\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 152}

\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=152}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0152}
\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=0.6044, line join=round, line cap=round]
\draw[line width=0.5pt] (8.827,6.753) -- (9.005,6.711) -- (9.183,6.673) -- (9.363,6.64) -- (9.543,6.611) -- (9.724,6.586) -- (9.906,6.565) -- (10.088,6.548) -- (10.27,6.536) -- (10.452,6.528) -- (10.635,6.525) -- (10.818,6.525) -- (11.0,6.53) -- (11.183,6.54) -- (11.365,6.553);
\draw[line width=0.5pt] (8.827,6.753) -- (8.82,6.557) -- (8.809,6.362) -- (8.792,6.167) -- (8.772,5.973) -- (8.746,5.779) -- (8.717,5.586) -- (8.682,5.393) -- (8.643,5.201) -- (8.6,5.01) -- (8.552,4.821) -- (8.5,4.632) -- (8.443,4.445) -- (8.382,4.259) -- (8.316,4.075);
\draw[line width=0.5pt] (11.365,6.553) -- (11.291,6.367) -- (11.221,6.179) -- (11.156,5.99) -- (11.096,5.799) -- (11.04,5.606) -- (10.988,5.413) -- (10.941,5.218) -- (10.899,5.022) -- (10.861,4.825) -- (10.828,4.627) -- (10.8,4.429) -- (10.776,4.23) -- (10.757,4.031) -- (10.743,3.831);
\draw[line width=0.5pt] (3.238,6.52) -- (3.42,6.535) -- (3.602,6.545) -- (3.784,6.551) -- (3.967,6.553) -- (4.15,6.55) -- (4.332,6.543) -- (4.514,6.532) -- (4.697,6.517) -- (4.878,6.497) -- (5.059,6.473) -- (5.24,6.445) -- (5.419,6.412) -- (5.598,6.376) -- (5.776,6.335);
\draw[line width=0.5pt] (3.238,6.52) -- (3.259,6.315) -- (3.285,6.111) -- (3.315,5.907) -- (3.351,5.704) -- (3.391,5.501) -- (3.436,5.3) -- (3.486,5.1) -- (3.541,4.901) -- (3.6,4.703) -- (3.663,4.507) -- (3.732,4.313) -- (3.805,4.12) -- (3.882,3.929) -- (3.964,3.74);
\draw[line width=0.5pt] (1.381,6.506) -- (1.512,6.527) -- (1.644,6.546) -- (1.776,6.561) -- (1.909,6.573) -- (2.042,6.581) -- (2.175,6.587) -- (2.309,6.59) -- (2.442,6.589) -- (2.575,6.585) -- (2.708,6.579) -- (2.841,6.569) -- (2.974,6.556) -- (3.106,6.539) -- (3.238,6.52);
\draw[line width=0.5pt] (1.381,6.506) -- (1.348,6.3) -- (1.32,6.093) -- (1.296,5.885) -- (1.278,5.677) -- (1.265,5.469) -- (1.256,5.26) -- (1.253,5.051) -- (1.254,4.842) -- (1.26,4.633) -- (1.271,4.424) -- (1.287,4.216) -- (1.308,4.008) -- (1.334,3.801) -- (1.365,3.595);
\draw[line width=0.5pt] (5.776,6.335) -- (5.988,6.398) -- (6.201,6.456) -- (6.415,6.509) -- (6.631,6.557) -- (6.848,6.599) -- (7.066,6.637) -- (7.284,6.67) -- (7.504,6.697) -- (7.724,6.719) -- (7.944,6.737) -- (8.165,6.748) -- (8.385,6.755) -- (8.606,6.757) -- (8.827,6.753);
\draw[line width=0.5pt] (5.776,6.335) -- (5.777,6.147) -- (5.783,5.959) -- (5.793,5.771) -- (5.808,5.584) -- (5.827,5.397) -- (5.85,5.21) -- (5.878,5.024) -- (5.91,4.838) -- (5.947,4.654) -- (5.988,4.47) -- (6.033,4.288) -- (6.082,4.106) -- (6.136,3.926) -- (6.194,3.747);
\draw[line width=0.5pt] (8.316,4.075) -- (8.491,4.084) -- (8.666,4.089) -- (8.841,4.09) -- (9.016,4.087) -- (9.19,4.08) -- (9.365,4.069) -- (9.54,4.053) -- (9.713,4.033) -- (9.887,4.01) -- (10.06,3.982) -- (10.232,3.95) -- (10.403,3.915) -- (10.573,3.875) -- (10.743,3.831);
\draw[line width=0.5pt] (6.194,3.747) -- (6.348,3.747) -- (6.502,3.751) -- (6.656,3.758) -- (6.809,3.769) -- (6.963,3.784) -- (7.116,3.802) -- (7.269,3.824) -- (7.421,3.849) -- (7.572,3.878) -- (7.723,3.91) -- (7.873,3.946) -- (8.022,3.986) -- (8.169,4.029) -- (8.316,4.075);
\draw[line width=0.5pt] (6.194,3.747) -- (6.191,3.589) -- (6.185,3.43) -- (6.176,3.272) -- (6.162,3.114) -- (6.145,2.956) -- (6.124,2.799) -- (6.099,2.642) -- (6.071,2.486) -- (6.039,2.331) -- (6.004,2.176) -- (5.964,2.023) -- (5.922,1.87) -- (5.875,1.718) -- (5.825,1.568);
\draw[line width=0.5pt] (3.964,3.74) -- (4.122,3.716) -- (4.281,3.696) -- (4.44,3.679) -- (4.599,3.667) -- (4.759,3.658) -- (4.919,3.653) -- (5.079,3.651) -- (5.239,3.654) -- (5.399,3.66) -- (5.559,3.67) -- (5.718,3.684) -- (5.877,3.701) -- (6.036,3.722) -- (6.194,3.747);
\draw[line width=0.5pt] (1.365,3.595) -- (1.547,3.633) -- (1.731,3.668) -- (1.915,3.698) -- (2.1,3.723) -- (2.286,3.745) -- (2.472,3.762) -- (2.658,3.774) -- (2.845,3.783) -- (3.032,3.786) -- (3.219,3.786) -- (3.405,3.781) -- (3.592,3.772) -- (3.778,3.758) -- (3.964,3.74);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.987,6.753) -- (8.966,6.833) -- (8.907,6.891) -- (8.827,6.913) -- (8.747,6.891) -- (8.689,6.833) -- (8.667,6.753) -- (8.689,6.673) -- (8.747,6.614) -- (8.827,6.593) -- (8.907,6.614) -- (8.966,6.673) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.525,6.553) -- (11.503,6.633) -- (11.445,6.692) -- (11.365,6.713) -- (11.285,6.692) -- (11.226,6.633) -- (11.205,6.553) -- (11.226,6.473) -- (11.285,6.415) -- (11.365,6.393) -- (11.445,6.415) -- (11.503,6.473) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.398,6.52) -- (3.376,6.6) -- (3.318,6.659) -- (3.238,6.68) -- (3.158,6.659) -- (3.099,6.6) -- (3.078,6.52) -- (3.099,6.44) -- (3.158,6.382) -- (3.238,6.36) -- (3.318,6.382) -- (3.376,6.44) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.541,6.506) -- (1.519,6.586) -- (1.461,6.645) -- (1.381,6.666) -- (1.301,6.645) -- (1.242,6.586) -- (1.221,6.506) -- (1.242,6.426) -- (1.301,6.368) -- (1.381,6.346) -- (1.461,6.368) -- (1.519,6.426) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.936,6.335) -- (5.914,6.415) -- (5.856,6.473) -- (5.776,6.495) -- (5.696,6.473) -- (5.637,6.415) -- (5.616,6.335) -- (5.637,6.255) -- (5.696,6.196) -- (5.776,6.175) -- (5.856,6.196) -- (5.914,6.255) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.476,4.075) -- (8.455,4.155) -- (8.396,4.214) -- (8.316,4.235) -- (8.236,4.214) -- (8.178,4.155) -- (8.156,4.075) -- (8.178,3.995) -- (8.236,3.936) -- (8.316,3.915) -- (8.396,3.936) -- (8.455,3.995) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.903,3.831) -- (10.881,3.911) -- (10.823,3.969) -- (10.743,3.991) -- (10.663,3.969) -- (10.604,3.911) -- (10.583,3.831) -- (10.604,3.751) -- (10.663,3.692) -- (10.743,3.671) -- (10.823,3.692) -- (10.881,3.751) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.354,3.747) -- (6.332,3.827) -- (6.274,3.886) -- (6.194,3.907) -- (6.114,3.886) -- (6.055,3.827) -- (6.034,3.747) -- (6.055,3.667) -- (6.114,3.609) -- (6.194,3.587) -- (6.274,3.609) -- (6.332,3.667) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.124,3.74) -- (4.102,3.82) -- (4.044,3.878) -- (3.964,3.9) -- (3.884,3.878) -- (3.825,3.82) -- (3.804,3.74) -- (3.825,3.66) -- (3.884,3.601) -- (3.964,3.58) -- (4.044,3.601) -- (4.102,3.66) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.525,3.595) -- (1.503,3.675) -- (1.445,3.733) -- (1.365,3.755) -- (1.285,3.733) -- (1.226,3.675) -- (1.205,3.595) -- (1.226,3.515) -- (1.285,3.456) -- (1.365,3.435) -- (1.445,3.456) -- (1.503,3.515) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.985,1.568) -- (5.964,1.648) -- (5.905,1.707) -- (5.825,1.728) -- (5.745,1.707) -- (5.687,1.648) -- (5.665,1.568) -- (5.687,1.488) -- (5.745,1.429) -- (5.825,1.408) -- (5.905,1.429) -- (5.964,1.488) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.465,6.528) -- (10.587,6.547) -- (10.696,6.603) -- (10.784,6.69) -- (10.839,6.8) -- (10.859,6.922) -- (10.839,7.044) -- (10.784,7.154) -- (10.696,7.241) -- (10.587,7.297) -- (10.465,7.316) -- (10.465,7.316) -- (10.343,7.297) -- (10.233,7.241) -- (10.146,7.154) -- (10.09,7.044) -- (10.071,6.922) -- (10.09,6.8) -- (10.146,6.69) -- (10.233,6.603) -- (10.343,6.547) -- (10.465,6.528) -- cycle;
\node at (10.465,6.922) {\scalebox{1.655}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.069,4.925) -- (9.191,4.944) -- (9.301,5.0) -- (9.388,5.088) -- (9.444,5.197) -- (9.463,5.319) -- (9.444,5.441) -- (9.388,5.551) -- (9.301,5.638) -- (9.191,5.694) -- (9.069,5.713) -- (9.069,5.713) -- (8.947,5.694) -- (8.838,5.638) -- (8.75,5.551) -- (8.694,5.441) -- (8.675,5.319) -- (8.694,5.197) -- (8.75,5.088) -- (8.838,5.0) -- (8.947,4.944) -- (9.069,4.925) -- cycle;
\node at (9.069,5.319) {\scalebox{1.655}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.325,4.736) -- (11.447,4.755) -- (11.557,4.811) -- (11.644,4.899) -- (11.7,5.008) -- (11.719,5.13) -- (11.7,5.252) -- (11.644,5.362) -- (11.557,5.449) -- (11.447,5.505) -- (11.325,5.524) -- (11.325,5.524) -- (11.204,5.505) -- (11.094,5.449) -- (11.007,5.362) -- (10.951,5.252) -- (10.931,5.13) -- (10.951,5.008) -- (11.007,4.899) -- (11.094,4.811) -- (11.204,4.755) -- (11.325,4.736) -- cycle;
\node at (11.325,5.13) {\scalebox{1.655}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.543,6.531) -- (4.665,6.55) -- (4.775,6.606) -- (4.862,6.694) -- (4.918,6.803) -- (4.937,6.925) -- (4.918,7.047) -- (4.862,7.157) -- (4.775,7.244) -- (4.665,7.3) -- (4.543,7.319) -- (4.543,7.319) -- (4.421,7.3) -- (4.312,7.244) -- (4.224,7.157) -- (4.169,7.047) -- (4.149,6.925) -- (4.169,6.803) -- (4.224,6.694) -- (4.312,6.606) -- (4.421,6.55) -- (4.543,6.531) -- cycle;
\node at (4.543,6.925) {\scalebox{1.655}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.867,4.806) -- (3.989,4.825) -- (4.099,4.881) -- (4.186,4.968) -- (4.242,5.078) -- (4.261,5.199) -- (4.242,5.321) -- (4.186,5.431) -- (4.099,5.518) -- (3.989,5.574) -- (3.867,5.593) -- (3.867,5.593) -- (3.746,5.574) -- (3.636,5.518) -- (3.549,5.431) -- (3.493,5.321) -- (3.473,5.199) -- (3.493,5.078) -- (3.549,4.968) -- (3.636,4.881) -- (3.746,4.825) -- (3.867,4.806) -- cycle;
\node at (3.867,5.199) {\scalebox{1.655}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.313,5.64) -- (2.435,5.659) -- (2.544,5.715) -- (2.632,5.802) -- (2.687,5.912) -- (2.707,6.034) -- (2.687,6.155) -- (2.632,6.265) -- (2.544,6.352) -- (2.435,6.408) -- (2.313,6.427) -- (2.313,6.427) -- (2.191,6.408) -- (2.081,6.352) -- (1.994,6.265) -- (1.938,6.155) -- (1.919,6.034) -- (1.938,5.912) -- (1.994,5.802) -- (2.081,5.715) -- (2.191,5.659) -- (2.313,5.64) -- cycle;
\node at (2.313,6.034) {\scalebox{1.655}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.646,4.655) -- (1.768,4.674) -- (1.878,4.73) -- (1.965,4.817) -- (2.021,4.927) -- (2.04,5.049) -- (2.021,5.17) -- (1.965,5.28) -- (1.878,5.367) -- (1.768,5.423) -- (1.646,5.443) -- (1.646,5.443) -- (1.525,5.423) -- (1.415,5.367) -- (1.328,5.28) -- (1.272,5.17) -- (1.253,5.049) -- (1.272,4.927) -- (1.328,4.817) -- (1.415,4.73) -- (1.525,4.674) -- (1.646,4.655) -- cycle;
\node at (1.646,5.049) {\scalebox{1.655}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.231,6.666) -- (7.353,6.685) -- (7.462,6.741) -- (7.549,6.828) -- (7.605,6.938) -- (7.625,7.06) -- (7.605,7.182) -- (7.549,7.292) -- (7.462,7.379) -- (7.353,7.435) -- (7.231,7.454) -- (7.231,7.454) -- (7.109,7.435) -- (6.999,7.379) -- (6.912,7.292) -- (6.856,7.182) -- (6.837,7.06) -- (6.856,6.938) -- (6.912,6.828) -- (6.999,6.741) -- (7.109,6.685) -- (7.231,6.666) -- cycle;
\node at (7.231,7.06) {\scalebox{1.655}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.267,4.693) -- (6.389,4.712) -- (6.498,4.768) -- (6.586,4.855) -- (6.642,4.965) -- (6.661,5.087) -- (6.642,5.208) -- (6.586,5.318) -- (6.498,5.405) -- (6.389,5.461) -- (6.267,5.48) -- (6.267,5.48) -- (6.145,5.461) -- (6.035,5.405) -- (5.948,5.318) -- (5.892,5.208) -- (5.873,5.087) -- (5.892,4.965) -- (5.948,4.855) -- (6.035,4.768) -- (6.145,4.712) -- (6.267,4.693) -- cycle;
\node at (6.267,5.087) {\scalebox{1.655}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.181,4.349) -- (10.303,4.369) -- (10.413,4.425) -- (10.5,4.512) -- (10.556,4.622) -- (10.575,4.743) -- (10.556,4.865) -- (10.5,4.975) -- (10.413,5.062) -- (10.303,5.118) -- (10.181,5.137) -- (9.81,5.137) -- (9.689,5.118) -- (9.579,5.062) -- (9.492,4.975) -- (9.436,4.865) -- (9.416,4.743) -- (9.436,4.622) -- (9.492,4.512) -- (9.579,4.425) -- (9.689,4.369) -- (9.81,4.349) -- cycle;
\node at (9.996,4.743) {\scalebox{1.655}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.394,3.819) -- (7.515,3.838) -- (7.625,3.894) -- (7.712,3.981) -- (7.768,4.091) -- (7.788,4.213) -- (7.768,4.335) -- (7.712,4.444) -- (7.625,4.532) -- (7.515,4.587) -- (7.394,4.607) -- (7.023,4.607) -- (6.901,4.587) -- (6.791,4.532) -- (6.704,4.444) -- (6.648,4.335) -- (6.629,4.213) -- (6.648,4.091) -- (6.704,3.981) -- (6.791,3.894) -- (6.901,3.838) -- (7.023,3.819) -- cycle;
\node at (7.208,4.213) {\scalebox{1.655}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.949,1.779) -- (7.071,1.798) -- (7.18,1.854) -- (7.268,1.942) -- (7.324,2.051) -- (7.343,2.173) -- (7.324,2.295) -- (7.268,2.405) -- (7.18,2.492) -- (7.071,2.548) -- (6.949,2.567) -- (6.578,2.567) -- (6.457,2.548) -- (6.347,2.492) -- (6.26,2.405) -- (6.204,2.295) -- (6.184,2.173) -- (6.204,2.051) -- (6.26,1.942) -- (6.347,1.854) -- (6.457,1.798) -- (6.578,1.779) -- cycle;
\node at (6.764,2.173) {\scalebox{1.655}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.263,3.651) -- (5.385,3.671) -- (5.495,3.727) -- (5.582,3.814) -- (5.638,3.924) -- (5.657,4.045) -- (5.638,4.167) -- (5.582,4.277) -- (5.495,4.364) -- (5.385,4.42) -- (5.263,4.439) -- (4.892,4.439) -- (4.771,4.42) -- (4.661,4.364) -- (4.574,4.277) -- (4.518,4.167) -- (4.499,4.045) -- (4.518,3.924) -- (4.574,3.814) -- (4.661,3.727) -- (4.771,3.671) -- (4.892,3.651) -- cycle;
\node at (5.078,4.045) {\scalebox{1.655}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.626,3.76) -- (2.748,3.78) -- (2.858,3.836) -- (2.945,3.923) -- (3.001,4.033) -- (3.02,4.154) -- (3.001,4.276) -- (2.945,4.386) -- (2.858,4.473) -- (2.748,4.529) -- (2.626,4.548) -- (2.255,4.548) -- (2.134,4.529) -- (2.024,4.473) -- (1.937,4.386) -- (1.881,4.276) -- (1.861,4.154) -- (1.881,4.033) -- (1.937,3.923) -- (2.024,3.836) -- (2.134,3.78) -- (2.255,3.76) -- cycle;
\node at (2.441,4.154) {\scalebox{1.655}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (11.065,6.253) -- (11.665,6.253) -- (11.665,6.853) -- (11.065,6.853) -- cycle;
\node at (11.365,6.553) {\scalebox{1.655}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.522,5.544) -- (10.122,5.544) -- (10.122,6.144) -- (9.522,6.144) -- cycle;
\node at (9.822,5.844) {\scalebox{1.655}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.047,3.131) -- (8.647,3.131) -- (8.647,3.731) -- (8.047,3.731) -- cycle;
\node at (8.347,3.431) {\scalebox{1.655}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (0.84,3.838) -- (1.44,3.838) -- (1.44,4.438) -- (0.84,4.438) -- cycle;
\node at (1.14,4.138) {\scalebox{1.655}{\small d}};
\node at (9.335,7.108) {\scalebox{1.655}{\small A}};
\node at (11.954,6.745) {\scalebox{1.655}{\small B}};
\node at (2.704,6.835) {\scalebox{1.655}{\small C}};
\node at (0.796,6.713) {\scalebox{1.655}{\small D}};
\node at (5.648,6.942) {\scalebox{1.655}{\small E}};
\node at (8.903,3.874) {\scalebox{1.655}{\small F}};
\node at (11.349,3.7) {\scalebox{1.655}{\small G}};
\node at (6.252,3.13) {\scalebox{1.655}{\small H}};
\node at (3.413,3.455) {\scalebox{1.655}{\small I}};
\node at (0.765,3.437) {\scalebox{1.655}{\small J}};
\node at (5.775,0.95) {\scalebox{1.655}{\small K}};
\draw[line width=0.5pt] (12.665,5.803) -- (12.665,6.903);
\draw[line width=0.5pt] (12.434,6.518) -- (12.665,6.903) -- (12.896,6.518);
\draw[line width=0.5pt] (12.362,6.353) -- (12.967,6.353);
\node at (12.665,7.288) {\scalebox{1.655}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour B, 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 avenue de la Gare & Ⓐ Carrefour du Lavoir & Ⓘ Rond-point des Vaches \\ \hline
① chemin de Ronde & ⑨ rue du Stade & Ⓑ Place du Marché & Ⓙ Place aux Herbes \\ \hline
② rue du Puits & ⑩ rue des Écoles & Ⓒ Rond-point de la Poste & Ⓚ Place du Vieux-Puits \\ \hline
③ rue du Pont & ⑪ allée des Peupliers & Ⓓ Place de la Fontaine & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ venelle du Chat & ⑫ impasse des Roses & Ⓔ Rond-point du Stade & \textbf{a} l'école \\ \hline
⑤ rue Haute & ⑬ rue du Moulin & Ⓕ Rond-point du Pont & \textbf{b} la gendarmerie \\ \hline
⑥ passage du Marché & ⑭ rue du Four & Ⓖ Fontaine Notre-Dame & \textbf{c} le marché couvert \\ \hline
⑦ allée des Tilleuls & \cellcolor{black!32}\bfseries Les carrefours & Ⓗ Carrefour Saint-Roch & \textbf{d} la salle des fêtes \\ \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=0.6191, line join=round, line cap=round]
\draw[line width=0.5pt] (8.827,6.753) -- (9.005,6.711) -- (9.183,6.673) -- (9.363,6.64) -- (9.543,6.611) -- (9.724,6.586) -- (9.906,6.565) -- (10.088,6.548) -- (10.27,6.536) -- (10.452,6.528) -- (10.635,6.525) -- (10.818,6.525) -- (11.0,6.53) -- (11.183,6.54) -- (11.365,6.553);
\draw[line width=0.5pt] (8.827,6.753) -- (8.82,6.557) -- (8.809,6.362) -- (8.792,6.167) -- (8.772,5.973) -- (8.746,5.779) -- (8.717,5.586) -- (8.682,5.393) -- (8.643,5.201) -- (8.6,5.01) -- (8.552,4.821) -- (8.5,4.632) -- (8.443,4.445) -- (8.382,4.259) -- (8.316,4.075);
\draw[line width=0.5pt] (11.365,6.553) -- (11.291,6.367) -- (11.221,6.179) -- (11.156,5.99) -- (11.096,5.799) -- (11.04,5.606) -- (10.988,5.413) -- (10.941,5.218) -- (10.899,5.022) -- (10.861,4.825) -- (10.828,4.627) -- (10.8,4.429) -- (10.776,4.23) -- (10.757,4.031) -- (10.743,3.831);
\draw[line width=0.5pt] (3.238,6.52) -- (3.42,6.535) -- (3.602,6.545) -- (3.784,6.551) -- (3.967,6.553) -- (4.15,6.55) -- (4.332,6.543) -- (4.514,6.532) -- (4.697,6.517) -- (4.878,6.497) -- (5.059,6.473) -- (5.24,6.445) -- (5.419,6.412) -- (5.598,6.376) -- (5.776,6.335);
\draw[line width=0.5pt] (3.238,6.52) -- (3.259,6.315) -- (3.285,6.111) -- (3.315,5.907) -- (3.351,5.704) -- (3.391,5.501) -- (3.436,5.3) -- (3.486,5.1) -- (3.541,4.901) -- (3.6,4.703) -- (3.663,4.507) -- (3.732,4.313) -- (3.805,4.12) -- (3.882,3.929) -- (3.964,3.74);
\draw[line width=0.5pt] (1.381,6.506) -- (1.512,6.527) -- (1.644,6.546) -- (1.776,6.561) -- (1.909,6.573) -- (2.042,6.581) -- (2.175,6.587) -- (2.309,6.59) -- (2.442,6.589) -- (2.575,6.585) -- (2.708,6.579) -- (2.841,6.569) -- (2.974,6.556) -- (3.106,6.539) -- (3.238,6.52);
\draw[line width=0.5pt] (1.381,6.506) -- (1.348,6.3) -- (1.32,6.093) -- (1.296,5.885) -- (1.278,5.677) -- (1.265,5.469) -- (1.256,5.26) -- (1.253,5.051) -- (1.254,4.842) -- (1.26,4.633) -- (1.271,4.424) -- (1.287,4.216) -- (1.308,4.008) -- (1.334,3.801) -- (1.365,3.595);
\draw[line width=0.5pt] (5.776,6.335) -- (5.988,6.398) -- (6.201,6.456) -- (6.415,6.509) -- (6.631,6.557) -- (6.848,6.599) -- (7.066,6.637) -- (7.284,6.67) -- (7.504,6.697) -- (7.724,6.719) -- (7.944,6.737) -- (8.165,6.748) -- (8.385,6.755) -- (8.606,6.757) -- (8.827,6.753);
\draw[line width=0.5pt] (5.776,6.335) -- (5.777,6.147) -- (5.783,5.959) -- (5.793,5.771) -- (5.808,5.584) -- (5.827,5.397) -- (5.85,5.21) -- (5.878,5.024) -- (5.91,4.838) -- (5.947,4.654) -- (5.988,4.47) -- (6.033,4.288) -- (6.082,4.106) -- (6.136,3.926) -- (6.194,3.747);
\draw[line width=0.5pt] (8.316,4.075) -- (8.491,4.084) -- (8.666,4.089) -- (8.841,4.09) -- (9.016,4.087) -- (9.19,4.08) -- (9.365,4.069) -- (9.54,4.053) -- (9.713,4.033) -- (9.887,4.01) -- (10.06,3.982) -- (10.232,3.95) -- (10.403,3.915) -- (10.573,3.875) -- (10.743,3.831);
\draw[line width=0.5pt] (6.194,3.747) -- (6.348,3.747) -- (6.502,3.751) -- (6.656,3.758) -- (6.809,3.769) -- (6.963,3.784) -- (7.116,3.802) -- (7.269,3.824) -- (7.421,3.849) -- (7.572,3.878) -- (7.723,3.91) -- (7.873,3.946) -- (8.022,3.986) -- (8.169,4.029) -- (8.316,4.075);
\draw[line width=0.5pt] (6.194,3.747) -- (6.191,3.589) -- (6.185,3.43) -- (6.176,3.272) -- (6.162,3.114) -- (6.145,2.956) -- (6.124,2.799) -- (6.099,2.642) -- (6.071,2.486) -- (6.039,2.331) -- (6.004,2.176) -- (5.964,2.023) -- (5.922,1.87) -- (5.875,1.718) -- (5.825,1.568);
\draw[line width=0.5pt] (3.964,3.74) -- (4.122,3.716) -- (4.281,3.696) -- (4.44,3.679) -- (4.599,3.667) -- (4.759,3.658) -- (4.919,3.653) -- (5.079,3.651) -- (5.239,3.654) -- (5.399,3.66) -- (5.559,3.67) -- (5.718,3.684) -- (5.877,3.701) -- (6.036,3.722) -- (6.194,3.747);
\draw[line width=0.5pt] (1.365,3.595) -- (1.547,3.633) -- (1.731,3.668) -- (1.915,3.698) -- (2.1,3.723) -- (2.286,3.745) -- (2.472,3.762) -- (2.658,3.774) -- (2.845,3.783) -- (3.032,3.786) -- (3.219,3.786) -- (3.405,3.781) -- (3.592,3.772) -- (3.778,3.758) -- (3.964,3.74);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (9.2,6.753) -- (9.172,6.896) -- (9.091,7.017) -- (8.97,7.098) -- (8.827,7.126) -- (8.684,7.098) -- (8.563,7.017) -- (8.482,6.896) -- (8.454,6.753) -- (8.482,6.61) -- (8.563,6.489) -- (8.684,6.408) -- (8.827,6.38) -- (8.97,6.408) -- (9.091,6.489) -- (9.172,6.61) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.738,6.553) -- (11.709,6.696) -- (11.628,6.817) -- (11.507,6.898) -- (11.365,6.927) -- (11.222,6.898) -- (11.101,6.817) -- (11.02,6.696) -- (10.991,6.553) -- (11.02,6.411) -- (11.101,6.289) -- (11.222,6.209) -- (11.365,6.18) -- (11.507,6.209) -- (11.628,6.289) -- (11.709,6.411) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (3.611,6.52) -- (3.583,6.663) -- (3.502,6.784) -- (3.381,6.865) -- (3.238,6.893) -- (3.095,6.865) -- (2.974,6.784) -- (2.893,6.663) -- (2.865,6.52) -- (2.893,6.377) -- (2.974,6.256) -- (3.095,6.175) -- (3.238,6.147) -- (3.381,6.175) -- (3.502,6.256) -- (3.583,6.377) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.754,6.506) -- (1.726,6.649) -- (1.645,6.77) -- (1.524,6.851) -- (1.381,6.879) -- (1.238,6.851) -- (1.117,6.77) -- (1.036,6.649) -- (1.007,6.506) -- (1.036,6.363) -- (1.117,6.242) -- (1.238,6.161) -- (1.381,6.133) -- (1.524,6.161) -- (1.645,6.242) -- (1.726,6.363) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.149,6.335) -- (6.121,6.478) -- (6.04,6.599) -- (5.919,6.68) -- (5.776,6.708) -- (5.633,6.68) -- (5.512,6.599) -- (5.431,6.478) -- (5.403,6.335) -- (5.431,6.192) -- (5.512,6.071) -- (5.633,5.99) -- (5.776,5.962) -- (5.919,5.99) -- (6.04,6.071) -- (6.121,6.192) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.689,4.075) -- (8.661,4.218) -- (8.58,4.339) -- (8.459,4.42) -- (8.316,4.448) -- (8.173,4.42) -- (8.052,4.339) -- (7.971,4.218) -- (7.943,4.075) -- (7.971,3.932) -- (8.052,3.811) -- (8.173,3.73) -- (8.316,3.702) -- (8.459,3.73) -- (8.58,3.811) -- (8.661,3.932) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.116,3.831) -- (11.087,3.974) -- (11.007,4.095) -- (10.885,4.176) -- (10.743,4.204) -- (10.6,4.176) -- (10.479,4.095) -- (10.398,3.974) -- (10.369,3.831) -- (10.398,3.688) -- (10.479,3.567) -- (10.6,3.486) -- (10.743,3.458) -- (10.885,3.486) -- (11.007,3.567) -- (11.087,3.688) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.567,3.747) -- (6.539,3.89) -- (6.458,4.011) -- (6.337,4.092) -- (6.194,4.12) -- (6.051,4.092) -- (5.93,4.011) -- (5.849,3.89) -- (5.82,3.747) -- (5.849,3.604) -- (5.93,3.483) -- (6.051,3.402) -- (6.194,3.374) -- (6.337,3.402) -- (6.458,3.483) -- (6.539,3.604) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.337,3.74) -- (4.309,3.882) -- (4.228,4.004) -- (4.107,4.084) -- (3.964,4.113) -- (3.821,4.084) -- (3.7,4.004) -- (3.619,3.882) -- (3.591,3.74) -- (3.619,3.597) -- (3.7,3.476) -- (3.821,3.395) -- (3.964,3.366) -- (4.107,3.395) -- (4.228,3.476) -- (4.309,3.597) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.738,3.595) -- (1.709,3.737) -- (1.628,3.858) -- (1.507,3.939) -- (1.365,3.968) -- (1.222,3.939) -- (1.101,3.858) -- (1.02,3.737) -- (0.991,3.595) -- (1.02,3.452) -- (1.101,3.331) -- (1.222,3.25) -- (1.365,3.221) -- (1.507,3.25) -- (1.628,3.331) -- (1.709,3.452) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.198,1.568) -- (6.17,1.711) -- (6.089,1.832) -- (5.968,1.913) -- (5.825,1.941) -- (5.682,1.913) -- (5.561,1.832) -- (5.48,1.711) -- (5.452,1.568) -- (5.48,1.425) -- (5.561,1.304) -- (5.682,1.223) -- (5.825,1.195) -- (5.968,1.223) -- (6.089,1.304) -- (6.17,1.425) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.465,6.528) -- (10.583,6.547) -- (10.691,6.602) -- (10.776,6.687) -- (10.83,6.794) -- (10.849,6.913) -- (10.83,7.031) -- (10.776,7.139) -- (10.691,7.224) -- (10.583,7.278) -- (10.465,7.297) -- (10.465,7.297) -- (10.346,7.278) -- (10.239,7.224) -- (10.153,7.139) -- (10.099,7.031) -- (10.08,6.913) -- (10.099,6.794) -- (10.153,6.687) -- (10.239,6.602) -- (10.346,6.547) -- (10.465,6.528) -- cycle;
\node at (10.465,6.913) {\scalebox{1.615}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.06,4.936) -- (9.179,4.955) -- (9.286,5.01) -- (9.371,5.095) -- (9.426,5.202) -- (9.444,5.321) -- (9.426,5.44) -- (9.371,5.547) -- (9.286,5.632) -- (9.179,5.687) -- (9.06,5.705) -- (9.06,5.705) -- (8.941,5.687) -- (8.834,5.632) -- (8.749,5.547) -- (8.694,5.44) -- (8.675,5.321) -- (8.694,5.202) -- (8.749,5.095) -- (8.834,5.01) -- (8.941,4.955) -- (9.06,4.936) -- cycle;
\node at (9.06,5.321) {\scalebox{1.615}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.316,4.748) -- (11.435,4.766) -- (11.542,4.821) -- (11.627,4.906) -- (11.682,5.013) -- (11.701,5.132) -- (11.682,5.251) -- (11.627,5.358) -- (11.542,5.443) -- (11.435,5.498) -- (11.316,5.517) -- (11.316,5.517) -- (11.197,5.498) -- (11.09,5.443) -- (11.005,5.358) -- (10.95,5.251) -- (10.932,5.132) -- (10.95,5.013) -- (11.005,4.906) -- (11.09,4.821) -- (11.197,4.766) -- (11.316,4.748) -- cycle;
\node at (11.316,5.132) {\scalebox{1.615}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.542,6.531) -- (4.661,6.55) -- (4.768,6.605) -- (4.854,6.69) -- (4.908,6.797) -- (4.927,6.916) -- (4.908,7.035) -- (4.854,7.142) -- (4.768,7.227) -- (4.661,7.281) -- (4.542,7.3) -- (4.542,7.3) -- (4.424,7.281) -- (4.316,7.227) -- (4.231,7.142) -- (4.177,7.035) -- (4.158,6.916) -- (4.177,6.797) -- (4.231,6.69) -- (4.316,6.605) -- (4.424,6.55) -- (4.542,6.531) -- cycle;
\node at (4.542,6.916) {\scalebox{1.615}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.858,4.813) -- (3.977,4.831) -- (4.084,4.886) -- (4.169,4.971) -- (4.224,5.078) -- (4.243,5.197) -- (4.224,5.316) -- (4.169,5.423) -- (4.084,5.508) -- (3.977,5.563) -- (3.858,5.582) -- (3.858,5.582) -- (3.739,5.563) -- (3.632,5.508) -- (3.547,5.423) -- (3.492,5.316) -- (3.474,5.197) -- (3.492,5.078) -- (3.547,4.971) -- (3.632,4.886) -- (3.739,4.831) -- (3.858,4.813) -- cycle;
\node at (3.858,5.197) {\scalebox{1.615}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.313,5.662) -- (2.432,5.681) -- (2.539,5.736) -- (2.624,5.821) -- (2.678,5.928) -- (2.697,6.047) -- (2.678,6.166) -- (2.624,6.273) -- (2.539,6.358) -- (2.432,6.413) -- (2.313,6.431) -- (2.313,6.431) -- (2.194,6.413) -- (2.087,6.358) -- (2.002,6.273) -- (1.947,6.166) -- (1.928,6.047) -- (1.947,5.928) -- (2.002,5.821) -- (2.087,5.736) -- (2.194,5.681) -- (2.313,5.662) -- cycle;
\node at (2.313,6.047) {\scalebox{1.615}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.637,4.664) -- (1.756,4.683) -- (1.863,4.738) -- (1.948,4.823) -- (2.003,4.93) -- (2.022,5.049) -- (2.003,5.168) -- (1.948,5.275) -- (1.863,5.36) -- (1.756,5.415) -- (1.637,5.433) -- (1.637,5.433) -- (1.518,5.415) -- (1.411,5.36) -- (1.326,5.275) -- (1.271,5.168) -- (1.253,5.049) -- (1.271,4.93) -- (1.326,4.823) -- (1.411,4.738) -- (1.518,4.683) -- (1.637,4.664) -- cycle;
\node at (1.637,5.049) {\scalebox{1.615}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.232,6.666) -- (7.351,6.685) -- (7.458,6.74) -- (7.543,6.825) -- (7.598,6.932) -- (7.617,7.051) -- (7.598,7.17) -- (7.543,7.277) -- (7.458,7.362) -- (7.351,7.416) -- (7.232,7.435) -- (7.232,7.435) -- (7.113,7.416) -- (7.006,7.362) -- (6.921,7.277) -- (6.866,7.17) -- (6.848,7.051) -- (6.866,6.932) -- (6.921,6.825) -- (7.006,6.74) -- (7.113,6.685) -- (7.232,6.666) -- cycle;
\node at (7.232,7.051) {\scalebox{1.615}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.258,4.701) -- (6.377,4.719) -- (6.484,4.774) -- (6.569,4.859) -- (6.623,4.966) -- (6.642,5.085) -- (6.623,5.204) -- (6.569,5.311) -- (6.484,5.396) -- (6.377,5.451) -- (6.258,5.47) -- (6.258,5.47) -- (6.139,5.451) -- (6.032,5.396) -- (5.947,5.311) -- (5.892,5.204) -- (5.873,5.085) -- (5.892,4.966) -- (5.947,4.859) -- (6.032,4.774) -- (6.139,4.719) -- (6.258,4.701) -- cycle;
\node at (6.258,5.085) {\scalebox{1.615}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.148,4.162) -- (10.266,4.181) -- (10.374,4.236) -- (10.459,4.321) -- (10.513,4.428) -- (10.532,4.547) -- (10.513,4.666) -- (10.459,4.773) -- (10.374,4.858) -- (10.266,4.913) -- (10.148,4.931) -- (9.786,4.931) -- (9.667,4.913) -- (9.56,4.858) -- (9.475,4.773) -- (9.42,4.666) -- (9.401,4.547) -- (9.42,4.428) -- (9.475,4.321) -- (9.56,4.236) -- (9.667,4.181) -- (9.786,4.162) -- cycle;
\node at (9.967,4.547) {\scalebox{1.615}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.391,3.819) -- (7.51,3.838) -- (7.617,3.892) -- (7.702,3.978) -- (7.756,4.085) -- (7.775,4.204) -- (7.756,4.322) -- (7.702,4.43) -- (7.617,4.515) -- (7.51,4.569) -- (7.391,4.588) -- (7.029,4.588) -- (6.91,4.569) -- (6.803,4.515) -- (6.718,4.43) -- (6.663,4.322) -- (6.644,4.204) -- (6.663,4.085) -- (6.718,3.978) -- (6.803,3.892) -- (6.91,3.838) -- (7.029,3.819) -- cycle;
\node at (7.21,4.204) {\scalebox{1.615}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.927,1.792) -- (7.046,1.811) -- (7.153,1.866) -- (7.238,1.951) -- (7.293,2.058) -- (7.312,2.177) -- (7.293,2.296) -- (7.238,2.403) -- (7.153,2.488) -- (7.046,2.543) -- (6.927,2.561) -- (6.565,2.561) -- (6.447,2.543) -- (6.339,2.488) -- (6.254,2.403) -- (6.2,2.296) -- (6.181,2.177) -- (6.2,2.058) -- (6.254,1.951) -- (6.339,1.866) -- (6.447,1.811) -- (6.565,1.792) -- cycle;
\node at (6.746,2.177) {\scalebox{1.615}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.259,3.651) -- (5.378,3.67) -- (5.485,3.725) -- (5.57,3.81) -- (5.624,3.917) -- (5.643,4.036) -- (5.624,4.155) -- (5.57,4.262) -- (5.485,4.347) -- (5.378,4.402) -- (5.259,4.42) -- (4.897,4.42) -- (4.778,4.402) -- (4.671,4.347) -- (4.586,4.262) -- (4.531,4.155) -- (4.512,4.036) -- (4.531,3.917) -- (4.586,3.81) -- (4.671,3.725) -- (4.778,3.67) -- (4.897,3.651) -- cycle;
\node at (5.078,4.036) {\scalebox{1.615}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.818,3.774) -- (2.937,3.793) -- (3.044,3.847) -- (3.129,3.932) -- (3.183,4.039) -- (3.202,4.158) -- (3.183,4.277) -- (3.129,4.384) -- (3.044,4.469) -- (2.937,4.524) -- (2.818,4.543) -- (2.456,4.543) -- (2.337,4.524) -- (2.23,4.469) -- (2.145,4.384) -- (2.09,4.277) -- (2.071,4.158) -- (2.09,4.039) -- (2.145,3.932) -- (2.23,3.847) -- (2.337,3.793) -- (2.456,3.774) -- cycle;
\node at (2.637,4.158) {\scalebox{1.615}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (11.065,6.253) -- (11.665,6.253) -- (11.665,6.853) -- (11.065,6.853) -- cycle;
\node at (11.365,6.553) {\scalebox{1.615}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.522,5.544) -- (10.122,5.544) -- (10.122,6.144) -- (9.522,6.144) -- cycle;
\node at (9.822,5.844) {\scalebox{1.615}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.047,3.131) -- (8.647,3.131) -- (8.647,3.731) -- (8.047,3.731) -- cycle;
\node at (8.347,3.431) {\scalebox{1.615}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (0.84,3.838) -- (1.44,3.838) -- (1.44,4.438) -- (0.84,4.438) -- cycle;
\node at (1.14,4.138) {\scalebox{1.615}{\small d}};
\node at (9.335,7.108) {\scalebox{1.615}{\small A}};
\node at (11.954,6.745) {\scalebox{1.615}{\small B}};
\node at (2.704,6.835) {\scalebox{1.615}{\small C}};
\node at (0.796,6.713) {\scalebox{1.615}{\small D}};
\node at (5.648,6.942) {\scalebox{1.615}{\small E}};
\node at (8.903,3.874) {\scalebox{1.615}{\small F}};
\node at (11.349,3.7) {\scalebox{1.615}{\small G}};
\node at (6.252,3.13) {\scalebox{1.615}{\small H}};
\node at (3.413,3.455) {\scalebox{1.615}{\small I}};
\node at (0.765,3.437) {\scalebox{1.615}{\small J}};
\node at (5.775,0.95) {\scalebox{1.615}{\small K}};
\draw[line width=0.5pt] (12.665,5.803) -- (12.665,6.903);
\draw[line width=0.5pt] (12.434,6.518) -- (12.665,6.903) -- (12.896,6.518);
\draw[line width=0.5pt] (12.362,6.353) -- (12.967,6.353);
\node at (12.665,7.288) {\scalebox{1.615}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & A, C, G, H, J & 5 \\ \hline
clairs & B, D, E, F, I, K & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{6} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\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 11 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 5 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{B} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{rue du Pont ③} vers le sud jusqu'à Fontaine Notre-Dame \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Écoles ⑩}, vers l'ouest, jusqu'à Rond-point du Pont \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Puits ②}, vers le nord, jusqu'à Carrefour du Lavoir \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{avenue de la Gare ⑧}, vers l'ouest, jusqu'à Rond-point du Stade \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{venelle du Chat ④}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{passage du Marché ⑥}, vers l'ouest, jusqu'à Place de la Fontaine \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{allée des Tilleuls ⑦}, vers le sud, jusqu'à Place aux Herbes \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Four ⑭}, vers l'est, jusqu'à Rond-point des Vaches \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Moulin ⑬}, vers l'est, jusqu'à Carrefour Saint-Roch \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{impasse des Roses ⑫}, vers le sud, jusqu'à Place du Vieux-Puits \textbf{(K)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 11 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}
