\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 219}

\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=219}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0219}
\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=0.6121, line join=round, line cap=round]
\draw[line width=0.5pt] (5.102,10.809) -- (5.074,10.598) -- (5.052,10.387) -- (5.034,10.175) -- (5.022,9.963) -- (5.015,9.751) -- (5.012,9.538) -- (5.015,9.325) -- (5.022,9.113) -- (5.035,8.901) -- (5.052,8.689) -- (5.075,8.478) -- (5.102,8.267) -- (5.134,8.057) -- (5.172,7.848);
\draw[line width=0.5pt] (1.194,10.556) -- (1.468,10.616) -- (1.744,10.67) -- (2.021,10.718) -- (2.299,10.759) -- (2.578,10.794) -- (2.857,10.822) -- (3.138,10.843) -- (3.418,10.858) -- (3.699,10.866) -- (3.98,10.868) -- (4.261,10.863) -- (4.542,10.852) -- (4.822,10.834) -- (5.102,10.809);
\draw[line width=0.5pt] (1.194,10.556) -- (1.178,10.339) -- (1.167,10.123) -- (1.161,9.906) -- (1.16,9.689) -- (1.165,9.472) -- (1.174,9.255) -- (1.189,9.038) -- (1.208,8.822) -- (1.233,8.607) -- (1.263,8.392) -- (1.297,8.177) -- (1.337,7.964) -- (1.382,7.752) -- (1.432,7.541);
\draw[line width=0.5pt] (11.194,10.228) -- (11.188,10.035) -- (11.178,9.843) -- (11.163,9.651) -- (11.143,9.459) -- (11.119,9.267) -- (11.09,9.076) -- (11.057,8.886) -- (11.019,8.697) -- (10.977,8.509) -- (10.931,8.322) -- (10.88,8.135) -- (10.825,7.951) -- (10.766,7.767) -- (10.702,7.585);
\draw[line width=0.5pt] (5.172,7.848) -- (5.349,7.807) -- (5.528,7.771) -- (5.708,7.738) -- (5.888,7.71) -- (6.069,7.686) -- (6.25,7.667) -- (6.432,7.651) -- (6.614,7.64) -- (6.797,7.633) -- (6.979,7.631) -- (7.162,7.632) -- (7.344,7.638) -- (7.526,7.649) -- (7.708,7.664);
\draw[line width=0.5pt] (5.172,7.848) -- (5.184,7.58) -- (5.189,7.311) -- (5.189,7.042) -- (5.182,6.774) -- (5.169,6.505) -- (5.149,6.237) -- (5.123,5.97) -- (5.091,5.703) -- (5.052,5.437) -- (5.008,5.171) -- (4.957,4.908) -- (4.9,4.645) -- (4.837,4.384) -- (4.767,4.125);
\draw[line width=0.5pt] (7.708,7.664) -- (7.919,7.625) -- (8.131,7.592) -- (8.344,7.564) -- (8.558,7.54) -- (8.772,7.522) -- (8.987,7.509) -- (9.202,7.501) -- (9.416,7.498) -- (9.631,7.5) -- (9.846,7.507) -- (10.061,7.519) -- (10.275,7.536) -- (10.489,7.558) -- (10.702,7.585);
\draw[line width=0.5pt] (7.708,7.664) -- (7.726,7.457) -- (7.74,7.25) -- (7.749,7.043) -- (7.752,6.836) -- (7.751,6.628) -- (7.745,6.421) -- (7.735,6.213) -- (7.719,6.007) -- (7.699,5.8) -- (7.673,5.594) -- (7.643,5.389) -- (7.608,5.184) -- (7.568,4.981) -- (7.524,4.779);
\draw[line width=0.5pt] (1.432,7.541) -- (1.7,7.522) -- (1.969,7.509) -- (2.238,7.503) -- (2.507,7.502) -- (2.777,7.509) -- (3.046,7.521) -- (3.314,7.54) -- (3.583,7.565) -- (3.85,7.597) -- (4.117,7.635) -- (4.382,7.679) -- (4.647,7.729) -- (4.91,7.785) -- (5.172,7.848);
\draw[line width=0.5pt] (1.432,7.541) -- (1.512,7.308) -- (1.587,7.074) -- (1.657,6.837) -- (1.721,6.599) -- (1.779,6.36) -- (1.832,6.119) -- (1.879,5.877) -- (1.92,5.634) -- (1.956,5.39) -- (1.986,5.146) -- (2.01,4.9) -- (2.028,4.655) -- (2.041,4.409) -- (2.047,4.163);
\draw[line width=0.5pt] (7.524,4.779) -- (7.475,4.55) -- (7.432,4.32) -- (7.395,4.089) -- (7.363,3.857) -- (7.336,3.624) -- (7.315,3.391) -- (7.3,3.158) -- (7.289,2.924) -- (7.285,2.69) -- (7.286,2.456) -- (7.292,2.222) -- (7.304,1.988) -- (7.321,1.755) -- (7.344,1.522);
\draw[line width=0.5pt] (2.047,4.163) -- (2.241,4.19) -- (2.434,4.212) -- (2.629,4.23) -- (2.823,4.243) -- (3.018,4.252) -- (3.214,4.256) -- (3.409,4.256) -- (3.604,4.251) -- (3.799,4.241) -- (3.994,4.227) -- (4.188,4.208) -- (4.382,4.185) -- (4.575,4.157) -- (4.767,4.125);
\draw[line width=0.5pt] (2.047,4.163) -- (1.977,4.004) -- (1.91,3.843) -- (1.847,3.681) -- (1.788,3.517) -- (1.732,3.352) -- (1.681,3.185) -- (1.633,3.018) -- (1.59,2.849) -- (1.55,2.679) -- (1.515,2.509) -- (1.483,2.338) -- (1.455,2.166) -- (1.432,1.993) -- (1.413,1.82);
\draw[line width=0.5pt] (4.767,4.125) -- (4.776,3.949) -- (4.78,3.773) -- (4.78,3.597) -- (4.776,3.421) -- (4.768,3.245) -- (4.756,3.069) -- (4.739,2.894) -- (4.719,2.719) -- (4.694,2.544) -- (4.665,2.37) -- (4.633,2.197) -- (4.596,2.025) -- (4.555,1.854) -- (4.51,1.684);
\draw[line width=0.5pt] (1.413,1.82) -- (1.631,1.777) -- (1.85,1.738) -- (2.07,1.705) -- (2.29,1.677) -- (2.512,1.654) -- (2.733,1.637) -- (2.956,1.624) -- (3.178,1.617) -- (3.401,1.615) -- (3.623,1.618) -- (3.845,1.627) -- (4.067,1.641) -- (4.289,1.66) -- (4.51,1.684);
\draw[line width=0.5pt] (4.51,1.684) -- (4.713,1.703) -- (4.916,1.718) -- (5.119,1.728) -- (5.323,1.733) -- (5.526,1.733) -- (5.73,1.729) -- (5.934,1.72) -- (6.137,1.706) -- (6.34,1.687) -- (6.542,1.663) -- (6.744,1.635) -- (6.945,1.602) -- (7.145,1.564) -- (7.344,1.522);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.262,10.809) -- (5.24,10.889) -- (5.182,10.947) -- (5.102,10.969) -- (5.022,10.947) -- (4.963,10.889) -- (4.942,10.809) -- (4.963,10.729) -- (5.022,10.67) -- (5.102,10.649) -- (5.182,10.67) -- (5.24,10.729) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.354,10.556) -- (1.333,10.636) -- (1.274,10.694) -- (1.194,10.716) -- (1.114,10.694) -- (1.056,10.636) -- (1.034,10.556) -- (1.056,10.476) -- (1.114,10.417) -- (1.194,10.396) -- (1.274,10.417) -- (1.333,10.476) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.354,10.228) -- (11.333,10.308) -- (11.274,10.367) -- (11.194,10.388) -- (11.114,10.367) -- (11.056,10.308) -- (11.034,10.228) -- (11.056,10.148) -- (11.114,10.089) -- (11.194,10.068) -- (11.274,10.089) -- (11.333,10.148) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.332,7.848) -- (5.31,7.928) -- (5.252,7.987) -- (5.172,8.008) -- (5.092,7.987) -- (5.033,7.928) -- (5.012,7.848) -- (5.033,7.768) -- (5.092,7.709) -- (5.172,7.688) -- (5.252,7.709) -- (5.31,7.768) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.868,7.664) -- (7.846,7.744) -- (7.788,7.802) -- (7.708,7.824) -- (7.628,7.802) -- (7.569,7.744) -- (7.548,7.664) -- (7.569,7.584) -- (7.628,7.525) -- (7.708,7.504) -- (7.788,7.525) -- (7.846,7.584) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.862,7.585) -- (10.84,7.665) -- (10.782,7.724) -- (10.702,7.745) -- (10.622,7.724) -- (10.563,7.665) -- (10.542,7.585) -- (10.563,7.505) -- (10.622,7.447) -- (10.702,7.425) -- (10.782,7.447) -- (10.84,7.505) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.592,7.541) -- (1.57,7.621) -- (1.512,7.679) -- (1.432,7.701) -- (1.352,7.679) -- (1.293,7.621) -- (1.272,7.541) -- (1.293,7.461) -- (1.352,7.402) -- (1.432,7.381) -- (1.512,7.402) -- (1.57,7.461) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.684,4.779) -- (7.662,4.859) -- (7.604,4.917) -- (7.524,4.939) -- (7.444,4.917) -- (7.385,4.859) -- (7.364,4.779) -- (7.385,4.699) -- (7.444,4.64) -- (7.524,4.619) -- (7.604,4.64) -- (7.662,4.699) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.207,4.163) -- (2.186,4.243) -- (2.127,4.301) -- (2.047,4.323) -- (1.967,4.301) -- (1.909,4.243) -- (1.887,4.163) -- (1.909,4.083) -- (1.967,4.024) -- (2.047,4.003) -- (2.127,4.024) -- (2.186,4.083) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.927,4.125) -- (4.906,4.205) -- (4.847,4.263) -- (4.767,4.285) -- (4.687,4.263) -- (4.629,4.205) -- (4.607,4.125) -- (4.629,4.045) -- (4.687,3.986) -- (4.767,3.965) -- (4.847,3.986) -- (4.906,4.045) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.573,1.82) -- (1.551,1.9) -- (1.493,1.959) -- (1.413,1.98) -- (1.333,1.959) -- (1.274,1.9) -- (1.253,1.82) -- (1.274,1.74) -- (1.333,1.682) -- (1.413,1.66) -- (1.493,1.682) -- (1.551,1.74) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.67,1.684) -- (4.648,1.764) -- (4.59,1.822) -- (4.51,1.844) -- (4.43,1.822) -- (4.371,1.764) -- (4.35,1.684) -- (4.371,1.604) -- (4.43,1.545) -- (4.51,1.524) -- (4.59,1.545) -- (4.648,1.604) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.504,1.522) -- (7.483,1.602) -- (7.424,1.66) -- (7.344,1.682) -- (7.264,1.66) -- (7.205,1.602) -- (7.184,1.522) -- (7.205,1.442) -- (7.264,1.383) -- (7.344,1.362) -- (7.424,1.383) -- (7.483,1.442) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.408,8.941) -- (5.53,8.96) -- (5.64,9.016) -- (5.727,9.103) -- (5.783,9.213) -- (5.802,9.335) -- (5.783,9.456) -- (5.727,9.566) -- (5.64,9.653) -- (5.53,9.709) -- (5.408,9.729) -- (5.408,9.729) -- (5.287,9.709) -- (5.177,9.653) -- (5.09,9.566) -- (5.034,9.456) -- (5.015,9.335) -- (5.034,9.213) -- (5.09,9.103) -- (5.177,9.016) -- (5.287,8.96) -- (5.408,8.941) -- cycle;
\node at (5.408,9.335) {\scalebox{1.634}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.112,10.842) -- (3.234,10.862) -- (3.344,10.918) -- (3.431,11.005) -- (3.487,11.115) -- (3.506,11.236) -- (3.487,11.358) -- (3.431,11.468) -- (3.344,11.555) -- (3.234,11.611) -- (3.112,11.63) -- (3.112,11.63) -- (2.99,11.611) -- (2.881,11.555) -- (2.793,11.468) -- (2.737,11.358) -- (2.718,11.236) -- (2.737,11.115) -- (2.793,11.005) -- (2.881,10.918) -- (2.99,10.862) -- (3.112,10.842) -- cycle;
\node at (3.112,11.236) {\scalebox{1.634}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.581,8.675) -- (1.703,8.695) -- (1.813,8.751) -- (1.9,8.838) -- (1.956,8.948) -- (1.975,9.069) -- (1.956,9.191) -- (1.9,9.301) -- (1.813,9.388) -- (1.703,9.444) -- (1.581,9.463) -- (1.581,9.463) -- (1.46,9.444) -- (1.35,9.388) -- (1.263,9.301) -- (1.207,9.191) -- (1.187,9.069) -- (1.207,8.948) -- (1.263,8.838) -- (1.35,8.751) -- (1.46,8.695) -- (1.581,8.675) -- cycle;
\node at (1.581,9.069) {\scalebox{1.634}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.444,8.42) -- (11.566,8.44) -- (11.676,8.496) -- (11.763,8.583) -- (11.819,8.692) -- (11.838,8.814) -- (11.819,8.936) -- (11.763,9.046) -- (11.676,9.133) -- (11.566,9.189) -- (11.444,9.208) -- (11.444,9.208) -- (11.322,9.189) -- (11.213,9.133) -- (11.126,9.046) -- (11.07,8.936) -- (11.05,8.814) -- (11.07,8.692) -- (11.126,8.583) -- (11.213,8.496) -- (11.322,8.44) -- (11.444,8.42) -- cycle;
\node at (11.444,8.814) {\scalebox{1.634}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.461,7.65) -- (6.582,7.669) -- (6.692,7.725) -- (6.779,7.812) -- (6.835,7.922) -- (6.855,8.044) -- (6.835,8.166) -- (6.779,8.276) -- (6.692,8.363) -- (6.582,8.419) -- (6.461,8.438) -- (6.461,8.438) -- (6.339,8.419) -- (6.229,8.363) -- (6.142,8.276) -- (6.086,8.166) -- (6.067,8.044) -- (6.086,7.922) -- (6.142,7.812) -- (6.229,7.725) -- (6.339,7.669) -- (6.461,7.65) -- cycle;
\node at (6.461,8.044) {\scalebox{1.634}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.515,5.533) -- (5.636,5.552) -- (5.746,5.608) -- (5.833,5.696) -- (5.889,5.805) -- (5.908,5.927) -- (5.889,6.049) -- (5.833,6.159) -- (5.746,6.246) -- (5.636,6.302) -- (5.515,6.321) -- (5.515,6.321) -- (5.393,6.302) -- (5.283,6.246) -- (5.196,6.159) -- (5.14,6.049) -- (5.121,5.927) -- (5.14,5.805) -- (5.196,5.696) -- (5.283,5.608) -- (5.393,5.552) -- (5.515,5.533) -- cycle;
\node at (5.515,5.927) {\scalebox{1.634}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.212,7.501) -- (9.333,7.52) -- (9.443,7.576) -- (9.53,7.663) -- (9.586,7.773) -- (9.606,7.895) -- (9.586,8.016) -- (9.53,8.126) -- (9.443,8.213) -- (9.333,8.269) -- (9.212,8.289) -- (9.212,8.289) -- (9.09,8.269) -- (8.98,8.213) -- (8.893,8.126) -- (8.837,8.016) -- (8.818,7.895) -- (8.837,7.773) -- (8.893,7.663) -- (8.98,7.576) -- (9.09,7.52) -- (9.212,7.501) -- cycle;
\node at (9.212,7.895) {\scalebox{1.634}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.128,5.794) -- (8.25,5.814) -- (8.359,5.87) -- (8.446,5.957) -- (8.502,6.067) -- (8.522,6.188) -- (8.502,6.31) -- (8.446,6.42) -- (8.359,6.507) -- (8.25,6.563) -- (8.128,6.582) -- (8.128,6.582) -- (8.006,6.563) -- (7.896,6.507) -- (7.809,6.42) -- (7.753,6.31) -- (7.734,6.188) -- (7.753,6.067) -- (7.809,5.957) -- (7.896,5.87) -- (8.006,5.814) -- (8.128,5.794) -- cycle;
\node at (8.128,6.188) {\scalebox{1.634}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.282,7.539) -- (3.404,7.558) -- (3.514,7.614) -- (3.601,7.701) -- (3.657,7.811) -- (3.676,7.933) -- (3.657,8.054) -- (3.601,8.164) -- (3.514,8.251) -- (3.404,8.307) -- (3.282,8.327) -- (3.282,8.327) -- (3.16,8.307) -- (3.051,8.251) -- (2.963,8.164) -- (2.908,8.054) -- (2.888,7.933) -- (2.908,7.811) -- (2.963,7.701) -- (3.051,7.614) -- (3.16,7.558) -- (3.282,7.539) -- cycle;
\node at (3.282,7.933) {\scalebox{1.634}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.452,5.554) -- (2.574,5.573) -- (2.683,5.629) -- (2.771,5.716) -- (2.826,5.826) -- (2.846,5.948) -- (2.826,6.07) -- (2.771,6.179) -- (2.683,6.266) -- (2.574,6.322) -- (2.452,6.342) -- (2.081,6.342) -- (1.959,6.322) -- (1.85,6.266) -- (1.762,6.179) -- (1.706,6.07) -- (1.687,5.948) -- (1.706,5.826) -- (1.762,5.716) -- (1.85,5.629) -- (1.959,5.573) -- (2.081,5.554) -- cycle;
\node at (2.266,5.948) {\scalebox{1.634}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.092,2.785) -- (7.213,2.805) -- (7.323,2.861) -- (7.41,2.948) -- (7.466,3.058) -- (7.486,3.179) -- (7.466,3.301) -- (7.41,3.411) -- (7.323,3.498) -- (7.213,3.554) -- (7.092,3.573) -- (6.721,3.573) -- (6.599,3.554) -- (6.489,3.498) -- (6.402,3.411) -- (6.346,3.301) -- (6.327,3.179) -- (6.346,3.058) -- (6.402,2.948) -- (6.489,2.861) -- (6.599,2.805) -- (6.721,2.785) -- cycle;
\node at (6.906,3.179) {\scalebox{1.634}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.6,4.256) -- (3.721,4.275) -- (3.831,4.331) -- (3.918,4.418) -- (3.974,4.528) -- (3.994,4.65) -- (3.974,4.771) -- (3.918,4.881) -- (3.831,4.968) -- (3.721,5.024) -- (3.6,5.044) -- (3.229,5.044) -- (3.107,5.024) -- (2.997,4.968) -- (2.91,4.881) -- (2.854,4.771) -- (2.835,4.65) -- (2.854,4.528) -- (2.91,4.418) -- (2.997,4.331) -- (3.107,4.275) -- (3.229,4.256) -- cycle;
\node at (3.414,4.65) {\scalebox{1.634}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.356,2.478) -- (2.477,2.498) -- (2.587,2.554) -- (2.674,2.641) -- (2.73,2.751) -- (2.749,2.872) -- (2.73,2.994) -- (2.674,3.104) -- (2.587,3.191) -- (2.477,3.247) -- (2.356,3.266) -- (1.985,3.266) -- (1.863,3.247) -- (1.753,3.191) -- (1.666,3.104) -- (1.61,2.994) -- (1.591,2.872) -- (1.61,2.751) -- (1.666,2.641) -- (1.753,2.554) -- (1.863,2.498) -- (1.985,2.478) -- cycle;
\node at (2.17,2.872) {\scalebox{1.634}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.429,2.066) -- (5.551,2.085) -- (5.661,2.141) -- (5.748,2.228) -- (5.804,2.338) -- (5.823,2.46) -- (5.804,2.582) -- (5.748,2.691) -- (5.661,2.779) -- (5.551,2.835) -- (5.429,2.854) -- (5.058,2.854) -- (4.937,2.835) -- (4.827,2.779) -- (4.74,2.691) -- (4.684,2.582) -- (4.664,2.46) -- (4.684,2.338) -- (4.74,2.228) -- (4.827,2.141) -- (4.937,2.085) -- (5.058,2.066) -- cycle;
\node at (5.244,2.46) {\scalebox{1.634}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.124,0.837) -- (3.245,0.856) -- (3.355,0.912) -- (3.442,0.999) -- (3.498,1.109) -- (3.518,1.231) -- (3.498,1.353) -- (3.442,1.462) -- (3.355,1.549) -- (3.245,1.605) -- (3.124,1.625) -- (2.753,1.625) -- (2.631,1.605) -- (2.521,1.549) -- (2.434,1.462) -- (2.378,1.353) -- (2.359,1.231) -- (2.378,1.109) -- (2.434,0.999) -- (2.521,0.912) -- (2.631,0.856) -- (2.753,0.837) -- cycle;
\node at (2.938,1.231) {\scalebox{1.634}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.137,1.645) -- (6.259,1.665) -- (6.369,1.721) -- (6.456,1.808) -- (6.512,1.917) -- (6.531,2.039) -- (6.512,2.161) -- (6.456,2.271) -- (6.369,2.358) -- (6.259,2.414) -- (6.137,2.433) -- (5.767,2.433) -- (5.645,2.414) -- (5.535,2.358) -- (5.448,2.271) -- (5.392,2.161) -- (5.373,2.039) -- (5.392,1.917) -- (5.448,1.808) -- (5.535,1.721) -- (5.645,1.665) -- (5.767,1.645) -- cycle;
\node at (5.952,2.039) {\scalebox{1.634}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.747,3.863) -- (2.347,3.863) -- (2.347,4.463) -- (1.747,4.463) -- cycle;
\node at (2.047,4.163) {\scalebox{1.634}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.271,0.884) -- (5.871,0.884) -- (5.871,1.484) -- (5.271,1.484) -- cycle;
\node at (5.571,1.184) {\scalebox{1.634}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.726,1.709) -- (1.326,1.709) -- (1.326,2.309) -- (0.726,2.309) -- cycle;
\node at (1.026,2.009) {\scalebox{1.634}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.343,2.472) -- (8.943,2.472) -- (8.943,3.072) -- (8.343,3.072) -- cycle;
\node at (8.643,2.772) {\scalebox{1.634}{\small d}};
\node at (5.063,11.427) {\scalebox{1.634}{\small A}};
\node at (0.765,11.003) {\scalebox{1.634}{\small B}};
\node at (11.703,10.583) {\scalebox{1.634}{\small C}};
\node at (5.09,8.463) {\scalebox{1.634}{\small D}};
\node at (8.23,7.998) {\scalebox{1.634}{\small E}};
\node at (11.301,7.744) {\scalebox{1.634}{\small F}};
\node at (0.845,7.743) {\scalebox{1.634}{\small G}};
\node at (8.042,4.438) {\scalebox{1.634}{\small H}};
\node at (1.516,3.843) {\scalebox{1.634}{\small I}};
\node at (4.586,3.531) {\scalebox{1.634}{\small J}};
\node at (0.995,1.363) {\scalebox{1.634}{\small K}};
\node at (4.39,1.075) {\scalebox{1.634}{\small L}};
\node at (7.584,0.95) {\scalebox{1.634}{\small M}};
\draw[line width=0.5pt] (12.494,9.859) -- (12.494,10.959);
\draw[line width=0.5pt] (12.263,10.574) -- (12.494,10.959) -- (12.725,10.574);
\draw[line width=0.5pt] (12.192,10.409) -- (12.797,10.409);
\node at (12.494,11.344) {\scalebox{1.634}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour I, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑨ chemin des Vignes & Ⓐ Place du Marché & Ⓙ Rond-point des Vaches \\ \hline
① rue des Lilas & ⑩ sentier des Prés & Ⓑ Place aux Herbes & Ⓚ Place Sainte-Croix \\ \hline
② rue Basse & ⑪ rue de la Mairie & Ⓒ Placette des Ormes & Ⓛ Rond-point du Stade \\ \hline
③ rue du Puits & ⑫ rue Neuve & Ⓓ Rond-point du Pont & Ⓜ Place des Tanneurs \\ \hline
④ impasse du Puits & ⑬ chemin du Lavoir & Ⓔ Carrefour de la Mairie & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
⑤ venelle du Chat & ⑭ rue du Pont & Ⓕ Place de l'Église & \textbf{a} l'école \\ \hline
⑥ rue des Écoles & ⑮ passage du Marché & Ⓖ Carrefour du Lavoir & \textbf{b} la pharmacie \\ \hline
⑦ boulevard du Parc & ⑯ rue Haute & Ⓗ Fontaine Notre-Dame & \textbf{c} le cimetière \\ \hline
⑧ rue de l'Église & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Carrefour de l'Hôpital & \textbf{d} le gymnase \\ \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=0.627, line join=round, line cap=round]
\draw[line width=0.5pt] (5.102,10.809) -- (5.074,10.598) -- (5.052,10.387) -- (5.034,10.175) -- (5.022,9.963) -- (5.015,9.751) -- (5.012,9.538) -- (5.015,9.325) -- (5.022,9.113) -- (5.035,8.901) -- (5.052,8.689) -- (5.075,8.478) -- (5.102,8.267) -- (5.134,8.057) -- (5.172,7.848);
\draw[line width=0.5pt] (1.194,10.556) -- (1.468,10.616) -- (1.744,10.67) -- (2.021,10.718) -- (2.299,10.759) -- (2.578,10.794) -- (2.857,10.822) -- (3.138,10.843) -- (3.418,10.858) -- (3.699,10.866) -- (3.98,10.868) -- (4.261,10.863) -- (4.542,10.852) -- (4.822,10.834) -- (5.102,10.809);
\draw[line width=0.5pt] (1.194,10.556) -- (1.178,10.339) -- (1.167,10.123) -- (1.161,9.906) -- (1.16,9.689) -- (1.165,9.472) -- (1.174,9.255) -- (1.189,9.038) -- (1.208,8.822) -- (1.233,8.607) -- (1.263,8.392) -- (1.297,8.177) -- (1.337,7.964) -- (1.382,7.752) -- (1.432,7.541);
\draw[line width=0.5pt] (11.194,10.228) -- (11.188,10.035) -- (11.178,9.843) -- (11.163,9.651) -- (11.143,9.459) -- (11.119,9.267) -- (11.09,9.076) -- (11.057,8.886) -- (11.019,8.697) -- (10.977,8.509) -- (10.931,8.322) -- (10.88,8.135) -- (10.825,7.951) -- (10.766,7.767) -- (10.702,7.585);
\draw[line width=0.5pt] (5.172,7.848) -- (5.349,7.807) -- (5.528,7.771) -- (5.708,7.738) -- (5.888,7.71) -- (6.069,7.686) -- (6.25,7.667) -- (6.432,7.651) -- (6.614,7.64) -- (6.797,7.633) -- (6.979,7.631) -- (7.162,7.632) -- (7.344,7.638) -- (7.526,7.649) -- (7.708,7.664);
\draw[line width=0.5pt] (5.172,7.848) -- (5.184,7.58) -- (5.189,7.311) -- (5.189,7.042) -- (5.182,6.774) -- (5.169,6.505) -- (5.149,6.237) -- (5.123,5.97) -- (5.091,5.703) -- (5.052,5.437) -- (5.008,5.171) -- (4.957,4.908) -- (4.9,4.645) -- (4.837,4.384) -- (4.767,4.125);
\draw[line width=0.5pt] (7.708,7.664) -- (7.919,7.625) -- (8.131,7.592) -- (8.344,7.564) -- (8.558,7.54) -- (8.772,7.522) -- (8.987,7.509) -- (9.202,7.501) -- (9.416,7.498) -- (9.631,7.5) -- (9.846,7.507) -- (10.061,7.519) -- (10.275,7.536) -- (10.489,7.558) -- (10.702,7.585);
\draw[line width=0.5pt] (7.708,7.664) -- (7.726,7.457) -- (7.74,7.25) -- (7.749,7.043) -- (7.752,6.836) -- (7.751,6.628) -- (7.745,6.421) -- (7.735,6.213) -- (7.719,6.007) -- (7.699,5.8) -- (7.673,5.594) -- (7.643,5.389) -- (7.608,5.184) -- (7.568,4.981) -- (7.524,4.779);
\draw[line width=0.5pt] (1.432,7.541) -- (1.7,7.522) -- (1.969,7.509) -- (2.238,7.503) -- (2.507,7.502) -- (2.777,7.509) -- (3.046,7.521) -- (3.314,7.54) -- (3.583,7.565) -- (3.85,7.597) -- (4.117,7.635) -- (4.382,7.679) -- (4.647,7.729) -- (4.91,7.785) -- (5.172,7.848);
\draw[line width=0.5pt] (1.432,7.541) -- (1.512,7.308) -- (1.587,7.074) -- (1.657,6.837) -- (1.721,6.599) -- (1.779,6.36) -- (1.832,6.119) -- (1.879,5.877) -- (1.92,5.634) -- (1.956,5.39) -- (1.986,5.146) -- (2.01,4.9) -- (2.028,4.655) -- (2.041,4.409) -- (2.047,4.163);
\draw[line width=0.5pt] (7.524,4.779) -- (7.475,4.55) -- (7.432,4.32) -- (7.395,4.089) -- (7.363,3.857) -- (7.336,3.624) -- (7.315,3.391) -- (7.3,3.158) -- (7.289,2.924) -- (7.285,2.69) -- (7.286,2.456) -- (7.292,2.222) -- (7.304,1.988) -- (7.321,1.755) -- (7.344,1.522);
\draw[line width=0.5pt] (2.047,4.163) -- (2.241,4.19) -- (2.434,4.212) -- (2.629,4.23) -- (2.823,4.243) -- (3.018,4.252) -- (3.214,4.256) -- (3.409,4.256) -- (3.604,4.251) -- (3.799,4.241) -- (3.994,4.227) -- (4.188,4.208) -- (4.382,4.185) -- (4.575,4.157) -- (4.767,4.125);
\draw[line width=0.5pt] (2.047,4.163) -- (1.977,4.004) -- (1.91,3.843) -- (1.847,3.681) -- (1.788,3.517) -- (1.732,3.352) -- (1.681,3.185) -- (1.633,3.018) -- (1.59,2.849) -- (1.55,2.679) -- (1.515,2.509) -- (1.483,2.338) -- (1.455,2.166) -- (1.432,1.993) -- (1.413,1.82);
\draw[line width=0.5pt] (4.767,4.125) -- (4.776,3.949) -- (4.78,3.773) -- (4.78,3.597) -- (4.776,3.421) -- (4.768,3.245) -- (4.756,3.069) -- (4.739,2.894) -- (4.719,2.719) -- (4.694,2.544) -- (4.665,2.37) -- (4.633,2.197) -- (4.596,2.025) -- (4.555,1.854) -- (4.51,1.684);
\draw[line width=0.5pt] (1.413,1.82) -- (1.631,1.777) -- (1.85,1.738) -- (2.07,1.705) -- (2.29,1.677) -- (2.512,1.654) -- (2.733,1.637) -- (2.956,1.624) -- (3.178,1.617) -- (3.401,1.615) -- (3.623,1.618) -- (3.845,1.627) -- (4.067,1.641) -- (4.289,1.66) -- (4.51,1.684);
\draw[line width=0.5pt] (4.51,1.684) -- (4.713,1.703) -- (4.916,1.718) -- (5.119,1.728) -- (5.323,1.733) -- (5.526,1.733) -- (5.73,1.729) -- (5.934,1.72) -- (6.137,1.706) -- (6.34,1.687) -- (6.542,1.663) -- (6.744,1.635) -- (6.945,1.602) -- (7.145,1.564) -- (7.344,1.522);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.475,10.809) -- (5.447,10.951) -- (5.366,11.073) -- (5.245,11.153) -- (5.102,11.182) -- (4.959,11.153) -- (4.838,11.073) -- (4.757,10.951) -- (4.729,10.809) -- (4.757,10.666) -- (4.838,10.545) -- (4.959,10.464) -- (5.102,10.435) -- (5.245,10.464) -- (5.366,10.545) -- (5.447,10.666) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.567,10.556) -- (1.539,10.698) -- (1.458,10.819) -- (1.337,10.9) -- (1.194,10.929) -- (1.051,10.9) -- (0.93,10.819) -- (0.849,10.698) -- (0.821,10.556) -- (0.849,10.413) -- (0.93,10.292) -- (1.051,10.211) -- (1.194,10.182) -- (1.337,10.211) -- (1.458,10.292) -- (1.539,10.413) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.567,10.228) -- (11.539,10.371) -- (11.458,10.492) -- (11.337,10.573) -- (11.194,10.601) -- (11.051,10.573) -- (10.93,10.492) -- (10.849,10.371) -- (10.821,10.228) -- (10.849,10.085) -- (10.93,9.964) -- (11.051,9.883) -- (11.194,9.855) -- (11.337,9.883) -- (11.458,9.964) -- (11.539,10.085) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.545,7.848) -- (5.516,7.991) -- (5.436,8.112) -- (5.314,8.193) -- (5.172,8.221) -- (5.029,8.193) -- (4.908,8.112) -- (4.827,7.991) -- (4.798,7.848) -- (4.827,7.705) -- (4.908,7.584) -- (5.029,7.503) -- (5.172,7.475) -- (5.314,7.503) -- (5.436,7.584) -- (5.516,7.705) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.081,7.664) -- (8.053,7.806) -- (7.972,7.927) -- (7.851,8.008) -- (7.708,8.037) -- (7.565,8.008) -- (7.444,7.927) -- (7.363,7.806) -- (7.334,7.664) -- (7.363,7.521) -- (7.444,7.4) -- (7.565,7.319) -- (7.708,7.29) -- (7.851,7.319) -- (7.972,7.4) -- (8.053,7.521) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.075,7.585) -- (11.047,7.728) -- (10.966,7.849) -- (10.845,7.93) -- (10.702,7.959) -- (10.559,7.93) -- (10.438,7.849) -- (10.357,7.728) -- (10.329,7.585) -- (10.357,7.443) -- (10.438,7.322) -- (10.559,7.241) -- (10.702,7.212) -- (10.845,7.241) -- (10.966,7.322) -- (11.047,7.443) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.805,7.541) -- (1.777,7.684) -- (1.696,7.805) -- (1.575,7.886) -- (1.432,7.914) -- (1.289,7.886) -- (1.168,7.805) -- (1.087,7.684) -- (1.059,7.541) -- (1.087,7.398) -- (1.168,7.277) -- (1.289,7.196) -- (1.432,7.168) -- (1.575,7.196) -- (1.696,7.277) -- (1.777,7.398) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.897,4.779) -- (7.869,4.921) -- (7.788,5.042) -- (7.667,5.123) -- (7.524,5.152) -- (7.381,5.123) -- (7.26,5.042) -- (7.179,4.921) -- (7.15,4.779) -- (7.179,4.636) -- (7.26,4.515) -- (7.381,4.434) -- (7.524,4.405) -- (7.667,4.434) -- (7.788,4.515) -- (7.869,4.636) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.421,4.163) -- (2.392,4.305) -- (2.311,4.427) -- (2.19,4.507) -- (2.047,4.536) -- (1.905,4.507) -- (1.784,4.427) -- (1.703,4.305) -- (1.674,4.163) -- (1.703,4.02) -- (1.784,3.899) -- (1.905,3.818) -- (2.047,3.789) -- (2.19,3.818) -- (2.311,3.899) -- (2.392,4.02) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.14,4.125) -- (5.112,4.267) -- (5.031,4.388) -- (4.91,4.469) -- (4.767,4.498) -- (4.624,4.469) -- (4.503,4.388) -- (4.422,4.267) -- (4.394,4.125) -- (4.422,3.982) -- (4.503,3.861) -- (4.624,3.78) -- (4.767,3.751) -- (4.91,3.78) -- (5.031,3.861) -- (5.112,3.982) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.786,1.82) -- (1.757,1.963) -- (1.677,2.084) -- (1.556,2.165) -- (1.413,2.194) -- (1.27,2.165) -- (1.149,2.084) -- (1.068,1.963) -- (1.039,1.82) -- (1.068,1.678) -- (1.149,1.556) -- (1.27,1.476) -- (1.413,1.447) -- (1.556,1.476) -- (1.677,1.556) -- (1.757,1.678) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.883,1.684) -- (4.855,1.827) -- (4.774,1.948) -- (4.653,2.029) -- (4.51,2.057) -- (4.367,2.029) -- (4.246,1.948) -- (4.165,1.827) -- (4.137,1.684) -- (4.165,1.541) -- (4.246,1.42) -- (4.367,1.339) -- (4.51,1.311) -- (4.653,1.339) -- (4.774,1.42) -- (4.855,1.541) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.717,1.522) -- (7.689,1.665) -- (7.608,1.786) -- (7.487,1.867) -- (7.344,1.895) -- (7.201,1.867) -- (7.08,1.786) -- (6.999,1.665) -- (6.971,1.522) -- (6.999,1.379) -- (7.08,1.258) -- (7.201,1.177) -- (7.344,1.149) -- (7.487,1.177) -- (7.608,1.258) -- (7.689,1.379) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.399,8.95) -- (5.518,8.969) -- (5.625,9.023) -- (5.71,9.109) -- (5.765,9.216) -- (5.784,9.335) -- (5.765,9.453) -- (5.71,9.561) -- (5.625,9.646) -- (5.518,9.7) -- (5.399,9.719) -- (5.399,9.719) -- (5.28,9.7) -- (5.173,9.646) -- (5.088,9.561) -- (5.033,9.453) -- (5.015,9.335) -- (5.033,9.216) -- (5.088,9.109) -- (5.173,9.023) -- (5.28,8.969) -- (5.399,8.95) -- cycle;
\node at (5.399,9.335) {\scalebox{1.595}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.113,10.842) -- (3.232,10.861) -- (3.339,10.916) -- (3.424,11.001) -- (3.478,11.108) -- (3.497,11.227) -- (3.478,11.346) -- (3.424,11.453) -- (3.339,11.538) -- (3.232,11.593) -- (3.113,11.612) -- (3.113,11.612) -- (2.994,11.593) -- (2.887,11.538) -- (2.802,11.453) -- (2.747,11.346) -- (2.728,11.227) -- (2.747,11.108) -- (2.802,11.001) -- (2.887,10.916) -- (2.994,10.861) -- (3.113,10.842) -- cycle;
\node at (3.113,11.227) {\scalebox{1.595}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.572,8.684) -- (1.691,8.703) -- (1.798,8.758) -- (1.883,8.843) -- (1.938,8.95) -- (1.956,9.069) -- (1.938,9.187) -- (1.883,9.295) -- (1.798,9.38) -- (1.691,9.434) -- (1.572,9.453) -- (1.572,9.453) -- (1.453,9.434) -- (1.346,9.38) -- (1.261,9.295) -- (1.206,9.187) -- (1.187,9.069) -- (1.206,8.95) -- (1.261,8.843) -- (1.346,8.758) -- (1.453,8.703) -- (1.572,8.684) -- cycle;
\node at (1.572,9.069) {\scalebox{1.595}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.435,8.431) -- (11.554,8.45) -- (11.661,8.505) -- (11.746,8.59) -- (11.801,8.697) -- (11.82,8.816) -- (11.801,8.935) -- (11.746,9.042) -- (11.661,9.127) -- (11.554,9.182) -- (11.435,9.2) -- (11.435,9.2) -- (11.316,9.182) -- (11.209,9.127) -- (11.124,9.042) -- (11.069,8.935) -- (11.05,8.816) -- (11.069,8.697) -- (11.124,8.59) -- (11.209,8.505) -- (11.316,8.45) -- (11.435,8.431) -- cycle;
\node at (11.435,8.816) {\scalebox{1.595}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.46,7.65) -- (6.579,7.669) -- (6.686,7.724) -- (6.771,7.809) -- (6.826,7.916) -- (6.844,8.035) -- (6.826,8.153) -- (6.771,8.261) -- (6.686,8.346) -- (6.579,8.4) -- (6.46,8.419) -- (6.46,8.419) -- (6.341,8.4) -- (6.234,8.346) -- (6.149,8.261) -- (6.094,8.153) -- (6.075,8.035) -- (6.094,7.916) -- (6.149,7.809) -- (6.234,7.724) -- (6.341,7.669) -- (6.46,7.65) -- cycle;
\node at (6.46,8.035) {\scalebox{1.595}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.505,5.544) -- (5.624,5.562) -- (5.731,5.617) -- (5.816,5.702) -- (5.871,5.809) -- (5.89,5.928) -- (5.871,6.047) -- (5.816,6.154) -- (5.731,6.239) -- (5.624,6.294) -- (5.505,6.313) -- (5.505,6.313) -- (5.386,6.294) -- (5.279,6.239) -- (5.194,6.154) -- (5.14,6.047) -- (5.121,5.928) -- (5.14,5.809) -- (5.194,5.702) -- (5.279,5.617) -- (5.386,5.562) -- (5.505,5.544) -- cycle;
\node at (5.505,5.928) {\scalebox{1.595}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.212,7.501) -- (9.33,7.52) -- (9.438,7.574) -- (9.523,7.659) -- (9.577,7.767) -- (9.596,7.885) -- (9.577,8.004) -- (9.523,8.111) -- (9.438,8.196) -- (9.33,8.251) -- (9.212,8.27) -- (9.212,8.27) -- (9.093,8.251) -- (8.986,8.196) -- (8.9,8.111) -- (8.846,8.004) -- (8.827,7.885) -- (8.846,7.767) -- (8.9,7.659) -- (8.986,7.574) -- (9.093,7.52) -- (9.212,7.501) -- cycle;
\node at (9.212,7.885) {\scalebox{1.595}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.118,5.804) -- (8.237,5.823) -- (8.344,5.878) -- (8.43,5.963) -- (8.484,6.07) -- (8.503,6.189) -- (8.484,6.308) -- (8.43,6.415) -- (8.344,6.5) -- (8.237,6.555) -- (8.118,6.573) -- (8.118,6.573) -- (8.0,6.555) -- (7.892,6.5) -- (7.807,6.415) -- (7.753,6.308) -- (7.734,6.189) -- (7.753,6.07) -- (7.807,5.963) -- (7.892,5.878) -- (8.0,5.823) -- (8.118,5.804) -- cycle;
\node at (8.118,6.189) {\scalebox{1.595}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.283,7.539) -- (3.402,7.558) -- (3.509,7.612) -- (3.594,7.697) -- (3.649,7.805) -- (3.667,7.923) -- (3.649,8.042) -- (3.594,8.149) -- (3.509,8.235) -- (3.402,8.289) -- (3.283,8.308) -- (3.283,8.308) -- (3.164,8.289) -- (3.057,8.235) -- (2.972,8.149) -- (2.917,8.042) -- (2.898,7.923) -- (2.917,7.805) -- (2.972,7.697) -- (3.057,7.612) -- (3.164,7.558) -- (3.283,7.539) -- cycle;
\node at (3.283,7.923) {\scalebox{1.595}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.438,5.562) -- (2.557,5.58) -- (2.664,5.635) -- (2.749,5.72) -- (2.804,5.827) -- (2.823,5.946) -- (2.804,6.065) -- (2.749,6.172) -- (2.664,6.257) -- (2.557,6.312) -- (2.438,6.331) -- (2.076,6.331) -- (1.957,6.312) -- (1.85,6.257) -- (1.765,6.172) -- (1.711,6.065) -- (1.692,5.946) -- (1.711,5.827) -- (1.765,5.72) -- (1.85,5.635) -- (1.957,5.58) -- (2.076,5.562) -- cycle;
\node at (2.257,5.946) {\scalebox{1.595}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.097,2.794) -- (7.215,2.813) -- (7.323,2.868) -- (7.408,2.953) -- (7.462,3.06) -- (7.481,3.179) -- (7.462,3.298) -- (7.408,3.405) -- (7.323,3.49) -- (7.215,3.544) -- (7.097,3.563) -- (6.735,3.563) -- (6.616,3.544) -- (6.509,3.49) -- (6.424,3.405) -- (6.369,3.298) -- (6.35,3.179) -- (6.369,3.06) -- (6.424,2.953) -- (6.509,2.868) -- (6.616,2.813) -- (6.735,2.794) -- cycle;
\node at (6.916,3.179) {\scalebox{1.595}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.595,4.256) -- (3.714,4.275) -- (3.821,4.329) -- (3.906,4.414) -- (3.961,4.521) -- (3.98,4.64) -- (3.961,4.759) -- (3.906,4.866) -- (3.821,4.951) -- (3.714,5.006) -- (3.595,5.025) -- (3.233,5.025) -- (3.114,5.006) -- (3.007,4.951) -- (2.922,4.866) -- (2.868,4.759) -- (2.849,4.64) -- (2.868,4.521) -- (2.922,4.414) -- (3.007,4.329) -- (3.114,4.275) -- (3.233,4.256) -- cycle;
\node at (3.414,4.64) {\scalebox{1.595}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.186,2.533) -- (2.304,2.551) -- (2.412,2.606) -- (2.497,2.691) -- (2.551,2.798) -- (2.57,2.917) -- (2.551,3.036) -- (2.497,3.143) -- (2.412,3.228) -- (2.304,3.283) -- (2.186,3.302) -- (1.824,3.302) -- (1.705,3.283) -- (1.598,3.228) -- (1.513,3.143) -- (1.458,3.036) -- (1.439,2.917) -- (1.458,2.798) -- (1.513,2.691) -- (1.598,2.606) -- (1.705,2.551) -- (1.824,2.533) -- cycle;
\node at (2.005,2.917) {\scalebox{1.595}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.412,2.077) -- (5.53,2.096) -- (5.638,2.151) -- (5.723,2.236) -- (5.777,2.343) -- (5.796,2.462) -- (5.777,2.581) -- (5.723,2.688) -- (5.638,2.773) -- (5.53,2.828) -- (5.412,2.846) -- (5.05,2.846) -- (4.931,2.828) -- (4.824,2.773) -- (4.739,2.688) -- (4.684,2.581) -- (4.665,2.462) -- (4.684,2.343) -- (4.739,2.236) -- (4.824,2.151) -- (4.931,2.096) -- (5.05,2.077) -- cycle;
\node at (5.231,2.462) {\scalebox{1.595}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.154,1.624) -- (3.272,1.643) -- (3.38,1.697) -- (3.465,1.782) -- (3.519,1.89) -- (3.538,2.008) -- (3.519,2.127) -- (3.465,2.234) -- (3.38,2.32) -- (3.272,2.374) -- (3.154,2.393) -- (2.792,2.393) -- (2.673,2.374) -- (2.566,2.32) -- (2.481,2.234) -- (2.426,2.127) -- (2.407,2.008) -- (2.426,1.89) -- (2.481,1.782) -- (2.566,1.697) -- (2.673,1.643) -- (2.792,1.624) -- cycle;
\node at (2.973,2.008) {\scalebox{1.595}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.133,1.655) -- (6.252,1.674) -- (6.359,1.728) -- (6.444,1.813) -- (6.499,1.92) -- (6.517,2.039) -- (6.499,2.158) -- (6.444,2.265) -- (6.359,2.35) -- (6.252,2.405) -- (6.133,2.424) -- (5.771,2.424) -- (5.652,2.405) -- (5.545,2.35) -- (5.46,2.265) -- (5.405,2.158) -- (5.386,2.039) -- (5.405,1.92) -- (5.46,1.813) -- (5.545,1.728) -- (5.652,1.674) -- (5.771,1.655) -- cycle;
\node at (5.952,2.039) {\scalebox{1.595}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.747,3.863) -- (2.347,3.863) -- (2.347,4.463) -- (1.747,4.463) -- cycle;
\node at (2.047,4.163) {\scalebox{1.595}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.271,0.884) -- (5.871,0.884) -- (5.871,1.484) -- (5.271,1.484) -- cycle;
\node at (5.571,1.184) {\scalebox{1.595}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.726,1.709) -- (1.326,1.709) -- (1.326,2.309) -- (0.726,2.309) -- cycle;
\node at (1.026,2.009) {\scalebox{1.595}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.343,2.472) -- (8.943,2.472) -- (8.943,3.072) -- (8.343,3.072) -- cycle;
\node at (8.643,2.772) {\scalebox{1.595}{\small d}};
\node at (5.063,11.427) {\scalebox{1.595}{\small A}};
\node at (0.765,11.003) {\scalebox{1.595}{\small B}};
\node at (11.703,10.583) {\scalebox{1.595}{\small C}};
\node at (5.09,8.463) {\scalebox{1.595}{\small D}};
\node at (8.23,7.998) {\scalebox{1.595}{\small E}};
\node at (11.301,7.744) {\scalebox{1.595}{\small F}};
\node at (0.845,7.743) {\scalebox{1.595}{\small G}};
\node at (8.042,4.438) {\scalebox{1.595}{\small H}};
\node at (1.516,3.843) {\scalebox{1.595}{\small I}};
\node at (4.586,3.531) {\scalebox{1.595}{\small J}};
\node at (0.995,1.363) {\scalebox{1.595}{\small K}};
\node at (4.39,1.075) {\scalebox{1.595}{\small L}};
\node at (7.584,0.95) {\scalebox{1.595}{\small M}};
\draw[line width=0.5pt] (12.494,9.859) -- (12.494,10.959);
\draw[line width=0.5pt] (12.263,10.574) -- (12.494,10.959) -- (12.725,10.574);
\draw[line width=0.5pt] (12.192,10.409) -- (12.797,10.409);
\node at (12.494,11.344) {\scalebox{1.595}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, C, E, G, J, K, M & 7 \\ \hline
clairs & B, D, F, H, I, L & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{7} carrefours sombres et \textbf{6} clairs. Il y en a donc \textbf{1 de plus} du côté des sombres.}\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 13 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{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 7 carrefours sombres et 6 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non plus.} Avec un carrefour sombre de plus, la tournée doit \textbf{partir et arriver} sur un carrefour sombre. Or l'école est en \textbf{I}, qui est clair.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Une tournée existe pourtant dans ce quartier : il suffirait d'installer l'école sur un carrefour sombre. Ce n'est pas le plan qui est en cause, c'est l'endroit d'où l'on part.}\par\addvspace{0.4em}
\end{document}
