\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 162}

\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=162}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0162}
\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=0.6054, line join=round, line cap=round]
\draw[line width=0.5pt] (5.084,11.706) -- (5.301,11.67) -- (5.519,11.639) -- (5.737,11.613) -- (5.956,11.592) -- (6.176,11.577) -- (6.396,11.567) -- (6.615,11.561) -- (6.836,11.561) -- (7.055,11.567) -- (7.275,11.577) -- (7.495,11.592) -- (7.714,11.613) -- (7.932,11.639) -- (8.149,11.67);
\draw[line width=0.5pt] (5.084,11.706) -- (5.106,11.456) -- (5.122,11.206) -- (5.132,10.956) -- (5.135,10.705) -- (5.133,10.454) -- (5.125,10.203) -- (5.112,9.953) -- (5.092,9.703) -- (5.066,9.453) -- (5.035,9.204) -- (4.998,8.956) -- (4.955,8.709) -- (4.906,8.464) -- (4.851,8.219);
\draw[line width=0.5pt] (8.149,11.67) -- (8.096,11.461) -- (8.048,11.251) -- (8.005,11.04) -- (7.967,10.828) -- (7.934,10.615) -- (7.906,10.402) -- (7.883,10.187) -- (7.864,9.973) -- (7.851,9.758) -- (7.843,9.542) -- (7.84,9.327) -- (7.843,9.111) -- (7.85,8.896) -- (7.862,8.681);
\draw[line width=0.5pt] (1.411,11.441) -- (1.668,11.5) -- (1.927,11.553) -- (2.187,11.6) -- (2.448,11.64) -- (2.71,11.675) -- (2.973,11.703) -- (3.237,11.725) -- (3.501,11.741) -- (3.765,11.751) -- (4.029,11.754) -- (4.293,11.751) -- (4.558,11.742) -- (4.821,11.727) -- (5.084,11.706);
\draw[line width=0.5pt] (1.411,11.441) -- (1.372,11.224) -- (1.339,11.006) -- (1.311,10.787) -- (1.288,10.568) -- (1.27,10.348) -- (1.257,10.128) -- (1.25,9.907) -- (1.248,9.686) -- (1.251,9.466) -- (1.259,9.245) -- (1.272,9.025) -- (1.29,8.805) -- (1.314,8.586) -- (1.343,8.368);
\draw[line width=0.5pt] (7.862,8.681) -- (7.91,8.469) -- (7.953,8.256) -- (7.99,8.041) -- (8.023,7.826) -- (8.051,7.61) -- (8.073,7.393) -- (8.09,7.176) -- (8.103,6.958) -- (8.11,6.74) -- (8.112,6.523) -- (8.109,6.305) -- (8.101,6.087) -- (8.087,5.87) -- (8.069,5.653);
\draw[line width=0.5pt] (1.343,8.368) -- (1.59,8.319) -- (1.838,8.276) -- (2.087,8.238) -- (2.337,8.207) -- (2.588,8.182) -- (2.839,8.162) -- (3.091,8.149) -- (3.343,8.141) -- (3.595,8.139) -- (3.847,8.143) -- (4.099,8.153) -- (4.35,8.169) -- (4.601,8.191) -- (4.851,8.219);
\draw[line width=0.5pt] (1.343,8.368) -- (1.425,8.141) -- (1.501,7.912) -- (1.572,7.681) -- (1.637,7.449) -- (1.697,7.216) -- (1.751,6.98) -- (1.8,6.744) -- (1.844,6.507) -- (1.881,6.268) -- (1.914,6.029) -- (1.94,5.79) -- (1.961,5.549) -- (1.976,5.309) -- (1.986,5.068);
\draw[line width=0.5pt] (4.851,8.219) -- (5.07,8.219) -- (5.288,8.224) -- (5.506,8.234) -- (5.724,8.249) -- (5.942,8.27) -- (6.159,8.295) -- (6.376,8.326) -- (6.592,8.362) -- (6.806,8.402) -- (7.02,8.448) -- (7.233,8.499) -- (7.444,8.555) -- (7.654,8.616) -- (7.862,8.681);
\draw[line width=0.5pt] (4.851,8.219) -- (4.873,7.997) -- (4.89,7.775) -- (4.901,7.552) -- (4.908,7.328) -- (4.909,7.105) -- (4.904,6.882) -- (4.895,6.659) -- (4.88,6.436) -- (4.86,6.213) -- (4.835,5.991) -- (4.805,5.77) -- (4.769,5.55) -- (4.728,5.33) -- (4.682,5.112);
\draw[line width=0.5pt] (8.069,5.653) -- (8.291,5.555) -- (8.516,5.462) -- (8.743,5.374) -- (8.972,5.291) -- (9.203,5.214) -- (9.435,5.143) -- (9.67,5.077) -- (9.905,5.016) -- (10.142,4.961) -- (10.381,4.912) -- (10.62,4.868) -- (10.86,4.83) -- (11.101,4.797) -- (11.343,4.77);
\draw[line width=0.5pt] (8.069,5.653) -- (8.104,5.378) -- (8.134,5.103) -- (8.156,4.826) -- (8.172,4.549) -- (8.181,4.272) -- (8.184,3.995) -- (8.181,3.717) -- (8.171,3.44) -- (8.154,3.163) -- (8.131,2.886) -- (8.102,2.611) -- (8.065,2.336) -- (8.023,2.062) -- (7.974,1.789);
\draw[line width=0.5pt] (4.682,5.112) -- (4.928,5.113) -- (5.174,5.121) -- (5.42,5.133) -- (5.665,5.152) -- (5.91,5.177) -- (6.154,5.207) -- (6.398,5.243) -- (6.641,5.285) -- (6.882,5.332) -- (7.123,5.385) -- (7.362,5.444) -- (7.599,5.508) -- (7.835,5.578) -- (8.069,5.653);
\draw[line width=0.5pt] (1.986,5.068) -- (2.176,5.1) -- (2.368,5.129) -- (2.56,5.152) -- (2.752,5.171) -- (2.945,5.186) -- (3.139,5.196) -- (3.332,5.201) -- (3.526,5.202) -- (3.719,5.198) -- (3.913,5.19) -- (4.106,5.178) -- (4.299,5.16) -- (4.491,5.138) -- (4.682,5.112);
\draw[line width=0.5pt] (1.986,5.068) -- (1.938,4.843) -- (1.897,4.617) -- (1.86,4.39) -- (1.829,4.162) -- (1.804,3.934) -- (1.783,3.704) -- (1.768,3.475) -- (1.759,3.245) -- (1.755,3.015) -- (1.756,2.785) -- (1.763,2.555) -- (1.775,2.326) -- (1.792,2.097) -- (1.815,1.868);
\draw[line width=0.5pt] (11.343,4.77) -- (11.315,4.568) -- (11.282,4.367) -- (11.244,4.166) -- (11.202,3.966) -- (11.155,3.767) -- (11.103,3.57) -- (11.047,3.373) -- (10.986,3.178) -- (10.921,2.985) -- (10.851,2.793) -- (10.776,2.603) -- (10.697,2.414) -- (10.614,2.228) -- (10.526,2.044);
\draw[line width=0.5pt] (7.974,1.789) -- (8.158,1.779) -- (8.341,1.773) -- (8.525,1.772) -- (8.71,1.776) -- (8.893,1.783) -- (9.077,1.795) -- (9.261,1.811) -- (9.444,1.831) -- (9.626,1.856) -- (9.808,1.885) -- (9.989,1.918) -- (10.169,1.956) -- (10.348,1.998) -- (10.526,2.044);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.244,11.706) -- (5.223,11.786) -- (5.164,11.844) -- (5.084,11.866) -- (5.004,11.844) -- (4.946,11.786) -- (4.924,11.706) -- (4.946,11.626) -- (5.004,11.567) -- (5.084,11.546) -- (5.164,11.567) -- (5.223,11.626) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.309,11.67) -- (8.288,11.75) -- (8.229,11.808) -- (8.149,11.83) -- (8.069,11.808) -- (8.011,11.75) -- (7.989,11.67) -- (8.011,11.59) -- (8.069,11.531) -- (8.149,11.51) -- (8.229,11.531) -- (8.288,11.59) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.571,11.441) -- (1.549,11.521) -- (1.491,11.58) -- (1.411,11.601) -- (1.331,11.58) -- (1.272,11.521) -- (1.251,11.441) -- (1.272,11.361) -- (1.331,11.303) -- (1.411,11.281) -- (1.491,11.303) -- (1.549,11.361) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.022,8.681) -- (8.001,8.761) -- (7.942,8.82) -- (7.862,8.841) -- (7.782,8.82) -- (7.724,8.761) -- (7.702,8.681) -- (7.724,8.601) -- (7.782,8.543) -- (7.862,8.521) -- (7.942,8.543) -- (8.001,8.601) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.503,8.368) -- (1.481,8.448) -- (1.423,8.506) -- (1.343,8.528) -- (1.263,8.506) -- (1.204,8.448) -- (1.183,8.368) -- (1.204,8.288) -- (1.263,8.229) -- (1.343,8.208) -- (1.423,8.229) -- (1.481,8.288) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.011,8.219) -- (4.99,8.299) -- (4.931,8.358) -- (4.851,8.379) -- (4.771,8.358) -- (4.713,8.299) -- (4.691,8.219) -- (4.713,8.139) -- (4.771,8.08) -- (4.851,8.059) -- (4.931,8.08) -- (4.99,8.139) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.229,5.653) -- (8.207,5.733) -- (8.149,5.792) -- (8.069,5.813) -- (7.989,5.792) -- (7.93,5.733) -- (7.909,5.653) -- (7.93,5.573) -- (7.989,5.515) -- (8.069,5.493) -- (8.149,5.515) -- (8.207,5.573) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.842,5.112) -- (4.821,5.192) -- (4.762,5.251) -- (4.682,5.272) -- (4.602,5.251) -- (4.544,5.192) -- (4.522,5.112) -- (4.544,5.032) -- (4.602,4.974) -- (4.682,4.952) -- (4.762,4.974) -- (4.821,5.032) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.146,5.068) -- (2.124,5.148) -- (2.066,5.206) -- (1.986,5.228) -- (1.906,5.206) -- (1.847,5.148) -- (1.826,5.068) -- (1.847,4.988) -- (1.906,4.929) -- (1.986,4.908) -- (2.066,4.929) -- (2.124,4.988) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.503,4.77) -- (11.481,4.85) -- (11.423,4.909) -- (11.343,4.93) -- (11.263,4.909) -- (11.204,4.85) -- (11.183,4.77) -- (11.204,4.69) -- (11.263,4.632) -- (11.343,4.61) -- (11.423,4.632) -- (11.481,4.69) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.686,2.044) -- (10.665,2.124) -- (10.606,2.182) -- (10.526,2.204) -- (10.446,2.182) -- (10.388,2.124) -- (10.366,2.044) -- (10.388,1.964) -- (10.446,1.905) -- (10.526,1.884) -- (10.606,1.905) -- (10.665,1.964) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.975,1.868) -- (1.954,1.948) -- (1.895,2.007) -- (1.815,2.028) -- (1.735,2.007) -- (1.677,1.948) -- (1.655,1.868) -- (1.677,1.788) -- (1.735,1.73) -- (1.815,1.708) -- (1.895,1.73) -- (1.954,1.788) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.134,1.789) -- (8.112,1.869) -- (8.054,1.927) -- (7.974,1.949) -- (7.894,1.927) -- (7.835,1.869) -- (7.814,1.789) -- (7.835,1.709) -- (7.894,1.65) -- (7.974,1.629) -- (8.054,1.65) -- (8.112,1.709) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.62,11.561) -- (6.742,11.581) -- (6.852,11.637) -- (6.939,11.724) -- (6.995,11.834) -- (7.014,11.955) -- (6.995,12.077) -- (6.939,12.187) -- (6.852,12.274) -- (6.742,12.33) -- (6.62,12.349) -- (6.62,12.349) -- (6.498,12.33) -- (6.389,12.274) -- (6.301,12.187) -- (6.245,12.077) -- (6.226,11.955) -- (6.245,11.834) -- (6.301,11.724) -- (6.389,11.637) -- (6.498,11.581) -- (6.62,11.561) -- cycle;
\node at (6.62,11.955) {\scalebox{1.652}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.505,9.533) -- (5.626,9.552) -- (5.736,9.608) -- (5.823,9.695) -- (5.879,9.805) -- (5.899,9.927) -- (5.879,10.048) -- (5.823,10.158) -- (5.736,10.245) -- (5.626,10.301) -- (5.505,10.32) -- (5.505,10.32) -- (5.383,10.301) -- (5.273,10.245) -- (5.186,10.158) -- (5.13,10.048) -- (5.111,9.927) -- (5.13,9.805) -- (5.186,9.695) -- (5.273,9.608) -- (5.383,9.552) -- (5.505,9.533) -- cycle;
\node at (5.505,9.927) {\scalebox{1.652}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.275,9.756) -- (8.396,9.775) -- (8.506,9.831) -- (8.593,9.918) -- (8.649,10.028) -- (8.669,10.15) -- (8.649,10.271) -- (8.593,10.381) -- (8.506,10.468) -- (8.396,10.524) -- (8.275,10.544) -- (8.275,10.544) -- (8.153,10.524) -- (8.043,10.468) -- (7.956,10.381) -- (7.9,10.271) -- (7.881,10.15) -- (7.9,10.028) -- (7.956,9.918) -- (8.043,9.831) -- (8.153,9.775) -- (8.275,9.756) -- cycle;
\node at (8.275,10.15) {\scalebox{1.652}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.208,11.724) -- (3.33,11.743) -- (3.44,11.799) -- (3.527,11.886) -- (3.583,11.996) -- (3.602,12.118) -- (3.583,12.24) -- (3.527,12.349) -- (3.44,12.437) -- (3.33,12.493) -- (3.208,12.512) -- (3.208,12.512) -- (3.087,12.493) -- (2.977,12.437) -- (2.89,12.349) -- (2.834,12.24) -- (2.814,12.118) -- (2.834,11.996) -- (2.89,11.886) -- (2.977,11.799) -- (3.087,11.743) -- (3.208,11.724) -- cycle;
\node at (3.208,12.118) {\scalebox{1.652}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.644,9.505) -- (1.766,9.524) -- (1.875,9.58) -- (1.962,9.667) -- (2.018,9.777) -- (2.038,9.898) -- (2.018,10.02) -- (1.962,10.13) -- (1.875,10.217) -- (1.766,10.273) -- (1.644,10.292) -- (1.644,10.292) -- (1.522,10.273) -- (1.412,10.217) -- (1.325,10.13) -- (1.269,10.02) -- (1.25,9.898) -- (1.269,9.777) -- (1.325,9.667) -- (1.412,9.58) -- (1.522,9.524) -- (1.644,9.505) -- cycle;
\node at (1.644,9.898) {\scalebox{1.652}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.483,6.809) -- (8.605,6.828) -- (8.715,6.884) -- (8.802,6.971) -- (8.858,7.081) -- (8.877,7.203) -- (8.858,7.324) -- (8.802,7.434) -- (8.715,7.521) -- (8.605,7.577) -- (8.483,7.596) -- (8.483,7.596) -- (8.362,7.577) -- (8.252,7.521) -- (8.165,7.434) -- (8.109,7.324) -- (8.09,7.203) -- (8.109,7.081) -- (8.165,6.971) -- (8.252,6.884) -- (8.362,6.828) -- (8.483,6.809) -- cycle;
\node at (8.483,7.203) {\scalebox{1.652}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.108,8.148) -- (3.229,8.167) -- (3.339,8.223) -- (3.426,8.311) -- (3.482,8.42) -- (3.501,8.542) -- (3.482,8.664) -- (3.426,8.774) -- (3.339,8.861) -- (3.229,8.917) -- (3.108,8.936) -- (3.108,8.936) -- (2.986,8.917) -- (2.876,8.861) -- (2.789,8.774) -- (2.733,8.664) -- (2.714,8.542) -- (2.733,8.42) -- (2.789,8.311) -- (2.876,8.223) -- (2.986,8.167) -- (3.108,8.148) -- cycle;
\node at (3.108,8.542) {\scalebox{1.652}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.187,6.426) -- (2.309,6.445) -- (2.419,6.501) -- (2.506,6.588) -- (2.562,6.698) -- (2.581,6.819) -- (2.562,6.941) -- (2.506,7.051) -- (2.419,7.138) -- (2.309,7.194) -- (2.187,7.213) -- (2.187,7.213) -- (2.065,7.194) -- (1.955,7.138) -- (1.868,7.051) -- (1.812,6.941) -- (1.793,6.819) -- (1.812,6.698) -- (1.868,6.588) -- (1.955,6.501) -- (2.065,6.445) -- (2.187,6.426) -- cycle;
\node at (2.187,6.819) {\scalebox{1.652}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.316,8.321) -- (6.438,8.341) -- (6.548,8.397) -- (6.635,8.484) -- (6.691,8.594) -- (6.71,8.715) -- (6.691,8.837) -- (6.635,8.947) -- (6.548,9.034) -- (6.438,9.09) -- (6.316,9.109) -- (6.316,9.109) -- (6.194,9.09) -- (6.084,9.034) -- (5.997,8.947) -- (5.941,8.837) -- (5.922,8.715) -- (5.941,8.594) -- (5.997,8.484) -- (6.084,8.397) -- (6.194,8.341) -- (6.316,8.321) -- cycle;
\node at (6.316,8.715) {\scalebox{1.652}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.687,6.286) -- (4.809,6.305) -- (4.919,6.361) -- (5.006,6.448) -- (5.062,6.558) -- (5.081,6.68) -- (5.062,6.802) -- (5.006,6.912) -- (4.919,6.999) -- (4.809,7.055) -- (4.687,7.074) -- (4.316,7.074) -- (4.195,7.055) -- (4.085,6.999) -- (3.998,6.912) -- (3.942,6.802) -- (3.922,6.68) -- (3.942,6.558) -- (3.998,6.448) -- (4.085,6.361) -- (4.195,6.305) -- (4.316,6.286) -- cycle;
\node at (4.502,6.68) {\scalebox{1.652}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.957,5.063) -- (10.079,5.082) -- (10.189,5.138) -- (10.276,5.225) -- (10.332,5.335) -- (10.351,5.457) -- (10.332,5.579) -- (10.276,5.689) -- (10.189,5.776) -- (10.079,5.832) -- (9.957,5.851) -- (9.587,5.851) -- (9.465,5.832) -- (9.355,5.776) -- (9.268,5.689) -- (9.212,5.579) -- (9.193,5.457) -- (9.212,5.335) -- (9.268,5.225) -- (9.355,5.138) -- (9.465,5.082) -- (9.587,5.063) -- cycle;
\node at (9.772,5.457) {\scalebox{1.652}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.76,3.314) -- (8.882,3.333) -- (8.991,3.389) -- (9.079,3.476) -- (9.135,3.586) -- (9.154,3.707) -- (9.135,3.829) -- (9.079,3.939) -- (8.991,4.026) -- (8.882,4.082) -- (8.76,4.101) -- (8.389,4.101) -- (8.268,4.082) -- (8.158,4.026) -- (8.071,3.939) -- (8.015,3.829) -- (7.995,3.707) -- (8.015,3.586) -- (8.071,3.476) -- (8.158,3.389) -- (8.268,3.333) -- (8.389,3.314) -- cycle;
\node at (8.575,3.707) {\scalebox{1.652}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.521,5.238) -- (6.643,5.257) -- (6.753,5.313) -- (6.84,5.4) -- (6.896,5.51) -- (6.915,5.632) -- (6.896,5.754) -- (6.84,5.863) -- (6.753,5.951) -- (6.643,6.007) -- (6.521,6.026) -- (6.15,6.026) -- (6.029,6.007) -- (5.919,5.951) -- (5.832,5.863) -- (5.776,5.754) -- (5.757,5.632) -- (5.776,5.51) -- (5.832,5.4) -- (5.919,5.313) -- (6.029,5.257) -- (6.15,5.238) -- cycle;
\node at (6.336,5.632) {\scalebox{1.652}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.511,5.201) -- (3.633,5.22) -- (3.743,5.276) -- (3.83,5.364) -- (3.886,5.473) -- (3.905,5.595) -- (3.886,5.717) -- (3.83,5.827) -- (3.743,5.914) -- (3.633,5.97) -- (3.511,5.989) -- (3.14,5.989) -- (3.019,5.97) -- (2.909,5.914) -- (2.822,5.827) -- (2.766,5.717) -- (2.746,5.595) -- (2.766,5.473) -- (2.822,5.364) -- (2.909,5.276) -- (3.019,5.22) -- (3.14,5.201) -- cycle;
\node at (3.326,5.595) {\scalebox{1.652}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.347,3.06) -- (2.469,3.079) -- (2.579,3.135) -- (2.666,3.223) -- (2.722,3.332) -- (2.741,3.454) -- (2.722,3.576) -- (2.666,3.686) -- (2.579,3.773) -- (2.469,3.829) -- (2.347,3.848) -- (1.976,3.848) -- (1.855,3.829) -- (1.745,3.773) -- (1.658,3.686) -- (1.602,3.576) -- (1.583,3.454) -- (1.602,3.332) -- (1.658,3.223) -- (1.745,3.135) -- (1.855,3.079) -- (1.976,3.06) -- cycle;
\node at (2.162,3.454) {\scalebox{1.652}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.61,2.866) -- (11.732,2.886) -- (11.841,2.942) -- (11.928,3.029) -- (11.984,3.139) -- (12.004,3.26) -- (11.984,3.382) -- (11.928,3.492) -- (11.841,3.579) -- (11.732,3.635) -- (11.61,3.654) -- (11.239,3.654) -- (11.117,3.635) -- (11.008,3.579) -- (10.92,3.492) -- (10.864,3.382) -- (10.845,3.26) -- (10.864,3.139) -- (10.92,3.029) -- (11.008,2.942) -- (11.117,2.886) -- (11.239,2.866) -- cycle;
\node at (11.424,3.26) {\scalebox{1.652}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.407,1.809) -- (9.529,1.828) -- (9.638,1.884) -- (9.726,1.971) -- (9.781,2.081) -- (9.801,2.203) -- (9.781,2.325) -- (9.726,2.434) -- (9.638,2.522) -- (9.529,2.578) -- (9.407,2.597) -- (9.036,2.597) -- (8.914,2.578) -- (8.805,2.522) -- (8.717,2.434) -- (8.662,2.325) -- (8.642,2.203) -- (8.662,2.081) -- (8.717,1.971) -- (8.805,1.884) -- (8.914,1.828) -- (9.036,1.809) -- cycle;
\node at (9.222,2.203) {\scalebox{1.652}{\small 17}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.674,1.489) -- (8.274,1.489) -- (8.274,2.089) -- (7.674,2.089) -- cycle;
\node at (7.974,1.789) {\scalebox{1.652}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (9.554,0.65) -- (10.154,0.65) -- (10.154,1.25) -- (9.554,1.25) -- cycle;
\node at (9.854,0.95) {\scalebox{1.652}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.728,2.511) -- (1.328,2.511) -- (1.328,3.111) -- (0.728,3.111) -- cycle;
\node at (1.028,2.811) {\scalebox{1.652}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.844,7.252) -- (5.444,7.252) -- (5.444,7.852) -- (4.844,7.852) -- cycle;
\node at (5.144,7.552) {\scalebox{1.652}{\small d}};
\node at (5.0,12.32) {\scalebox{1.652}{\small A}};
\node at (8.414,12.23) {\scalebox{1.652}{\small B}};
\node at (0.993,11.9) {\scalebox{1.652}{\small C}};
\node at (8.306,9.114) {\scalebox{1.652}{\small D}};
\node at (0.765,8.592) {\scalebox{1.652}{\small E}};
\node at (4.537,8.753) {\scalebox{1.652}{\small F}};
\node at (8.638,5.407) {\scalebox{1.652}{\small G}};
\node at (4.322,4.607) {\scalebox{1.652}{\small H}};
\node at (1.413,4.83) {\scalebox{1.652}{\small I}};
\node at (11.93,4.572) {\scalebox{1.652}{\small J}};
\node at (10.972,1.612) {\scalebox{1.652}{\small K}};
\node at (1.42,1.391) {\scalebox{1.652}{\small L}};
\node at (8.229,1.224) {\scalebox{1.652}{\small M}};
\draw[line width=0.5pt] (12.643,10.756) -- (12.643,11.856);
\draw[line width=0.5pt] (12.412,11.471) -- (12.643,11.856) -- (12.874,11.471);
\draw[line width=0.5pt] (12.34,11.306) -- (12.945,11.306);
\node at (12.643,12.241) {\scalebox{1.652}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour M, 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 & ⑩ allée des Peupliers & Ⓑ Coin du Lavoir & Ⓛ Place Sainte-Croix \\ \hline
① boulevard du Parc & ⑪ rue des Jardins & Ⓒ Place aux Herbes & Ⓜ Place du Marché \\ \hline
② rue du Pont & ⑫ chemin des Vignes & Ⓓ Carrefour des Tilleuls & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ rue du Stade & ⑬ rue Verte & Ⓔ Rond-point du Pont & \textbf{a} l'école \\ \hline
④ rue Basse & ⑭ quai aux Fleurs & Ⓕ Rond-point de la Poste & \textbf{b} l'église \\ \hline
⑤ rue du Puits & ⑮ impasse du Puits & Ⓖ Carrefour de la Mairie & \textbf{c} le stade \\ \hline
⑥ rue de la Poste & ⑯ rue de l'Église & Ⓗ Carrefour Saint-Roch & \textbf{d} la bibliothèque \\ \hline
⑦ rue du Moulin & ⑰ passage du Marché & Ⓘ Rond-point du Stade &  \\ \hline
⑧ rue Neuve & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Place de l'Église &  \\ \hline
⑨ sentier des Prés & Ⓐ Carrefour du Lavoir & Ⓚ Placette des Ormes &  \\ \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=0.6201, line join=round, line cap=round]
\draw[line width=0.5pt] (5.084,11.706) -- (5.301,11.67) -- (5.519,11.639) -- (5.737,11.613) -- (5.956,11.592) -- (6.176,11.577) -- (6.396,11.567) -- (6.615,11.561) -- (6.836,11.561) -- (7.055,11.567) -- (7.275,11.577) -- (7.495,11.592) -- (7.714,11.613) -- (7.932,11.639) -- (8.149,11.67);
\draw[line width=0.5pt] (5.084,11.706) -- (5.106,11.456) -- (5.122,11.206) -- (5.132,10.956) -- (5.135,10.705) -- (5.133,10.454) -- (5.125,10.203) -- (5.112,9.953) -- (5.092,9.703) -- (5.066,9.453) -- (5.035,9.204) -- (4.998,8.956) -- (4.955,8.709) -- (4.906,8.464) -- (4.851,8.219);
\draw[line width=0.5pt] (8.149,11.67) -- (8.096,11.461) -- (8.048,11.251) -- (8.005,11.04) -- (7.967,10.828) -- (7.934,10.615) -- (7.906,10.402) -- (7.883,10.187) -- (7.864,9.973) -- (7.851,9.758) -- (7.843,9.542) -- (7.84,9.327) -- (7.843,9.111) -- (7.85,8.896) -- (7.862,8.681);
\draw[line width=0.5pt] (1.411,11.441) -- (1.668,11.5) -- (1.927,11.553) -- (2.187,11.6) -- (2.448,11.64) -- (2.71,11.675) -- (2.973,11.703) -- (3.237,11.725) -- (3.501,11.741) -- (3.765,11.751) -- (4.029,11.754) -- (4.293,11.751) -- (4.558,11.742) -- (4.821,11.727) -- (5.084,11.706);
\draw[line width=0.5pt] (1.411,11.441) -- (1.372,11.224) -- (1.339,11.006) -- (1.311,10.787) -- (1.288,10.568) -- (1.27,10.348) -- (1.257,10.128) -- (1.25,9.907) -- (1.248,9.686) -- (1.251,9.466) -- (1.259,9.245) -- (1.272,9.025) -- (1.29,8.805) -- (1.314,8.586) -- (1.343,8.368);
\draw[line width=0.5pt] (7.862,8.681) -- (7.91,8.469) -- (7.953,8.256) -- (7.99,8.041) -- (8.023,7.826) -- (8.051,7.61) -- (8.073,7.393) -- (8.09,7.176) -- (8.103,6.958) -- (8.11,6.74) -- (8.112,6.523) -- (8.109,6.305) -- (8.101,6.087) -- (8.087,5.87) -- (8.069,5.653);
\draw[line width=0.5pt] (1.343,8.368) -- (1.59,8.319) -- (1.838,8.276) -- (2.087,8.238) -- (2.337,8.207) -- (2.588,8.182) -- (2.839,8.162) -- (3.091,8.149) -- (3.343,8.141) -- (3.595,8.139) -- (3.847,8.143) -- (4.099,8.153) -- (4.35,8.169) -- (4.601,8.191) -- (4.851,8.219);
\draw[line width=0.5pt] (1.343,8.368) -- (1.425,8.141) -- (1.501,7.912) -- (1.572,7.681) -- (1.637,7.449) -- (1.697,7.216) -- (1.751,6.98) -- (1.8,6.744) -- (1.844,6.507) -- (1.881,6.268) -- (1.914,6.029) -- (1.94,5.79) -- (1.961,5.549) -- (1.976,5.309) -- (1.986,5.068);
\draw[line width=0.5pt] (4.851,8.219) -- (5.07,8.219) -- (5.288,8.224) -- (5.506,8.234) -- (5.724,8.249) -- (5.942,8.27) -- (6.159,8.295) -- (6.376,8.326) -- (6.592,8.362) -- (6.806,8.402) -- (7.02,8.448) -- (7.233,8.499) -- (7.444,8.555) -- (7.654,8.616) -- (7.862,8.681);
\draw[line width=0.5pt] (4.851,8.219) -- (4.873,7.997) -- (4.89,7.775) -- (4.901,7.552) -- (4.908,7.328) -- (4.909,7.105) -- (4.904,6.882) -- (4.895,6.659) -- (4.88,6.436) -- (4.86,6.213) -- (4.835,5.991) -- (4.805,5.77) -- (4.769,5.55) -- (4.728,5.33) -- (4.682,5.112);
\draw[line width=0.5pt] (8.069,5.653) -- (8.291,5.555) -- (8.516,5.462) -- (8.743,5.374) -- (8.972,5.291) -- (9.203,5.214) -- (9.435,5.143) -- (9.67,5.077) -- (9.905,5.016) -- (10.142,4.961) -- (10.381,4.912) -- (10.62,4.868) -- (10.86,4.83) -- (11.101,4.797) -- (11.343,4.77);
\draw[line width=0.5pt] (8.069,5.653) -- (8.104,5.378) -- (8.134,5.103) -- (8.156,4.826) -- (8.172,4.549) -- (8.181,4.272) -- (8.184,3.995) -- (8.181,3.717) -- (8.171,3.44) -- (8.154,3.163) -- (8.131,2.886) -- (8.102,2.611) -- (8.065,2.336) -- (8.023,2.062) -- (7.974,1.789);
\draw[line width=0.5pt] (4.682,5.112) -- (4.928,5.113) -- (5.174,5.121) -- (5.42,5.133) -- (5.665,5.152) -- (5.91,5.177) -- (6.154,5.207) -- (6.398,5.243) -- (6.641,5.285) -- (6.882,5.332) -- (7.123,5.385) -- (7.362,5.444) -- (7.599,5.508) -- (7.835,5.578) -- (8.069,5.653);
\draw[line width=0.5pt] (1.986,5.068) -- (2.176,5.1) -- (2.368,5.129) -- (2.56,5.152) -- (2.752,5.171) -- (2.945,5.186) -- (3.139,5.196) -- (3.332,5.201) -- (3.526,5.202) -- (3.719,5.198) -- (3.913,5.19) -- (4.106,5.178) -- (4.299,5.16) -- (4.491,5.138) -- (4.682,5.112);
\draw[line width=0.5pt] (1.986,5.068) -- (1.938,4.843) -- (1.897,4.617) -- (1.86,4.39) -- (1.829,4.162) -- (1.804,3.934) -- (1.783,3.704) -- (1.768,3.475) -- (1.759,3.245) -- (1.755,3.015) -- (1.756,2.785) -- (1.763,2.555) -- (1.775,2.326) -- (1.792,2.097) -- (1.815,1.868);
\draw[line width=0.5pt] (11.343,4.77) -- (11.315,4.568) -- (11.282,4.367) -- (11.244,4.166) -- (11.202,3.966) -- (11.155,3.767) -- (11.103,3.57) -- (11.047,3.373) -- (10.986,3.178) -- (10.921,2.985) -- (10.851,2.793) -- (10.776,2.603) -- (10.697,2.414) -- (10.614,2.228) -- (10.526,2.044);
\draw[line width=0.5pt] (7.974,1.789) -- (8.158,1.779) -- (8.341,1.773) -- (8.525,1.772) -- (8.71,1.776) -- (8.893,1.783) -- (9.077,1.795) -- (9.261,1.811) -- (9.444,1.831) -- (9.626,1.856) -- (9.808,1.885) -- (9.989,1.918) -- (10.169,1.956) -- (10.348,1.998) -- (10.526,2.044);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.458,11.706) -- (5.429,11.849) -- (5.348,11.97) -- (5.227,12.051) -- (5.084,12.079) -- (4.942,12.051) -- (4.821,11.97) -- (4.74,11.849) -- (4.711,11.706) -- (4.74,11.563) -- (4.821,11.442) -- (4.942,11.361) -- (5.084,11.333) -- (5.227,11.361) -- (5.348,11.442) -- (5.429,11.563) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.523,11.67) -- (8.494,11.813) -- (8.413,11.934) -- (8.292,12.015) -- (8.149,12.043) -- (8.007,12.015) -- (7.886,11.934) -- (7.805,11.813) -- (7.776,11.67) -- (7.805,11.527) -- (7.886,11.406) -- (8.007,11.325) -- (8.149,11.297) -- (8.292,11.325) -- (8.413,11.406) -- (8.494,11.527) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.784,11.441) -- (1.755,11.584) -- (1.674,11.705) -- (1.553,11.786) -- (1.411,11.814) -- (1.268,11.786) -- (1.147,11.705) -- (1.066,11.584) -- (1.037,11.441) -- (1.066,11.298) -- (1.147,11.177) -- (1.268,11.096) -- (1.411,11.068) -- (1.553,11.096) -- (1.674,11.177) -- (1.755,11.298) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.235,8.681) -- (8.207,8.824) -- (8.126,8.945) -- (8.005,9.026) -- (7.862,9.054) -- (7.719,9.026) -- (7.598,8.945) -- (7.517,8.824) -- (7.489,8.681) -- (7.517,8.538) -- (7.598,8.417) -- (7.719,8.336) -- (7.862,8.308) -- (8.005,8.336) -- (8.126,8.417) -- (8.207,8.538) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.716,8.368) -- (1.688,8.51) -- (1.607,8.631) -- (1.486,8.712) -- (1.343,8.741) -- (1.2,8.712) -- (1.079,8.631) -- (0.998,8.51) -- (0.97,8.368) -- (0.998,8.225) -- (1.079,8.104) -- (1.2,8.023) -- (1.343,7.994) -- (1.486,8.023) -- (1.607,8.104) -- (1.688,8.225) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.224,8.219) -- (5.196,8.362) -- (5.115,8.483) -- (4.994,8.564) -- (4.851,8.592) -- (4.708,8.564) -- (4.587,8.483) -- (4.506,8.362) -- (4.478,8.219) -- (4.506,8.076) -- (4.587,7.955) -- (4.708,7.874) -- (4.851,7.846) -- (4.994,7.874) -- (5.115,7.955) -- (5.196,8.076) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.442,5.653) -- (8.414,5.796) -- (8.333,5.917) -- (8.212,5.998) -- (8.069,6.026) -- (7.926,5.998) -- (7.805,5.917) -- (7.724,5.796) -- (7.696,5.653) -- (7.724,5.51) -- (7.805,5.389) -- (7.926,5.308) -- (8.069,5.28) -- (8.212,5.308) -- (8.333,5.389) -- (8.414,5.51) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.056,5.112) -- (5.027,5.255) -- (4.946,5.376) -- (4.825,5.457) -- (4.682,5.485) -- (4.54,5.457) -- (4.419,5.376) -- (4.338,5.255) -- (4.309,5.112) -- (4.338,4.969) -- (4.419,4.848) -- (4.54,4.767) -- (4.682,4.739) -- (4.825,4.767) -- (4.946,4.848) -- (5.027,4.969) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.359,5.068) -- (2.33,5.211) -- (2.249,5.332) -- (2.128,5.413) -- (1.986,5.441) -- (1.843,5.413) -- (1.722,5.332) -- (1.641,5.211) -- (1.612,5.068) -- (1.641,4.925) -- (1.722,4.804) -- (1.843,4.723) -- (1.986,4.695) -- (2.128,4.723) -- (2.249,4.804) -- (2.33,4.925) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.716,4.77) -- (11.688,4.913) -- (11.607,5.034) -- (11.486,5.115) -- (11.343,5.144) -- (11.2,5.115) -- (11.079,5.034) -- (10.998,4.913) -- (10.97,4.77) -- (10.998,4.628) -- (11.079,4.506) -- (11.2,4.426) -- (11.343,4.397) -- (11.486,4.426) -- (11.607,4.506) -- (11.688,4.628) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.9,2.044) -- (10.871,2.186) -- (10.79,2.308) -- (10.669,2.388) -- (10.526,2.417) -- (10.384,2.388) -- (10.262,2.308) -- (10.182,2.186) -- (10.153,2.044) -- (10.182,1.901) -- (10.262,1.78) -- (10.384,1.699) -- (10.526,1.67) -- (10.669,1.699) -- (10.79,1.78) -- (10.871,1.901) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.189,1.868) -- (2.16,2.011) -- (2.079,2.132) -- (1.958,2.213) -- (1.815,2.241) -- (1.673,2.213) -- (1.551,2.132) -- (1.471,2.011) -- (1.442,1.868) -- (1.471,1.725) -- (1.551,1.604) -- (1.673,1.523) -- (1.815,1.495) -- (1.958,1.523) -- (2.079,1.604) -- (2.16,1.725) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.347,1.789) -- (8.319,1.932) -- (8.238,2.053) -- (8.117,2.134) -- (7.974,2.162) -- (7.831,2.134) -- (7.71,2.053) -- (7.629,1.932) -- (7.601,1.789) -- (7.629,1.646) -- (7.71,1.525) -- (7.831,1.444) -- (7.974,1.416) -- (8.117,1.444) -- (8.238,1.525) -- (8.319,1.646) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.62,11.561) -- (6.739,11.58) -- (6.846,11.635) -- (6.931,11.72) -- (6.986,11.827) -- (7.005,11.946) -- (6.986,12.065) -- (6.931,12.172) -- (6.846,12.257) -- (6.739,12.312) -- (6.62,12.33) -- (6.62,12.33) -- (6.501,12.312) -- (6.394,12.257) -- (6.309,12.172) -- (6.254,12.065) -- (6.235,11.946) -- (6.254,11.827) -- (6.309,11.72) -- (6.394,11.635) -- (6.501,11.58) -- (6.62,11.561) -- cycle;
\node at (6.62,11.946) {\scalebox{1.613}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.495,9.543) -- (5.614,9.561) -- (5.721,9.616) -- (5.806,9.701) -- (5.861,9.808) -- (5.88,9.927) -- (5.861,10.046) -- (5.806,10.153) -- (5.721,10.238) -- (5.614,10.293) -- (5.495,10.312) -- (5.495,10.312) -- (5.376,10.293) -- (5.269,10.238) -- (5.184,10.153) -- (5.13,10.046) -- (5.111,9.927) -- (5.13,9.808) -- (5.184,9.701) -- (5.269,9.616) -- (5.376,9.561) -- (5.495,9.543) -- cycle;
\node at (5.495,9.927) {\scalebox{1.613}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.265,9.766) -- (8.384,9.785) -- (8.491,9.84) -- (8.576,9.925) -- (8.631,10.032) -- (8.65,10.151) -- (8.631,10.269) -- (8.576,10.377) -- (8.491,10.462) -- (8.384,10.516) -- (8.265,10.535) -- (8.265,10.535) -- (8.147,10.516) -- (8.039,10.462) -- (7.954,10.377) -- (7.9,10.269) -- (7.881,10.151) -- (7.9,10.032) -- (7.954,9.925) -- (8.039,9.84) -- (8.147,9.785) -- (8.265,9.766) -- cycle;
\node at (8.265,10.151) {\scalebox{1.613}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.209,11.724) -- (3.328,11.743) -- (3.435,11.797) -- (3.52,11.883) -- (3.575,11.99) -- (3.594,12.109) -- (3.575,12.227) -- (3.52,12.335) -- (3.435,12.42) -- (3.328,12.474) -- (3.209,12.493) -- (3.209,12.493) -- (3.09,12.474) -- (2.983,12.42) -- (2.898,12.335) -- (2.843,12.227) -- (2.824,12.109) -- (2.843,11.99) -- (2.898,11.883) -- (2.983,11.797) -- (3.09,11.743) -- (3.209,11.724) -- cycle;
\node at (3.209,12.109) {\scalebox{1.613}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.634,9.514) -- (1.753,9.533) -- (1.86,9.588) -- (1.945,9.673) -- (2.0,9.78) -- (2.019,9.899) -- (2.0,10.017) -- (1.945,10.125) -- (1.86,10.21) -- (1.753,10.264) -- (1.634,10.283) -- (1.634,10.283) -- (1.516,10.264) -- (1.408,10.21) -- (1.323,10.125) -- (1.269,10.017) -- (1.25,9.899) -- (1.269,9.78) -- (1.323,9.673) -- (1.408,9.588) -- (1.516,9.533) -- (1.634,9.514) -- cycle;
\node at (1.634,9.899) {\scalebox{1.613}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.474,6.817) -- (8.593,6.836) -- (8.7,6.891) -- (8.785,6.976) -- (8.84,7.083) -- (8.859,7.202) -- (8.84,7.321) -- (8.785,7.428) -- (8.7,7.513) -- (8.593,7.568) -- (8.474,7.586) -- (8.474,7.586) -- (8.355,7.568) -- (8.248,7.513) -- (8.163,7.428) -- (8.108,7.321) -- (8.09,7.202) -- (8.108,7.083) -- (8.163,6.976) -- (8.248,6.891) -- (8.355,6.836) -- (8.474,6.817) -- cycle;
\node at (8.474,7.202) {\scalebox{1.613}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.107,8.148) -- (3.226,8.167) -- (3.333,8.222) -- (3.418,8.307) -- (3.473,8.414) -- (3.492,8.533) -- (3.473,8.652) -- (3.418,8.759) -- (3.333,8.844) -- (3.226,8.898) -- (3.107,8.917) -- (3.107,8.917) -- (2.988,8.898) -- (2.881,8.844) -- (2.796,8.759) -- (2.741,8.652) -- (2.723,8.533) -- (2.741,8.414) -- (2.796,8.307) -- (2.881,8.222) -- (2.988,8.167) -- (3.107,8.148) -- cycle;
\node at (3.107,8.533) {\scalebox{1.613}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.178,6.433) -- (2.297,6.452) -- (2.404,6.507) -- (2.489,6.592) -- (2.543,6.699) -- (2.562,6.818) -- (2.543,6.936) -- (2.489,7.044) -- (2.404,7.129) -- (2.297,7.183) -- (2.178,7.202) -- (2.178,7.202) -- (2.059,7.183) -- (1.952,7.129) -- (1.867,7.044) -- (1.812,6.936) -- (1.793,6.818) -- (1.812,6.699) -- (1.867,6.592) -- (1.952,6.507) -- (2.059,6.452) -- (2.178,6.433) -- cycle;
\node at (2.178,6.818) {\scalebox{1.613}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.317,8.321) -- (6.436,8.34) -- (6.543,8.395) -- (6.629,8.48) -- (6.683,8.587) -- (6.702,8.706) -- (6.683,8.825) -- (6.629,8.932) -- (6.543,9.017) -- (6.436,9.072) -- (6.317,9.091) -- (6.317,9.091) -- (6.199,9.072) -- (6.091,9.017) -- (6.006,8.932) -- (5.952,8.825) -- (5.933,8.706) -- (5.952,8.587) -- (6.006,8.48) -- (6.091,8.395) -- (6.199,8.34) -- (6.317,8.321) -- cycle;
\node at (6.317,8.706) {\scalebox{1.613}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.692,6.295) -- (4.811,6.314) -- (4.918,6.368) -- (5.003,6.453) -- (5.058,6.561) -- (5.076,6.679) -- (5.058,6.798) -- (5.003,6.905) -- (4.918,6.991) -- (4.811,7.045) -- (4.692,7.064) -- (4.33,7.064) -- (4.211,7.045) -- (4.104,6.991) -- (4.019,6.905) -- (3.964,6.798) -- (3.946,6.679) -- (3.964,6.561) -- (4.019,6.453) -- (4.104,6.368) -- (4.211,6.314) -- (4.33,6.295) -- cycle;
\node at (4.511,6.679) {\scalebox{1.613}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.951,5.063) -- (10.069,5.082) -- (10.177,5.137) -- (10.262,5.222) -- (10.316,5.329) -- (10.335,5.448) -- (10.316,5.567) -- (10.262,5.674) -- (10.177,5.759) -- (10.069,5.814) -- (9.951,5.832) -- (9.589,5.832) -- (9.47,5.814) -- (9.363,5.759) -- (9.278,5.674) -- (9.223,5.567) -- (9.204,5.448) -- (9.223,5.329) -- (9.278,5.222) -- (9.363,5.137) -- (9.47,5.082) -- (9.589,5.063) -- cycle;
\node at (9.77,5.448) {\scalebox{1.613}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.746,3.323) -- (8.865,3.342) -- (8.972,3.397) -- (9.057,3.482) -- (9.112,3.589) -- (9.131,3.708) -- (9.112,3.827) -- (9.057,3.934) -- (8.972,4.019) -- (8.865,4.073) -- (8.746,4.092) -- (8.384,4.092) -- (8.265,4.073) -- (8.158,4.019) -- (8.073,3.934) -- (8.019,3.827) -- (8.0,3.708) -- (8.019,3.589) -- (8.073,3.482) -- (8.158,3.397) -- (8.265,3.342) -- (8.384,3.323) -- cycle;
\node at (8.565,3.708) {\scalebox{1.613}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.518,5.238) -- (6.637,5.257) -- (6.744,5.312) -- (6.829,5.397) -- (6.884,5.504) -- (6.903,5.623) -- (6.884,5.741) -- (6.829,5.849) -- (6.744,5.934) -- (6.637,5.988) -- (6.518,6.007) -- (6.156,6.007) -- (6.038,5.988) -- (5.93,5.934) -- (5.845,5.849) -- (5.791,5.741) -- (5.772,5.623) -- (5.791,5.504) -- (5.845,5.397) -- (5.93,5.312) -- (6.038,5.257) -- (6.156,5.238) -- cycle;
\node at (6.337,5.623) {\scalebox{1.613}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.507,5.201) -- (3.626,5.22) -- (3.733,5.275) -- (3.818,5.36) -- (3.872,5.467) -- (3.891,5.586) -- (3.872,5.704) -- (3.818,5.812) -- (3.733,5.897) -- (3.626,5.951) -- (3.507,5.97) -- (3.145,5.97) -- (3.026,5.951) -- (2.919,5.897) -- (2.834,5.812) -- (2.779,5.704) -- (2.76,5.586) -- (2.779,5.467) -- (2.834,5.36) -- (2.919,5.275) -- (3.026,5.22) -- (3.145,5.201) -- cycle;
\node at (3.326,5.586) {\scalebox{1.613}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.333,3.07) -- (2.452,3.089) -- (2.559,3.143) -- (2.644,3.229) -- (2.699,3.336) -- (2.718,3.455) -- (2.699,3.573) -- (2.644,3.681) -- (2.559,3.766) -- (2.452,3.82) -- (2.333,3.839) -- (1.971,3.839) -- (1.853,3.82) -- (1.745,3.766) -- (1.66,3.681) -- (1.606,3.573) -- (1.587,3.455) -- (1.606,3.336) -- (1.66,3.229) -- (1.745,3.143) -- (1.853,3.089) -- (1.971,3.07) -- cycle;
\node at (2.152,3.455) {\scalebox{1.613}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.596,2.878) -- (11.715,2.897) -- (11.822,2.952) -- (11.908,3.037) -- (11.962,3.144) -- (11.981,3.263) -- (11.962,3.382) -- (11.908,3.489) -- (11.822,3.574) -- (11.715,3.629) -- (11.596,3.647) -- (11.235,3.647) -- (11.116,3.629) -- (11.008,3.574) -- (10.923,3.489) -- (10.869,3.382) -- (10.85,3.263) -- (10.869,3.144) -- (10.923,3.037) -- (11.008,2.952) -- (11.116,2.897) -- (11.235,2.878) -- cycle;
\node at (11.415,3.263) {\scalebox{1.613}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.403,1.809) -- (9.522,1.828) -- (9.629,1.883) -- (9.714,1.968) -- (9.769,2.075) -- (9.788,2.194) -- (9.769,2.312) -- (9.714,2.42) -- (9.629,2.505) -- (9.522,2.559) -- (9.403,2.578) -- (9.041,2.578) -- (8.923,2.559) -- (8.815,2.505) -- (8.73,2.42) -- (8.676,2.312) -- (8.657,2.194) -- (8.676,2.075) -- (8.73,1.968) -- (8.815,1.883) -- (8.923,1.828) -- (9.041,1.809) -- cycle;
\node at (9.222,2.194) {\scalebox{1.613}{\small 17}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.674,1.489) -- (8.274,1.489) -- (8.274,2.089) -- (7.674,2.089) -- cycle;
\node at (7.974,1.789) {\scalebox{1.613}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (9.554,0.65) -- (10.154,0.65) -- (10.154,1.25) -- (9.554,1.25) -- cycle;
\node at (9.854,0.95) {\scalebox{1.613}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (0.728,2.511) -- (1.328,2.511) -- (1.328,3.111) -- (0.728,3.111) -- cycle;
\node at (1.028,2.811) {\scalebox{1.613}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.844,7.252) -- (5.444,7.252) -- (5.444,7.852) -- (4.844,7.852) -- cycle;
\node at (5.144,7.552) {\scalebox{1.613}{\small d}};
\node at (5.0,12.32) {\scalebox{1.613}{\small A}};
\node at (8.414,12.23) {\scalebox{1.613}{\small B}};
\node at (0.993,11.9) {\scalebox{1.613}{\small C}};
\node at (8.306,9.114) {\scalebox{1.613}{\small D}};
\node at (0.765,8.592) {\scalebox{1.613}{\small E}};
\node at (4.537,8.753) {\scalebox{1.613}{\small F}};
\node at (8.638,5.407) {\scalebox{1.613}{\small G}};
\node at (4.322,4.607) {\scalebox{1.613}{\small H}};
\node at (1.413,4.83) {\scalebox{1.613}{\small I}};
\node at (11.93,4.572) {\scalebox{1.613}{\small J}};
\node at (10.972,1.612) {\scalebox{1.613}{\small K}};
\node at (1.42,1.391) {\scalebox{1.613}{\small L}};
\node at (8.229,1.224) {\scalebox{1.613}{\small M}};
\draw[line width=0.5pt] (12.643,10.756) -- (12.643,11.856);
\draw[line width=0.5pt] (12.412,11.471) -- (12.643,11.856) -- (12.874,11.471);
\draw[line width=0.5pt] (12.34,11.306) -- (12.945,11.306);
\node at (12.643,12.241) {\scalebox{1.613}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, D, E, H, J, L, M & 7 \\ \hline
clairs & B, C, F, G, I, K & 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{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{M} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend le \textbf{passage du Marché ⑰} vers l'est jusqu'à Placette des Ormes \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de l'Église ⑯}, vers le nord, jusqu'à Place de l'Église \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Jardins ⑪}, vers l'ouest, jusqu'à Carrefour de la Mairie \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Poste ⑥}, vers le nord, jusqu'à Carrefour des Tilleuls \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Stade ③}, vers le nord, jusqu'à Coin du Lavoir \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{boulevard du Parc ①}, vers l'ouest, jusqu'à Carrefour du Lavoir \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Basse ④}, vers l'ouest, jusqu'à Place aux Herbes \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ⑤}, vers le sud, jusqu'à Rond-point du Pont \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Moulin ⑦}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Peupliers ⑩}, vers le sud, jusqu'à Carrefour Saint-Roch \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{quai aux Fleurs ⑭}, vers l'ouest, jusqu'à Rond-point du Stade \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse du Puits ⑮}, vers le sud, jusqu'à Place Sainte-Croix \textbf{(L)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 13 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
