\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 20}

\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=20}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0020}
\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=0.6069, line join=round, line cap=round]
\draw[line width=0.5pt] (11.112,6.94) -- (11.097,6.75) -- (11.086,6.559) -- (11.08,6.367) -- (11.078,6.176) -- (11.081,5.985) -- (11.089,5.793) -- (11.101,5.602) -- (11.117,5.412) -- (11.138,5.221) -- (11.163,5.032) -- (11.193,4.842) -- (11.227,4.654) -- (11.266,4.467) -- (11.309,4.281);
\draw[line width=0.5pt] (6.257,6.863) -- (6.456,6.865) -- (6.654,6.863) -- (6.853,6.856) -- (7.051,6.845) -- (7.249,6.828) -- (7.447,6.808) -- (7.644,6.782) -- (7.841,6.752) -- (8.036,6.717) -- (8.231,6.678) -- (8.425,6.634) -- (8.618,6.586) -- (8.809,6.533) -- (8.999,6.475);
\draw[line width=0.5pt] (6.257,6.863) -- (6.243,6.675) -- (6.233,6.487) -- (6.228,6.299) -- (6.227,6.111) -- (6.23,5.923) -- (6.238,5.734) -- (6.251,5.547) -- (6.268,5.359) -- (6.289,5.172) -- (6.314,4.985) -- (6.344,4.799) -- (6.379,4.614) -- (6.417,4.43) -- (6.461,4.247);
\draw[line width=0.5pt] (3.899,6.797) -- (4.066,6.827) -- (4.233,6.854) -- (4.4,6.876) -- (4.569,6.895) -- (4.737,6.91) -- (4.906,6.92) -- (5.075,6.927) -- (5.245,6.93) -- (5.414,6.929) -- (5.583,6.923) -- (5.752,6.914) -- (5.921,6.901) -- (6.089,6.884) -- (6.257,6.863);
\draw[line width=0.5pt] (3.899,6.797) -- (3.862,6.628) -- (3.829,6.459) -- (3.8,6.289) -- (3.775,6.118) -- (3.754,5.946) -- (3.737,5.775) -- (3.724,5.602) -- (3.715,5.43) -- (3.71,5.257) -- (3.709,5.084) -- (3.713,4.912) -- (3.72,4.739) -- (3.731,4.567) -- (3.747,4.395);
\draw[line width=0.5pt] (8.999,6.475) -- (9.144,6.531) -- (9.29,6.584) -- (9.437,6.633) -- (9.585,6.679) -- (9.735,6.721) -- (9.885,6.76) -- (10.036,6.795) -- (10.188,6.827) -- (10.341,6.855) -- (10.494,6.879) -- (10.648,6.9) -- (10.802,6.917) -- (10.957,6.93) -- (11.112,6.94);
\draw[line width=0.5pt] (8.999,6.475) -- (8.991,6.321) -- (8.987,6.167) -- (8.986,6.012) -- (8.989,5.858) -- (8.996,5.704) -- (9.006,5.549) -- (9.02,5.396) -- (9.038,5.242) -- (9.059,5.089) -- (9.084,4.936) -- (9.112,4.785) -- (9.144,4.634) -- (9.179,4.483) -- (9.218,4.334);
\draw[line width=0.5pt] (3.747,4.395) -- (3.941,4.414) -- (4.135,4.429) -- (4.33,4.439) -- (4.525,4.444) -- (4.72,4.445) -- (4.915,4.441) -- (5.11,4.433) -- (5.304,4.42) -- (5.499,4.403) -- (5.693,4.381) -- (5.886,4.354) -- (6.078,4.323) -- (6.27,4.287) -- (6.461,4.247);
\draw[line width=0.5pt] (3.747,4.395) -- (3.725,4.184) -- (3.708,3.973) -- (3.696,3.762) -- (3.689,3.55) -- (3.687,3.338) -- (3.69,3.126) -- (3.698,2.914) -- (3.711,2.702) -- (3.729,2.491) -- (3.752,2.28) -- (3.78,2.07) -- (3.813,1.861) -- (3.851,1.653) -- (3.893,1.445);
\draw[line width=0.5pt] (1.482,4.368) -- (1.642,4.395) -- (1.803,4.418) -- (1.965,4.437) -- (2.126,4.452) -- (2.289,4.464) -- (2.451,4.471) -- (2.613,4.475) -- (2.776,4.475) -- (2.939,4.471) -- (3.101,4.464) -- (3.263,4.452) -- (3.425,4.437) -- (3.586,4.418) -- (3.747,4.395);
\draw[line width=0.5pt] (1.482,4.368) -- (1.442,4.189) -- (1.406,4.009) -- (1.374,3.828) -- (1.347,3.646) -- (1.324,3.464) -- (1.305,3.281) -- (1.29,3.098) -- (1.28,2.914) -- (1.274,2.73) -- (1.272,2.546) -- (1.275,2.363) -- (1.282,2.179) -- (1.293,1.995) -- (1.309,1.812);
\draw[line width=0.5pt] (9.218,4.334) -- (9.367,4.353) -- (9.516,4.369) -- (9.666,4.381) -- (9.816,4.389) -- (9.966,4.394) -- (10.116,4.396) -- (10.266,4.393) -- (10.416,4.388) -- (10.566,4.379) -- (10.715,4.366) -- (10.864,4.35) -- (11.013,4.33) -- (11.161,4.307) -- (11.309,4.281);
\draw[line width=0.5pt] (6.461,4.247) -- (6.657,4.223) -- (6.854,4.204) -- (7.051,4.189) -- (7.249,4.179) -- (7.447,4.174) -- (7.645,4.173) -- (7.843,4.177) -- (8.041,4.185) -- (8.238,4.199) -- (8.436,4.216) -- (8.632,4.239) -- (8.828,4.266) -- (9.024,4.298) -- (9.218,4.334);
\draw[line width=0.5pt] (6.461,4.247) -- (6.488,4.071) -- (6.51,3.894) -- (6.529,3.716) -- (6.543,3.538) -- (6.554,3.36) -- (6.56,3.181) -- (6.561,3.003) -- (6.559,2.824) -- (6.552,2.646) -- (6.542,2.467) -- (6.527,2.289) -- (6.508,2.112) -- (6.484,1.935) -- (6.457,1.759);
\draw[line width=0.5pt] (1.309,1.812) -- (1.488,1.758) -- (1.668,1.708) -- (1.85,1.662) -- (2.033,1.621) -- (2.216,1.584) -- (2.401,1.551) -- (2.586,1.522) -- (2.772,1.498) -- (2.958,1.478) -- (3.145,1.463) -- (3.332,1.452) -- (3.519,1.445) -- (3.706,1.443) -- (3.893,1.445);
\draw[line width=0.5pt] (3.893,1.445) -- (4.071,1.495) -- (4.251,1.541) -- (4.431,1.583) -- (4.613,1.621) -- (4.795,1.654) -- (4.978,1.683) -- (5.162,1.708) -- (5.346,1.728) -- (5.531,1.744) -- (5.716,1.756) -- (5.901,1.763) -- (6.087,1.766) -- (6.272,1.764) -- (6.457,1.759);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.272,6.94) -- (11.25,7.02) -- (11.192,7.079) -- (11.112,7.1) -- (11.032,7.079) -- (10.973,7.02) -- (10.952,6.94) -- (10.973,6.86) -- (11.032,6.802) -- (11.112,6.78) -- (11.192,6.802) -- (11.25,6.86) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.417,6.863) -- (6.396,6.943) -- (6.337,7.001) -- (6.257,7.023) -- (6.177,7.001) -- (6.118,6.943) -- (6.097,6.863) -- (6.118,6.783) -- (6.177,6.724) -- (6.257,6.703) -- (6.337,6.724) -- (6.396,6.783) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.059,6.797) -- (4.038,6.877) -- (3.979,6.935) -- (3.899,6.957) -- (3.819,6.935) -- (3.761,6.877) -- (3.739,6.797) -- (3.761,6.717) -- (3.819,6.658) -- (3.899,6.637) -- (3.979,6.658) -- (4.038,6.717) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.159,6.475) -- (9.138,6.555) -- (9.079,6.614) -- (8.999,6.635) -- (8.919,6.614) -- (8.861,6.555) -- (8.839,6.475) -- (8.861,6.395) -- (8.919,6.337) -- (8.999,6.315) -- (9.079,6.337) -- (9.138,6.395) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.907,4.395) -- (3.885,4.475) -- (3.827,4.534) -- (3.747,4.555) -- (3.667,4.534) -- (3.608,4.475) -- (3.587,4.395) -- (3.608,4.315) -- (3.667,4.257) -- (3.747,4.235) -- (3.827,4.257) -- (3.885,4.315) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.642,4.368) -- (1.621,4.448) -- (1.562,4.507) -- (1.482,4.528) -- (1.402,4.507) -- (1.344,4.448) -- (1.322,4.368) -- (1.344,4.288) -- (1.402,4.23) -- (1.482,4.208) -- (1.562,4.23) -- (1.621,4.288) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.378,4.334) -- (9.357,4.414) -- (9.298,4.472) -- (9.218,4.494) -- (9.138,4.472) -- (9.08,4.414) -- (9.058,4.334) -- (9.08,4.254) -- (9.138,4.195) -- (9.218,4.174) -- (9.298,4.195) -- (9.357,4.254) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.469,4.281) -- (11.447,4.361) -- (11.389,4.419) -- (11.309,4.441) -- (11.229,4.419) -- (11.17,4.361) -- (11.149,4.281) -- (11.17,4.201) -- (11.229,4.142) -- (11.309,4.121) -- (11.389,4.142) -- (11.447,4.201) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.621,4.247) -- (6.599,4.327) -- (6.541,4.386) -- (6.461,4.407) -- (6.381,4.386) -- (6.322,4.327) -- (6.301,4.247) -- (6.322,4.167) -- (6.381,4.109) -- (6.461,4.087) -- (6.541,4.109) -- (6.599,4.167) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.469,1.812) -- (1.447,1.892) -- (1.389,1.951) -- (1.309,1.972) -- (1.229,1.951) -- (1.17,1.892) -- (1.149,1.812) -- (1.17,1.732) -- (1.229,1.674) -- (1.309,1.652) -- (1.389,1.674) -- (1.447,1.732) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.617,1.759) -- (6.595,1.839) -- (6.537,1.897) -- (6.457,1.919) -- (6.377,1.897) -- (6.318,1.839) -- (6.297,1.759) -- (6.318,1.679) -- (6.377,1.62) -- (6.457,1.599) -- (6.537,1.62) -- (6.595,1.679) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.053,1.445) -- (4.032,1.525) -- (3.973,1.584) -- (3.893,1.605) -- (3.813,1.584) -- (3.755,1.525) -- (3.733,1.445) -- (3.755,1.365) -- (3.813,1.307) -- (3.893,1.285) -- (3.973,1.307) -- (4.032,1.365) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.493,5.238) -- (11.615,5.257) -- (11.725,5.313) -- (11.812,5.4) -- (11.868,5.51) -- (11.887,5.631) -- (11.868,5.753) -- (11.812,5.863) -- (11.725,5.95) -- (11.615,6.006) -- (11.493,6.025) -- (11.493,6.025) -- (11.372,6.006) -- (11.262,5.95) -- (11.175,5.863) -- (11.119,5.753) -- (11.1,5.631) -- (11.119,5.51) -- (11.175,5.4) -- (11.262,5.313) -- (11.372,5.257) -- (11.493,5.238) -- cycle;
\node at (11.493,5.631) {\scalebox{1.648}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.699,6.778) -- (7.821,6.798) -- (7.931,6.853) -- (8.018,6.941) -- (8.074,7.05) -- (8.093,7.172) -- (8.074,7.294) -- (8.018,7.404) -- (7.931,7.491) -- (7.821,7.547) -- (7.699,7.566) -- (7.699,7.566) -- (7.577,7.547) -- (7.468,7.491) -- (7.38,7.404) -- (7.325,7.294) -- (7.305,7.172) -- (7.325,7.05) -- (7.38,6.941) -- (7.468,6.853) -- (7.577,6.798) -- (7.699,6.778) -- cycle;
\node at (7.699,7.172) {\scalebox{1.648}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.644,5.183) -- (6.765,5.202) -- (6.875,5.258) -- (6.962,5.346) -- (7.018,5.455) -- (7.037,5.577) -- (7.018,5.699) -- (6.962,5.809) -- (6.875,5.896) -- (6.765,5.952) -- (6.644,5.971) -- (6.644,5.971) -- (6.522,5.952) -- (6.412,5.896) -- (6.325,5.809) -- (6.269,5.699) -- (6.25,5.577) -- (6.269,5.455) -- (6.325,5.346) -- (6.412,5.258) -- (6.522,5.202) -- (6.644,5.183) -- cycle;
\node at (6.644,5.577) {\scalebox{1.648}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.064,6.927) -- (5.186,6.946) -- (5.296,7.002) -- (5.383,7.089) -- (5.439,7.199) -- (5.458,7.321) -- (5.439,7.442) -- (5.383,7.552) -- (5.296,7.639) -- (5.186,7.695) -- (5.064,7.715) -- (5.064,7.715) -- (4.943,7.695) -- (4.833,7.639) -- (4.746,7.552) -- (4.69,7.442) -- (4.671,7.321) -- (4.69,7.199) -- (4.746,7.089) -- (4.833,7.002) -- (4.943,6.946) -- (5.064,6.927) -- cycle;
\node at (5.064,7.321) {\scalebox{1.648}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.117,5.183) -- (4.239,5.203) -- (4.349,5.259) -- (4.436,5.346) -- (4.492,5.456) -- (4.511,5.577) -- (4.492,5.699) -- (4.436,5.809) -- (4.349,5.896) -- (4.239,5.952) -- (4.117,5.971) -- (4.117,5.971) -- (3.995,5.952) -- (3.886,5.896) -- (3.798,5.809) -- (3.743,5.699) -- (3.723,5.577) -- (3.743,5.456) -- (3.798,5.346) -- (3.886,5.259) -- (3.995,5.203) -- (4.117,5.183) -- cycle;
\node at (4.117,5.577) {\scalebox{1.648}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.435,5.913) -- (10.557,5.932) -- (10.666,5.988) -- (10.754,6.075) -- (10.81,6.185) -- (10.829,6.306) -- (10.81,6.428) -- (10.754,6.538) -- (10.666,6.625) -- (10.557,6.681) -- (10.435,6.7) -- (10.435,6.7) -- (10.313,6.681) -- (10.203,6.625) -- (10.116,6.538) -- (10.06,6.428) -- (10.041,6.306) -- (10.06,6.185) -- (10.116,6.075) -- (10.203,5.988) -- (10.313,5.932) -- (10.435,5.913) -- cycle;
\node at (10.435,6.306) {\scalebox{1.648}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.412,5.042) -- (9.534,5.061) -- (9.644,5.117) -- (9.731,5.204) -- (9.787,5.314) -- (9.806,5.436) -- (9.787,5.557) -- (9.731,5.667) -- (9.644,5.754) -- (9.534,5.81) -- (9.412,5.83) -- (9.412,5.83) -- (9.291,5.81) -- (9.181,5.754) -- (9.094,5.667) -- (9.038,5.557) -- (9.018,5.436) -- (9.038,5.314) -- (9.094,5.204) -- (9.181,5.117) -- (9.291,5.061) -- (9.412,5.042) -- cycle;
\node at (9.412,5.436) {\scalebox{1.648}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.131,4.433) -- (5.253,4.452) -- (5.363,4.508) -- (5.45,4.595) -- (5.506,4.705) -- (5.525,4.826) -- (5.506,4.948) -- (5.45,5.058) -- (5.363,5.145) -- (5.253,5.201) -- (5.131,5.22) -- (5.131,5.22) -- (5.01,5.201) -- (4.9,5.145) -- (4.813,5.058) -- (4.757,4.948) -- (4.737,4.826) -- (4.757,4.705) -- (4.813,4.595) -- (4.9,4.508) -- (5.01,4.452) -- (5.131,4.433) -- cycle;
\node at (5.131,4.826) {\scalebox{1.648}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.092,2.54) -- (4.214,2.559) -- (4.323,2.615) -- (4.41,2.702) -- (4.466,2.812) -- (4.486,2.934) -- (4.466,3.055) -- (4.41,3.165) -- (4.323,3.252) -- (4.214,3.308) -- (4.092,3.327) -- (4.092,3.327) -- (3.97,3.308) -- (3.86,3.252) -- (3.773,3.165) -- (3.717,3.055) -- (3.698,2.934) -- (3.717,2.812) -- (3.773,2.702) -- (3.86,2.615) -- (3.97,2.559) -- (4.092,2.54) -- cycle;
\node at (4.092,2.934) {\scalebox{1.648}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.61,4.818) -- (2.732,4.838) -- (2.842,4.894) -- (2.929,4.981) -- (2.985,5.091) -- (3.004,5.212) -- (2.985,5.334) -- (2.929,5.444) -- (2.842,5.531) -- (2.732,5.587) -- (2.61,5.606) -- (2.239,5.606) -- (2.118,5.587) -- (2.008,5.531) -- (1.921,5.444) -- (1.865,5.334) -- (1.845,5.212) -- (1.865,5.091) -- (1.921,4.981) -- (2.008,4.894) -- (2.118,4.838) -- (2.239,4.818) -- cycle;
\node at (2.425,5.212) {\scalebox{1.648}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.868,2.677) -- (1.99,2.696) -- (2.1,2.752) -- (2.187,2.839) -- (2.243,2.949) -- (2.262,3.071) -- (2.243,3.193) -- (2.187,3.302) -- (2.1,3.39) -- (1.99,3.446) -- (1.868,3.465) -- (1.498,3.465) -- (1.376,3.446) -- (1.266,3.39) -- (1.179,3.302) -- (1.123,3.193) -- (1.104,3.071) -- (1.123,2.949) -- (1.179,2.839) -- (1.266,2.752) -- (1.376,2.696) -- (1.498,2.677) -- cycle;
\node at (1.683,3.071) {\scalebox{1.648}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.682,3.037) -- (10.804,3.057) -- (10.914,3.113) -- (11.001,3.2) -- (11.057,3.31) -- (11.076,3.431) -- (11.057,3.553) -- (11.001,3.663) -- (10.914,3.75) -- (10.804,3.806) -- (10.682,3.825) -- (10.312,3.825) -- (10.19,3.806) -- (10.08,3.75) -- (9.993,3.663) -- (9.937,3.553) -- (9.918,3.431) -- (9.937,3.31) -- (9.993,3.2) -- (10.08,3.113) -- (10.19,3.057) -- (10.312,3.037) -- cycle;
\node at (10.497,3.431) {\scalebox{1.648}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.016,4.177) -- (8.138,4.196) -- (8.247,4.252) -- (8.335,4.339) -- (8.391,4.449) -- (8.41,4.57) -- (8.391,4.692) -- (8.335,4.802) -- (8.247,4.889) -- (8.138,4.945) -- (8.016,4.964) -- (7.645,4.964) -- (7.523,4.945) -- (7.414,4.889) -- (7.327,4.802) -- (7.271,4.692) -- (7.251,4.57) -- (7.271,4.449) -- (7.327,4.339) -- (7.414,4.252) -- (7.523,4.196) -- (7.645,4.177) -- cycle;
\node at (7.831,4.57) {\scalebox{1.648}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.353,2.609) -- (6.475,2.629) -- (6.584,2.685) -- (6.671,2.772) -- (6.727,2.882) -- (6.747,3.003) -- (6.727,3.125) -- (6.671,3.235) -- (6.584,3.322) -- (6.475,3.378) -- (6.353,3.397) -- (5.982,3.397) -- (5.86,3.378) -- (5.751,3.322) -- (5.663,3.235) -- (5.607,3.125) -- (5.588,3.003) -- (5.607,2.882) -- (5.663,2.772) -- (5.751,2.685) -- (5.86,2.629) -- (5.982,2.609) -- cycle;
\node at (6.167,3.003) {\scalebox{1.648}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.827,1.518) -- (2.948,1.538) -- (3.058,1.593) -- (3.145,1.681) -- (3.201,1.79) -- (3.221,1.912) -- (3.201,2.034) -- (3.145,2.144) -- (3.058,2.231) -- (2.948,2.287) -- (2.827,2.306) -- (2.456,2.306) -- (2.334,2.287) -- (2.224,2.231) -- (2.137,2.144) -- (2.081,2.034) -- (2.062,1.912) -- (2.081,1.79) -- (2.137,1.681) -- (2.224,1.593) -- (2.334,1.538) -- (2.456,1.518) -- cycle;
\node at (2.641,1.912) {\scalebox{1.648}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.3,1.705) -- (5.421,1.724) -- (5.531,1.78) -- (5.618,1.867) -- (5.674,1.977) -- (5.694,2.099) -- (5.674,2.22) -- (5.618,2.33) -- (5.531,2.417) -- (5.421,2.473) -- (5.3,2.492) -- (4.929,2.492) -- (4.807,2.473) -- (4.697,2.417) -- (4.61,2.33) -- (4.554,2.22) -- (4.535,2.099) -- (4.554,1.977) -- (4.61,1.867) -- (4.697,1.78) -- (4.807,1.724) -- (4.929,1.705) -- cycle;
\node at (5.114,2.099) {\scalebox{1.648}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (8.699,6.175) -- (9.299,6.175) -- (9.299,6.775) -- (8.699,6.775) -- cycle;
\node at (8.999,6.475) {\scalebox{1.648}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (10.145,7.646) -- (10.745,7.646) -- (10.745,8.246) -- (10.145,8.246) -- cycle;
\node at (10.445,7.946) {\scalebox{1.648}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (3.121,6.246) -- (3.721,6.246) -- (3.721,6.846) -- (3.121,6.846) -- cycle;
\node at (3.421,6.546) {\scalebox{1.648}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (7.397,8.102) -- (7.997,8.102) -- (7.997,8.702) -- (7.397,8.702) -- cycle;
\node at (7.697,8.402) {\scalebox{1.648}{\small d}};
\node at (11.666,7.217) {\scalebox{1.648}{\small A}};
\node at (6.277,7.482) {\scalebox{1.648}{\small B}};
\node at (3.465,7.239) {\scalebox{1.648}{\small C}};
\node at (9.505,6.834) {\scalebox{1.648}{\small D}};
\node at (3.127,4.374) {\scalebox{1.648}{\small E}};
\node at (0.862,4.354) {\scalebox{1.648}{\small F}};
\node at (9.838,4.305) {\scalebox{1.648}{\small G}};
\node at (11.928,4.257) {\scalebox{1.648}{\small H}};
\node at (6.942,3.856) {\scalebox{1.648}{\small I}};
\node at (0.765,1.515) {\scalebox{1.648}{\small J}};
\node at (6.52,1.142) {\scalebox{1.648}{\small K}};
\node at (3.52,0.95) {\scalebox{1.648}{\small L}};
\draw[line width=0.5pt] (12.609,5.99) -- (12.609,7.09);
\draw[line width=0.5pt] (12.378,6.705) -- (12.609,7.09) -- (12.84,6.705);
\draw[line width=0.5pt] (12.306,6.54) -- (12.911,6.54);
\node at (12.609,7.475) {\scalebox{1.648}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour D, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ rue de la Mairie & Ⓐ Place des Tanneurs & Ⓙ Coin du Lavoir \\ \hline
① chemin de Ronde & ⑩ chemin des Vignes & Ⓑ Carrefour du Moulin & Ⓚ Place aux Herbes \\ \hline
② rue Verte & ⑪ rue des Jardins & Ⓒ Place de l'Église & Ⓛ Rond-point du Stade \\ \hline
③ chemin du Lavoir & ⑫ boulevard du Parc & Ⓓ Rond-point du Pont & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ sentier des Prés & ⑬ rue Pasteur & Ⓔ Rond-point des Vaches & \textbf{a} l'école \\ \hline
⑤ allée des Peupliers & ⑭ chemin Creux & Ⓕ Placette des Ormes & \textbf{b} le cimetière \\ \hline
⑥ rue du Stade & ⑮ rue de la Poste & Ⓖ Carrefour des Tilleuls & \textbf{c} l'église \\ \hline
⑦ rue Basse & ⑯ rue des Lilas & Ⓗ Place de la Fontaine & \textbf{d} la mairie \\ \hline
⑧ rue de l'Église & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour de la Mairie &  \\ \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=0.6217, line join=round, line cap=round]
\draw[line width=0.5pt] (11.112,6.94) -- (11.097,6.75) -- (11.086,6.559) -- (11.08,6.367) -- (11.078,6.176) -- (11.081,5.985) -- (11.089,5.793) -- (11.101,5.602) -- (11.117,5.412) -- (11.138,5.221) -- (11.163,5.032) -- (11.193,4.842) -- (11.227,4.654) -- (11.266,4.467) -- (11.309,4.281);
\draw[line width=0.5pt] (6.257,6.863) -- (6.456,6.865) -- (6.654,6.863) -- (6.853,6.856) -- (7.051,6.845) -- (7.249,6.828) -- (7.447,6.808) -- (7.644,6.782) -- (7.841,6.752) -- (8.036,6.717) -- (8.231,6.678) -- (8.425,6.634) -- (8.618,6.586) -- (8.809,6.533) -- (8.999,6.475);
\draw[line width=0.5pt] (6.257,6.863) -- (6.243,6.675) -- (6.233,6.487) -- (6.228,6.299) -- (6.227,6.111) -- (6.23,5.923) -- (6.238,5.734) -- (6.251,5.547) -- (6.268,5.359) -- (6.289,5.172) -- (6.314,4.985) -- (6.344,4.799) -- (6.379,4.614) -- (6.417,4.43) -- (6.461,4.247);
\draw[line width=0.5pt] (3.899,6.797) -- (4.066,6.827) -- (4.233,6.854) -- (4.4,6.876) -- (4.569,6.895) -- (4.737,6.91) -- (4.906,6.92) -- (5.075,6.927) -- (5.245,6.93) -- (5.414,6.929) -- (5.583,6.923) -- (5.752,6.914) -- (5.921,6.901) -- (6.089,6.884) -- (6.257,6.863);
\draw[line width=0.5pt] (3.899,6.797) -- (3.862,6.628) -- (3.829,6.459) -- (3.8,6.289) -- (3.775,6.118) -- (3.754,5.946) -- (3.737,5.775) -- (3.724,5.602) -- (3.715,5.43) -- (3.71,5.257) -- (3.709,5.084) -- (3.713,4.912) -- (3.72,4.739) -- (3.731,4.567) -- (3.747,4.395);
\draw[line width=0.5pt] (8.999,6.475) -- (9.144,6.531) -- (9.29,6.584) -- (9.437,6.633) -- (9.585,6.679) -- (9.735,6.721) -- (9.885,6.76) -- (10.036,6.795) -- (10.188,6.827) -- (10.341,6.855) -- (10.494,6.879) -- (10.648,6.9) -- (10.802,6.917) -- (10.957,6.93) -- (11.112,6.94);
\draw[line width=0.5pt] (8.999,6.475) -- (8.991,6.321) -- (8.987,6.167) -- (8.986,6.012) -- (8.989,5.858) -- (8.996,5.704) -- (9.006,5.549) -- (9.02,5.396) -- (9.038,5.242) -- (9.059,5.089) -- (9.084,4.936) -- (9.112,4.785) -- (9.144,4.634) -- (9.179,4.483) -- (9.218,4.334);
\draw[line width=0.5pt] (3.747,4.395) -- (3.941,4.414) -- (4.135,4.429) -- (4.33,4.439) -- (4.525,4.444) -- (4.72,4.445) -- (4.915,4.441) -- (5.11,4.433) -- (5.304,4.42) -- (5.499,4.403) -- (5.693,4.381) -- (5.886,4.354) -- (6.078,4.323) -- (6.27,4.287) -- (6.461,4.247);
\draw[line width=0.5pt] (3.747,4.395) -- (3.725,4.184) -- (3.708,3.973) -- (3.696,3.762) -- (3.689,3.55) -- (3.687,3.338) -- (3.69,3.126) -- (3.698,2.914) -- (3.711,2.702) -- (3.729,2.491) -- (3.752,2.28) -- (3.78,2.07) -- (3.813,1.861) -- (3.851,1.653) -- (3.893,1.445);
\draw[line width=0.5pt] (1.482,4.368) -- (1.642,4.395) -- (1.803,4.418) -- (1.965,4.437) -- (2.126,4.452) -- (2.289,4.464) -- (2.451,4.471) -- (2.613,4.475) -- (2.776,4.475) -- (2.939,4.471) -- (3.101,4.464) -- (3.263,4.452) -- (3.425,4.437) -- (3.586,4.418) -- (3.747,4.395);
\draw[line width=0.5pt] (1.482,4.368) -- (1.442,4.189) -- (1.406,4.009) -- (1.374,3.828) -- (1.347,3.646) -- (1.324,3.464) -- (1.305,3.281) -- (1.29,3.098) -- (1.28,2.914) -- (1.274,2.73) -- (1.272,2.546) -- (1.275,2.363) -- (1.282,2.179) -- (1.293,1.995) -- (1.309,1.812);
\draw[line width=0.5pt] (9.218,4.334) -- (9.367,4.353) -- (9.516,4.369) -- (9.666,4.381) -- (9.816,4.389) -- (9.966,4.394) -- (10.116,4.396) -- (10.266,4.393) -- (10.416,4.388) -- (10.566,4.379) -- (10.715,4.366) -- (10.864,4.35) -- (11.013,4.33) -- (11.161,4.307) -- (11.309,4.281);
\draw[line width=0.5pt] (6.461,4.247) -- (6.657,4.223) -- (6.854,4.204) -- (7.051,4.189) -- (7.249,4.179) -- (7.447,4.174) -- (7.645,4.173) -- (7.843,4.177) -- (8.041,4.185) -- (8.238,4.199) -- (8.436,4.216) -- (8.632,4.239) -- (8.828,4.266) -- (9.024,4.298) -- (9.218,4.334);
\draw[line width=0.5pt] (6.461,4.247) -- (6.488,4.071) -- (6.51,3.894) -- (6.529,3.716) -- (6.543,3.538) -- (6.554,3.36) -- (6.56,3.181) -- (6.561,3.003) -- (6.559,2.824) -- (6.552,2.646) -- (6.542,2.467) -- (6.527,2.289) -- (6.508,2.112) -- (6.484,1.935) -- (6.457,1.759);
\draw[line width=0.5pt] (1.309,1.812) -- (1.488,1.758) -- (1.668,1.708) -- (1.85,1.662) -- (2.033,1.621) -- (2.216,1.584) -- (2.401,1.551) -- (2.586,1.522) -- (2.772,1.498) -- (2.958,1.478) -- (3.145,1.463) -- (3.332,1.452) -- (3.519,1.445) -- (3.706,1.443) -- (3.893,1.445);
\draw[line width=0.5pt] (3.893,1.445) -- (4.071,1.495) -- (4.251,1.541) -- (4.431,1.583) -- (4.613,1.621) -- (4.795,1.654) -- (4.978,1.683) -- (5.162,1.708) -- (5.346,1.728) -- (5.531,1.744) -- (5.716,1.756) -- (5.901,1.763) -- (6.087,1.766) -- (6.272,1.764) -- (6.457,1.759);
\filldraw[fill=white, draw=black, line width=0.5pt] (11.485,6.94) -- (11.457,7.083) -- (11.376,7.204) -- (11.255,7.285) -- (11.112,7.314) -- (10.969,7.285) -- (10.848,7.204) -- (10.767,7.083) -- (10.739,6.94) -- (10.767,6.798) -- (10.848,6.676) -- (10.969,6.596) -- (11.112,6.567) -- (11.255,6.596) -- (11.376,6.676) -- (11.457,6.798) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.63,6.863) -- (6.602,7.006) -- (6.521,7.127) -- (6.4,7.207) -- (6.257,7.236) -- (6.114,7.207) -- (5.993,7.127) -- (5.912,7.006) -- (5.884,6.863) -- (5.912,6.72) -- (5.993,6.599) -- (6.114,6.518) -- (6.257,6.489) -- (6.4,6.518) -- (6.521,6.599) -- (6.602,6.72) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.273,6.797) -- (4.244,6.94) -- (4.163,7.061) -- (4.042,7.142) -- (3.899,7.17) -- (3.756,7.142) -- (3.635,7.061) -- (3.554,6.94) -- (3.526,6.797) -- (3.554,6.654) -- (3.635,6.533) -- (3.756,6.452) -- (3.899,6.424) -- (4.042,6.452) -- (4.163,6.533) -- (4.244,6.654) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.372,6.475) -- (9.344,6.618) -- (9.263,6.739) -- (9.142,6.82) -- (8.999,6.849) -- (8.856,6.82) -- (8.735,6.739) -- (8.654,6.618) -- (8.626,6.475) -- (8.654,6.332) -- (8.735,6.211) -- (8.856,6.13) -- (8.999,6.102) -- (9.142,6.13) -- (9.263,6.211) -- (9.344,6.332) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.12,4.395) -- (4.092,4.538) -- (4.011,4.659) -- (3.89,4.74) -- (3.747,4.768) -- (3.604,4.74) -- (3.483,4.659) -- (3.402,4.538) -- (3.374,4.395) -- (3.402,4.252) -- (3.483,4.131) -- (3.604,4.05) -- (3.747,4.022) -- (3.89,4.05) -- (4.011,4.131) -- (4.092,4.252) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.855,4.368) -- (1.827,4.511) -- (1.746,4.632) -- (1.625,4.713) -- (1.482,4.742) -- (1.339,4.713) -- (1.218,4.632) -- (1.137,4.511) -- (1.109,4.368) -- (1.137,4.226) -- (1.218,4.104) -- (1.339,4.024) -- (1.482,3.995) -- (1.625,4.024) -- (1.746,4.104) -- (1.827,4.226) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.591,4.334) -- (9.563,4.477) -- (9.482,4.598) -- (9.361,4.679) -- (9.218,4.707) -- (9.075,4.679) -- (8.954,4.598) -- (8.873,4.477) -- (8.845,4.334) -- (8.873,4.191) -- (8.954,4.07) -- (9.075,3.989) -- (9.218,3.961) -- (9.361,3.989) -- (9.482,4.07) -- (9.563,4.191) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.682,4.281) -- (11.654,4.423) -- (11.573,4.544) -- (11.452,4.625) -- (11.309,4.654) -- (11.166,4.625) -- (11.045,4.544) -- (10.964,4.423) -- (10.936,4.281) -- (10.964,4.138) -- (11.045,4.017) -- (11.166,3.936) -- (11.309,3.907) -- (11.452,3.936) -- (11.573,4.017) -- (11.654,4.138) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.834,4.247) -- (6.805,4.39) -- (6.724,4.511) -- (6.603,4.592) -- (6.461,4.62) -- (6.318,4.592) -- (6.197,4.511) -- (6.116,4.39) -- (6.087,4.247) -- (6.116,4.104) -- (6.197,3.983) -- (6.318,3.902) -- (6.461,3.874) -- (6.603,3.902) -- (6.724,3.983) -- (6.805,4.104) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.682,1.812) -- (1.654,1.955) -- (1.573,2.076) -- (1.452,2.157) -- (1.309,2.186) -- (1.166,2.157) -- (1.045,2.076) -- (0.964,1.955) -- (0.936,1.812) -- (0.964,1.67) -- (1.045,1.549) -- (1.166,1.468) -- (1.309,1.439) -- (1.452,1.468) -- (1.573,1.549) -- (1.654,1.67) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.83,1.759) -- (6.802,1.901) -- (6.721,2.023) -- (6.6,2.103) -- (6.457,2.132) -- (6.314,2.103) -- (6.193,2.023) -- (6.112,1.901) -- (6.084,1.759) -- (6.112,1.616) -- (6.193,1.495) -- (6.314,1.414) -- (6.457,1.385) -- (6.6,1.414) -- (6.721,1.495) -- (6.802,1.616) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.266,1.445) -- (4.238,1.588) -- (4.157,1.709) -- (4.036,1.79) -- (3.893,1.818) -- (3.75,1.79) -- (3.629,1.709) -- (3.548,1.588) -- (3.52,1.445) -- (3.548,1.302) -- (3.629,1.181) -- (3.75,1.1) -- (3.893,1.072) -- (4.036,1.1) -- (4.157,1.181) -- (4.238,1.302) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.484,5.246) -- (11.603,5.265) -- (11.71,5.32) -- (11.795,5.405) -- (11.85,5.512) -- (11.869,5.631) -- (11.85,5.75) -- (11.795,5.857) -- (11.71,5.942) -- (11.603,5.996) -- (11.484,6.015) -- (11.484,6.015) -- (11.365,5.996) -- (11.258,5.942) -- (11.173,5.857) -- (11.118,5.75) -- (11.1,5.631) -- (11.118,5.512) -- (11.173,5.405) -- (11.258,5.32) -- (11.365,5.265) -- (11.484,5.246) -- cycle;
\node at (11.484,5.631) {\scalebox{1.608}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.698,6.778) -- (7.817,6.797) -- (7.924,6.852) -- (8.009,6.937) -- (8.064,7.044) -- (8.082,7.163) -- (8.064,7.282) -- (8.009,7.389) -- (7.924,7.474) -- (7.817,7.529) -- (7.698,7.547) -- (7.698,7.547) -- (7.579,7.529) -- (7.472,7.474) -- (7.387,7.389) -- (7.332,7.282) -- (7.313,7.163) -- (7.332,7.044) -- (7.387,6.937) -- (7.472,6.852) -- (7.579,6.797) -- (7.698,6.778) -- cycle;
\node at (7.698,7.163) {\scalebox{1.608}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.634,5.192) -- (6.753,5.211) -- (6.86,5.265) -- (6.945,5.35) -- (7.0,5.458) -- (7.019,5.576) -- (7.0,5.695) -- (6.945,5.802) -- (6.86,5.887) -- (6.753,5.942) -- (6.634,5.961) -- (6.634,5.961) -- (6.515,5.942) -- (6.408,5.887) -- (6.323,5.802) -- (6.269,5.695) -- (6.25,5.576) -- (6.269,5.458) -- (6.323,5.35) -- (6.408,5.265) -- (6.515,5.211) -- (6.634,5.192) -- cycle;
\node at (6.634,5.576) {\scalebox{1.608}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.065,6.927) -- (5.184,6.946) -- (5.291,7.0) -- (5.376,7.085) -- (5.43,7.193) -- (5.449,7.311) -- (5.43,7.43) -- (5.376,7.537) -- (5.291,7.622) -- (5.184,7.677) -- (5.065,7.696) -- (5.065,7.696) -- (4.946,7.677) -- (4.839,7.622) -- (4.754,7.537) -- (4.699,7.43) -- (4.68,7.311) -- (4.699,7.193) -- (4.754,7.085) -- (4.839,7.0) -- (4.946,6.946) -- (5.065,6.927) -- cycle;
\node at (5.065,7.311) {\scalebox{1.608}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.108,5.193) -- (4.227,5.212) -- (4.334,5.267) -- (4.419,5.352) -- (4.473,5.459) -- (4.492,5.578) -- (4.473,5.697) -- (4.419,5.804) -- (4.334,5.889) -- (4.227,5.944) -- (4.108,5.962) -- (4.108,5.962) -- (3.989,5.944) -- (3.882,5.889) -- (3.797,5.804) -- (3.742,5.697) -- (3.723,5.578) -- (3.742,5.459) -- (3.797,5.352) -- (3.882,5.267) -- (3.989,5.212) -- (4.108,5.193) -- cycle;
\node at (4.108,5.578) {\scalebox{1.608}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.153,5.88) -- (10.272,5.899) -- (10.379,5.954) -- (10.464,6.039) -- (10.519,6.146) -- (10.538,6.265) -- (10.519,6.384) -- (10.464,6.491) -- (10.379,6.576) -- (10.272,6.63) -- (10.153,6.649) -- (10.153,6.649) -- (10.034,6.63) -- (9.927,6.576) -- (9.842,6.491) -- (9.787,6.384) -- (9.768,6.265) -- (9.787,6.146) -- (9.842,6.039) -- (9.927,5.954) -- (10.034,5.899) -- (10.153,5.88) -- cycle;
\node at (10.153,6.265) {\scalebox{1.608}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.403,5.05) -- (9.522,5.069) -- (9.629,5.124) -- (9.714,5.209) -- (9.769,5.316) -- (9.787,5.435) -- (9.769,5.553) -- (9.714,5.661) -- (9.629,5.746) -- (9.522,5.8) -- (9.403,5.819) -- (9.403,5.819) -- (9.284,5.8) -- (9.177,5.746) -- (9.092,5.661) -- (9.037,5.553) -- (9.018,5.435) -- (9.037,5.316) -- (9.092,5.209) -- (9.177,5.124) -- (9.284,5.069) -- (9.403,5.05) -- cycle;
\node at (9.403,5.435) {\scalebox{1.608}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.131,4.433) -- (5.25,4.451) -- (5.357,4.506) -- (5.442,4.591) -- (5.496,4.698) -- (5.515,4.817) -- (5.496,4.936) -- (5.442,5.043) -- (5.357,5.128) -- (5.25,5.183) -- (5.131,5.202) -- (5.131,5.202) -- (5.012,5.183) -- (4.905,5.128) -- (4.82,5.043) -- (4.765,4.936) -- (4.746,4.817) -- (4.765,4.698) -- (4.82,4.591) -- (4.905,4.506) -- (5.012,4.451) -- (5.131,4.433) -- cycle;
\node at (5.131,4.817) {\scalebox{1.608}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.082,2.549) -- (4.201,2.567) -- (4.308,2.622) -- (4.394,2.707) -- (4.448,2.814) -- (4.467,2.933) -- (4.448,3.052) -- (4.394,3.159) -- (4.308,3.244) -- (4.201,3.299) -- (4.082,3.318) -- (4.082,3.318) -- (3.964,3.299) -- (3.856,3.244) -- (3.771,3.159) -- (3.717,3.052) -- (3.698,2.933) -- (3.717,2.814) -- (3.771,2.707) -- (3.856,2.622) -- (3.964,2.567) -- (4.082,2.549) -- cycle;
\node at (4.082,2.933) {\scalebox{1.608}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.606,4.81) -- (2.725,4.829) -- (2.832,4.884) -- (2.917,4.969) -- (2.972,5.076) -- (2.991,5.195) -- (2.972,5.314) -- (2.917,5.421) -- (2.832,5.506) -- (2.725,5.56) -- (2.606,5.579) -- (2.244,5.579) -- (2.126,5.56) -- (2.018,5.506) -- (1.933,5.421) -- (1.879,5.314) -- (1.86,5.195) -- (1.879,5.076) -- (1.933,4.969) -- (2.018,4.884) -- (2.126,4.829) -- (2.244,4.81) -- cycle;
\node at (2.425,5.195) {\scalebox{1.608}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.855,2.687) -- (1.974,2.706) -- (2.081,2.76) -- (2.166,2.846) -- (2.22,2.953) -- (2.239,3.072) -- (2.22,3.19) -- (2.166,3.298) -- (2.081,3.383) -- (1.974,3.437) -- (1.855,3.456) -- (1.493,3.456) -- (1.374,3.437) -- (1.267,3.383) -- (1.182,3.298) -- (1.127,3.19) -- (1.108,3.072) -- (1.127,2.953) -- (1.182,2.846) -- (1.267,2.76) -- (1.374,2.706) -- (1.493,2.687) -- cycle;
\node at (1.674,3.072) {\scalebox{1.608}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.679,3.069) -- (10.798,3.088) -- (10.905,3.143) -- (10.991,3.228) -- (11.045,3.335) -- (11.064,3.454) -- (11.045,3.573) -- (10.991,3.68) -- (10.905,3.765) -- (10.798,3.82) -- (10.679,3.838) -- (10.318,3.838) -- (10.199,3.82) -- (10.092,3.765) -- (10.006,3.68) -- (9.952,3.573) -- (9.933,3.454) -- (9.952,3.335) -- (10.006,3.228) -- (10.092,3.143) -- (10.199,3.088) -- (10.318,3.069) -- cycle;
\node at (10.499,3.454) {\scalebox{1.608}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.012,4.177) -- (8.131,4.195) -- (8.238,4.25) -- (8.323,4.335) -- (8.378,4.442) -- (8.396,4.561) -- (8.378,4.68) -- (8.323,4.787) -- (8.238,4.872) -- (8.131,4.927) -- (8.012,4.946) -- (7.65,4.946) -- (7.531,4.927) -- (7.424,4.872) -- (7.339,4.787) -- (7.284,4.68) -- (7.265,4.561) -- (7.284,4.442) -- (7.339,4.335) -- (7.424,4.25) -- (7.531,4.195) -- (7.65,4.177) -- cycle;
\node at (7.831,4.561) {\scalebox{1.608}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.358,2.619) -- (6.477,2.638) -- (6.584,2.692) -- (6.669,2.777) -- (6.723,2.884) -- (6.742,3.003) -- (6.723,3.122) -- (6.669,3.229) -- (6.584,3.314) -- (6.477,3.369) -- (6.358,3.388) -- (5.996,3.388) -- (5.877,3.369) -- (5.77,3.314) -- (5.685,3.229) -- (5.63,3.122) -- (5.611,3.003) -- (5.63,2.884) -- (5.685,2.777) -- (5.77,2.692) -- (5.877,2.638) -- (5.996,2.619) -- cycle;
\node at (6.177,3.003) {\scalebox{1.608}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.821,1.518) -- (2.94,1.537) -- (3.047,1.592) -- (3.132,1.677) -- (3.187,1.784) -- (3.206,1.903) -- (3.187,2.022) -- (3.132,2.129) -- (3.047,2.214) -- (2.94,2.269) -- (2.821,2.287) -- (2.459,2.287) -- (2.34,2.269) -- (2.233,2.214) -- (2.148,2.129) -- (2.093,2.022) -- (2.075,1.903) -- (2.093,1.784) -- (2.148,1.677) -- (2.233,1.592) -- (2.34,1.537) -- (2.459,1.518) -- cycle;
\node at (2.64,1.903) {\scalebox{1.608}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.296,1.705) -- (5.415,1.724) -- (5.522,1.778) -- (5.607,1.863) -- (5.662,1.97) -- (5.681,2.089) -- (5.662,2.208) -- (5.607,2.315) -- (5.522,2.4) -- (5.415,2.455) -- (5.296,2.474) -- (4.934,2.474) -- (4.816,2.455) -- (4.708,2.4) -- (4.623,2.315) -- (4.569,2.208) -- (4.55,2.089) -- (4.569,1.97) -- (4.623,1.863) -- (4.708,1.778) -- (4.816,1.724) -- (4.934,1.705) -- cycle;
\node at (5.115,2.089) {\scalebox{1.608}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (8.699,6.175) -- (9.299,6.175) -- (9.299,6.775) -- (8.699,6.775) -- cycle;
\node at (8.999,6.475) {\scalebox{1.608}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (10.145,7.646) -- (10.745,7.646) -- (10.745,8.246) -- (10.145,8.246) -- cycle;
\node at (10.445,7.946) {\scalebox{1.608}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (3.121,6.246) -- (3.721,6.246) -- (3.721,6.846) -- (3.121,6.846) -- cycle;
\node at (3.421,6.546) {\scalebox{1.608}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (7.397,8.102) -- (7.997,8.102) -- (7.997,8.702) -- (7.397,8.702) -- cycle;
\node at (7.697,8.402) {\scalebox{1.608}{\small d}};
\node at (11.666,7.217) {\scalebox{1.608}{\small A}};
\node at (6.277,7.482) {\scalebox{1.608}{\small B}};
\node at (3.465,7.239) {\scalebox{1.608}{\small C}};
\node at (9.505,6.834) {\scalebox{1.608}{\small D}};
\node at (3.127,4.374) {\scalebox{1.608}{\small E}};
\node at (0.862,4.354) {\scalebox{1.608}{\small F}};
\node at (9.838,4.305) {\scalebox{1.608}{\small G}};
\node at (11.928,4.257) {\scalebox{1.608}{\small H}};
\node at (6.942,3.856) {\scalebox{1.608}{\small I}};
\node at (0.765,1.515) {\scalebox{1.608}{\small J}};
\node at (6.52,1.142) {\scalebox{1.608}{\small K}};
\node at (3.52,0.95) {\scalebox{1.608}{\small L}};
\draw[line width=0.5pt] (12.609,5.99) -- (12.609,7.09);
\draw[line width=0.5pt] (12.378,6.705) -- (12.609,7.09) -- (12.84,6.705);
\draw[line width=0.5pt] (12.306,6.54) -- (12.911,6.54);
\node at (12.609,7.475) {\scalebox{1.608}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & C, D, F, H, I, L & 6 \\ \hline
clairs & A, B, E, G, J, K & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{6} clairs. Il y en a \textbf{autant} de chaque côté.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 12 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui.} Un circuit existe :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{rue du Stade ⑥} vers l'est jusqu'à Place des Tanneurs \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin de Ronde ①}, vers le sud, jusqu'à Place de la Fontaine \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{boulevard du Parc ⑫}, vers l'ouest, jusqu'à Carrefour des Tilleuls \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Pasteur ⑬}, vers l'ouest, jusqu'à Carrefour de la Mairie \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin Creux ⑭}, vers le sud, jusqu'à Place aux Herbes \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Lilas ⑯}, vers l'ouest, jusqu'à Rond-point du Stade \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de la Poste ⑮}, vers l'ouest, jusqu'à Coin du Lavoir \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Jardins ⑪}, vers le nord, jusqu'à Placette des Ormes \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin des Vignes ⑩}, vers l'est, jusqu'à Rond-point des Vaches \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{allée des Peupliers ⑤}, vers le nord, jusqu'à Place de l'Église \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ④}, vers l'est, jusqu'à Carrefour du Moulin \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Verte ②}, vers l'est, jusqu'à Rond-point du Pont \textbf{(D)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Le car est passé une fois par chaque carrefour et se retrouve à l'école.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, et c'est déjà fait : le trajet ci-dessus dessert tous les carrefours. Il suffit au car de s'arrêter avant le dernier trajet de retour.}\par\addvspace{0.4em}
\end{document}
