (1.316,5.407) -- (1.347,5.181) -- (1.383,4.956);
\draw[line width=0.5pt] (4.648,5.53) -- (4.846,5.542) -- (5.045,5.55) -- (5.243,5.553) -- (5.442,5.551) -- (5.64,5.545) -- (5.838,5.534) -- (6.036,5.518) -- (6.234,5.498) -- (6.431,5.473) -- (6.627,5.443) -- (6.823,5.409) -- (7.018,5.37) -- (7.211,5.327) -- (7.404,5.279);
\draw[line width=0.5pt] (11.36,5.385) -- (11.313,5.15) -- (11.272,4.914) -- (11.237,4.676) -- (11.207,4.438) -- (11.183,4.199) -- (11.164,3.96) -- (11.151,3.72) -- (11.144,3.48) -- (11.142,3.24) -- (11.146,3.0) -- (11.156,2.76) -- (11.171,2.521) -- (11.192,2.282) -- (11.218,2.043);
\draw[line width=0.5pt] (7.404,5.279) -- (7.683,5.33) -- (7.963,5.374) -- (8.245,5.411) -- (8.527,5.442) -- (8.81,5.467) -- (9.094,5.484) -- (9.377,5.495) -- (9.661,5.5) -- (9.946,5.497) -- (10.229,5.488) -- (10.513,5.473) -- (10.796,5.45) -- (11.078,5.421) -- (11.36,5.385);
\draw[line width=0.5pt] (7.404,5.279) -- (7.41,5.005) -- (7.422,4.73) -- (7.441,4.457) -- (7.466,4.183) -- (7.498,3.91) -- (7.537,3.638) -- (7.581,3.367) -- (7.632,3.098) -- (7.689,2.829) -- (7.753,2.562) -- (7.823,2.296) -- (7.899,2.032) -- (7.981,1.771) -- (8.069,1.511);
\draw[line width=0.5pt] (1.383,4.956) -- (1.609,5.032) -- (1.835,5.103) -- (2.064,5.169) -- (2.294,5.229) -- (2.526,5.284) -- (2.758,5.333) -- (2.992,5.377) -- (3.227,5.416) -- (3.463,5.449) -- (3.699,5.476) -- (3.936,5.498) -- (4.173,5.514) -- (4.411,5.525) -- (4.648,5.53);
\draw[line width=0.5pt] (1.383,4.956) -- (1.37,4.763) -- (1.36,4.569) -- (1.355,4.376) -- (1.355,4.182) -- (1.359,3.988) -- (1.368,3.795) -- (1.382,3.601) -- (1.4,3.408) -- (1.422,3.216) -- (1.449,3.024) -- (1.481,2.833) -- (1.517,2.642) -- (1.557,2.453) -- (1.602,2.265);
\draw[line width=0.5pt] (1.602,2.265) -- (1.811,2.266) -- (2.019,2.263) -- (2.228,2.254) -- (2.436,2.241) -- (2.644,2.222) -- (2.852,2.199) -- (3.059,2.171) -- (3.265,2.138) -- (3.471,2.1) -- (3.675,2.058) -- (3.878,2.01) -- (4.081,1.958) -- (4.281,1.901) -- (4.481,1.84);
\draw[line width=0.5pt] (4.481,1.84) -- (4.731,1.777) -- (4.984,1.721) -- (5.237,1.67) -- (5.492,1.625) -- (5.748,1.587) -- (6.004,1.554) -- (6.262,1.527) -- (6.519,1.507) -- (6.778,1.492) -- (7.036,1.484) -- (7.295,1.481) -- (7.553,1.485) -- (7.812,1.495) -- (8.069,1.511);
\draw[line width=0.5pt] (8.069,1.511) -- (8.287,1.583) -- (8.506,1.65) -- (8.726,1.712) -- (8.948,1.769) -- (9.172,1.82) -- (9.396,1.866) -- (9.622,1.907) -- (9.848,1.943) -- (10.076,1.973) -- (10.304,1.998) -- (10.532,2.017) -- (10.761,2.031) -- (10.989,2.04) -- (11.218,2.043);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.411,9.088) -- (5.383,9.23) -- (5.302,9.352) -- (5.181,9.432) -- (5.038,9.461) -- (4.895,9.432) -- (4.774,9.352) -- (4.693,9.23) -- (4.664,9.088) -- (4.693,8.945) -- (4.774,8.824) -- (4.895,8.743) -- (5.038,8.714) -- (5.181,8.743) -- (5.302,8.824) -- (5.383,8.945) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.678,8.839) -- (11.65,8.982) -- (11.569,9.103) -- (11.448,9.184) -- (11.305,9.212) -- (11.162,9.184) -- (11.041,9.103) -- (10.96,8.982) -- (10.932,8.839) -- (10.96,8.696) -- (11.041,8.575) -- (11.162,8.494) -- (11.305,8.466) -- (11.448,8.494) -- (11.569,8.575) -- (11.65,8.696) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.916,8.732) -- (7.887,8.874) -- (7.806,8.996) -- (7.685,9.076) -- (7.543,9.105) -- (7.4,9.076) -- (7.279,8.996) -- (7.198,8.874) -- (7.169,8.732) -- (7.198,8.589) -- (7.279,8.468) -- (7.4,8.387) -- (7.543,8.358) -- (7.685,8.387) -- (7.806,8.468) -- (7.887,8.589) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.733,8.136) -- (1.705,8.279) -- (1.624,8.4) -- (1.503,8.481) -- (1.36,8.509) -- (1.217,8.481) -- (1.096,8.4) -- (1.015,8.279) -- (0.987,8.136) -- (1.015,7.993) -- (1.096,7.872) -- (1.217,7.791) -- (1.36,7.763) -- (1.503,7.791) -- (1.624,7.872) -- (1.705,7.993) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.022,5.53) -- (4.993,5.673) -- (4.912,5.794) -- (4.791,5.875) -- (4.648,5.903) -- (4.506,5.875) -- (4.385,5.794) -- (4.304,5.673) -- (4.275,5.53) -- (4.304,5.387) -- (4.385,5.266) -- (4.506,5.185) -- (4.648,5.157) -- (4.791,5.185) -- (4.912,5.266) -- (4.993,5.387) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.733,5.385) -- (11.705,5.528) -- (11.624,5.649) -- (11.503,5.73) -- (11.36,5.759) -- (11.217,5.73) -- (11.096,5.649) -- (11.015,5.528) -- (10.987,5.385) -- (11.015,5.243) -- (11.096,5.122) -- (11.217,5.041) -- (11.36,5.012) -- (11.503,5.041) -- (11.624,5.122) -- (11.705,5.243) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.777,5.279) -- (7.748,5.422) -- (7.668,5.543) -- (7.547,5.624) -- (7.404,5.652) -- (7.261,5.624) -- (7.14,5.543) -- (7.059,5.422) -- (7.03,5.279) -- (7.059,5.136) -- (7.14,5.015) -- (7.261,4.934) -- (7.404,4.906) -- (7.547,4.934) -- (7.668,5.015) -- (7.748,5.136) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.757,4.956) -- (1.728,5.098) -- (1.647,5.22) -- (1.526,5.3) -- (1.383,5.329) -- (1.241,5.3) -- (1.12,5.22) -- (1.039,5.098) -- (1.01,4.956) -- (1.039,4.813) -- (1.12,4.692) -- (1.241,4.611) -- (1.383,4.582) -- (1.526,4.611) -- (1.647,4.692) -- (1.728,4.813) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.975,2.265) -- (1.947,2.408) -- (1.866,2.529) -- (1.745,2.61) -- (1.602,2.638) -- (1.459,2.61) -- (1.338,2.529) -- (1.257,2.408) -- (1.229,2.265) -- (1.257,2.122) -- (1.338,2.001) -- (1.459,1.92) -- (1.602,1.892) -- (1.745,1.92) -- (1.866,2.001) -- (1.947,2.122) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.592,2.043) -- (11.563,2.186) -- (11.482,2.307) -- (11.361,2.388) -- (11.218,2.417) -- (11.076,2.388) -- (10.955,2.307) -- (10.874,2.186) -- (10.845,2.043) -- (10.874,1.901) -- (10.955,1.78) -- (11.076,1.699) -- (11.218,1.67) -- (11.361,1.699) -- (11.482,1.78) -- (11.563,1.901) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.854,1.84) -- (4.826,1.983) -- (4.745,2.104) -- (4.624,2.185) -- (4.481,2.213) -- (4.338,2.185) -- (4.217,2.104) -- (4.136,1.983) -- (4.108,1.84) -- (4.136,1.697) -- (4.217,1.576) -- (4.338,1.495) -- (4.481,1.467) -- (4.624,1.495) -- (4.745,1.576) -- (4.826,1.697) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.443,1.511) -- (8.414,1.654) -- (8.333,1.775) -- (8.212,1.856) -- (8.069,1.884) -- (7.927,1.856) -- (7.806,1.775) -- (7.725,1.654) -- (7.696,1.511) -- (7.725,1.368) -- (7.806,1.247) -- (7.927,1.166) -- (8.069,1.138) -- (8.212,1.166) -- (8.333,1.247) -- (8.414,1.368) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.33,8.802) -- (6.448,8.821) -- (6.556,8.876) -- (6.641,8.961) -- (6.695,9.068) -- (6.714,9.187) -- (6.695,9.306) -- (6.641,9.413) -- (6.556,9.498) -- (6.448,9.553) -- (6.33,9.572) -- (6.33,9.572) -- (6.211,9.553) -- (6.104,9.498) -- (6.018,9.413) -- (5.964,9.306) -- (5.945,9.187) -- (5.964,9.068) -- (6.018,8.961) -- (6.104,8.876) -- (6.211,8.821) -- (6.33,8.802) -- cycle;
\node at (6.33,9.187) {\scalebox{1.632}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.372,6.866) -- (5.491,6.885) -- (5.598,6.94) -- (5.683,7.025) -- (5.738,7.132) -- (5.757,7.251) -- (5.738,7.37) -- (5.683,7.477) -- (5.598,7.562) -- (5.491,7.617) -- (5.372,7.635) -- (5.372,7.635) -- (5.253,7.617) -- (5.146,7.562) -- (5.061,7.477) -- (5.006,7.37) -- (4.988,7.251) -- (5.006,7.132) -- (5.061,7.025) -- (5.146,6.94) -- (5.253,6.885) -- (5.372,6.866) -- cycle;
\node at (5.372,7.251) {\scalebox{1.632}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.575,6.732) -- (11.693,6.75) -- (11.801,6.805) -- (11.886,6.89) -- (11.94,6.997) -- (11.959,7.116) -- (11.94,7.235) -- (11.886,7.342) -- (11.801,7.427) -- (11.693,7.482) -- (11.575,7.501) -- (11.575,7.501) -- (11.456,7.482) -- (11.348,7.427) -- (11.263,7.342) -- (11.209,7.235) -- (11.19,7.116) -- (11.209,6.997) -- (11.263,6.89) -- (11.348,6.805) -- (11.456,6.75) -- (11.575,6.732) -- cycle;
\node at (11.575,7.116) {\scalebox{1.632}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.408,8.94) -- (9.527,8.959) -- (9.634,9.014) -- (9.719,9.099) -- (9.774,9.206) -- (9.793,9.325) -- (9.774,9.444) -- (9.719,9.551) -- (9.634,9.636) -- (9.527,9.691) -- (9.408,9.71) -- (9.408,9.71) -- (9.29,9.691) -- (9.182,9.636) -- (9.097,9.551) -- (9.043,9.444) -- (9.024,9.325) -- (9.043,9.206) -- (9.097,9.099) -- (9.182,9.014) -- (9.29,8.959) -- (9.408,8.94) -- cycle;
\node at (9.408,9.325) {\scalebox{1.632}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.715,6.611) -- (7.834,6.63) -- (7.941,6.684) -- (8.026,6.769) -- (8.081,6.877) -- (8.099,6.996) -- (8.081,7.114) -- (8.026,7.222) -- (7.941,7.307) -- (7.834,7.361) -- (7.715,7.38) -- (7.715,7.38) -- (7.596,7.361) -- (7.489,7.307) -- (7.404,7.222) -- (7.349,7.114) -- (7.33,6.996) -- (7.349,6.877) -- (7.404,6.769) -- (7.489,6.684) -- (7.596,6.63) -- (7.715,6.611) -- cycle;
\node at (7.715,6.996) {\scalebox{1.632}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.063,8.751) -- (3.182,8.77) -- (3.289,8.825) -- (3.374,8.91) -- (3.429,9.017) -- (3.448,9.136) -- (3.429,9.255) -- (3.374,9.362) -- (3.289,9.447) -- (3.182,9.502) -- (3.063,9.52) -- (3.063,9.52) -- (2.944,9.502) -- (2.837,9.447) -- (2.752,9.362) -- (2.698,9.255) -- (2.679,9.136) -- (2.698,9.017) -- (2.752,8.91) -- (2.837,8.825) -- (2.944,8.77) -- (3.063,8.751) -- cycle;
\node at (3.063,9.136) {\scalebox{1.632}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.625,6.163) -- (1.744,6.182) -- (1.851,6.237) -- (1.936,6.322) -- (1.991,6.429) -- (2.01,6.548) -- (1.991,6.667) -- (1.936,6.774) -- (1.851,6.859) -- (1.744,6.913) -- (1.625,6.932) -- (1.625,6.932) -- (1.506,6.913) -- (1.399,6.859) -- (1.314,6.774) -- (1.259,6.667) -- (1.241,6.548) -- (1.259,6.429) -- (1.314,6.322) -- (1.399,6.237) -- (1.506,6.182) -- (1.625,6.163) -- cycle;
\node at (1.625,6.548) {\scalebox{1.632}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.071,5.516) -- (6.19,5.535) -- (6.297,5.59) -- (6.382,5.675) -- (6.437,5.782) -- (6.456,5.901) -- (6.437,6.02) -- (6.382,6.127) -- (6.297,6.212) -- (6.19,6.267) -- (6.071,6.285) -- (6.071,6.285) -- (5.952,6.267) -- (5.845,6.212) -- (5.76,6.127) -- (5.706,6.02) -- (5.687,5.901) -- (5.706,5.782) -- (5.76,5.675) -- (5.845,5.59) -- (5.952,5.535) -- (6.071,5.516) -- cycle;
\node at (6.071,5.901) {\scalebox{1.632}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.536,3.319) -- (11.654,3.338) -- (11.762,3.393) -- (11.847,3.478) -- (11.901,3.585) -- (11.92,3.704) -- (11.901,3.823) -- (11.847,3.93) -- (11.762,4.015) -- (11.654,4.07) -- (11.536,4.089) -- (11.536,4.089) -- (11.417,4.07) -- (11.31,4.015) -- (11.224,3.93) -- (11.17,3.823) -- (11.151,3.704) -- (11.17,3.585) -- (11.224,3.478) -- (11.31,3.393) -- (11.417,3.338) -- (11.536,3.319) -- cycle;
\node at (11.536,3.704) {\scalebox{1.632}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.548,5.495) -- (9.667,5.514) -- (9.774,5.569) -- (9.859,5.654) -- (9.914,5.761) -- (9.933,5.88) -- (9.914,5.999) -- (9.859,6.106) -- (9.774,6.191) -- (9.667,6.245) -- (9.548,6.264) -- (9.186,6.264) -- (9.067,6.245) -- (8.96,6.191) -- (8.875,6.106) -- (8.82,5.999) -- (8.802,5.88) -- (8.82,5.761) -- (8.875,5.654) -- (8.96,5.569) -- (9.067,5.514) -- (9.186,5.495) -- cycle;
\node at (9.367,5.88) {\scalebox{1.632}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.141,3.05) -- (8.26,3.069) -- (8.367,3.123) -- (8.452,3.208) -- (8.506,3.315) -- (8.525,3.434) -- (8.506,3.553) -- (8.452,3.66) -- (8.367,3.745) -- (8.26,3.8) -- (8.141,3.819) -- (7.779,3.819) -- (7.66,3.8) -- (7.553,3.745) -- (7.468,3.66) -- (7.413,3.553) -- (7.394,3.434) -- (7.413,3.315) -- (7.468,3.208) -- (7.553,3.123) -- (7.66,3.069) -- (7.779,3.05) -- cycle;
\node at (7.96,3.434) {\scalebox{1.632}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.107,5.372) -- (3.225,5.39) -- (3.333,5.445) -- (3.418,5.53) -- (3.472,5.637) -- (3.491,5.756) -- (3.472,5.875) -- (3.418,5.982) -- (3.333,6.067) -- (3.225,6.122) -- (3.107,6.141) -- (2.745,6.141) -- (2.626,6.122) -- (2.519,6.067) -- (2.434,5.982) -- (2.379,5.875) -- (2.36,5.756) -- (2.379,5.637) -- (2.434,5.53) -- (2.519,5.445) -- (2.626,5.39) -- (2.745,5.372) -- cycle;
\node at (2.926,5.756) {\scalebox{1.632}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.946,3.248) -- (2.065,3.267) -- (2.172,3.321) -- (2.257,3.406) -- (2.312,3.514) -- (2.331,3.632) -- (2.312,3.751) -- (2.257,3.858) -- (2.172,3.943) -- (2.065,3.998) -- (1.946,4.017) -- (1.584,4.017) -- (1.465,3.998) -- (1.358,3.943) -- (1.273,3.858) -- (1.218,3.751) -- (1.2,3.632) -- (1.218,3.514) -- (1.273,3.406) -- (1.358,3.321) -- (1.465,3.267) -- (1.584,3.248) -- cycle;
\node at (1.765,3.632) {\scalebox{1.632}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.296,2.167) -- (3.415,2.186) -- (3.522,2.24) -- (3.607,2.325) -- (3.662,2.433) -- (3.681,2.551) -- (3.662,2.67) -- (3.607,2.777) -- (3.522,2.863) -- (3.415,2.917) -- (3.296,2.936) -- (2.934,2.936) -- (2.815,2.917) -- (2.708,2.863) -- (2.623,2.777) -- (2.568,2.67) -- (2.55,2.551) -- (2.568,2.433) -- (2.623,2.325) -- (2.708,2.24) -- (2.815,2.186) -- (2.934,2.167) -- cycle;
\node at (3.115,2.551) {\scalebox{1.632}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.478,1.526) -- (6.596,1.545) -- (6.704,1.599) -- (6.789,1.684) -- (6.843,1.791) -- (6.862,1.91) -- (6.843,2.029) -- (6.789,2.136) -- (6.704,2.221) -- (6.596,2.276) -- (6.478,2.295) -- (6.116,2.295) -- (5.997,2.276) -- (5.89,2.221) -- (5.805,2.136) -- (5.75,2.029) -- (5.731,1.91) -- (5.75,1.791) -- (5.805,1.684) -- (5.89,1.599) -- (5.997,1.545) -- (6.116,1.526) -- cycle;
\node at (6.297,1.91) {\scalebox{1.632}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.739,1.902) -- (9.858,1.92) -- (9.965,1.975) -- (10.05,2.06) -- (10.105,2.167) -- (10.123,2.286) -- (10.105,2.405) -- (10.05,2.512) -- (9.965,2.597) -- (9.858,2.652) -- (9.739,2.671) -- (9.377,2.671) -- (9.258,2.652) -- (9.151,2.597) -- (9.066,2.512) -- (9.011,2.405) -- (8.992,2.286) -- (9.011,2.167) -- (9.066,2.06) -- (9.151,1.975) -- (9.258,1.92) -- (9.377,1.902) -- cycle;
\node at (9.558,2.286) {\scalebox{1.632}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.348,5.23) -- (4.948,5.23) -- (4.948,5.83) -- (4.348,5.83) -- cycle;
\node at (4.648,5.53) {\scalebox{1.632}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (12.507,3.623) -- (13.107,3.623) -- (13.107,4.223) -- (12.507,4.223) -- cycle;
\node at (12.807,3.923) {\scalebox{1.632}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.114,10.098) -- (9.714,10.098) -- (9.714,10.698) -- (9.114,10.698) -- cycle;
\node at (9.414,10.398) {\scalebox{1.632}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (7.941,7.597) -- (8.541,7.597) -- (8.541,8.197) -- (7.941,8.197) -- cycle;
\node at (8.241,7.897) {\scalebox{1.632}{\small d}};
\node at (4.844,9.677) {\scalebox{1.632}{\small A}};
\node at (11.812,9.196) {\scalebox{1.632}{\small B}};
\node at (7.756,9.314) {\scalebox{1.632}{\small C}};
\node at (0.823,8.445) {\scalebox{1.632}{\small D}};
\node at (4.035,5.616) {\scalebox{1.632}{\small E}};
\node at (11.98,5.396) {\scalebox{1.632}{\small F}};
\node at (8.024,5.267) {\scalebox{1.632}{\small G}};
\node at (0.765,4.912) {\scalebox{1.632}{\small H}};
\node at (1.082,1.928) {\scalebox{1.632}{\small I}};
\node at (11.736,1.702) {\scalebox{1.632}{\small J}};
\node at (4.194,1.29) {\scalebox{1.632}{\small K}};
\node at (8.334,0.95) {\scalebox{1.632}{\small L}};
\draw[line width=0.5pt] (12.66,8.138) -- (12.66,9.238);
\draw[line width=0.5pt] (12.429,8.853) -- (12.66,9.238) -- (12.891,8.853);
\draw[line width=0.5pt] (12.357,8.688) -- (12.962,8.688);
\node at (12.66,9.623) {\scalebox{1.632}{\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, B, G, H, J, K & 6 \\ \hline
clairs & C, D, E, F, I, 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 ②} vers le nord jusqu'à Carrefour Saint-Roch \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Jardins ⑥}, vers l'ouest, jusqu'à Place Sainte-Croix \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de l'Église ⑦}, vers le sud, jusqu'à Rond-point du Stade \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{quai aux Fleurs ⑬}, vers le sud, jusqu'à Fontaine Notre-Dame \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de la Mairie ⑭}, vers l'est, jusqu'à Coin du Lavoir \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de la Poste ⑮}, vers l'est, jusqu'à Carrefour de l'Hôpital \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Basse ⑯}, vers l'est, jusqu'à Placette des Ormes \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Stade ⑨}, vers le nord, jusqu'à Rond-point de la Gare \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Haute ③}, vers le nord, jusqu'à Place de la Fontaine \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin de Ronde ④}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{allée des Peupliers ⑤}, vers le sud, jusqu'à Carrefour de la Mairie \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Lilas ⑧}, vers l'ouest, jusqu'à Rond-point du Pont \textbf{(E)}.\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}
