\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 203}

\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=203}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0203}
\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=0.6203, line join=round, line cap=round]
\draw[line width=0.5pt] (5.233,11.382) -- (5.396,11.33) -- (5.562,11.281) -- (5.728,11.237) -- (5.895,11.197) -- (6.063,11.16) -- (6.232,11.127) -- (6.402,11.099) -- (6.573,11.074) -- (6.744,11.053) -- (6.915,11.037) -- (7.087,11.024) -- (7.259,11.016) -- (7.431,11.011) -- (7.603,11.011);
\draw[line width=0.5pt] (5.233,11.382) -- (5.25,11.129) -- (5.262,10.875) -- (5.268,10.621) -- (5.268,10.367) -- (5.261,10.113) -- (5.249,9.859) -- (5.231,9.606) -- (5.207,9.353) -- (5.177,9.1) -- (5.141,8.849) -- (5.099,8.598) -- (5.052,8.349) -- (4.998,8.101) -- (4.939,7.854);
\draw[line width=0.5pt] (7.603,11.011) -- (7.571,10.834) -- (7.544,10.657) -- (7.521,10.479) -- (7.502,10.301) -- (7.488,10.122) -- (7.478,9.943) -- (7.471,9.764) -- (7.469,9.585) -- (7.472,9.405) -- (7.478,9.226) -- (7.489,9.047) -- (7.504,8.868) -- (7.523,8.69) -- (7.546,8.513);
\draw[line width=0.5pt] (1.203,10.755) -- (1.482,10.843) -- (1.762,10.925) -- (2.045,11.0) -- (2.33,11.069) -- (2.616,11.131) -- (2.903,11.186) -- (3.192,11.235) -- (3.482,11.276) -- (3.772,11.311) -- (4.064,11.339) -- (4.356,11.36) -- (4.648,11.374) -- (4.94,11.382) -- (5.233,11.382);
\draw[line width=0.5pt] (1.203,10.755) -- (1.201,10.556) -- (1.204,10.358) -- (1.211,10.159) -- (1.223,9.961) -- (1.24,9.763) -- (1.261,9.566) -- (1.287,9.369) -- (1.318,9.173) -- (1.353,8.978) -- (1.393,8.783) -- (1.437,8.589) -- (1.486,8.397) -- (1.54,8.206) -- (1.598,8.016);
\draw[line width=0.5pt] (7.546,8.513) -- (7.618,8.236) -- (7.683,7.958) -- (7.741,7.679) -- (7.793,7.398) -- (7.838,7.116) -- (7.877,6.833) -- (7.909,6.549) -- (7.934,6.264) -- (7.952,5.979) -- (7.964,5.694) -- (7.969,5.409) -- (7.968,5.123) -- (7.959,4.838) -- (7.944,4.553);
\draw[line width=0.5pt] (1.598,8.016) -- (1.833,7.968) -- (2.069,7.926) -- (2.306,7.889) -- (2.544,7.858) -- (2.783,7.832) -- (3.022,7.812) -- (3.261,7.797) -- (3.501,7.788) -- (3.741,7.785) -- (3.981,7.788) -- (4.221,7.796) -- (4.461,7.809) -- (4.7,7.829) -- (4.939,7.854);
\draw[line width=0.5pt] (1.598,8.016) -- (1.62,7.792) -- (1.636,7.567) -- (1.648,7.341) -- (1.654,7.115) -- (1.654,6.889) -- (1.65,6.664) -- (1.64,6.438) -- (1.624,6.212) -- (1.604,5.987) -- (1.578,5.763) -- (1.547,5.539) -- (1.511,5.316) -- (1.469,5.095) -- (1.422,4.874);
\draw[line width=0.5pt] (4.939,7.854) -- (5.131,7.872) -- (5.322,7.895) -- (5.513,7.922) -- (5.704,7.954) -- (5.893,7.99) -- (6.082,8.03) -- (6.27,8.076) -- (6.456,8.125) -- (6.642,8.179) -- (6.826,8.237) -- (7.008,8.3) -- (7.189,8.366) -- (7.369,8.437) -- (7.546,8.513);
\draw[line width=0.5pt] (4.939,7.854) -- (4.99,7.617) -- (5.035,7.38) -- (5.075,7.141) -- (5.109,6.902) -- (5.137,6.661) -- (5.16,6.42) -- (5.177,6.179) -- (5.188,5.937) -- (5.194,5.695) -- (5.194,5.453) -- (5.189,5.211) -- (5.177,4.969) -- (5.16,4.728) -- (5.137,4.487);
\draw[line width=0.5pt] (11.016,5.427) -- (10.95,5.144) -- (10.891,4.859) -- (10.838,4.573) -- (10.793,4.285) -- (10.754,3.997) -- (10.722,3.707) -- (10.696,3.417) -- (10.678,3.127) -- (10.666,2.836) -- (10.661,2.545) -- (10.663,2.253) -- (10.672,1.963) -- (10.688,1.672) -- (10.71,1.382);
\draw[line width=0.5pt] (1.422,4.874) -- (1.69,4.887) -- (1.958,4.894) -- (2.226,4.895) -- (2.494,4.889) -- (2.761,4.877) -- (3.029,4.858) -- (3.296,4.834) -- (3.562,4.803) -- (3.828,4.766) -- (4.092,4.722) -- (4.356,4.673) -- (4.618,4.617) -- (4.878,4.555) -- (5.137,4.487);
\draw[line width=0.5pt] (1.422,4.874) -- (1.385,4.679) -- (1.352,4.482) -- (1.323,4.286) -- (1.299,4.088) -- (1.28,3.89) -- (1.266,3.691) -- (1.256,3.493) -- (1.251,3.294) -- (1.25,3.095) -- (1.254,2.896) -- (1.263,2.697) -- (1.277,2.498) -- (1.295,2.3) -- (1.318,2.103);
\draw[line width=0.5pt] (7.944,4.553) -- (8.152,4.648) -- (8.363,4.739) -- (8.575,4.825) -- (8.79,4.905) -- (9.006,4.981) -- (9.224,5.051) -- (9.444,5.117) -- (9.665,5.177) -- (9.888,5.232) -- (10.111,5.281) -- (10.336,5.326) -- (10.562,5.365) -- (10.789,5.399) -- (11.016,5.427);
\draw[line width=0.5pt] (7.944,4.553) -- (7.879,4.369) -- (7.818,4.184) -- (7.761,3.997) -- (7.709,3.809) -- (7.662,3.62) -- (7.618,3.43) -- (7.58,3.239) -- (7.545,3.047) -- (7.515,2.854) -- (7.49,2.661) -- (7.47,2.467) -- (7.453,2.272) -- (7.442,2.078) -- (7.435,1.883);
\draw[line width=0.5pt] (5.137,4.487) -- (5.336,4.522) -- (5.535,4.553) -- (5.734,4.579) -- (5.935,4.6) -- (6.136,4.617) -- (6.337,4.629) -- (6.538,4.636) -- (6.74,4.638) -- (6.941,4.636) -- (7.142,4.629) -- (7.344,4.617) -- (7.544,4.6) -- (7.745,4.579) -- (7.944,4.553);
\draw[line width=0.5pt] (7.435,1.883) -- (7.662,1.812) -- (7.89,1.746) -- (8.12,1.685) -- (8.351,1.63) -- (8.584,1.58) -- (8.817,1.536) -- (9.052,1.497) -- (9.288,1.464) -- (9.524,1.437) -- (9.76,1.415) -- (9.998,1.398) -- (10.235,1.387) -- (10.473,1.382) -- (10.71,1.382);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.393,11.382) -- (5.371,11.462) -- (5.313,11.521) -- (5.233,11.542) -- (5.153,11.521) -- (5.094,11.462) -- (5.073,11.382) -- (5.094,11.302) -- (5.153,11.244) -- (5.233,11.222) -- (5.313,11.244) -- (5.371,11.302) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.763,11.011) -- (7.741,11.091) -- (7.683,11.149) -- (7.603,11.171) -- (7.523,11.149) -- (7.464,11.091) -- (7.443,11.011) -- (7.464,10.931) -- (7.523,10.872) -- (7.603,10.851) -- (7.683,10.872) -- (7.741,10.931) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.363,10.755) -- (1.342,10.835) -- (1.283,10.893) -- (1.203,10.915) -- (1.123,10.893) -- (1.064,10.835) -- (1.043,10.755) -- (1.064,10.675) -- (1.123,10.616) -- (1.203,10.595) -- (1.283,10.616) -- (1.342,10.675) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.706,8.513) -- (7.685,8.593) -- (7.626,8.651) -- (7.546,8.673) -- (7.466,8.651) -- (7.408,8.593) -- (7.386,8.513) -- (7.408,8.433) -- (7.466,8.374) -- (7.546,8.353) -- (7.626,8.374) -- (7.685,8.433) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.758,8.016) -- (1.736,8.096) -- (1.678,8.155) -- (1.598,8.176) -- (1.518,8.155) -- (1.459,8.096) -- (1.438,8.016) -- (1.459,7.936) -- (1.518,7.878) -- (1.598,7.856) -- (1.678,7.878) -- (1.736,7.936) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.099,7.854) -- (5.077,7.934) -- (5.019,7.992) -- (4.939,8.014) -- (4.859,7.992) -- (4.8,7.934) -- (4.779,7.854) -- (4.8,7.774) -- (4.859,7.715) -- (4.939,7.694) -- (5.019,7.715) -- (5.077,7.774) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.176,5.427) -- (11.154,5.507) -- (11.096,5.566) -- (11.016,5.587) -- (10.936,5.566) -- (10.877,5.507) -- (10.856,5.427) -- (10.877,5.347) -- (10.936,5.288) -- (11.016,5.267) -- (11.096,5.288) -- (11.154,5.347) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.582,4.874) -- (1.561,4.954) -- (1.502,5.012) -- (1.422,5.034) -- (1.342,5.012) -- (1.284,4.954) -- (1.262,4.874) -- (1.284,4.794) -- (1.342,4.735) -- (1.422,4.714) -- (1.502,4.735) -- (1.561,4.794) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.104,4.553) -- (8.083,4.633) -- (8.024,4.691) -- (7.944,4.713) -- (7.864,4.691) -- (7.806,4.633) -- (7.784,4.553) -- (7.806,4.473) -- (7.864,4.414) -- (7.944,4.393) -- (8.024,4.414) -- (8.083,4.473) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.297,4.487) -- (5.276,4.567) -- (5.217,4.626) -- (5.137,4.647) -- (5.057,4.626) -- (4.999,4.567) -- (4.977,4.487) -- (4.999,4.407) -- (5.057,4.349) -- (5.137,4.327) -- (5.217,4.349) -- (5.276,4.407) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.478,2.103) -- (1.456,2.183) -- (1.398,2.241) -- (1.318,2.263) -- (1.238,2.241) -- (1.179,2.183) -- (1.158,2.103) -- (1.179,2.023) -- (1.238,1.964) -- (1.318,1.943) -- (1.398,1.964) -- (1.456,2.023) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.595,1.883) -- (7.574,1.963) -- (7.515,2.022) -- (7.435,2.043) -- (7.355,2.022) -- (7.297,1.963) -- (7.275,1.883) -- (7.297,1.803) -- (7.355,1.744) -- (7.435,1.723) -- (7.515,1.744) -- (7.574,1.803) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.87,1.382) -- (10.849,1.462) -- (10.79,1.521) -- (10.71,1.542) -- (10.63,1.521) -- (10.572,1.462) -- (10.55,1.382) -- (10.572,1.302) -- (10.63,1.244) -- (10.71,1.222) -- (10.79,1.244) -- (10.849,1.302) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.463,11.094) -- (6.585,11.113) -- (6.695,11.169) -- (6.782,11.256) -- (6.838,11.366) -- (6.857,11.488) -- (6.838,11.61) -- (6.782,11.719) -- (6.695,11.806) -- (6.585,11.862) -- (6.463,11.882) -- (6.463,11.882) -- (6.342,11.862) -- (6.232,11.806) -- (6.145,11.719) -- (6.089,11.61) -- (6.069,11.488) -- (6.089,11.366) -- (6.145,11.256) -- (6.232,11.169) -- (6.342,11.113) -- (6.463,11.094) -- cycle;
\node at (6.463,11.488) {\scalebox{1.612}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.624,9.179) -- (5.745,9.199) -- (5.855,9.254) -- (5.942,9.342) -- (5.998,9.451) -- (6.018,9.573) -- (5.998,9.695) -- (5.942,9.805) -- (5.855,9.892) -- (5.745,9.948) -- (5.624,9.967) -- (5.624,9.967) -- (5.502,9.948) -- (5.392,9.892) -- (5.305,9.805) -- (5.249,9.695) -- (5.23,9.573) -- (5.249,9.451) -- (5.305,9.342) -- (5.392,9.254) -- (5.502,9.199) -- (5.624,9.179) -- cycle;
\node at (5.624,9.573) {\scalebox{1.612}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.078,9.379) -- (7.199,9.398) -- (7.309,9.454) -- (7.396,9.541) -- (7.452,9.651) -- (7.472,9.773) -- (7.452,9.894) -- (7.396,10.004) -- (7.309,10.091) -- (7.199,10.147) -- (7.078,10.167) -- (7.078,10.167) -- (6.956,10.147) -- (6.846,10.091) -- (6.759,10.004) -- (6.703,9.894) -- (6.684,9.773) -- (6.703,9.651) -- (6.759,9.541) -- (6.846,9.454) -- (6.956,9.398) -- (7.078,9.379) -- cycle;
\node at (7.078,9.773) {\scalebox{1.612}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.131,11.23) -- (3.253,11.249) -- (3.363,11.305) -- (3.45,11.392) -- (3.506,11.502) -- (3.525,11.624) -- (3.506,11.745) -- (3.45,11.855) -- (3.363,11.942) -- (3.253,11.998) -- (3.131,12.018) -- (3.131,12.018) -- (3.01,11.998) -- (2.9,11.942) -- (2.813,11.855) -- (2.757,11.745) -- (2.737,11.624) -- (2.757,11.502) -- (2.813,11.392) -- (2.9,11.305) -- (3.01,11.249) -- (3.131,11.23) -- cycle;
\node at (3.131,11.624) {\scalebox{1.612}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.677,9.031) -- (1.799,9.051) -- (1.909,9.107) -- (1.996,9.194) -- (2.052,9.304) -- (2.071,9.425) -- (2.052,9.547) -- (1.996,9.657) -- (1.909,9.744) -- (1.799,9.8) -- (1.677,9.819) -- (1.677,9.819) -- (1.556,9.8) -- (1.446,9.744) -- (1.359,9.657) -- (1.303,9.547) -- (1.283,9.425) -- (1.303,9.304) -- (1.359,9.194) -- (1.446,9.107) -- (1.556,9.051) -- (1.677,9.031) -- cycle;
\node at (1.677,9.425) {\scalebox{1.612}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.301,6.195) -- (8.422,6.214) -- (8.532,6.27) -- (8.619,6.357) -- (8.675,6.467) -- (8.694,6.588) -- (8.675,6.71) -- (8.619,6.82) -- (8.532,6.907) -- (8.422,6.963) -- (8.301,6.982) -- (8.301,6.982) -- (8.179,6.963) -- (8.069,6.907) -- (7.982,6.82) -- (7.926,6.71) -- (7.907,6.588) -- (7.926,6.467) -- (7.982,6.357) -- (8.069,6.27) -- (8.179,6.214) -- (8.301,6.195) -- cycle;
\node at (8.301,6.588) {\scalebox{1.612}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.281,7.797) -- (3.402,7.816) -- (3.512,7.872) -- (3.599,7.959) -- (3.655,8.069) -- (3.674,8.191) -- (3.655,8.312) -- (3.599,8.422) -- (3.512,8.509) -- (3.402,8.565) -- (3.281,8.585) -- (3.281,8.585) -- (3.159,8.565) -- (3.049,8.509) -- (2.962,8.422) -- (2.906,8.312) -- (2.887,8.191) -- (2.906,8.069) -- (2.962,7.959) -- (3.049,7.872) -- (3.159,7.816) -- (3.281,7.797) -- cycle;
\node at (3.281,8.191) {\scalebox{1.612}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.033,6.022) -- (2.155,6.041) -- (2.264,6.097) -- (2.352,6.184) -- (2.408,6.294) -- (2.427,6.416) -- (2.408,6.538) -- (2.352,6.647) -- (2.264,6.735) -- (2.155,6.791) -- (2.033,6.81) -- (2.033,6.81) -- (1.911,6.791) -- (1.801,6.735) -- (1.714,6.647) -- (1.658,6.538) -- (1.639,6.416) -- (1.658,6.294) -- (1.714,6.184) -- (1.801,6.097) -- (1.911,6.041) -- (2.033,6.022) -- cycle;
\node at (2.033,6.416) {\scalebox{1.612}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.366,7.3) -- (6.488,7.319) -- (6.598,7.375) -- (6.685,7.462) -- (6.741,7.572) -- (6.76,7.694) -- (6.741,7.815) -- (6.685,7.925) -- (6.598,8.012) -- (6.488,8.068) -- (6.366,8.088) -- (6.366,8.088) -- (6.244,8.068) -- (6.135,8.012) -- (6.047,7.925) -- (5.991,7.815) -- (5.972,7.694) -- (5.991,7.572) -- (6.047,7.462) -- (6.135,7.375) -- (6.244,7.319) -- (6.366,7.3) -- cycle;
\node at (6.366,7.694) {\scalebox{1.612}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.755,5.808) -- (5.877,5.827) -- (5.987,5.883) -- (6.074,5.97) -- (6.13,6.08) -- (6.149,6.202) -- (6.13,6.324) -- (6.074,6.433) -- (5.987,6.521) -- (5.877,6.577) -- (5.755,6.596) -- (5.385,6.596) -- (5.263,6.577) -- (5.153,6.521) -- (5.066,6.433) -- (5.01,6.324) -- (4.991,6.202) -- (5.01,6.08) -- (5.066,5.97) -- (5.153,5.883) -- (5.263,5.827) -- (5.385,5.808) -- cycle;
\node at (5.57,6.202) {\scalebox{1.612}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.274,2.994) -- (11.396,3.013) -- (11.506,3.069) -- (11.593,3.156) -- (11.649,3.266) -- (11.668,3.387) -- (11.649,3.509) -- (11.593,3.619) -- (11.506,3.706) -- (11.396,3.762) -- (11.274,3.781) -- (10.904,3.781) -- (10.782,3.762) -- (10.672,3.706) -- (10.585,3.619) -- (10.529,3.509) -- (10.51,3.387) -- (10.529,3.266) -- (10.585,3.156) -- (10.672,3.069) -- (10.782,3.013) -- (10.904,2.994) -- cycle;
\node at (11.089,3.387) {\scalebox{1.612}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.522,4.832) -- (3.644,4.851) -- (3.754,4.907) -- (3.841,4.994) -- (3.897,5.104) -- (3.916,5.225) -- (3.897,5.347) -- (3.841,5.457) -- (3.754,5.544) -- (3.644,5.6) -- (3.522,5.619) -- (3.151,5.619) -- (3.03,5.6) -- (2.92,5.544) -- (2.833,5.457) -- (2.777,5.347) -- (2.757,5.225) -- (2.777,5.104) -- (2.833,4.994) -- (2.92,4.907) -- (3.03,4.851) -- (3.151,4.832) -- cycle;
\node at (3.337,5.225) {\scalebox{1.612}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.835,3.084) -- (1.957,3.103) -- (2.066,3.159) -- (2.153,3.246) -- (2.209,3.356) -- (2.229,3.478) -- (2.209,3.599) -- (2.153,3.709) -- (2.066,3.796) -- (1.957,3.852) -- (1.835,3.872) -- (1.464,3.872) -- (1.342,3.852) -- (1.233,3.796) -- (1.145,3.709) -- (1.089,3.599) -- (1.07,3.478) -- (1.089,3.356) -- (1.145,3.246) -- (1.233,3.159) -- (1.342,3.103) -- (1.464,3.084) -- cycle;
\node at (1.649,3.478) {\scalebox{1.612}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.521,5.102) -- (9.643,5.121) -- (9.753,5.177) -- (9.84,5.264) -- (9.896,5.374) -- (9.915,5.495) -- (9.896,5.617) -- (9.84,5.727) -- (9.753,5.814) -- (9.643,5.87) -- (9.521,5.889) -- (9.151,5.889) -- (9.029,5.87) -- (8.919,5.814) -- (8.832,5.727) -- (8.776,5.617) -- (8.757,5.495) -- (8.776,5.374) -- (8.832,5.264) -- (8.919,5.177) -- (9.029,5.121) -- (9.151,5.102) -- cycle;
\node at (9.336,5.495) {\scalebox{1.612}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.152,2.771) -- (8.274,2.79) -- (8.383,2.846) -- (8.47,2.934) -- (8.526,3.043) -- (8.546,3.165) -- (8.526,3.287) -- (8.47,3.397) -- (8.383,3.484) -- (8.274,3.54) -- (8.152,3.559) -- (7.781,3.559) -- (7.659,3.54) -- (7.55,3.484) -- (7.462,3.397) -- (7.406,3.287) -- (7.387,3.165) -- (7.406,3.043) -- (7.462,2.934) -- (7.55,2.846) -- (7.659,2.79) -- (7.781,2.771) -- cycle;
\node at (7.966,3.165) {\scalebox{1.612}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.714,4.636) -- (6.836,4.655) -- (6.946,4.711) -- (7.033,4.798) -- (7.089,4.908) -- (7.108,5.03) -- (7.089,5.151) -- (7.033,5.261) -- (6.946,5.348) -- (6.836,5.404) -- (6.714,5.423) -- (6.344,5.423) -- (6.222,5.404) -- (6.112,5.348) -- (6.025,5.261) -- (5.969,5.151) -- (5.95,5.03) -- (5.969,4.908) -- (6.025,4.798) -- (6.112,4.711) -- (6.222,4.655) -- (6.344,4.636) -- cycle;
\node at (6.529,5.03) {\scalebox{1.612}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.297,1.493) -- (9.419,1.512) -- (9.528,1.568) -- (9.616,1.655) -- (9.672,1.765) -- (9.691,1.887) -- (9.672,2.009) -- (9.616,2.118) -- (9.528,2.206) -- (9.419,2.261) -- (9.297,2.281) -- (8.926,2.281) -- (8.804,2.261) -- (8.695,2.206) -- (8.608,2.118) -- (8.552,2.009) -- (8.532,1.887) -- (8.552,1.765) -- (8.608,1.655) -- (8.695,1.568) -- (8.804,1.512) -- (8.926,1.493) -- cycle;
\node at (9.112,1.887) {\scalebox{1.612}{\small 17}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (10.716,5.127) -- (11.316,5.127) -- (11.316,5.727) -- (10.716,5.727) -- cycle;
\node at (11.016,5.427) {\scalebox{1.612}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.607,7.062) -- (9.207,7.062) -- (9.207,7.662) -- (8.607,7.662) -- cycle;
\node at (8.907,7.362) {\scalebox{1.612}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (12.013,3.202) -- (12.613,3.202) -- (12.613,3.802) -- (12.013,3.802) -- cycle;
\node at (12.313,3.502) {\scalebox{1.612}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (5.484,8.229) -- (6.084,8.229) -- (6.084,8.829) -- (5.484,8.829) -- cycle;
\node at (5.784,8.529) {\scalebox{1.612}{\small d}};
\node at (5.185,12.0) {\scalebox{1.612}{\small A}};
\node at (7.844,11.582) {\scalebox{1.612}{\small B}};
\node at (0.765,11.193) {\scalebox{1.612}{\small C}};
\node at (7.956,8.978) {\scalebox{1.612}{\small D}};
\node at (1.026,8.256) {\scalebox{1.612}{\small E}};
\node at (4.685,8.42) {\scalebox{1.612}{\small F}};
\node at (11.627,5.325) {\scalebox{1.612}{\small G}};
\node at (0.836,4.671) {\scalebox{1.612}{\small H}};
\node at (8.437,4.176) {\scalebox{1.612}{\small I}};
\node at (4.979,3.888) {\scalebox{1.612}{\small J}};
\node at (0.875,1.669) {\scalebox{1.612}{\small K}};
\node at (7.669,1.309) {\scalebox{1.612}{\small L}};
\node at (11.155,0.95) {\scalebox{1.612}{\small M}};
\draw[line width=0.5pt] (12.316,10.432) -- (12.316,11.532);
\draw[line width=0.5pt] (12.085,11.147) -- (12.316,11.532) -- (12.547,11.147);
\draw[line width=0.5pt] (12.013,10.982) -- (12.618,10.982);
\node at (12.316,11.917) {\scalebox{1.612}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour G, 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 rue Verte & Ⓑ Place aux Herbes & Ⓛ Fontaine Notre-Dame \\ \hline
① passage du Marché & ⑪ rue Pasteur & Ⓒ Place du Vieux-Puits & Ⓜ Place de la Fontaine \\ \hline
② rue du Stade & ⑫ rue du Four & Ⓓ Rond-point de la Gare & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ sentier des Prés & ⑬ impasse du Puits & Ⓔ Carrefour Saint-Roch & \textbf{a} l'école \\ \hline
④ rue Neuve & ⑭ rue des Écoles & Ⓕ Rond-point du Stade & \textbf{b} le cimetière \\ \hline
⑤ rue de la Poste & ⑮ rue de la Mairie & Ⓖ Placette des Ormes & \textbf{c} la bibliothèque \\ \hline
⑥ venelle du Chat & ⑯ rue des Lilas & Ⓗ Carrefour de l'Hôpital & \textbf{d} la salle des fêtes \\ \hline
⑦ chemin des Vignes & ⑰ rue Haute & Ⓘ Carrefour de la Mairie &  \\ \hline
⑧ avenue de la Gare & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Carrefour des Tilleuls &  \\ \hline
⑨ chemin de Ronde & Ⓐ Rond-point de la Poste & Ⓚ Place de l'Église &  \\ \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=0.6355, line join=round, line cap=round]
\draw[line width=0.5pt] (5.233,11.382) -- (5.396,11.33) -- (5.562,11.281) -- (5.728,11.237) -- (5.895,11.197) -- (6.063,11.16) -- (6.232,11.127) -- (6.402,11.099) -- (6.573,11.074) -- (6.744,11.053) -- (6.915,11.037) -- (7.087,11.024) -- (7.259,11.016) -- (7.431,11.011) -- (7.603,11.011);
\draw[line width=0.5pt] (5.233,11.382) -- (5.25,11.129) -- (5.262,10.875) -- (5.268,10.621) -- (5.268,10.367) -- (5.261,10.113) -- (5.249,9.859) -- (5.231,9.606) -- (5.207,9.353) -- (5.177,9.1) -- (5.141,8.849) -- (5.099,8.598) -- (5.052,8.349) -- (4.998,8.101) -- (4.939,7.854);
\draw[line width=0.5pt] (7.603,11.011) -- (7.571,10.834) -- (7.544,10.657) -- (7.521,10.479) -- (7.502,10.301) -- (7.488,10.122) -- (7.478,9.943) -- (7.471,9.764) -- (7.469,9.585) -- (7.472,9.405) -- (7.478,9.226) -- (7.489,9.047) -- (7.504,8.868) -- (7.523,8.69) -- (7.546,8.513);
\draw[line width=0.5pt] (1.203,10.755) -- (1.482,10.843) -- (1.762,10.925) -- (2.045,11.0) -- (2.33,11.069) -- (2.616,11.131) -- (2.903,11.186) -- (3.192,11.235) -- (3.482,11.276) -- (3.772,11.311) -- (4.064,11.339) -- (4.356,11.36) -- (4.648,11.374) -- (4.94,11.382) -- (5.233,11.382);
\draw[line width=0.5pt] (1.203,10.755) -- (1.201,10.556) -- (1.204,10.358) -- (1.211,10.159) -- (1.223,9.961) -- (1.24,9.763) -- (1.261,9.566) -- (1.287,9.369) -- (1.318,9.173) -- (1.353,8.978) -- (1.393,8.783) -- (1.437,8.589) -- (1.486,8.397) -- (1.54,8.206) -- (1.598,8.016);
\draw[line width=0.5pt] (7.546,8.513) -- (7.618,8.236) -- (7.683,7.958) -- (7.741,7.679) -- (7.793,7.398) -- (7.838,7.116) -- (7.877,6.833) -- (7.909,6.549) -- (7.934,6.264) -- (7.952,5.979) -- (7.964,5.694) -- (7.969,5.409) -- (7.968,5.123) -- (7.959,4.838) -- (7.944,4.553);
\draw[line width=0.5pt] (1.598,8.016) -- (1.833,7.968) -- (2.069,7.926) -- (2.306,7.889) -- (2.544,7.858) -- (2.783,7.832) -- (3.022,7.812) -- (3.261,7.797) -- (3.501,7.788) -- (3.741,7.785) -- (3.981,7.788) -- (4.221,7.796) -- (4.461,7.809) -- (4.7,7.829) -- (4.939,7.854);
\draw[line width=0.5pt] (1.598,8.016) -- (1.62,7.792) -- (1.636,7.567) -- (1.648,7.341) -- (1.654,7.115) -- (1.654,6.889) -- (1.65,6.664) -- (1.64,6.438) -- (1.624,6.212) -- (1.604,5.987) -- (1.578,5.763) -- (1.547,5.539) -- (1.511,5.316) -- (1.469,5.095) -- (1.422,4.874);
\draw[line width=0.5pt] (4.939,7.854) -- (5.131,7.872) -- (5.322,7.895) -- (5.513,7.922) -- (5.704,7.954) -- (5.893,7.99) -- (6.082,8.03) -- (6.27,8.076) -- (6.456,8.125) -- (6.642,8.179) -- (6.826,8.237) -- (7.008,8.3) -- (7.189,8.366) -- (7.369,8.437) -- (7.546,8.513);
\draw[line width=0.5pt] (4.939,7.854) -- (4.99,7.617) -- (5.035,7.38) -- (5.075,7.141) -- (5.109,6.902) -- (5.137,6.661) -- (5.16,6.42) -- (5.177,6.179) -- (5.188,5.937) -- (5.194,5.695) -- (5.194,5.453) -- (5.189,5.211) -- (5.177,4.969) -- (5.16,4.728) -- (5.137,4.487);
\draw[line width=0.5pt] (11.016,5.427) -- (10.95,5.144) -- (10.891,4.859) -- (10.838,4.573) -- (10.793,4.285) -- (10.754,3.997) -- (10.722,3.707) -- (10.696,3.417) -- (10.678,3.127) -- (10.666,2.836) -- (10.661,2.545) -- (10.663,2.253) -- (10.672,1.963) -- (10.688,1.672) -- (10.71,1.382);
\draw[line width=0.5pt] (1.422,4.874) -- (1.69,4.887) -- (1.958,4.894) -- (2.226,4.895) -- (2.494,4.889) -- (2.761,4.877) -- (3.029,4.858) -- (3.296,4.834) -- (3.562,4.803) -- (3.828,4.766) -- (4.092,4.722) -- (4.356,4.673) -- (4.618,4.617) -- (4.878,4.555) -- (5.137,4.487);
\draw[line width=0.5pt] (1.422,4.874) -- (1.385,4.679) -- (1.352,4.482) -- (1.323,4.286) -- (1.299,4.088) -- (1.28,3.89) -- (1.266,3.691) -- (1.256,3.493) -- (1.251,3.294) -- (1.25,3.095) -- (1.254,2.896) -- (1.263,2.697) -- (1.277,2.498) -- (1.295,2.3) -- (1.318,2.103);
\draw[line width=0.5pt] (7.944,4.553) -- (8.152,4.648) -- (8.363,4.739) -- (8.575,4.825) -- (8.79,4.905) -- (9.006,4.981) -- (9.224,5.051) -- (9.444,5.117) -- (9.665,5.177) -- (9.888,5.232) -- (10.111,5.281) -- (10.336,5.326) -- (10.562,5.365) -- (10.789,5.399) -- (11.016,5.427);
\draw[line width=0.5pt] (7.944,4.553) -- (7.879,4.369) -- (7.818,4.184) -- (7.761,3.997) -- (7.709,3.809) -- (7.662,3.62) -- (7.618,3.43) -- (7.58,3.239) -- (7.545,3.047) -- (7.515,2.854) -- (7.49,2.661) -- (7.47,2.467) -- (7.453,2.272) -- (7.442,2.078) -- (7.435,1.883);
\draw[line width=0.5pt] (5.137,4.487) -- (5.336,4.522) -- (5.535,4.553) -- (5.734,4.579) -- (5.935,4.6) -- (6.136,4.617) -- (6.337,4.629) -- (6.538,4.636) -- (6.74,4.638) -- (6.941,4.636) -- (7.142,4.629) -- (7.344,4.617) -- (7.544,4.6) -- (7.745,4.579) -- (7.944,4.553);
\draw[line width=0.5pt] (7.435,1.883) -- (7.662,1.812) -- (7.89,1.746) -- (8.12,1.685) -- (8.351,1.63) -- (8.584,1.58) -- (8.817,1.536) -- (9.052,1.497) -- (9.288,1.464) -- (9.524,1.437) -- (9.76,1.415) -- (9.998,1.398) -- (10.235,1.387) -- (10.473,1.382) -- (10.71,1.382);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.606,11.382) -- (5.577,11.525) -- (5.496,11.646) -- (5.375,11.727) -- (5.233,11.755) -- (5.09,11.727) -- (4.969,11.646) -- (4.888,11.525) -- (4.859,11.382) -- (4.888,11.239) -- (4.969,11.118) -- (5.09,11.037) -- (5.233,11.009) -- (5.375,11.037) -- (5.496,11.118) -- (5.577,11.239) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.976,11.011) -- (7.947,11.153) -- (7.866,11.275) -- (7.745,11.355) -- (7.603,11.384) -- (7.46,11.355) -- (7.339,11.275) -- (7.258,11.153) -- (7.229,11.011) -- (7.258,10.868) -- (7.339,10.747) -- (7.46,10.666) -- (7.603,10.637) -- (7.745,10.666) -- (7.866,10.747) -- (7.947,10.868) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.576,10.755) -- (1.548,10.897) -- (1.467,11.018) -- (1.346,11.099) -- (1.203,11.128) -- (1.06,11.099) -- (0.939,11.018) -- (0.858,10.897) -- (0.83,10.755) -- (0.858,10.612) -- (0.939,10.491) -- (1.06,10.41) -- (1.203,10.381) -- (1.346,10.41) -- (1.467,10.491) -- (1.548,10.612) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.92,8.513) -- (7.891,8.655) -- (7.81,8.776) -- (7.689,8.857) -- (7.546,8.886) -- (7.404,8.857) -- (7.282,8.776) -- (7.202,8.655) -- (7.173,8.513) -- (7.202,8.37) -- (7.282,8.249) -- (7.404,8.168) -- (7.546,8.139) -- (7.689,8.168) -- (7.81,8.249) -- (7.891,8.37) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.971,8.016) -- (1.943,8.159) -- (1.862,8.28) -- (1.741,8.361) -- (1.598,8.39) -- (1.455,8.361) -- (1.334,8.28) -- (1.253,8.159) -- (1.224,8.016) -- (1.253,7.874) -- (1.334,7.752) -- (1.455,7.672) -- (1.598,7.643) -- (1.741,7.672) -- (1.862,7.752) -- (1.943,7.874) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.312,7.854) -- (5.283,7.997) -- (5.202,8.118) -- (5.081,8.199) -- (4.939,8.227) -- (4.796,8.199) -- (4.675,8.118) -- (4.594,7.997) -- (4.565,7.854) -- (4.594,7.711) -- (4.675,7.59) -- (4.796,7.509) -- (4.939,7.481) -- (5.081,7.509) -- (5.202,7.59) -- (5.283,7.711) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.389,5.427) -- (11.361,5.57) -- (11.28,5.691) -- (11.159,5.772) -- (11.016,5.8) -- (10.873,5.772) -- (10.752,5.691) -- (10.671,5.57) -- (10.643,5.427) -- (10.671,5.284) -- (10.752,5.163) -- (10.873,5.082) -- (11.016,5.054) -- (11.159,5.082) -- (11.28,5.163) -- (11.361,5.284) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.796,4.874) -- (1.767,5.017) -- (1.686,5.138) -- (1.565,5.219) -- (1.422,5.247) -- (1.28,5.219) -- (1.158,5.138) -- (1.078,5.017) -- (1.049,4.874) -- (1.078,4.731) -- (1.158,4.61) -- (1.28,4.529) -- (1.422,4.501) -- (1.565,4.529) -- (1.686,4.61) -- (1.767,4.731) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.317,4.553) -- (8.289,4.696) -- (8.208,4.817) -- (8.087,4.898) -- (7.944,4.926) -- (7.801,4.898) -- (7.68,4.817) -- (7.599,4.696) -- (7.571,4.553) -- (7.599,4.41) -- (7.68,4.289) -- (7.801,4.208) -- (7.944,4.18) -- (8.087,4.208) -- (8.208,4.289) -- (8.289,4.41) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.511,4.487) -- (5.482,4.63) -- (5.401,4.751) -- (5.28,4.832) -- (5.137,4.86) -- (4.995,4.832) -- (4.874,4.751) -- (4.793,4.63) -- (4.764,4.487) -- (4.793,4.344) -- (4.874,4.223) -- (4.995,4.142) -- (5.137,4.114) -- (5.28,4.142) -- (5.401,4.223) -- (5.482,4.344) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.691,2.103) -- (1.663,2.246) -- (1.582,2.367) -- (1.461,2.447) -- (1.318,2.476) -- (1.175,2.447) -- (1.054,2.367) -- (0.973,2.246) -- (0.945,2.103) -- (0.973,1.96) -- (1.054,1.839) -- (1.175,1.758) -- (1.318,1.729) -- (1.461,1.758) -- (1.582,1.839) -- (1.663,1.96) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.808,1.883) -- (7.78,2.026) -- (7.699,2.147) -- (7.578,2.228) -- (7.435,2.256) -- (7.292,2.228) -- (7.171,2.147) -- (7.09,2.026) -- (7.062,1.883) -- (7.09,1.74) -- (7.171,1.619) -- (7.292,1.538) -- (7.435,1.51) -- (7.578,1.538) -- (7.699,1.619) -- (7.78,1.74) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.083,1.382) -- (11.055,1.525) -- (10.974,1.646) -- (10.853,1.727) -- (10.71,1.755) -- (10.567,1.727) -- (10.446,1.646) -- (10.365,1.525) -- (10.337,1.382) -- (10.365,1.239) -- (10.446,1.118) -- (10.567,1.037) -- (10.71,1.009) -- (10.853,1.037) -- (10.974,1.118) -- (11.055,1.239) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.462,11.094) -- (6.581,11.113) -- (6.688,11.167) -- (6.773,11.253) -- (6.827,11.36) -- (6.846,11.479) -- (6.827,11.597) -- (6.773,11.705) -- (6.688,11.79) -- (6.581,11.844) -- (6.462,11.863) -- (6.462,11.863) -- (6.343,11.844) -- (6.236,11.79) -- (6.151,11.705) -- (6.096,11.597) -- (6.077,11.479) -- (6.096,11.36) -- (6.151,11.253) -- (6.236,11.167) -- (6.343,11.113) -- (6.462,11.094) -- cycle;
\node at (6.462,11.479) {\scalebox{1.574}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.614,9.189) -- (5.733,9.208) -- (5.84,9.263) -- (5.925,9.348) -- (5.98,9.455) -- (5.999,9.574) -- (5.98,9.693) -- (5.925,9.8) -- (5.84,9.885) -- (5.733,9.94) -- (5.614,9.958) -- (5.614,9.958) -- (5.495,9.94) -- (5.388,9.885) -- (5.303,9.8) -- (5.249,9.693) -- (5.23,9.574) -- (5.249,9.455) -- (5.303,9.348) -- (5.388,9.263) -- (5.495,9.208) -- (5.614,9.189) -- cycle;
\node at (5.614,9.574) {\scalebox{1.574}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.087,9.388) -- (7.206,9.407) -- (7.313,9.461) -- (7.398,9.547) -- (7.453,9.654) -- (7.472,9.773) -- (7.453,9.891) -- (7.398,9.999) -- (7.313,10.084) -- (7.206,10.138) -- (7.087,10.157) -- (7.087,10.157) -- (6.968,10.138) -- (6.861,10.084) -- (6.776,9.999) -- (6.721,9.891) -- (6.702,9.773) -- (6.721,9.654) -- (6.776,9.547) -- (6.861,9.461) -- (6.968,9.407) -- (7.087,9.388) -- cycle;
\node at (7.087,9.773) {\scalebox{1.574}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.133,11.23) -- (3.252,11.249) -- (3.359,11.303) -- (3.444,11.388) -- (3.498,11.496) -- (3.517,11.614) -- (3.498,11.733) -- (3.444,11.84) -- (3.359,11.926) -- (3.252,11.98) -- (3.133,11.999) -- (3.133,11.999) -- (3.014,11.98) -- (2.907,11.926) -- (2.822,11.84) -- (2.767,11.733) -- (2.748,11.614) -- (2.767,11.496) -- (2.822,11.388) -- (2.907,11.303) -- (3.014,11.249) -- (3.133,11.23) -- cycle;
\node at (3.133,11.614) {\scalebox{1.574}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.668,9.039) -- (1.787,9.058) -- (1.894,9.113) -- (1.979,9.198) -- (2.034,9.305) -- (2.053,9.424) -- (2.034,9.543) -- (1.979,9.65) -- (1.894,9.735) -- (1.787,9.79) -- (1.668,9.809) -- (1.668,9.809) -- (1.549,9.79) -- (1.442,9.735) -- (1.357,9.65) -- (1.302,9.543) -- (1.283,9.424) -- (1.302,9.305) -- (1.357,9.198) -- (1.442,9.113) -- (1.549,9.058) -- (1.668,9.039) -- cycle;
\node at (1.668,9.424) {\scalebox{1.574}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.291,6.203) -- (8.41,6.222) -- (8.517,6.276) -- (8.602,6.361) -- (8.657,6.469) -- (8.676,6.587) -- (8.657,6.706) -- (8.602,6.813) -- (8.517,6.899) -- (8.41,6.953) -- (8.291,6.972) -- (8.291,6.972) -- (8.172,6.953) -- (8.065,6.899) -- (7.98,6.813) -- (7.925,6.706) -- (7.907,6.587) -- (7.925,6.469) -- (7.98,6.361) -- (8.065,6.276) -- (8.172,6.222) -- (8.291,6.203) -- cycle;
\node at (8.291,6.587) {\scalebox{1.574}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.28,7.797) -- (3.399,7.816) -- (3.506,7.87) -- (3.591,7.955) -- (3.646,8.063) -- (3.665,8.181) -- (3.646,8.3) -- (3.591,8.407) -- (3.506,8.492) -- (3.399,8.547) -- (3.28,8.566) -- (3.28,8.566) -- (3.161,8.547) -- (3.054,8.492) -- (2.969,8.407) -- (2.914,8.3) -- (2.896,8.181) -- (2.914,8.063) -- (2.969,7.955) -- (3.054,7.87) -- (3.161,7.816) -- (3.28,7.797) -- cycle;
\node at (3.28,8.181) {\scalebox{1.574}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.024,6.032) -- (2.142,6.051) -- (2.25,6.105) -- (2.335,6.19) -- (2.389,6.298) -- (2.408,6.416) -- (2.389,6.535) -- (2.335,6.642) -- (2.25,6.728) -- (2.142,6.782) -- (2.024,6.801) -- (2.024,6.801) -- (1.905,6.782) -- (1.798,6.728) -- (1.713,6.642) -- (1.658,6.535) -- (1.639,6.416) -- (1.658,6.298) -- (1.713,6.19) -- (1.798,6.105) -- (1.905,6.051) -- (2.024,6.032) -- cycle;
\node at (2.024,6.416) {\scalebox{1.574}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.364,7.318) -- (6.483,7.337) -- (6.59,7.392) -- (6.675,7.477) -- (6.73,7.584) -- (6.748,7.703) -- (6.73,7.822) -- (6.675,7.929) -- (6.59,8.014) -- (6.483,8.068) -- (6.364,8.087) -- (6.364,8.087) -- (6.245,8.068) -- (6.138,8.014) -- (6.053,7.929) -- (5.998,7.822) -- (5.979,7.703) -- (5.998,7.584) -- (6.053,7.477) -- (6.138,7.392) -- (6.245,7.337) -- (6.364,7.318) -- cycle;
\node at (6.364,7.703) {\scalebox{1.574}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.742,5.817) -- (5.861,5.836) -- (5.968,5.89) -- (6.053,5.975) -- (6.107,6.083) -- (6.126,6.201) -- (6.107,6.32) -- (6.053,6.427) -- (5.968,6.512) -- (5.861,6.567) -- (5.742,6.586) -- (5.38,6.586) -- (5.261,6.567) -- (5.154,6.512) -- (5.069,6.427) -- (5.014,6.32) -- (4.995,6.201) -- (5.014,6.083) -- (5.069,5.975) -- (5.154,5.89) -- (5.261,5.836) -- (5.38,5.817) -- cycle;
\node at (5.561,6.201) {\scalebox{1.574}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.261,3.004) -- (11.379,3.022) -- (11.487,3.077) -- (11.572,3.162) -- (11.626,3.269) -- (11.645,3.388) -- (11.626,3.507) -- (11.572,3.614) -- (11.487,3.699) -- (11.379,3.754) -- (11.261,3.773) -- (10.899,3.773) -- (10.78,3.754) -- (10.673,3.699) -- (10.588,3.614) -- (10.533,3.507) -- (10.514,3.388) -- (10.533,3.269) -- (10.588,3.162) -- (10.673,3.077) -- (10.78,3.022) -- (10.899,3.004) -- cycle;
\node at (11.08,3.388) {\scalebox{1.574}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.517,4.832) -- (3.635,4.85) -- (3.743,4.905) -- (3.828,4.99) -- (3.882,5.097) -- (3.901,5.216) -- (3.882,5.335) -- (3.828,5.442) -- (3.743,5.527) -- (3.635,5.582) -- (3.517,5.601) -- (3.155,5.601) -- (3.036,5.582) -- (2.929,5.527) -- (2.844,5.442) -- (2.789,5.335) -- (2.77,5.216) -- (2.789,5.097) -- (2.844,4.99) -- (2.929,4.905) -- (3.036,4.85) -- (3.155,4.832) -- cycle;
\node at (3.336,5.216) {\scalebox{1.574}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.821,3.094) -- (1.94,3.112) -- (2.047,3.167) -- (2.132,3.252) -- (2.187,3.359) -- (2.206,3.478) -- (2.187,3.597) -- (2.132,3.704) -- (2.047,3.789) -- (1.94,3.844) -- (1.821,3.863) -- (1.459,3.863) -- (1.34,3.844) -- (1.233,3.789) -- (1.148,3.704) -- (1.093,3.597) -- (1.075,3.478) -- (1.093,3.359) -- (1.148,3.252) -- (1.233,3.167) -- (1.34,3.112) -- (1.459,3.094) -- cycle;
\node at (1.64,3.478) {\scalebox{1.574}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.52,5.102) -- (9.638,5.121) -- (9.746,5.175) -- (9.831,5.26) -- (9.885,5.368) -- (9.904,5.486) -- (9.885,5.605) -- (9.831,5.712) -- (9.746,5.797) -- (9.638,5.852) -- (9.52,5.871) -- (9.158,5.871) -- (9.039,5.852) -- (8.932,5.797) -- (8.847,5.712) -- (8.792,5.605) -- (8.773,5.486) -- (8.792,5.368) -- (8.847,5.26) -- (8.932,5.175) -- (9.039,5.121) -- (9.158,5.102) -- cycle;
\node at (9.339,5.486) {\scalebox{1.574}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.138,2.782) -- (8.257,2.801) -- (8.364,2.856) -- (8.449,2.941) -- (8.504,3.048) -- (8.523,3.167) -- (8.504,3.286) -- (8.449,3.393) -- (8.364,3.478) -- (8.257,3.533) -- (8.138,3.551) -- (7.776,3.551) -- (7.657,3.533) -- (7.55,3.478) -- (7.465,3.393) -- (7.411,3.286) -- (7.392,3.167) -- (7.411,3.048) -- (7.465,2.941) -- (7.55,2.856) -- (7.657,2.801) -- (7.776,2.782) -- cycle;
\node at (7.957,3.167) {\scalebox{1.574}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.71,4.636) -- (6.829,4.654) -- (6.936,4.709) -- (7.021,4.794) -- (7.076,4.901) -- (7.095,5.02) -- (7.076,5.139) -- (7.021,5.246) -- (6.936,5.331) -- (6.829,5.386) -- (6.71,5.405) -- (6.348,5.405) -- (6.229,5.386) -- (6.122,5.331) -- (6.037,5.246) -- (5.982,5.139) -- (5.964,5.02) -- (5.982,4.901) -- (6.037,4.794) -- (6.122,4.709) -- (6.229,4.654) -- (6.348,4.636) -- cycle;
\node at (6.529,5.02) {\scalebox{1.574}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.291,1.493) -- (9.41,1.512) -- (9.517,1.566) -- (9.602,1.652) -- (9.657,1.759) -- (9.676,1.878) -- (9.657,1.996) -- (9.602,2.104) -- (9.517,2.189) -- (9.41,2.243) -- (9.291,2.262) -- (8.929,2.262) -- (8.81,2.243) -- (8.703,2.189) -- (8.618,2.104) -- (8.563,1.996) -- (8.545,1.878) -- (8.563,1.759) -- (8.618,1.652) -- (8.703,1.566) -- (8.81,1.512) -- (8.929,1.493) -- cycle;
\node at (9.11,1.878) {\scalebox{1.574}{\small 17}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (10.716,5.127) -- (11.316,5.127) -- (11.316,5.727) -- (10.716,5.727) -- cycle;
\node at (11.016,5.427) {\scalebox{1.574}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.607,7.062) -- (9.207,7.062) -- (9.207,7.662) -- (8.607,7.662) -- cycle;
\node at (8.907,7.362) {\scalebox{1.574}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (12.013,3.202) -- (12.613,3.202) -- (12.613,3.802) -- (12.013,3.802) -- cycle;
\node at (12.313,3.502) {\scalebox{1.574}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (5.484,8.229) -- (6.084,8.229) -- (6.084,8.829) -- (5.484,8.829) -- cycle;
\node at (5.784,8.529) {\scalebox{1.574}{\small d}};
\node at (5.185,12.0) {\scalebox{1.574}{\small A}};
\node at (7.844,11.582) {\scalebox{1.574}{\small B}};
\node at (0.765,11.193) {\scalebox{1.574}{\small C}};
\node at (7.956,8.978) {\scalebox{1.574}{\small D}};
\node at (1.026,8.256) {\scalebox{1.574}{\small E}};
\node at (4.685,8.42) {\scalebox{1.574}{\small F}};
\node at (11.627,5.325) {\scalebox{1.574}{\small G}};
\node at (0.836,4.671) {\scalebox{1.574}{\small H}};
\node at (8.437,4.176) {\scalebox{1.574}{\small I}};
\node at (4.979,3.888) {\scalebox{1.574}{\small J}};
\node at (0.875,1.669) {\scalebox{1.574}{\small K}};
\node at (7.669,1.309) {\scalebox{1.574}{\small L}};
\node at (11.155,0.95) {\scalebox{1.574}{\small M}};
\draw[line width=0.5pt] (12.316,10.432) -- (12.316,11.532);
\draw[line width=0.5pt] (12.085,11.147) -- (12.316,11.532) -- (12.547,11.147);
\draw[line width=0.5pt] (12.013,10.982) -- (12.618,10.982);
\node at (12.316,11.917) {\scalebox{1.574}{\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, G, J, K, L & 7 \\ \hline
clairs & B, C, F, H, I, M & 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{G} :}\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 ⑪} vers le sud jusqu'à Place de la Fontaine \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Haute ⑰}, vers l'ouest, jusqu'à Fontaine Notre-Dame \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Mairie ⑮}, vers le nord, jusqu'à Carrefour de la Mairie \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{venelle du Chat ⑥}, vers le nord, jusqu'à Rond-point de la Gare \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{sentier des Prés ③}, vers le nord, jusqu'à Place aux Herbes \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{passage du Marché ①}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Neuve ④}, vers l'ouest, jusqu'à Place du Vieux-Puits \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de la Poste ⑤}, vers le sud, jusqu'à Carrefour Saint-Roch \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin des Vignes ⑦}, vers l'est, jusqu'à Rond-point du Stade \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Verte ⑩}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Four ⑫}, vers l'ouest, jusqu'à Carrefour de l'Hôpital \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{impasse du Puits ⑬}, vers le sud, jusqu'à Place de l'Église \textbf{(K)}.\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}
