\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 110}

\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=110}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0110}
\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=0.6066, line join=round, line cap=round]
\draw[line width=0.5pt] (1.324,8.702) -- (1.535,8.638) -- (1.748,8.579) -- (1.962,8.525) -- (2.178,8.476) -- (2.394,8.432) -- (2.612,8.394) -- (2.83,8.36) -- (3.05,8.332) -- (3.269,8.309) -- (3.49,8.29) -- (3.71,8.278) -- (3.931,8.27) -- (4.152,8.267) -- (4.372,8.27);
\draw[line width=0.5pt] (1.324,8.702) -- (1.398,8.426) -- (1.465,8.149) -- (1.526,7.87) -- (1.58,7.59) -- (1.628,7.309) -- (1.669,7.027) -- (1.703,6.743) -- (1.731,6.459) -- (1.752,6.175) -- (1.766,5.89) -- (1.774,5.604) -- (1.775,5.319) -- (1.769,5.034) -- (1.756,4.749);
\draw[line width=0.5pt] (4.372,8.27) -- (4.339,8.052) -- (4.311,7.834) -- (4.288,7.615) -- (4.27,7.395) -- (4.257,7.175) -- (4.25,6.955) -- (4.248,6.734) -- (4.25,6.514) -- (4.258,6.293) -- (4.272,6.073) -- (4.29,5.853) -- (4.314,5.634) -- (4.342,5.416) -- (4.376,5.198);
\draw[line width=0.5pt] (8.598,5.44) -- (8.794,5.446) -- (8.99,5.448) -- (9.187,5.446) -- (9.383,5.438) -- (9.579,5.426) -- (9.775,5.41) -- (9.97,5.389) -- (10.165,5.363) -- (10.359,5.333) -- (10.552,5.298) -- (10.744,5.259) -- (10.936,5.215) -- (11.126,5.167) -- (11.315,5.114);
\draw[line width=0.5pt] (8.598,5.44) -- (8.509,5.218) -- (8.425,4.994) -- (8.347,4.768) -- (8.274,4.54) -- (8.206,4.311) -- (8.143,4.08) -- (8.086,3.848) -- (8.035,3.614) -- (7.989,3.379) -- (7.948,3.143) -- (7.914,2.907) -- (7.884,2.669) -- (7.861,2.431) -- (7.843,2.193);
\draw[line width=0.5pt] (4.376,5.198) -- (4.673,5.261) -- (4.971,5.318) -- (5.27,5.367) -- (5.571,5.409) -- (5.872,5.444) -- (6.174,5.472) -- (6.477,5.493) -- (6.78,5.507) -- (7.084,5.513) -- (7.387,5.513) -- (7.691,5.505) -- (7.994,5.491) -- (8.296,5.469) -- (8.598,5.44);
\draw[line width=0.5pt] (4.376,5.198) -- (4.366,4.935) -- (4.362,4.671) -- (4.364,4.407) -- (4.372,4.143) -- (4.386,3.88) -- (4.407,3.616) -- (4.434,3.354) -- (4.467,3.092) -- (4.506,2.831) -- (4.551,2.57) -- (4.603,2.311) -- (4.66,2.054) -- (4.724,1.798) -- (4.793,1.543);
\draw[line width=0.5pt] (11.315,5.114) -- (11.348,4.881) -- (11.375,4.647) -- (11.397,4.412) -- (11.413,4.177) -- (11.424,3.942) -- (11.429,3.707) -- (11.429,3.471) -- (11.423,3.235) -- (11.411,3.0) -- (11.395,2.765) -- (11.372,2.531) -- (11.344,2.297) -- (11.311,2.064) -- (11.272,1.832);
\draw[line width=0.5pt] (1.756,4.749) -- (1.947,4.752) -- (2.137,4.76) -- (2.328,4.772) -- (2.518,4.789) -- (2.707,4.81) -- (2.896,4.836) -- (3.085,4.866) -- (3.272,4.9) -- (3.459,4.939) -- (3.645,4.982) -- (3.83,5.03) -- (4.013,5.082) -- (4.195,5.138) -- (4.376,5.198);
\draw[line width=0.5pt] (1.756,4.749) -- (1.752,4.567) -- (1.744,4.384) -- (1.732,4.202) -- (1.715,4.02) -- (1.694,3.839) -- (1.668,3.658) -- (1.639,3.478) -- (1.605,3.298) -- (1.567,3.119) -- (1.525,2.942) -- (1.479,2.765) -- (1.428,2.589) -- (1.373,2.415) -- (1.315,2.242);
\draw[line width=0.5pt] (1.315,2.242) -- (1.554,2.155) -- (1.794,2.073) -- (2.037,1.997) -- (2.282,1.926) -- (2.528,1.861) -- (2.776,1.802) -- (3.025,1.749) -- (3.275,1.702) -- (3.527,1.661) -- (3.779,1.625) -- (4.032,1.596) -- (4.285,1.572) -- (4.539,1.555) -- (4.793,1.543);
\draw[line width=0.5pt] (7.843,2.193) -- (8.09,2.205) -- (8.337,2.211) -- (8.584,2.211) -- (8.832,2.206) -- (9.079,2.194) -- (9.326,2.177) -- (9.572,2.154) -- (9.818,2.125) -- (10.063,2.091) -- (10.307,2.05) -- (10.55,2.004) -- (10.792,1.952) -- (11.033,1.895) -- (11.272,1.832);
\draw[line width=0.5pt] (4.793,1.543) -- (5.016,1.556) -- (5.239,1.574) -- (5.462,1.597) -- (5.684,1.626) -- (5.905,1.66) -- (6.125,1.698) -- (6.345,1.742) -- (6.563,1.792) -- (6.78,1.846) -- (6.996,1.905) -- (7.21,1.97) -- (7.423,2.039) -- (7.634,2.114) -- (7.843,2.193);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.484,8.702) -- (1.463,8.782) -- (1.404,8.84) -- (1.324,8.862) -- (1.244,8.84) -- (1.186,8.782) -- (1.164,8.702) -- (1.186,8.622) -- (1.244,8.563) -- (1.324,8.542) -- (1.404,8.563) -- (1.463,8.622) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.532,8.27) -- (4.511,8.35) -- (4.452,8.409) -- (4.372,8.43) -- (4.292,8.409) -- (4.234,8.35) -- (4.212,8.27) -- (4.234,8.19) -- (4.292,8.132) -- (4.372,8.11) -- (4.452,8.132) -- (4.511,8.19) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.758,5.44) -- (8.737,5.52) -- (8.678,5.578) -- (8.598,5.6) -- (8.518,5.578) -- (8.459,5.52) -- (8.438,5.44) -- (8.459,5.36) -- (8.518,5.301) -- (8.598,5.28) -- (8.678,5.301) -- (8.737,5.36) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.536,5.198) -- (4.515,5.278) -- (4.456,5.337) -- (4.376,5.358) -- (4.296,5.337) -- (4.238,5.278) -- (4.216,5.198) -- (4.238,5.118) -- (4.296,5.06) -- (4.376,5.038) -- (4.456,5.06) -- (4.515,5.118) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.475,5.114) -- (11.453,5.194) -- (11.395,5.252) -- (11.315,5.274) -- (11.235,5.252) -- (11.176,5.194) -- (11.155,5.114) -- (11.176,5.034) -- (11.235,4.975) -- (11.315,4.954) -- (11.395,4.975) -- (11.453,5.034) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.916,4.749) -- (1.895,4.829) -- (1.836,4.888) -- (1.756,4.909) -- (1.676,4.888) -- (1.618,4.829) -- (1.596,4.749) -- (1.618,4.669) -- (1.676,4.611) -- (1.756,4.589) -- (1.836,4.611) -- (1.895,4.669) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.475,2.242) -- (1.453,2.322) -- (1.395,2.381) -- (1.315,2.402) -- (1.235,2.381) -- (1.176,2.322) -- (1.155,2.242) -- (1.176,2.162) -- (1.235,2.104) -- (1.315,2.082) -- (1.395,2.104) -- (1.453,2.162) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.003,2.193) -- (7.981,2.273) -- (7.923,2.332) -- (7.843,2.353) -- (7.763,2.332) -- (7.704,2.273) -- (7.683,2.193) -- (7.704,2.113) -- (7.763,2.055) -- (7.843,2.033) -- (7.923,2.055) -- (7.981,2.113) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.432,1.832) -- (11.41,1.912) -- (11.352,1.97) -- (11.272,1.992) -- (11.192,1.97) -- (11.133,1.912) -- (11.112,1.832) -- (11.133,1.752) -- (11.192,1.693) -- (11.272,1.672) -- (11.352,1.693) -- (11.41,1.752) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.953,1.543) -- (4.932,1.623) -- (4.873,1.682) -- (4.793,1.703) -- (4.713,1.682) -- (4.655,1.623) -- (4.633,1.543) -- (4.655,1.463) -- (4.713,1.405) -- (4.793,1.383) -- (4.873,1.405) -- (4.932,1.463) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.886,8.356) -- (3.007,8.376) -- (3.117,8.431) -- (3.204,8.519) -- (3.26,8.628) -- (3.279,8.75) -- (3.26,8.872) -- (3.204,8.982) -- (3.117,9.069) -- (3.007,9.125) -- (2.886,9.144) -- (2.886,9.144) -- (2.764,9.125) -- (2.654,9.069) -- (2.567,8.982) -- (2.511,8.872) -- (2.492,8.75) -- (2.511,8.628) -- (2.567,8.519) -- (2.654,8.431) -- (2.764,8.376) -- (2.886,8.356) -- cycle;
\node at (2.886,8.75) {\scalebox{1.649}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.095,6.392) -- (2.216,6.411) -- (2.326,6.467) -- (2.413,6.555) -- (2.469,6.664) -- (2.489,6.786) -- (2.469,6.908) -- (2.413,7.018) -- (2.326,7.105) -- (2.216,7.161) -- (2.095,7.18) -- (2.095,7.18) -- (1.973,7.161) -- (1.863,7.105) -- (1.776,7.018) -- (1.72,6.908) -- (1.701,6.786) -- (1.72,6.664) -- (1.776,6.555) -- (1.863,6.467) -- (1.973,6.411) -- (2.095,6.392) -- cycle;
\node at (2.095,6.786) {\scalebox{1.649}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.641,6.341) -- (4.763,6.36) -- (4.873,6.416) -- (4.96,6.503) -- (5.016,6.613) -- (5.035,6.735) -- (5.016,6.856) -- (4.96,6.966) -- (4.873,7.053) -- (4.763,7.109) -- (4.641,7.128) -- (4.641,7.128) -- (4.52,7.109) -- (4.41,7.053) -- (4.323,6.966) -- (4.267,6.856) -- (4.248,6.735) -- (4.267,6.613) -- (4.323,6.503) -- (4.41,6.416) -- (4.52,6.36) -- (4.641,6.341) -- cycle;
\node at (4.641,6.735) {\scalebox{1.649}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.923,4.604) -- (10.045,4.623) -- (10.154,4.679) -- (10.242,4.766) -- (10.297,4.876) -- (10.317,4.998) -- (10.297,5.119) -- (10.242,5.229) -- (10.154,5.316) -- (10.045,5.372) -- (9.923,5.392) -- (9.923,5.392) -- (9.801,5.372) -- (9.691,5.316) -- (9.604,5.229) -- (9.548,5.119) -- (9.529,4.998) -- (9.548,4.876) -- (9.604,4.766) -- (9.691,4.679) -- (9.801,4.623) -- (9.923,4.604) -- cycle;
\node at (9.923,4.998) {\scalebox{1.649}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.47,3.364) -- (8.592,3.384) -- (8.702,3.44) -- (8.789,3.527) -- (8.845,3.637) -- (8.864,3.758) -- (8.845,3.88) -- (8.789,3.99) -- (8.702,4.077) -- (8.592,4.133) -- (8.47,4.152) -- (8.47,4.152) -- (8.348,4.133) -- (8.238,4.077) -- (8.151,3.99) -- (8.095,3.88) -- (8.076,3.758) -- (8.095,3.637) -- (8.151,3.527) -- (8.238,3.44) -- (8.348,3.384) -- (8.47,3.364) -- cycle;
\node at (8.47,3.758) {\scalebox{1.649}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.455,5.492) -- (6.576,5.512) -- (6.686,5.568) -- (6.773,5.655) -- (6.829,5.765) -- (6.849,5.886) -- (6.829,6.008) -- (6.773,6.118) -- (6.686,6.205) -- (6.576,6.261) -- (6.455,6.28) -- (6.455,6.28) -- (6.333,6.261) -- (6.223,6.205) -- (6.136,6.118) -- (6.08,6.008) -- (6.061,5.886) -- (6.08,5.765) -- (6.136,5.655) -- (6.223,5.568) -- (6.333,5.512) -- (6.455,5.492) -- cycle;
\node at (6.455,5.886) {\scalebox{1.649}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.825,3.004) -- (4.947,3.024) -- (5.057,3.08) -- (5.144,3.167) -- (5.2,3.277) -- (5.219,3.398) -- (5.2,3.52) -- (5.144,3.63) -- (5.057,3.717) -- (4.947,3.773) -- (4.825,3.792) -- (4.825,3.792) -- (4.704,3.773) -- (4.594,3.717) -- (4.507,3.63) -- (4.451,3.52) -- (4.431,3.398) -- (4.451,3.277) -- (4.507,3.167) -- (4.594,3.08) -- (4.704,3.024) -- (4.825,3.004) -- cycle;
\node at (4.825,3.398) {\scalebox{1.649}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.822,3.072) -- (11.944,3.091) -- (12.054,3.147) -- (12.141,3.234) -- (12.197,3.344) -- (12.216,3.466) -- (12.197,3.588) -- (12.141,3.697) -- (12.054,3.784) -- (11.944,3.84) -- (11.822,3.86) -- (11.822,3.86) -- (11.701,3.84) -- (11.591,3.784) -- (11.504,3.697) -- (11.448,3.588) -- (11.429,3.466) -- (11.448,3.344) -- (11.504,3.234) -- (11.591,3.147) -- (11.701,3.091) -- (11.822,3.072) -- cycle;
\node at (11.822,3.466) {\scalebox{1.649}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.151,4.084) -- (3.273,4.103) -- (3.383,4.159) -- (3.47,4.246) -- (3.526,4.356) -- (3.545,4.477) -- (3.526,4.599) -- (3.47,4.709) -- (3.383,4.796) -- (3.273,4.852) -- (3.151,4.871) -- (3.151,4.871) -- (3.029,4.852) -- (2.92,4.796) -- (2.833,4.709) -- (2.777,4.599) -- (2.757,4.477) -- (2.777,4.356) -- (2.833,4.246) -- (2.92,4.159) -- (3.029,4.103) -- (3.151,4.084) -- cycle;
\node at (3.151,4.477) {\scalebox{1.649}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.212,3.015) -- (2.334,3.035) -- (2.444,3.091) -- (2.531,3.178) -- (2.587,3.288) -- (2.606,3.409) -- (2.587,3.531) -- (2.531,3.641) -- (2.444,3.728) -- (2.334,3.784) -- (2.212,3.803) -- (1.841,3.803) -- (1.72,3.784) -- (1.61,3.728) -- (1.523,3.641) -- (1.467,3.531) -- (1.447,3.409) -- (1.467,3.288) -- (1.523,3.178) -- (1.61,3.091) -- (1.72,3.035) -- (1.841,3.015) -- cycle;
\node at (2.027,3.409) {\scalebox{1.649}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.288,1.742) -- (3.41,1.761) -- (3.52,1.817) -- (3.607,1.904) -- (3.663,2.014) -- (3.682,2.136) -- (3.663,2.257) -- (3.607,2.367) -- (3.52,2.454) -- (3.41,2.51) -- (3.288,2.529) -- (2.917,2.529) -- (2.796,2.51) -- (2.686,2.454) -- (2.599,2.367) -- (2.543,2.257) -- (2.523,2.136) -- (2.543,2.014) -- (2.599,1.904) -- (2.686,1.817) -- (2.796,1.761) -- (2.917,1.742) -- cycle;
\node at (3.103,2.136) {\scalebox{1.649}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.799,2.152) -- (9.92,2.171) -- (10.03,2.227) -- (10.117,2.314) -- (10.173,2.424) -- (10.193,2.546) -- (10.173,2.667) -- (10.117,2.777) -- (10.03,2.864) -- (9.92,2.92) -- (9.799,2.94) -- (9.428,2.94) -- (9.306,2.92) -- (9.196,2.864) -- (9.109,2.777) -- (9.053,2.667) -- (9.034,2.546) -- (9.053,2.424) -- (9.109,2.314) -- (9.196,2.227) -- (9.306,2.171) -- (9.428,2.152) -- cycle;
\node at (9.613,2.546) {\scalebox{1.649}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.448,1.734) -- (6.57,1.753) -- (6.68,1.809) -- (6.767,1.896) -- (6.823,2.006) -- (6.842,2.128) -- (6.823,2.249) -- (6.767,2.359) -- (6.68,2.446) -- (6.57,2.502) -- (6.448,2.522) -- (6.077,2.522) -- (5.956,2.502) -- (5.846,2.446) -- (5.759,2.359) -- (5.703,2.249) -- (5.683,2.128) -- (5.703,2.006) -- (5.759,1.896) -- (5.846,1.809) -- (5.956,1.753) -- (6.077,1.734) -- cycle;
\node at (6.263,2.128) {\scalebox{1.649}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.456,4.449) -- (2.056,4.449) -- (2.056,5.049) -- (1.456,5.049) -- cycle;
\node at (1.756,4.749) {\scalebox{1.649}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.33,7.371) -- (4.93,7.371) -- (4.93,7.971) -- (4.33,7.971) -- cycle;
\node at (4.63,7.671) {\scalebox{1.649}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.444,7.441) -- (2.044,7.441) -- (2.044,8.041) -- (1.444,8.041) -- cycle;
\node at (1.744,7.741) {\scalebox{1.649}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (10.351,6.056) -- (10.951,6.056) -- (10.951,6.656) -- (10.351,6.656) -- cycle;
\node at (10.651,6.356) {\scalebox{1.649}{\small d}};
\node at (0.876,9.13) {\scalebox{1.649}{\small A}};
\node at (4.165,8.855) {\scalebox{1.649}{\small B}};
\node at (9.19,5.625) {\scalebox{1.649}{\small C}};
\node at (3.823,5.479) {\scalebox{1.649}{\small D}};
\node at (11.931,5.178) {\scalebox{1.649}{\small E}};
\node at (1.137,4.784) {\scalebox{1.649}{\small F}};
\node at (0.765,1.956) {\scalebox{1.649}{\small G}};
\node at (8.262,1.737) {\scalebox{1.649}{\small H}};
\node at (11.83,1.562) {\scalebox{1.649}{\small I}};
\node at (4.614,0.95) {\scalebox{1.649}{\small J}};
\draw[line width=0.5pt] (12.615,7.752) -- (12.615,8.852);
\draw[line width=0.5pt] (12.384,8.467) -- (12.615,8.852) -- (12.846,8.467);
\draw[line width=0.5pt] (12.312,8.302) -- (12.917,8.302);
\node at (12.615,9.237) {\scalebox{1.649}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour F, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑧ rue du Puits & Ⓑ Place de la Fontaine & Ⓙ Rond-point des Vaches \\ \hline
① rue des Jardins & ⑨ allée des Peupliers & Ⓒ Rond-point de la Poste & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
② rue Pasteur & ⑩ passage du Marché & Ⓓ Carrefour du Moulin & \textbf{a} l'école \\ \hline
③ rue Verte & ⑪ boulevard du Parc & Ⓔ Place du Vieux-Puits & \textbf{b} la salle des fêtes \\ \hline
④ rue Basse & ⑫ rue du Stade & Ⓕ Rond-point du Pont & \textbf{c} la mairie \\ \hline
⑤ rue du Pont & ⑬ rue du Moulin & Ⓖ Fontaine Notre-Dame & \textbf{d} le cimetière \\ \hline
⑥ avenue de la Gare & \cellcolor{black!32}\bfseries Les carrefours & Ⓗ Carrefour Saint-Roch &  \\ \hline
⑦ chemin du Lavoir & Ⓐ Place aux Herbes & Ⓘ Place des Tanneurs &  \\ \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=0.6214, line join=round, line cap=round]
\draw[line width=0.5pt] (1.324,8.702) -- (1.535,8.638) -- (1.748,8.579) -- (1.962,8.525) -- (2.178,8.476) -- (2.394,8.432) -- (2.612,8.394) -- (2.83,8.36) -- (3.05,8.332) -- (3.269,8.309) -- (3.49,8.29) -- (3.71,8.278) -- (3.931,8.27) -- (4.152,8.267) -- (4.372,8.27);
\draw[line width=0.5pt] (1.324,8.702) -- (1.398,8.426) -- (1.465,8.149) -- (1.526,7.87) -- (1.58,7.59) -- (1.628,7.309) -- (1.669,7.027) -- (1.703,6.743) -- (1.731,6.459) -- (1.752,6.175) -- (1.766,5.89) -- (1.774,5.604) -- (1.775,5.319) -- (1.769,5.034) -- (1.756,4.749);
\draw[line width=0.5pt] (4.372,8.27) -- (4.339,8.052) -- (4.311,7.834) -- (4.288,7.615) -- (4.27,7.395) -- (4.257,7.175) -- (4.25,6.955) -- (4.248,6.734) -- (4.25,6.514) -- (4.258,6.293) -- (4.272,6.073) -- (4.29,5.853) -- (4.314,5.634) -- (4.342,5.416) -- (4.376,5.198);
\draw[line width=0.5pt] (8.598,5.44) -- (8.794,5.446) -- (8.99,5.448) -- (9.187,5.446) -- (9.383,5.438) -- (9.579,5.426) -- (9.775,5.41) -- (9.97,5.389) -- (10.165,5.363) -- (10.359,5.333) -- (10.552,5.298) -- (10.744,5.259) -- (10.936,5.215) -- (11.126,5.167) -- (11.315,5.114);
\draw[line width=0.5pt] (8.598,5.44) -- (8.509,5.218) -- (8.425,4.994) -- (8.347,4.768) -- (8.274,4.54) -- (8.206,4.311) -- (8.143,4.08) -- (8.086,3.848) -- (8.035,3.614) -- (7.989,3.379) -- (7.948,3.143) -- (7.914,2.907) -- (7.884,2.669) -- (7.861,2.431) -- (7.843,2.193);
\draw[line width=0.5pt] (4.376,5.198) -- (4.673,5.261) -- (4.971,5.318) -- (5.27,5.367) -- (5.571,5.409) -- (5.872,5.444) -- (6.174,5.472) -- (6.477,5.493) -- (6.78,5.507) -- (7.084,5.513) -- (7.387,5.513) -- (7.691,5.505) -- (7.994,5.491) -- (8.296,5.469) -- (8.598,5.44);
\draw[line width=0.5pt] (4.376,5.198) -- (4.366,4.935) -- (4.362,4.671) -- (4.364,4.407) -- (4.372,4.143) -- (4.386,3.88) -- (4.407,3.616) -- (4.434,3.354) -- (4.467,3.092) -- (4.506,2.831) -- (4.551,2.57) -- (4.603,2.311) -- (4.66,2.054) -- (4.724,1.798) -- (4.793,1.543);
\draw[line width=0.5pt] (11.315,5.114) -- (11.348,4.881) -- (11.375,4.647) -- (11.397,4.412) -- (11.413,4.177) -- (11.424,3.942) -- (11.429,3.707) -- (11.429,3.471) -- (11.423,3.235) -- (11.411,3.0) -- (11.395,2.765) -- (11.372,2.531) -- (11.344,2.297) -- (11.311,2.064) -- (11.272,1.832);
\draw[line width=0.5pt] (1.756,4.749) -- (1.947,4.752) -- (2.137,4.76) -- (2.328,4.772) -- (2.518,4.789) -- (2.707,4.81) -- (2.896,4.836) -- (3.085,4.866) -- (3.272,4.9) -- (3.459,4.939) -- (3.645,4.982) -- (3.83,5.03) -- (4.013,5.082) -- (4.195,5.138) -- (4.376,5.198);
\draw[line width=0.5pt] (1.756,4.749) -- (1.752,4.567) -- (1.744,4.384) -- (1.732,4.202) -- (1.715,4.02) -- (1.694,3.839) -- (1.668,3.658) -- (1.639,3.478) -- (1.605,3.298) -- (1.567,3.119) -- (1.525,2.942) -- (1.479,2.765) -- (1.428,2.589) -- (1.373,2.415) -- (1.315,2.242);
\draw[line width=0.5pt] (1.315,2.242) -- (1.554,2.155) -- (1.794,2.073) -- (2.037,1.997) -- (2.282,1.926) -- (2.528,1.861) -- (2.776,1.802) -- (3.025,1.749) -- (3.275,1.702) -- (3.527,1.661) -- (3.779,1.625) -- (4.032,1.596) -- (4.285,1.572) -- (4.539,1.555) -- (4.793,1.543);
\draw[line width=0.5pt] (7.843,2.193) -- (8.09,2.205) -- (8.337,2.211) -- (8.584,2.211) -- (8.832,2.206) -- (9.079,2.194) -- (9.326,2.177) -- (9.572,2.154) -- (9.818,2.125) -- (10.063,2.091) -- (10.307,2.05) -- (10.55,2.004) -- (10.792,1.952) -- (11.033,1.895) -- (11.272,1.832);
\draw[line width=0.5pt] (4.793,1.543) -- (5.016,1.556) -- (5.239,1.574) -- (5.462,1.597) -- (5.684,1.626) -- (5.905,1.66) -- (6.125,1.698) -- (6.345,1.742) -- (6.563,1.792) -- (6.78,1.846) -- (6.996,1.905) -- (7.21,1.97) -- (7.423,2.039) -- (7.634,2.114) -- (7.843,2.193);
\filldraw[fill=white, draw=black, line width=0.5pt] (1.697,8.702) -- (1.669,8.844) -- (1.588,8.965) -- (1.467,9.046) -- (1.324,9.075) -- (1.181,9.046) -- (1.06,8.965) -- (0.979,8.844) -- (0.951,8.702) -- (0.979,8.559) -- (1.06,8.438) -- (1.181,8.357) -- (1.324,8.328) -- (1.467,8.357) -- (1.588,8.438) -- (1.669,8.559) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.746,8.27) -- (4.717,8.413) -- (4.636,8.534) -- (4.515,8.615) -- (4.372,8.643) -- (4.229,8.615) -- (4.108,8.534) -- (4.027,8.413) -- (3.999,8.27) -- (4.027,8.127) -- (4.108,8.006) -- (4.229,7.925) -- (4.372,7.897) -- (4.515,7.925) -- (4.636,8.006) -- (4.717,8.127) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.971,5.44) -- (8.943,5.582) -- (8.862,5.704) -- (8.741,5.784) -- (8.598,5.813) -- (8.455,5.784) -- (8.334,5.704) -- (8.253,5.582) -- (8.225,5.44) -- (8.253,5.297) -- (8.334,5.176) -- (8.455,5.095) -- (8.598,5.066) -- (8.741,5.095) -- (8.862,5.176) -- (8.943,5.297) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.749,5.198) -- (4.721,5.341) -- (4.64,5.462) -- (4.519,5.543) -- (4.376,5.571) -- (4.233,5.543) -- (4.112,5.462) -- (4.031,5.341) -- (4.003,5.198) -- (4.031,5.055) -- (4.112,4.934) -- (4.233,4.853) -- (4.376,4.825) -- (4.519,4.853) -- (4.64,4.934) -- (4.721,5.055) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.688,5.114) -- (11.659,5.257) -- (11.579,5.378) -- (11.458,5.459) -- (11.315,5.487) -- (11.172,5.459) -- (11.051,5.378) -- (10.97,5.257) -- (10.941,5.114) -- (10.97,4.971) -- (11.051,4.85) -- (11.172,4.769) -- (11.315,4.74) -- (11.458,4.769) -- (11.579,4.85) -- (11.659,4.971) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.129,4.749) -- (2.101,4.892) -- (2.02,5.013) -- (1.899,5.094) -- (1.756,5.123) -- (1.613,5.094) -- (1.492,5.013) -- (1.411,4.892) -- (1.383,4.749) -- (1.411,4.606) -- (1.492,4.485) -- (1.613,4.404) -- (1.756,4.376) -- (1.899,4.404) -- (2.02,4.485) -- (2.101,4.606) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.688,2.242) -- (1.659,2.385) -- (1.579,2.506) -- (1.458,2.587) -- (1.315,2.616) -- (1.172,2.587) -- (1.051,2.506) -- (0.97,2.385) -- (0.941,2.242) -- (0.97,2.099) -- (1.051,1.978) -- (1.172,1.897) -- (1.315,1.869) -- (1.458,1.897) -- (1.579,1.978) -- (1.659,2.099) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.216,2.193) -- (8.187,2.336) -- (8.106,2.457) -- (7.985,2.538) -- (7.843,2.566) -- (7.7,2.538) -- (7.579,2.457) -- (7.498,2.336) -- (7.469,2.193) -- (7.498,2.05) -- (7.579,1.929) -- (7.7,1.848) -- (7.843,1.82) -- (7.985,1.848) -- (8.106,1.929) -- (8.187,2.05) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.645,1.832) -- (11.617,1.975) -- (11.536,2.096) -- (11.415,2.177) -- (11.272,2.205) -- (11.129,2.177) -- (11.008,2.096) -- (10.927,1.975) -- (10.899,1.832) -- (10.927,1.689) -- (11.008,1.568) -- (11.129,1.487) -- (11.272,1.459) -- (11.415,1.487) -- (11.536,1.568) -- (11.617,1.689) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.166,1.543) -- (5.138,1.686) -- (5.057,1.807) -- (4.936,1.888) -- (4.793,1.917) -- (4.65,1.888) -- (4.529,1.807) -- (4.448,1.686) -- (4.42,1.543) -- (4.448,1.401) -- (4.529,1.28) -- (4.65,1.199) -- (4.793,1.17) -- (4.936,1.199) -- (5.057,1.28) -- (5.138,1.401) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.884,8.356) -- (3.003,8.375) -- (3.11,8.43) -- (3.195,8.515) -- (3.25,8.622) -- (3.269,8.741) -- (3.25,8.86) -- (3.195,8.967) -- (3.11,9.052) -- (3.003,9.107) -- (2.884,9.125) -- (2.884,9.125) -- (2.765,9.107) -- (2.658,9.052) -- (2.573,8.967) -- (2.519,8.86) -- (2.5,8.741) -- (2.519,8.622) -- (2.573,8.515) -- (2.658,8.43) -- (2.765,8.375) -- (2.884,8.356) -- cycle;
\node at (2.884,8.741) {\scalebox{1.609}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.085,6.401) -- (2.204,6.419) -- (2.311,6.474) -- (2.396,6.559) -- (2.451,6.666) -- (2.47,6.785) -- (2.451,6.904) -- (2.396,7.011) -- (2.311,7.096) -- (2.204,7.151) -- (2.085,7.17) -- (2.085,7.17) -- (1.967,7.151) -- (1.859,7.096) -- (1.774,7.011) -- (1.72,6.904) -- (1.701,6.785) -- (1.72,6.666) -- (1.774,6.559) -- (1.859,6.474) -- (1.967,6.419) -- (2.085,6.401) -- cycle;
\node at (2.085,6.785) {\scalebox{1.609}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.632,6.35) -- (4.751,6.369) -- (4.858,6.423) -- (4.943,6.509) -- (4.998,6.616) -- (5.017,6.735) -- (4.998,6.853) -- (4.943,6.961) -- (4.858,7.046) -- (4.751,7.1) -- (4.632,7.119) -- (4.632,7.119) -- (4.513,7.1) -- (4.406,7.046) -- (4.321,6.961) -- (4.266,6.853) -- (4.248,6.735) -- (4.266,6.616) -- (4.321,6.509) -- (4.406,6.423) -- (4.513,6.369) -- (4.632,6.35) -- cycle;
\node at (4.632,6.735) {\scalebox{1.609}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.924,4.622) -- (10.043,4.641) -- (10.15,4.696) -- (10.235,4.781) -- (10.29,4.888) -- (10.309,5.007) -- (10.29,5.126) -- (10.235,5.233) -- (10.15,5.318) -- (10.043,5.373) -- (9.924,5.391) -- (9.924,5.391) -- (9.805,5.373) -- (9.698,5.318) -- (9.613,5.233) -- (9.558,5.126) -- (9.539,5.007) -- (9.558,4.888) -- (9.613,4.781) -- (9.698,4.696) -- (9.805,4.641) -- (9.924,4.622) -- cycle;
\node at (9.924,5.007) {\scalebox{1.609}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.461,3.376) -- (8.58,3.395) -- (8.687,3.449) -- (8.772,3.534) -- (8.827,3.642) -- (8.845,3.76) -- (8.827,3.879) -- (8.772,3.986) -- (8.687,4.071) -- (8.58,4.126) -- (8.461,4.145) -- (8.461,4.145) -- (8.342,4.126) -- (8.235,4.071) -- (8.15,3.986) -- (8.095,3.879) -- (8.076,3.76) -- (8.095,3.642) -- (8.15,3.534) -- (8.235,3.449) -- (8.342,3.395) -- (8.461,3.376) -- cycle;
\node at (8.461,3.76) {\scalebox{1.609}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.455,5.492) -- (6.574,5.511) -- (6.681,5.566) -- (6.766,5.651) -- (6.821,5.758) -- (6.84,5.877) -- (6.821,5.996) -- (6.766,6.103) -- (6.681,6.188) -- (6.574,6.243) -- (6.455,6.261) -- (6.455,6.261) -- (6.336,6.243) -- (6.229,6.188) -- (6.144,6.103) -- (6.089,5.996) -- (6.071,5.877) -- (6.089,5.758) -- (6.144,5.651) -- (6.229,5.566) -- (6.336,5.511) -- (6.455,5.492) -- cycle;
\node at (6.455,5.877) {\scalebox{1.609}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.816,3.013) -- (4.935,3.032) -- (5.042,3.086) -- (5.127,3.171) -- (5.182,3.278) -- (5.201,3.397) -- (5.182,3.516) -- (5.127,3.623) -- (5.042,3.708) -- (4.935,3.763) -- (4.816,3.782) -- (4.816,3.782) -- (4.697,3.763) -- (4.59,3.708) -- (4.505,3.623) -- (4.45,3.516) -- (4.431,3.397) -- (4.45,3.278) -- (4.505,3.171) -- (4.59,3.086) -- (4.697,3.032) -- (4.816,3.013) -- cycle;
\node at (4.816,3.397) {\scalebox{1.609}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.813,3.081) -- (11.932,3.1) -- (12.039,3.155) -- (12.124,3.24) -- (12.179,3.347) -- (12.198,3.466) -- (12.179,3.585) -- (12.124,3.692) -- (12.039,3.777) -- (11.932,3.832) -- (11.813,3.85) -- (11.813,3.85) -- (11.694,3.832) -- (11.587,3.777) -- (11.502,3.692) -- (11.447,3.585) -- (11.429,3.466) -- (11.447,3.347) -- (11.502,3.24) -- (11.587,3.155) -- (11.694,3.1) -- (11.813,3.081) -- cycle;
\node at (11.813,3.466) {\scalebox{1.609}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.15,4.102) -- (3.268,4.121) -- (3.376,4.176) -- (3.461,4.261) -- (3.515,4.368) -- (3.534,4.487) -- (3.515,4.605) -- (3.461,4.713) -- (3.376,4.798) -- (3.268,4.852) -- (3.15,4.871) -- (3.15,4.871) -- (3.031,4.852) -- (2.924,4.798) -- (2.839,4.713) -- (2.784,4.605) -- (2.765,4.487) -- (2.784,4.368) -- (2.839,4.261) -- (2.924,4.176) -- (3.031,4.121) -- (3.15,4.102) -- cycle;
\node at (3.15,4.487) {\scalebox{1.609}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.198,3.026) -- (2.317,3.045) -- (2.424,3.1) -- (2.51,3.185) -- (2.564,3.292) -- (2.583,3.411) -- (2.564,3.53) -- (2.51,3.637) -- (2.424,3.722) -- (2.317,3.777) -- (2.198,3.795) -- (1.837,3.795) -- (1.718,3.777) -- (1.611,3.722) -- (1.525,3.637) -- (1.471,3.53) -- (1.452,3.411) -- (1.471,3.292) -- (1.525,3.185) -- (1.611,3.1) -- (1.718,3.045) -- (1.837,3.026) -- cycle;
\node at (2.018,3.411) {\scalebox{1.609}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.282,1.742) -- (3.401,1.761) -- (3.508,1.815) -- (3.593,1.9) -- (3.648,2.008) -- (3.666,2.126) -- (3.648,2.245) -- (3.593,2.352) -- (3.508,2.437) -- (3.401,2.492) -- (3.282,2.511) -- (2.92,2.511) -- (2.801,2.492) -- (2.694,2.437) -- (2.609,2.352) -- (2.554,2.245) -- (2.535,2.126) -- (2.554,2.008) -- (2.609,1.9) -- (2.694,1.815) -- (2.801,1.761) -- (2.92,1.742) -- cycle;
\node at (3.101,2.126) {\scalebox{1.609}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.793,2.152) -- (9.912,2.171) -- (10.019,2.225) -- (10.104,2.31) -- (10.159,2.417) -- (10.178,2.536) -- (10.159,2.655) -- (10.104,2.762) -- (10.019,2.847) -- (9.912,2.902) -- (9.793,2.921) -- (9.431,2.921) -- (9.313,2.902) -- (9.205,2.847) -- (9.12,2.762) -- (9.066,2.655) -- (9.047,2.536) -- (9.066,2.417) -- (9.12,2.31) -- (9.205,2.225) -- (9.313,2.171) -- (9.431,2.152) -- cycle;
\node at (9.612,2.536) {\scalebox{1.609}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.446,1.734) -- (6.564,1.753) -- (6.672,1.807) -- (6.757,1.893) -- (6.811,2.0) -- (6.83,2.119) -- (6.811,2.237) -- (6.757,2.345) -- (6.672,2.43) -- (6.564,2.484) -- (6.446,2.503) -- (6.084,2.503) -- (5.965,2.484) -- (5.858,2.43) -- (5.773,2.345) -- (5.718,2.237) -- (5.699,2.119) -- (5.718,2.0) -- (5.773,1.893) -- (5.858,1.807) -- (5.965,1.753) -- (6.084,1.734) -- cycle;
\node at (6.265,2.119) {\scalebox{1.609}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.456,4.449) -- (2.056,4.449) -- (2.056,5.049) -- (1.456,5.049) -- cycle;
\node at (1.756,4.749) {\scalebox{1.609}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.33,7.371) -- (4.93,7.371) -- (4.93,7.971) -- (4.33,7.971) -- cycle;
\node at (4.63,7.671) {\scalebox{1.609}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.444,7.441) -- (2.044,7.441) -- (2.044,8.041) -- (1.444,8.041) -- cycle;
\node at (1.744,7.741) {\scalebox{1.609}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (10.351,6.056) -- (10.951,6.056) -- (10.951,6.656) -- (10.351,6.656) -- cycle;
\node at (10.651,6.356) {\scalebox{1.609}{\small d}};
\node at (0.876,9.13) {\scalebox{1.609}{\small A}};
\node at (4.165,8.855) {\scalebox{1.609}{\small B}};
\node at (9.19,5.625) {\scalebox{1.609}{\small C}};
\node at (3.823,5.479) {\scalebox{1.609}{\small D}};
\node at (11.931,5.178) {\scalebox{1.609}{\small E}};
\node at (1.137,4.784) {\scalebox{1.609}{\small F}};
\node at (0.765,1.956) {\scalebox{1.609}{\small G}};
\node at (8.262,1.737) {\scalebox{1.609}{\small H}};
\node at (11.83,1.562) {\scalebox{1.609}{\small I}};
\node at (4.614,0.95) {\scalebox{1.609}{\small J}};
\draw[line width=0.5pt] (12.615,7.752) -- (12.615,8.852);
\draw[line width=0.5pt] (12.384,8.467) -- (12.615,8.852) -- (12.846,8.467);
\draw[line width=0.5pt] (12.312,8.302) -- (12.917,8.302);
\node at (12.615,9.237) {\scalebox{1.609}{\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 & B, C, F, I, J & 5 \\ \hline
clairs & A, D, E, G, H & 5 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{5} clairs. Il y en a \textbf{autant} de chaque côté.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 10 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui.} Un circuit existe :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{rue Pasteur ②} vers le nord jusqu'à Place aux Herbes \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Jardins ①}, vers l'est, jusqu'à Place de la Fontaine \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Verte ③}, vers le sud, jusqu'à Carrefour du Moulin \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{avenue de la Gare ⑥}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Basse ④}, vers l'est, jusqu'à Place du Vieux-Puits \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Puits ⑧}, vers le sud, jusqu'à Place des Tanneurs \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Stade ⑫}, vers l'ouest, jusqu'à Carrefour Saint-Roch \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Moulin ⑬}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{boulevard du Parc ⑪}, vers l'ouest, jusqu'à Fontaine Notre-Dame \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{passage du Marché ⑩}, vers le nord, jusqu'à Rond-point du Pont \textbf{(F)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Le car est passé une fois par chaque carrefour et se retrouve à l'école.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, et c'est déjà fait : le trajet ci-dessus dessert tous les carrefours. Il suffit au car de s'arrêter avant le dernier trajet de retour.}\par\addvspace{0.4em}
\end{document}
