\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}{La tournée du facteur}
\newcommand{\soustitre}{Probl\`eme --- Fiche n\textdegree 277}

\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/facteur_sans_retour.php?x=277}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0277}
\end{minipage}%
}
}

\begin{document}

\begin{center}
    \small Un facteur veut parcourir toutes les rues de son quartier en n'empruntant chacune qu'une seule fois. Selon le plan, c'est tantôt facile, tantôt impossible — et l'on peut le dire d'avance, sans essayer, en regardant simplement combien de rues aboutissent à chaque carrefour.
\end{center}
\vspace{0.5em}
\sloppy
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{La tournée du facteur}}
\noindent Voici le plan d'un quartier.\par\addvspace{0.35em}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.7, line join=round, line cap=round]
\draw[line width=0.5pt] (5.467,10.966) -- (5.378,10.883) -- (5.291,10.798) -- (5.206,10.711) -- (5.123,10.622) -- (5.042,10.531) -- (4.963,10.438) -- (4.886,10.343) -- (4.812,10.247) -- (4.74,10.148) -- (4.67,10.049) -- (4.603,9.947) -- (4.538,9.844) -- (4.476,9.74) -- (4.416,9.634);
\draw[line width=0.5pt] (5.467,10.966) -- (5.715,10.825) -- (5.96,10.679) -- (6.201,10.527) -- (6.439,10.37) -- (6.674,10.206) -- (6.904,10.038) -- (7.13,9.864) -- (7.352,9.684) -- (7.57,9.5) -- (7.784,9.31) -- (7.993,9.116) -- (8.197,8.917) -- (8.396,8.713) -- (8.591,8.504) -- (8.705,8.357) -- (8.823,8.214) -- (8.944,8.072) -- (9.068,7.934) -- (9.196,7.799) -- (9.327,7.667) -- (9.461,7.538) -- (9.598,7.412) -- (9.738,7.289) -- (9.881,7.17) -- (10.026,7.054) -- (10.174,6.941) -- (10.325,6.833) -- (10.478,6.727);
\draw[line width=0.5pt] (5.467,10.966) -- (5.423,10.658) -- (5.387,10.349) -- (5.358,10.04) -- (5.336,9.73) -- (5.322,9.419) -- (5.315,9.108) -- (5.315,8.797) -- (5.322,8.486) -- (5.337,8.176) -- (5.359,7.866) -- (5.388,7.556) -- (5.425,7.248) -- (5.469,6.94) -- (5.52,6.634);
\draw[line width=0.5pt] (4.416,9.634) -- (4.527,9.433) -- (4.633,9.23) -- (4.735,9.025) -- (4.831,8.817) -- (4.923,8.606) -- (5.01,8.394) -- (5.092,8.179) -- (5.169,7.963) -- (5.24,7.745) -- (5.307,7.525) -- (5.368,7.304) -- (5.424,7.082) -- (5.475,6.858) -- (5.52,6.634);
\draw[line width=0.5pt] (10.478,6.727) -- (10.126,6.775) -- (9.773,6.814) -- (9.418,6.845) -- (9.063,6.868) -- (8.707,6.882) -- (8.351,6.888) -- (7.995,6.885) -- (7.64,6.874) -- (7.284,6.855) -- (6.929,6.827) -- (6.575,6.791) -- (6.222,6.747) -- (5.871,6.695) -- (5.52,6.634);
\draw[line width=0.5pt] (10.478,6.727) -- (10.325,6.5) -- (10.166,6.276) -- (10.001,6.056) -- (9.832,5.84) -- (9.657,5.628) -- (9.477,5.421) -- (9.293,5.217) -- (9.104,5.018) -- (8.91,4.823) -- (8.711,4.633) -- (8.509,4.447) -- (8.302,4.267) -- (8.091,4.091) -- (7.876,3.921);
\draw[line width=0.5pt] (10.478,6.727) -- (10.494,6.414) -- (10.518,6.1) -- (10.549,5.787) -- (10.587,5.475) -- (10.633,5.164) -- (10.686,4.854) -- (10.747,4.545) -- (10.814,4.237) -- (10.889,3.932) -- (10.971,3.628) -- (11.06,3.326) -- (11.156,3.027) -- (11.259,2.73) -- (11.369,2.436);
\draw[line width=0.5pt] (1.369,6.652) -- (1.663,6.696) -- (1.959,6.733) -- (2.255,6.764) -- (2.552,6.787) -- (2.849,6.803) -- (3.147,6.812) -- (3.445,6.814) -- (3.743,6.809) -- (4.041,6.798) -- (4.338,6.779) -- (4.635,6.753) -- (4.931,6.72) -- (5.226,6.68) -- (5.52,6.634);
\draw[line width=0.5pt] (1.369,6.652) -- (1.325,6.277) -- (1.29,5.9) -- (1.264,5.523) -- (1.247,5.145) -- (1.239,4.767) -- (1.24,4.388) -- (1.25,4.01) -- (1.268,3.632) -- (1.296,3.254) -- (1.332,2.878) -- (1.377,2.502) -- (1.431,2.128) -- (1.494,1.755) -- (1.565,1.384);
\draw[line width=0.5pt] (5.52,6.634) -- (5.717,6.467) -- (5.909,6.296) -- (6.098,6.12) -- (6.282,5.94) -- (6.463,5.756) -- (6.638,5.567) -- (6.81,5.375) -- (6.977,5.178) -- (7.139,4.977) -- (7.296,4.773) -- (7.449,4.565) -- (7.596,4.354) -- (7.738,4.139) -- (7.876,3.921);
\draw[line width=0.5pt] (5.52,6.634) -- (5.586,6.334) -- (5.658,6.036) -- (5.738,5.74) -- (5.825,5.445) -- (5.918,5.153) -- (6.019,4.863) -- (6.126,4.575) -- (6.24,4.29) -- (6.361,4.007) -- (6.488,3.728) -- (6.622,3.452) -- (6.762,3.179) -- (6.908,2.909) -- (7.061,2.643);
\draw[line width=0.5pt] (3.614,5.024) -- (3.884,4.893) -- (4.152,4.755) -- (4.416,4.612) -- (4.676,4.462) -- (4.933,4.306) -- (5.186,4.144) -- (5.436,3.976) -- (5.681,3.802) -- (5.923,3.622) -- (6.159,3.437) -- (6.392,3.247) -- (6.62,3.051) -- (6.843,2.85) -- (7.061,2.643);
\draw[line width=0.5pt] (3.614,5.024) -- (3.429,4.788) -- (3.25,4.548) -- (3.076,4.304) -- (2.908,4.056) -- (2.746,3.804) -- (2.59,3.548) -- (2.44,3.288) -- (2.296,3.025) -- (2.158,2.759) -- (2.026,2.49) -- (1.901,2.217) -- (1.783,1.942) -- (1.671,1.664) -- (1.565,1.384);
\draw[line width=0.5pt] (7.876,3.921) -- (7.832,3.822) -- (7.786,3.723) -- (7.737,3.626) -- (7.687,3.53) -- (7.634,3.435) -- (7.578,3.341) -- (7.521,3.249) -- (7.462,3.158) -- (7.4,3.068) -- (7.336,2.98) -- (7.27,2.893) -- (7.203,2.808) -- (7.133,2.725) -- (7.061,2.643);
\draw[line width=0.5pt] (7.061,2.643) -- (7.364,2.581) -- (7.669,2.527) -- (7.974,2.479) -- (8.281,2.439) -- (8.589,2.406) -- (8.897,2.38) -- (9.206,2.362) -- (9.516,2.35) -- (9.825,2.346) -- (10.135,2.35) -- (10.444,2.36) -- (10.753,2.378) -- (11.061,2.403) -- (11.369,2.436);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.627,10.966) -- (5.605,11.046) -- (5.547,11.104) -- (5.467,11.126) -- (5.387,11.104) -- (5.328,11.046) -- (5.307,10.966) -- (5.328,10.886) -- (5.387,10.827) -- (5.467,10.806) -- (5.547,10.827) -- (5.605,10.886) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.576,9.634) -- (4.554,9.714) -- (4.496,9.773) -- (4.416,9.794) -- (4.336,9.773) -- (4.277,9.714) -- (4.256,9.634) -- (4.277,9.554) -- (4.336,9.495) -- (4.416,9.474) -- (4.496,9.495) -- (4.554,9.554) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.638,6.727) -- (10.617,6.807) -- (10.558,6.866) -- (10.478,6.887) -- (10.398,6.866) -- (10.34,6.807) -- (10.318,6.727) -- (10.34,6.647) -- (10.398,6.589) -- (10.478,6.567) -- (10.558,6.589) -- (10.617,6.647) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.529,6.652) -- (1.507,6.732) -- (1.449,6.791) -- (1.369,6.812) -- (1.289,6.791) -- (1.23,6.732) -- (1.209,6.652) -- (1.23,6.572) -- (1.289,6.514) -- (1.369,6.492) -- (1.449,6.514) -- (1.507,6.572) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.68,6.634) -- (5.659,6.714) -- (5.6,6.772) -- (5.52,6.794) -- (5.44,6.772) -- (5.382,6.714) -- (5.36,6.634) -- (5.382,6.554) -- (5.44,6.495) -- (5.52,6.474) -- (5.6,6.495) -- (5.659,6.554) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.774,5.024) -- (3.753,5.104) -- (3.694,5.162) -- (3.614,5.184) -- (3.534,5.162) -- (3.476,5.104) -- (3.454,5.024) -- (3.476,4.944) -- (3.534,4.885) -- (3.614,4.864) -- (3.694,4.885) -- (3.753,4.944) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.036,3.921) -- (8.014,4.001) -- (7.956,4.06) -- (7.876,4.081) -- (7.796,4.06) -- (7.737,4.001) -- (7.716,3.921) -- (7.737,3.841) -- (7.796,3.783) -- (7.876,3.761) -- (7.956,3.783) -- (8.014,3.841) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.221,2.643) -- (7.2,2.723) -- (7.141,2.782) -- (7.061,2.803) -- (6.981,2.782) -- (6.923,2.723) -- (6.901,2.643) -- (6.923,2.563) -- (6.981,2.505) -- (7.061,2.483) -- (7.141,2.505) -- (7.2,2.563) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.529,2.436) -- (11.507,2.516) -- (11.449,2.574) -- (11.369,2.596) -- (11.289,2.574) -- (11.23,2.516) -- (11.209,2.436) -- (11.23,2.356) -- (11.289,2.297) -- (11.369,2.276) -- (11.449,2.297) -- (11.507,2.356) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.725,1.384) -- (1.704,1.464) -- (1.645,1.522) -- (1.565,1.544) -- (1.485,1.522) -- (1.427,1.464) -- (1.405,1.384) -- (1.427,1.304) -- (1.485,1.245) -- (1.565,1.224) -- (1.645,1.245) -- (1.704,1.304) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.478,10.145) -- (5.432,10.315) -- (5.308,10.439) -- (5.138,10.485) -- (4.968,10.439) -- (4.843,10.315) -- (4.798,10.145) -- (4.843,9.975) -- (4.968,9.85) -- (5.138,9.805) -- (5.308,9.85) -- (5.432,9.975) -- cycle;
\node at (5.138,10.145) {\scalebox{1.429}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.172,8.714) -- (9.126,8.884) -- (9.002,9.008) -- (8.832,9.054) -- (8.662,9.008) -- (8.538,8.884) -- (8.492,8.714) -- (8.538,8.544) -- (8.662,8.419) -- (8.832,8.374) -- (9.002,8.419) -- (9.126,8.544) -- cycle;
\node at (8.832,8.714) {\scalebox{1.429}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.975,8.801) -- (5.929,8.971) -- (5.805,9.096) -- (5.635,9.141) -- (5.465,9.096) -- (5.34,8.971) -- (5.295,8.801) -- (5.34,8.631) -- (5.465,8.507) -- (5.635,8.461) -- (5.805,8.507) -- (5.929,8.631) -- cycle;
\node at (5.635,8.801) {\scalebox{1.429}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.132,8.069) -- (5.086,8.239) -- (4.962,8.363) -- (4.792,8.409) -- (4.622,8.363) -- (4.497,8.239) -- (4.452,8.069) -- (4.497,7.899) -- (4.622,7.774) -- (4.792,7.729) -- (4.962,7.774) -- (5.086,7.899) -- cycle;
\node at (4.792,8.069) {\scalebox{1.429}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.341,6.565) -- (8.296,6.735) -- (8.171,6.86) -- (8.001,6.905) -- (7.831,6.86) -- (7.707,6.735) -- (7.661,6.565) -- (7.707,6.395) -- (7.831,6.271) -- (8.001,6.225) -- (8.171,6.271) -- (8.296,6.395) -- cycle;
\node at (8.001,6.565) {\scalebox{1.429}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.867,4.999) -- (9.822,5.169) -- (9.697,5.294) -- (9.527,5.339) -- (9.357,5.294) -- (9.233,5.169) -- (9.187,4.999) -- (9.233,4.829) -- (9.357,4.705) -- (9.527,4.659) -- (9.697,4.705) -- (9.822,4.829) -- cycle;
\node at (9.527,4.999) {\scalebox{1.429}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.4,4.61) -- (11.354,4.78) -- (11.23,4.904) -- (11.06,4.95) -- (10.89,4.904) -- (10.765,4.78) -- (10.72,4.61) -- (10.765,4.44) -- (10.89,4.315) -- (11.06,4.27) -- (11.23,4.315) -- (11.354,4.44) -- cycle;
\node at (11.06,4.61) {\scalebox{1.429}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.787,7.134) -- (3.741,7.304) -- (3.617,7.429) -- (3.447,7.474) -- (3.277,7.429) -- (3.152,7.304) -- (3.107,7.134) -- (3.152,6.964) -- (3.277,6.84) -- (3.447,6.794) -- (3.617,6.84) -- (3.741,6.964) -- cycle;
\node at (3.447,7.134) {\scalebox{1.429}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.909,4.022) -- (1.864,4.192) -- (1.739,4.316) -- (1.569,4.362) -- (1.399,4.316) -- (1.275,4.192) -- (1.229,4.022) -- (1.275,3.852) -- (1.399,3.727) -- (1.569,3.682) -- (1.739,3.727) -- (1.864,3.852) -- cycle;
\node at (1.569,4.022) {\scalebox{1.429}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.391,5.584) -- (7.346,5.754) -- (7.221,5.879) -- (7.051,5.924) -- (6.881,5.879) -- (6.757,5.754) -- (6.711,5.584) -- (6.757,5.414) -- (6.881,5.29) -- (7.051,5.244) -- (7.221,5.29) -- (7.346,5.414) -- cycle;
\node at (7.051,5.584) {\scalebox{1.429}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.765,4.69) -- (6.719,4.86) -- (6.595,4.985) -- (6.425,5.03) -- (6.255,4.985) -- (6.13,4.86) -- (6.085,4.69) -- (6.13,4.52) -- (6.255,4.396) -- (6.425,4.35) -- (6.595,4.396) -- (6.719,4.52) -- cycle;
\node at (6.425,4.69) {\scalebox{1.429}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.958,4.239) -- (5.912,4.409) -- (5.788,4.533) -- (5.618,4.579) -- (5.448,4.533) -- (5.323,4.409) -- (5.278,4.239) -- (5.323,4.069) -- (5.448,3.944) -- (5.618,3.899) -- (5.788,3.944) -- (5.912,4.069) -- cycle;
\node at (5.618,4.239) {\scalebox{1.429}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.058,3.131) -- (3.013,3.301) -- (2.888,3.426) -- (2.718,3.471) -- (2.548,3.426) -- (2.424,3.301) -- (2.378,3.131) -- (2.424,2.961) -- (2.548,2.837) -- (2.718,2.791) -- (2.888,2.837) -- (3.013,2.961) -- cycle;
\node at (2.718,3.131) {\scalebox{1.429}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.131,3.077) -- (8.085,3.247) -- (7.961,3.371) -- (7.791,3.417) -- (7.621,3.371) -- (7.496,3.247) -- (7.451,3.077) -- (7.496,2.907) -- (7.621,2.782) -- (7.791,2.737) -- (7.961,2.782) -- (8.085,2.907) -- cycle;
\node at (7.791,3.077) {\scalebox{1.429}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.562,2.681) -- (9.516,2.851) -- (9.392,2.976) -- (9.222,3.021) -- (9.052,2.976) -- (8.927,2.851) -- (8.882,2.681) -- (8.927,2.511) -- (9.052,2.387) -- (9.222,2.341) -- (9.392,2.387) -- (9.516,2.511) -- cycle;
\node at (9.222,2.681) {\scalebox{1.429}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.576,3.621) -- (8.176,3.621) -- (8.176,4.221) -- (7.576,4.221) -- cycle;
\node at (7.876,3.921) {\scalebox{1.429}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (5.565,8.504) -- (6.165,8.504) -- (6.165,9.104) -- (5.565,9.104) -- cycle;
\node at (5.865,8.804) {\scalebox{1.429}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.88,10.794) -- (5.48,10.794) -- (5.48,11.394) -- (4.88,11.394) -- cycle;
\node at (5.18,11.094) {\scalebox{1.429}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.617,5.602) -- (9.217,5.602) -- (9.217,6.202) -- (8.617,6.202) -- cycle;
\node at (8.917,5.902) {\scalebox{1.429}{\small d}};
\node at (5.42,11.584) {\scalebox{1.429}{\small A}};
\node at (4.205,10.217) {\scalebox{1.429}{\small B}};
\node at (11.081,6.875) {\scalebox{1.429}{\small C}};
\node at (0.765,6.793) {\scalebox{1.429}{\small D}};
\node at (5.319,7.22) {\scalebox{1.429}{\small E}};
\node at (3.014,4.87) {\scalebox{1.429}{\small F}};
\node at (8.35,3.522) {\scalebox{1.429}{\small G}};
\node at (7.292,2.068) {\scalebox{1.429}{\small H}};
\node at (11.906,2.126) {\scalebox{1.429}{\small I}};
\node at (1.122,0.95) {\scalebox{1.429}{\small J}};
\draw[line width=0.5pt] (12.669,10.016) -- (12.669,11.116);
\draw[line width=0.5pt] (12.438,10.731) -- (12.669,11.116) -- (12.9,10.731);
\draw[line width=0.5pt] (12.366,10.566) -- (12.971,10.566);
\node at (12.669,11.501) {\scalebox{1.429}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le facteur part de la poste et doit emprunter chaque rue exactement une fois. Il n'est pas obligé d'y revenir : il termine sa tournée où il veut.}\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Les rues & \cellcolor{blue!7}\bfseries  & \cellcolor{blue!7}\bfseries Les carrefours & \cellcolor{blue!7}\bfseries Les bâtiments \\ \hline
① passage du Marché & ⑨ rue du Four & Ⓐ Carrefour Saint-Roch & \textbf{a} la poste \\ \hline
② rue du Stade & ⑩ allée des Tilleuls & Ⓑ Place Sainte-Croix & \textbf{b} la bibliothèque \\ \hline
③ rue Neuve & ⑪ rue des Jardins & Ⓒ Rond-point des Vaches & \textbf{c} la mairie \\ \hline
④ quai aux Fleurs & ⑫ rue du Puits & Ⓓ Place aux Herbes & \textbf{d} la salle des fêtes \\ \hline
⑤ boulevard du Parc & ⑬ chemin des Vignes & Ⓔ Rond-point de la Gare &  \\ \hline
⑥ rue des Lilas & ⑭ rue Basse & Ⓕ Place de l'Église &  \\ \hline
⑦ rue du Pont & ⑮ rue des Écoles & Ⓖ Carrefour de la Mairie &  \\ \hline
⑧ rue de la Poste &  & Ⓗ Rond-point de la Poste &  \\ \hline
 &  & Ⓘ Place du Vieux-Puits &  \\ \hline
 &  & Ⓙ Fontaine Notre-Dame &  \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\noindent Les questions se suivent : la première n'est qu'un comptage, la dernière demande une preuve.\par\addvspace{0.45em}
\begin{enumerate}
  \item Pour \textbf{chaque carrefour}, compter le nombre de rues qui y aboutissent. Présenter les résultats dans un tableau.
  \item Le facteur imagine sa tournée. À chaque fois qu'il \textbf{traverse} un carrefour, combien de rues utilise-t-il ? Qu'en déduire pour les carrefours où aboutit un nombre \textbf{impair} de rues ?
  \item \textbf{Le facteur peut-il faire sa tournée en partant de la poste ?} Si oui, décrire un trajet possible ; sinon, démontrer que c'est impossible.
  \item Si ce n'est pas possible : une tournée existe-t-elle \textbf{malgré tout} dans ce quartier, en partant d'ailleurs ? Si oui, où faudrait-il installer la poste — et le facteur y reviendrait-il ?
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{La tournée du facteur — Correction}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6263, line join=round, line cap=round]
\draw[line width=0.5pt] (5.467,10.966) -- (5.378,10.883) -- (5.291,10.798) -- (5.206,10.711) -- (5.123,10.622) -- (5.042,10.531) -- (4.963,10.438) -- (4.886,10.343) -- (4.812,10.247) -- (4.74,10.148) -- (4.67,10.049) -- (4.603,9.947) -- (4.538,9.844) -- (4.476,9.74) -- (4.416,9.634);
\draw[line width=0.5pt] (5.467,10.966) -- (5.715,10.825) -- (5.96,10.679) -- (6.201,10.527) -- (6.439,10.37) -- (6.674,10.206) -- (6.904,10.038) -- (7.13,9.864) -- (7.352,9.684) -- (7.57,9.5) -- (7.784,9.31) -- (7.993,9.116) -- (8.197,8.917) -- (8.396,8.713) -- (8.591,8.504) -- (8.705,8.357) -- (8.823,8.214) -- (8.944,8.072) -- (9.068,7.934) -- (9.196,7.799) -- (9.327,7.667) -- (9.461,7.538) -- (9.598,7.412) -- (9.738,7.289) -- (9.881,7.17) -- (10.026,7.054) -- (10.174,6.941) -- (10.325,6.833) -- (10.478,6.727);
\draw[line width=0.5pt] (5.467,10.966) -- (5.423,10.658) -- (5.387,10.349) -- (5.358,10.04) -- (5.336,9.73) -- (5.322,9.419) -- (5.315,9.108) -- (5.315,8.797) -- (5.322,8.486) -- (5.337,8.176) -- (5.359,7.866) -- (5.388,7.556) -- (5.425,7.248) -- (5.469,6.94) -- (5.52,6.634);
\draw[line width=0.5pt] (4.416,9.634) -- (4.527,9.433) -- (4.633,9.23) -- (4.735,9.025) -- (4.831,8.817) -- (4.923,8.606) -- (5.01,8.394) -- (5.092,8.179) -- (5.169,7.963) -- (5.24,7.745) -- (5.307,7.525) -- (5.368,7.304) -- (5.424,7.082) -- (5.475,6.858) -- (5.52,6.634);
\draw[line width=0.5pt] (10.478,6.727) -- (10.126,6.775) -- (9.773,6.814) -- (9.418,6.845) -- (9.063,6.868) -- (8.707,6.882) -- (8.351,6.888) -- (7.995,6.885) -- (7.64,6.874) -- (7.284,6.855) -- (6.929,6.827) -- (6.575,6.791) -- (6.222,6.747) -- (5.871,6.695) -- (5.52,6.634);
\draw[line width=0.5pt] (10.478,6.727) -- (10.325,6.5) -- (10.166,6.276) -- (10.001,6.056) -- (9.832,5.84) -- (9.657,5.628) -- (9.477,5.421) -- (9.293,5.217) -- (9.104,5.018) -- (8.91,4.823) -- (8.711,4.633) -- (8.509,4.447) -- (8.302,4.267) -- (8.091,4.091) -- (7.876,3.921);
\draw[line width=0.5pt] (10.478,6.727) -- (10.494,6.414) -- (10.518,6.1) -- (10.549,5.787) -- (10.587,5.475) -- (10.633,5.164) -- (10.686,4.854) -- (10.747,4.545) -- (10.814,4.237) -- (10.889,3.932) -- (10.971,3.628) -- (11.06,3.326) -- (11.156,3.027) -- (11.259,2.73) -- (11.369,2.436);
\draw[line width=0.5pt] (1.369,6.652) -- (1.663,6.696) -- (1.959,6.733) -- (2.255,6.764) -- (2.552,6.787) -- (2.849,6.803) -- (3.147,6.812) -- (3.445,6.814) -- (3.743,6.809) -- (4.041,6.798) -- (4.338,6.779) -- (4.635,6.753) -- (4.931,6.72) -- (5.226,6.68) -- (5.52,6.634);
\draw[line width=0.5pt] (1.369,6.652) -- (1.325,6.277) -- (1.29,5.9) -- (1.264,5.523) -- (1.247,5.145) -- (1.239,4.767) -- (1.24,4.388) -- (1.25,4.01) -- (1.268,3.632) -- (1.296,3.254) -- (1.332,2.878) -- (1.377,2.502) -- (1.431,2.128) -- (1.494,1.755) -- (1.565,1.384);
\draw[line width=0.5pt] (5.52,6.634) -- (5.717,6.467) -- (5.909,6.296) -- (6.098,6.12) -- (6.282,5.94) -- (6.463,5.756) -- (6.638,5.567) -- (6.81,5.375) -- (6.977,5.178) -- (7.139,4.977) -- (7.296,4.773) -- (7.449,4.565) -- (7.596,4.354) -- (7.738,4.139) -- (7.876,3.921);
\draw[line width=0.5pt] (5.52,6.634) -- (5.586,6.334) -- (5.658,6.036) -- (5.738,5.74) -- (5.825,5.445) -- (5.918,5.153) -- (6.019,4.863) -- (6.126,4.575) -- (6.24,4.29) -- (6.361,4.007) -- (6.488,3.728) -- (6.622,3.452) -- (6.762,3.179) -- (6.908,2.909) -- (7.061,2.643);
\draw[line width=0.5pt] (3.614,5.024) -- (3.884,4.893) -- (4.152,4.755) -- (4.416,4.612) -- (4.676,4.462) -- (4.933,4.306) -- (5.186,4.144) -- (5.436,3.976) -- (5.681,3.802) -- (5.923,3.622) -- (6.159,3.437) -- (6.392,3.247) -- (6.62,3.051) -- (6.843,2.85) -- (7.061,2.643);
\draw[line width=0.5pt] (3.614,5.024) -- (3.429,4.788) -- (3.25,4.548) -- (3.076,4.304) -- (2.908,4.056) -- (2.746,3.804) -- (2.59,3.548) -- (2.44,3.288) -- (2.296,3.025) -- (2.158,2.759) -- (2.026,2.49) -- (1.901,2.217) -- (1.783,1.942) -- (1.671,1.664) -- (1.565,1.384);
\draw[line width=0.5pt] (7.876,3.921) -- (7.832,3.822) -- (7.786,3.723) -- (7.737,3.626) -- (7.687,3.53) -- (7.634,3.435) -- (7.578,3.341) -- (7.521,3.249) -- (7.462,3.158) -- (7.4,3.068) -- (7.336,2.98) -- (7.27,2.893) -- (7.203,2.808) -- (7.133,2.725) -- (7.061,2.643);
\draw[line width=0.5pt] (7.061,2.643) -- (7.364,2.581) -- (7.669,2.527) -- (7.974,2.479) -- (8.281,2.439) -- (8.589,2.406) -- (8.897,2.38) -- (9.206,2.362) -- (9.516,2.35) -- (9.825,2.346) -- (10.135,2.35) -- (10.444,2.36) -- (10.753,2.378) -- (11.061,2.403) -- (11.369,2.436);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.627,10.966) -- (5.605,11.046) -- (5.547,11.104) -- (5.467,11.126) -- (5.387,11.104) -- (5.328,11.046) -- (5.307,10.966) -- (5.328,10.886) -- (5.387,10.827) -- (5.467,10.806) -- (5.547,10.827) -- (5.605,10.886) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.576,9.634) -- (4.554,9.714) -- (4.496,9.773) -- (4.416,9.794) -- (4.336,9.773) -- (4.277,9.714) -- (4.256,9.634) -- (4.277,9.554) -- (4.336,9.495) -- (4.416,9.474) -- (4.496,9.495) -- (4.554,9.554) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.638,6.727) -- (10.617,6.807) -- (10.558,6.866) -- (10.478,6.887) -- (10.398,6.866) -- (10.34,6.807) -- (10.318,6.727) -- (10.34,6.647) -- (10.398,6.589) -- (10.478,6.567) -- (10.558,6.589) -- (10.617,6.647) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.529,6.652) -- (1.507,6.732) -- (1.449,6.791) -- (1.369,6.812) -- (1.289,6.791) -- (1.23,6.732) -- (1.209,6.652) -- (1.23,6.572) -- (1.289,6.514) -- (1.369,6.492) -- (1.449,6.514) -- (1.507,6.572) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.68,6.634) -- (5.659,6.714) -- (5.6,6.772) -- (5.52,6.794) -- (5.44,6.772) -- (5.382,6.714) -- (5.36,6.634) -- (5.382,6.554) -- (5.44,6.495) -- (5.52,6.474) -- (5.6,6.495) -- (5.659,6.554) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.774,5.024) -- (3.753,5.104) -- (3.694,5.162) -- (3.614,5.184) -- (3.534,5.162) -- (3.476,5.104) -- (3.454,5.024) -- (3.476,4.944) -- (3.534,4.885) -- (3.614,4.864) -- (3.694,4.885) -- (3.753,4.944) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.036,3.921) -- (8.014,4.001) -- (7.956,4.06) -- (7.876,4.081) -- (7.796,4.06) -- (7.737,4.001) -- (7.716,3.921) -- (7.737,3.841) -- (7.796,3.783) -- (7.876,3.761) -- (7.956,3.783) -- (8.014,3.841) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.221,2.643) -- (7.2,2.723) -- (7.141,2.782) -- (7.061,2.803) -- (6.981,2.782) -- (6.923,2.723) -- (6.901,2.643) -- (6.923,2.563) -- (6.981,2.505) -- (7.061,2.483) -- (7.141,2.505) -- (7.2,2.563) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.529,2.436) -- (11.507,2.516) -- (11.449,2.574) -- (11.369,2.596) -- (11.289,2.574) -- (11.23,2.516) -- (11.209,2.436) -- (11.23,2.356) -- (11.289,2.297) -- (11.369,2.276) -- (11.449,2.297) -- (11.507,2.356) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.725,1.384) -- (1.704,1.464) -- (1.645,1.522) -- (1.565,1.544) -- (1.485,1.522) -- (1.427,1.464) -- (1.405,1.384) -- (1.427,1.304) -- (1.485,1.245) -- (1.565,1.224) -- (1.645,1.245) -- (1.704,1.304) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.478,10.145) -- (5.432,10.315) -- (5.308,10.439) -- (5.138,10.485) -- (4.968,10.439) -- (4.843,10.315) -- (4.798,10.145) -- (4.843,9.975) -- (4.968,9.85) -- (5.138,9.805) -- (5.308,9.85) -- (5.432,9.975) -- cycle;
\node at (5.138,10.145) {\scalebox{1.597}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.172,8.714) -- (9.126,8.884) -- (9.002,9.008) -- (8.832,9.054) -- (8.662,9.008) -- (8.538,8.884) -- (8.492,8.714) -- (8.538,8.544) -- (8.662,8.419) -- (8.832,8.374) -- (9.002,8.419) -- (9.126,8.544) -- cycle;
\node at (8.832,8.714) {\scalebox{1.597}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.975,8.801) -- (5.929,8.971) -- (5.805,9.096) -- (5.635,9.141) -- (5.465,9.096) -- (5.34,8.971) -- (5.295,8.801) -- (5.34,8.631) -- (5.465,8.507) -- (5.635,8.461) -- (5.805,8.507) -- (5.929,8.631) -- cycle;
\node at (5.635,8.801) {\scalebox{1.597}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.132,8.069) -- (5.086,8.239) -- (4.962,8.363) -- (4.792,8.409) -- (4.622,8.363) -- (4.497,8.239) -- (4.452,8.069) -- (4.497,7.899) -- (4.622,7.774) -- (4.792,7.729) -- (4.962,7.774) -- (5.086,7.899) -- cycle;
\node at (4.792,8.069) {\scalebox{1.597}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.341,6.565) -- (8.296,6.735) -- (8.171,6.86) -- (8.001,6.905) -- (7.831,6.86) -- (7.707,6.735) -- (7.661,6.565) -- (7.707,6.395) -- (7.831,6.271) -- (8.001,6.225) -- (8.171,6.271) -- (8.296,6.395) -- cycle;
\node at (8.001,6.565) {\scalebox{1.597}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.867,4.999) -- (9.822,5.169) -- (9.697,5.294) -- (9.527,5.339) -- (9.357,5.294) -- (9.233,5.169) -- (9.187,4.999) -- (9.233,4.829) -- (9.357,4.705) -- (9.527,4.659) -- (9.697,4.705) -- (9.822,4.829) -- cycle;
\node at (9.527,4.999) {\scalebox{1.597}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.4,4.61) -- (11.354,4.78) -- (11.23,4.904) -- (11.06,4.95) -- (10.89,4.904) -- (10.765,4.78) -- (10.72,4.61) -- (10.765,4.44) -- (10.89,4.315) -- (11.06,4.27) -- (11.23,4.315) -- (11.354,4.44) -- cycle;
\node at (11.06,4.61) {\scalebox{1.597}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.787,7.134) -- (3.741,7.304) -- (3.617,7.429) -- (3.447,7.474) -- (3.277,7.429) -- (3.152,7.304) -- (3.107,7.134) -- (3.152,6.964) -- (3.277,6.84) -- (3.447,6.794) -- (3.617,6.84) -- (3.741,6.964) -- cycle;
\node at (3.447,7.134) {\scalebox{1.597}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.909,4.022) -- (1.864,4.192) -- (1.739,4.316) -- (1.569,4.362) -- (1.399,4.316) -- (1.275,4.192) -- (1.229,4.022) -- (1.275,3.852) -- (1.399,3.727) -- (1.569,3.682) -- (1.739,3.727) -- (1.864,3.852) -- cycle;
\node at (1.569,4.022) {\scalebox{1.597}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.391,5.584) -- (7.346,5.754) -- (7.221,5.879) -- (7.051,5.924) -- (6.881,5.879) -- (6.757,5.754) -- (6.711,5.584) -- (6.757,5.414) -- (6.881,5.29) -- (7.051,5.244) -- (7.221,5.29) -- (7.346,5.414) -- cycle;
\node at (7.051,5.584) {\scalebox{1.597}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.765,4.69) -- (6.719,4.86) -- (6.595,4.985) -- (6.425,5.03) -- (6.255,4.985) -- (6.13,4.86) -- (6.085,4.69) -- (6.13,4.52) -- (6.255,4.396) -- (6.425,4.35) -- (6.595,4.396) -- (6.719,4.52) -- cycle;
\node at (6.425,4.69) {\scalebox{1.597}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.958,4.239) -- (5.912,4.409) -- (5.788,4.533) -- (5.618,4.579) -- (5.448,4.533) -- (5.323,4.409) -- (5.278,4.239) -- (5.323,4.069) -- (5.448,3.944) -- (5.618,3.899) -- (5.788,3.944) -- (5.912,4.069) -- cycle;
\node at (5.618,4.239) {\scalebox{1.597}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.058,3.131) -- (3.013,3.301) -- (2.888,3.426) -- (2.718,3.471) -- (2.548,3.426) -- (2.424,3.301) -- (2.378,3.131) -- (2.424,2.961) -- (2.548,2.837) -- (2.718,2.791) -- (2.888,2.837) -- (3.013,2.961) -- cycle;
\node at (2.718,3.131) {\scalebox{1.597}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.131,3.077) -- (8.085,3.247) -- (7.961,3.371) -- (7.791,3.417) -- (7.621,3.371) -- (7.496,3.247) -- (7.451,3.077) -- (7.496,2.907) -- (7.621,2.782) -- (7.791,2.737) -- (7.961,2.782) -- (8.085,2.907) -- cycle;
\node at (7.791,3.077) {\scalebox{1.597}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.562,2.681) -- (9.516,2.851) -- (9.392,2.976) -- (9.222,3.021) -- (9.052,2.976) -- (8.927,2.851) -- (8.882,2.681) -- (8.927,2.511) -- (9.052,2.387) -- (9.222,2.341) -- (9.392,2.387) -- (9.516,2.511) -- cycle;
\node at (9.222,2.681) {\scalebox{1.597}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (7.576,3.621) -- (8.176,3.621) -- (8.176,4.221) -- (7.576,4.221) -- cycle;
\node at (7.876,3.921) {\scalebox{1.597}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (5.565,8.504) -- (6.165,8.504) -- (6.165,9.104) -- (5.565,9.104) -- cycle;
\node at (5.865,8.804) {\scalebox{1.597}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (4.88,10.794) -- (5.48,10.794) -- (5.48,11.394) -- (4.88,11.394) -- cycle;
\node at (5.18,11.094) {\scalebox{1.597}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.617,5.602) -- (9.217,5.602) -- (9.217,6.202) -- (8.617,6.202) -- cycle;
\node at (8.917,5.902) {\scalebox{1.597}{\small d}};
\node at (5.42,11.584) {\scalebox{1.597}{\small A}};
\node at (4.205,10.217) {\scalebox{1.597}{\small B}};
\node at (11.081,6.875) {\scalebox{1.597}{\small C}};
\node at (0.765,6.793) {\scalebox{1.597}{\small D}};
\node at (5.319,7.22) {\scalebox{1.597}{\small E}};
\node at (3.014,4.87) {\scalebox{1.597}{\small F}};
\node at (8.35,3.522) {\scalebox{1.597}{\small G}};
\node at (7.292,2.068) {\scalebox{1.597}{\small H}};
\node at (11.906,2.126) {\scalebox{1.597}{\small I}};
\node at (1.122,0.95) {\scalebox{1.597}{\small J}};
\draw[line width=0.5pt] (12.669,10.016) -- (12.669,11.116);
\draw[line width=0.5pt] (12.438,10.731) -- (12.669,11.116) -- (12.9,10.731);
\draw[line width=0.5pt] (12.366,10.566) -- (12.971,10.566);
\node at (12.669,11.501) {\scalebox{1.597}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{1.} On compte les rues aboutissant à chaque carrefour :\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Carrefour & \cellcolor{blue!7}\bfseries A & \cellcolor{blue!7}\bfseries B & \cellcolor{blue!7}\bfseries C & \cellcolor{blue!7}\bfseries D & \cellcolor{blue!7}\bfseries E & \cellcolor{blue!7}\bfseries F & \cellcolor{blue!7}\bfseries G & \cellcolor{blue!7}\bfseries H & \cellcolor{blue!7}\bfseries I & \cellcolor{blue!7}\bfseries J \\ \hline
Nombre de rues & 3 & 2 & 4 & 2 & 6 & 2 & 3 & 4 & 2 & 2 \\ \hline
Parité & \textbf{impair} & pair & pair & pair & pair & pair & \textbf{impair} & pair & pair & pair \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 30 = 2 \(\times\) 15, car chaque rue est comptée à ses \textbf{deux} extrémités.\par\addvspace{0.3em}
\noindent{\color{red!80!black}\bfseries Les carrefours à nombre \textbf{impair} de rues sont : \textbf{A}, \textbf{G} — il y en a 2.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Quand le facteur \textbf{traverse} un carrefour, il y arrive par une rue et en repart par une autre : il consomme les rues \textbf{deux par deux}. Un carrefour traversé 3 fois voit donc 6 de ses rues utilisées — toujours un nombre \textbf{pair}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Un carrefour où aboutit un nombre \textbf{impair} de rues ne peut donc pas être seulement traversé : il doit être le \textbf{départ} ou l'\textbf{arrivée} de la tournée. Là, une rue reste « non appariée ».}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Ici la poste est \textbf{sur un carrefour}, en \textbf{G}, et le facteur n'a pas à y revenir. Ce carrefour est donc le \textbf{départ} — il a droit d'être impair. Mais il n'y a qu'\textbf{une seule arrivée} : tous les autres carrefours doivent être pairs, sauf un au plus.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui.} Deux carrefours sont impairs, \textbf{A} et \textbf{G}, et la poste est justement sur l'un des deux (\textbf{G}). Il part donc de la poste et \textbf{termine en A}, l'autre carrefour impair.}\par\addvspace{0.4em}
\newpage
\noindent Voici une tournée possible :\par\addvspace{0.35em}
\begin{enumerate}
  \item Il part de Carrefour de la Mairie \textbf{(G)}, prend la \textbf{rue des Lilas ⑥} vers le nord-est et atteint Rond-point des Vaches \textbf{(C)}.
  \item Il tourne à gauche dans la \textbf{rue du Stade ②}, vers le nord-ouest, jusqu'à Carrefour Saint-Roch \textbf{(A)}.
  \item Il tourne à gauche dans le \textbf{passage du Marché ①}, vers le sud-ouest, jusqu'à Place Sainte-Croix \textbf{(B)}.
  \item Il tourne à gauche dans le \textbf{quai aux Fleurs ④}, vers le sud, jusqu'à Rond-point de la Gare \textbf{(E)}.
  \item Il tourne à gauche dans le \textbf{boulevard du Parc ⑤}, vers l'est, jusqu'à Rond-point des Vaches \textbf{(C)}.
  \item Il tourne à droite dans la \textbf{rue du Pont ⑦}, vers le sud, jusqu'à Place du Vieux-Puits \textbf{(I)}.
  \item Il tourne à droite dans la \textbf{rue des Écoles ⑮}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(H)}.
  \item Il tourne à droite dans la \textbf{rue des Jardins ⑪}, vers le nord, jusqu'à Rond-point de la Gare \textbf{(E)}.
  \item Il tourne à gauche dans la \textbf{rue de la Poste ⑧}, vers l'ouest, jusqu'à Place aux Herbes \textbf{(D)}.
  \item Il tourne à gauche dans la \textbf{rue du Four ⑨}, vers le sud, jusqu'à Fontaine Notre-Dame \textbf{(J)}.
  \item Il tourne à gauche dans le \textbf{chemin des Vignes ⑬}, vers le nord-est, jusqu'à Place de l'Église \textbf{(F)}.
  \item Il tourne à droite dans la \textbf{rue du Puits ⑫}, vers le sud-est, jusqu'à Rond-point de la Poste \textbf{(H)}.
  \item Il tourne à gauche dans la \textbf{rue Basse ⑭}, vers le nord-est, jusqu'à Carrefour de la Mairie \textbf{(G)}.
  \item Il tourne à gauche dans l'\textbf{allée des Tilleuls ⑩}, vers le nord-ouest, jusqu'à Rond-point de la Gare \textbf{(E)}.
  \item Il tourne à droite dans la \textbf{rue Neuve ③}, vers le nord, jusqu'à Carrefour Saint-Roch \textbf{(A)}.
\end{enumerate}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries C'était déjà possible. Notons seulement que la poste devait être en \textbf{A} ou en \textbf{G} : ailleurs, la tournée n'aurait pas pu commencer.}\par\addvspace{0.4em}
\end{document}
