\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 398}

\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=398}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0398}
\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=0.6993, line join=round, line cap=round]
\draw[line width=0.5pt] (10.171,9.673) -- (9.905,9.702) -- (9.637,9.726) -- (9.37,9.743) -- (9.102,9.754) -- (8.833,9.759) -- (8.565,9.757) -- (8.297,9.749) -- (8.029,9.735) -- (7.761,9.714) -- (7.494,9.687) -- (7.228,9.654) -- (6.963,9.615) -- (6.698,9.569) -- (6.435,9.517);
\draw[line width=0.5pt] (10.171,9.673) -- (10.295,9.44) -- (10.413,9.205) -- (10.526,8.966) -- (10.633,8.725) -- (10.735,8.482) -- (10.83,8.236) -- (10.92,7.988) -- (11.004,7.738) -- (11.083,7.486) -- (11.155,7.233) -- (11.221,6.978) -- (11.281,6.721) -- (11.335,6.463) -- (11.383,6.204);
\draw[line width=0.5pt] (6.435,9.517) -- (6.296,9.462) -- (6.158,9.404) -- (6.022,9.343) -- (5.886,9.278) -- (5.753,9.21) -- (5.621,9.139) -- (5.491,9.065) -- (5.362,8.988) -- (5.236,8.908) -- (5.111,8.825) -- (4.988,8.739) -- (4.867,8.65) -- (4.749,8.559) -- (4.633,8.464);
\draw[line width=0.5pt] (6.435,9.517) -- (6.479,9.308) -- (6.518,9.097) -- (6.552,8.885) -- (6.581,8.673) -- (6.605,8.46) -- (6.624,8.247) -- (6.638,8.033) -- (6.647,7.819) -- (6.651,7.604) -- (6.649,7.39) -- (6.643,7.176) -- (6.632,6.962) -- (6.616,6.749) -- (6.594,6.536);
\draw[line width=0.5pt] (6.435,9.517) -- (6.603,9.418) -- (6.773,9.323) -- (6.945,9.231) -- (7.12,9.144) -- (7.296,9.061) -- (7.475,8.982) -- (7.655,8.908) -- (7.837,8.837) -- (8.02,8.771) -- (8.205,8.71) -- (8.392,8.652) -- (8.579,8.599) -- (8.768,8.551) -- (8.958,8.507) -- (9.155,8.37) -- (9.349,8.229) -- (9.539,8.083) -- (9.726,7.933) -- (9.91,7.778) -- (10.09,7.619) -- (10.266,7.455) -- (10.438,7.288) -- (10.606,7.117) -- (10.77,6.942) -- (10.93,6.763) -- (11.085,6.58) -- (11.237,6.394) -- (11.383,6.204);
\draw[line width=0.5pt] (4.633,8.464) -- (4.793,8.349) -- (4.95,8.23) -- (5.105,8.107) -- (5.256,7.981) -- (5.405,7.851) -- (5.551,7.718) -- (5.693,7.581) -- (5.832,7.441) -- (5.968,7.297) -- (6.1,7.151) -- (6.229,7.002) -- (6.355,6.849) -- (6.476,6.694) -- (6.594,6.536);
\draw[line width=0.5pt] (4.633,8.464) -- (4.378,8.339) -- (4.126,8.208) -- (3.877,8.071) -- (3.631,7.927) -- (3.388,7.779) -- (3.15,7.624) -- (2.914,7.464) -- (2.683,7.298) -- (2.456,7.127) -- (2.233,6.951) -- (2.013,6.77) -- (1.799,6.583) -- (1.589,6.392) -- (1.383,6.195);
\draw[line width=0.5pt] (6.594,6.536) -- (6.937,6.565) -- (7.281,6.585) -- (7.625,6.598) -- (7.969,6.602) -- (8.314,6.599) -- (8.658,6.587) -- (9.002,6.567) -- (9.346,6.54) -- (9.689,6.504) -- (10.03,6.46) -- (10.371,6.408) -- (10.71,6.348) -- (11.048,6.28) -- (11.383,6.204);
\draw[line width=0.5pt] (6.594,6.536) -- (6.221,6.569) -- (5.847,6.593) -- (5.473,6.608) -- (5.098,6.614) -- (4.724,6.612) -- (4.349,6.6) -- (3.975,6.58) -- (3.601,6.552) -- (3.228,6.514) -- (2.856,6.468) -- (2.485,6.413) -- (2.116,6.349) -- (1.749,6.276) -- (1.383,6.195);
\draw[line width=0.5pt] (6.594,6.536) -- (6.759,6.351) -- (6.92,6.162) -- (7.076,5.97) -- (7.228,5.774) -- (7.375,5.574) -- (7.518,5.371) -- (7.655,5.164) -- (7.788,4.955) -- (7.916,4.742) -- (8.038,4.526) -- (8.156,4.308) -- (8.268,4.087) -- (8.375,3.864) -- (8.477,3.638);
\draw[line width=0.5pt] (6.594,6.536) -- (6.491,6.19) -- (6.396,5.841) -- (6.31,5.49) -- (6.232,5.138) -- (6.162,4.783) -- (6.1,4.427) -- (6.047,4.07) -- (6.002,3.711) -- (5.966,3.351) -- (5.938,2.991) -- (5.918,2.63) -- (5.907,2.269) -- (5.905,1.908) -- (5.911,1.547);
\draw[line width=0.5pt] (11.383,6.204) -- (11.403,5.865) -- (11.415,5.525) -- (11.419,5.185) -- (11.415,4.845) -- (11.403,4.505) -- (11.383,4.165) -- (11.355,3.826) -- (11.319,3.487) -- (11.275,3.15) -- (11.224,2.814) -- (11.164,2.479) -- (11.096,2.146) -- (11.021,1.814) -- (10.938,1.484);
\draw[line width=0.5pt] (1.383,6.195) -- (1.617,6.034) -- (1.854,5.879) -- (2.095,5.729) -- (2.34,5.585) -- (2.588,5.446) -- (2.839,5.314) -- (3.094,5.187) -- (3.351,5.066) -- (3.611,4.952) -- (3.873,4.843) -- (4.138,4.741) -- (4.405,4.645) -- (4.675,4.556) -- (4.946,4.473);
\draw[line width=0.5pt] (4.946,4.473) -- (5.047,4.276) -- (5.142,4.077) -- (5.233,3.876) -- (5.32,3.672) -- (5.401,3.466) -- (5.478,3.259) -- (5.549,3.05) -- (5.616,2.839) -- (5.678,2.627) -- (5.735,2.413) -- (5.786,2.198) -- (5.833,1.982) -- (5.875,1.765) -- (5.911,1.547);
\draw[line width=0.5pt] (8.477,3.638) -- (8.272,3.518) -- (8.071,3.393) -- (7.872,3.263) -- (7.676,3.129) -- (7.483,2.99) -- (7.294,2.846) -- (7.108,2.698) -- (6.925,2.546) -- (6.747,2.39) -- (6.572,2.229) -- (6.4,2.064) -- (6.233,1.896) -- (6.07,1.723) -- (5.911,1.547);
\draw[line width=0.5pt] (8.477,3.638) -- (8.675,3.512) -- (8.87,3.382) -- (9.062,3.247) -- (9.251,3.107) -- (9.436,2.963) -- (9.618,2.815) -- (9.796,2.662) -- (9.971,2.506) -- (10.142,2.345) -- (10.309,2.181) -- (10.473,2.012) -- (10.632,1.84) -- (10.787,1.664) -- (10.938,1.484);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.331,9.673) -- (10.31,9.753) -- (10.251,9.811) -- (10.171,9.833) -- (10.091,9.811) -- (10.032,9.753) -- (10.011,9.673) -- (10.032,9.593) -- (10.091,9.534) -- (10.171,9.513) -- (10.251,9.534) -- (10.31,9.593) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.595,9.517) -- (6.574,9.597) -- (6.515,9.656) -- (6.435,9.677) -- (6.355,9.656) -- (6.297,9.597) -- (6.275,9.517) -- (6.297,9.437) -- (6.355,9.378) -- (6.435,9.357) -- (6.515,9.378) -- (6.574,9.437) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.793,8.464) -- (4.771,8.544) -- (4.713,8.603) -- (4.633,8.624) -- (4.553,8.603) -- (4.494,8.544) -- (4.473,8.464) -- (4.494,8.384) -- (4.553,8.326) -- (4.633,8.304) -- (4.713,8.326) -- (4.771,8.384) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.754,6.536) -- (6.733,6.616) -- (6.674,6.674) -- (6.594,6.696) -- (6.514,6.674) -- (6.456,6.616) -- (6.434,6.536) -- (6.456,6.456) -- (6.514,6.397) -- (6.594,6.376) -- (6.674,6.397) -- (6.733,6.456) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.543,6.204) -- (11.522,6.284) -- (11.463,6.343) -- (11.383,6.364) -- (11.303,6.343) -- (11.245,6.284) -- (11.223,6.204) -- (11.245,6.124) -- (11.303,6.065) -- (11.383,6.044) -- (11.463,6.065) -- (11.522,6.124) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.543,6.195) -- (1.522,6.275) -- (1.463,6.334) -- (1.383,6.355) -- (1.303,6.334) -- (1.245,6.275) -- (1.223,6.195) -- (1.245,6.115) -- (1.303,6.057) -- (1.383,6.035) -- (1.463,6.057) -- (1.522,6.115) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.106,4.473) -- (5.085,4.553) -- (5.026,4.611) -- (4.946,4.633) -- (4.866,4.611) -- (4.808,4.553) -- (4.786,4.473) -- (4.808,4.393) -- (4.866,4.334) -- (4.946,4.313) -- (5.026,4.334) -- (5.085,4.393) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.637,3.638) -- (8.616,3.718) -- (8.557,3.776) -- (8.477,3.798) -- (8.397,3.776) -- (8.339,3.718) -- (8.317,3.638) -- (8.339,3.558) -- (8.397,3.499) -- (8.477,3.478) -- (8.557,3.499) -- (8.616,3.558) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.071,1.547) -- (6.05,1.627) -- (5.991,1.686) -- (5.911,1.707) -- (5.831,1.686) -- (5.772,1.627) -- (5.751,1.547) -- (5.772,1.467) -- (5.831,1.409) -- (5.911,1.387) -- (5.991,1.409) -- (6.05,1.467) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.098,1.484) -- (11.076,1.564) -- (11.018,1.623) -- (10.938,1.644) -- (10.858,1.623) -- (10.799,1.564) -- (10.778,1.484) -- (10.799,1.404) -- (10.858,1.346) -- (10.938,1.324) -- (11.018,1.346) -- (11.076,1.404) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.65,9.429) -- (8.604,9.599) -- (8.48,9.724) -- (8.31,9.769) -- (8.14,9.724) -- (8.016,9.599) -- (7.97,9.429) -- (8.016,9.259) -- (8.14,9.135) -- (8.31,9.089) -- (8.48,9.135) -- (8.604,9.259) -- cycle;
\node at (8.31,9.429) {\scalebox{1.43}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.562,8.094) -- (11.517,8.264) -- (11.392,8.388) -- (11.222,8.434) -- (11.052,8.388) -- (10.928,8.264) -- (10.882,8.094) -- (10.928,7.924) -- (11.052,7.799) -- (11.222,7.754) -- (11.392,7.799) -- (11.517,7.924) -- cycle;
\node at (11.222,8.094) {\scalebox{1.43}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.992,8.789) -- (5.946,8.959) -- (5.822,9.083) -- (5.652,9.129) -- (5.482,9.083) -- (5.358,8.959) -- (5.312,8.789) -- (5.358,8.619) -- (5.482,8.494) -- (5.652,8.449) -- (5.822,8.494) -- (5.946,8.619) -- cycle;
\node at (5.652,8.789) {\scalebox{1.43}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.297,8.05) -- (7.252,8.22) -- (7.127,8.344) -- (6.957,8.39) -- (6.787,8.344) -- (6.663,8.22) -- (6.617,8.05) -- (6.663,7.88) -- (6.787,7.755) -- (6.957,7.71) -- (7.127,7.755) -- (7.252,7.88) -- cycle;
\node at (6.957,8.05) {\scalebox{1.43}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.433,8.797) -- (9.388,8.967) -- (9.263,9.091) -- (9.093,9.137) -- (8.923,9.091) -- (8.799,8.967) -- (8.753,8.797) -- (8.799,8.627) -- (8.923,8.502) -- (9.093,8.457) -- (9.263,8.502) -- (9.388,8.627) -- cycle;
\node at (9.093,8.797) {\scalebox{1.43}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.257,7.809) -- (6.212,7.979) -- (6.087,8.103) -- (5.917,8.149) -- (5.747,8.103) -- (5.623,7.979) -- (5.577,7.809) -- (5.623,7.639) -- (5.747,7.515) -- (5.917,7.469) -- (6.087,7.515) -- (6.212,7.639) -- cycle;
\node at (5.917,7.809) {\scalebox{1.43}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.438,7.201) -- (3.392,7.371) -- (3.268,7.496) -- (3.098,7.541) -- (2.928,7.496) -- (2.803,7.371) -- (2.758,7.201) -- (2.803,7.031) -- (2.928,6.907) -- (3.098,6.861) -- (3.268,6.907) -- (3.392,7.031) -- cycle;
\node at (3.098,7.201) {\scalebox{1.43}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.365,6.887) -- (9.319,7.057) -- (9.195,7.181) -- (9.025,7.227) -- (8.855,7.181) -- (8.73,7.057) -- (8.685,6.887) -- (8.73,6.717) -- (8.855,6.592) -- (9.025,6.547) -- (9.195,6.592) -- (9.319,6.717) -- cycle;
\node at (9.025,6.887) {\scalebox{1.43}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.336,6.261) -- (4.29,6.431) -- (4.166,6.556) -- (3.996,6.601) -- (3.826,6.556) -- (3.701,6.431) -- (3.656,6.261) -- (3.701,6.091) -- (3.826,5.967) -- (3.996,5.921) -- (4.166,5.967) -- (4.29,6.091) -- cycle;
\node at (3.996,6.261) {\scalebox{1.43}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.264,5.339) -- (8.218,5.509) -- (8.094,5.633) -- (7.924,5.679) -- (7.754,5.633) -- (7.629,5.509) -- (7.584,5.339) -- (7.629,5.169) -- (7.754,5.044) -- (7.924,4.999) -- (8.094,5.044) -- (8.218,5.169) -- cycle;
\node at (7.924,5.339) {\scalebox{1.43}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.704,4.026) -- (6.658,4.196) -- (6.534,4.321) -- (6.364,4.366) -- (6.194,4.321) -- (6.069,4.196) -- (6.024,4.026) -- (6.069,3.856) -- (6.194,3.732) -- (6.364,3.686) -- (6.534,3.732) -- (6.658,3.856) -- cycle;
\node at (6.364,4.026) {\scalebox{1.43}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (12.014,3.796) -- (11.968,3.966) -- (11.844,4.09) -- (11.674,4.136) -- (11.504,4.09) -- (11.379,3.966) -- (11.334,3.796) -- (11.379,3.626) -- (11.504,3.501) -- (11.674,3.456) -- (11.844,3.501) -- (11.968,3.626) -- cycle;
\node at (11.674,3.796) {\scalebox{1.43}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.573,5.475) -- (3.527,5.645) -- (3.403,5.77) -- (3.233,5.815) -- (3.063,5.77) -- (2.939,5.645) -- (2.893,5.475) -- (2.939,5.305) -- (3.063,5.181) -- (3.233,5.135) -- (3.403,5.181) -- (3.527,5.305) -- cycle;
\node at (3.233,5.475) {\scalebox{1.43}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.193,3.15) -- (6.148,3.32) -- (6.023,3.445) -- (5.853,3.49) -- (5.683,3.445) -- (5.559,3.32) -- (5.513,3.15) -- (5.559,2.98) -- (5.683,2.856) -- (5.853,2.81) -- (6.023,2.856) -- (6.148,2.98) -- cycle;
\node at (5.853,3.15) {\scalebox{1.43}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.65,2.45) -- (7.604,2.62) -- (7.48,2.745) -- (7.31,2.79) -- (7.14,2.745) -- (7.016,2.62) -- (6.97,2.45) -- (7.016,2.28) -- (7.14,2.156) -- (7.31,2.11) -- (7.48,2.156) -- (7.604,2.28) -- cycle;
\node at (7.31,2.45) {\scalebox{1.43}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.347,2.903) -- (10.301,3.073) -- (10.177,3.198) -- (10.007,3.243) -- (9.837,3.198) -- (9.713,3.073) -- (9.667,2.903) -- (9.713,2.733) -- (9.837,2.609) -- (10.007,2.563) -- (10.177,2.609) -- (10.301,2.733) -- cycle;
\node at (10.007,2.903) {\scalebox{1.43}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (6.294,6.236) -- (6.894,6.236) -- (6.894,6.836) -- (6.294,6.836) -- cycle;
\node at (6.594,6.536) {\scalebox{1.43}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.647,7.439) -- (10.247,7.439) -- (10.247,8.039) -- (9.647,8.039) -- cycle;
\node at (9.947,7.739) {\scalebox{1.43}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.27,0.878) -- (7.87,0.878) -- (7.87,1.478) -- (7.27,1.478) -- cycle;
\node at (7.57,1.178) {\scalebox{1.43}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.392,9.934) -- (8.992,9.934) -- (8.992,10.534) -- (8.392,10.534) -- cycle;
\node at (8.692,10.234) {\scalebox{1.43}{\small d}};
\node at (10.556,10.159) {\scalebox{1.43}{\small A}};
\node at (6.329,10.128) {\scalebox{1.43}{\small B}};
\node at (4.215,8.922) {\scalebox{1.43}{\small C}};
\node at (6.258,7.056) {\scalebox{1.43}{\small D}};
\node at (12.0,6.266) {\scalebox{1.43}{\small E}};
\node at (0.765,6.241) {\scalebox{1.43}{\small F}};
\node at (4.416,4.151) {\scalebox{1.43}{\small G}};
\node at (8.815,3.118) {\scalebox{1.43}{\small H}};
\node at (5.745,0.95) {\scalebox{1.43}{\small I}};
\node at (11.352,1.023) {\scalebox{1.43}{\small J}};
\draw[line width=0.5pt] (12.683,8.723) -- (12.683,9.823);
\draw[line width=0.5pt] (12.452,9.438) -- (12.683,9.823) -- (12.914,9.438);
\draw[line width=0.5pt] (12.381,9.273) -- (12.986,9.273);
\node at (12.683,10.208) {\scalebox{1.43}{\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 chemin Creux & Ⓐ Place de la Fontaine & \textbf{a} la poste \\ \hline
② rue de l'Église & ⑩ rue Haute & Ⓑ Carrefour Saint-Roch & \textbf{b} la salle des fêtes \\ \hline
③ quai aux Fleurs & ⑪ venelle du Chat & Ⓒ Carrefour des Tilleuls & \textbf{c} le cimetière \\ \hline
④ rue de la Mairie & ⑫ avenue de la Gare & Ⓓ Carrefour du Lavoir & \textbf{d} la pharmacie \\ \hline
⑤ allée des Peupliers & ⑬ rue de la Poste & Ⓔ Rond-point du Stade &  \\ \hline
⑥ rue du Puits & ⑭ rue Verte & Ⓕ Rond-point des Vaches &  \\ \hline
⑦ allée des Tilleuls & ⑮ boulevard du Parc & Ⓖ Place Sainte-Croix &  \\ \hline
⑧ rue Basse & ⑯ chemin de Ronde & Ⓗ Rond-point de la Gare &  \\ \hline
 &  & Ⓘ Rond-point du Pont &  \\ \hline
 &  & Ⓙ Place aux Herbes &  \\ \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=0.6257, line join=round, line cap=round]
\draw[line width=0.5pt] (10.171,9.673) -- (9.905,9.702) -- (9.637,9.726) -- (9.37,9.743) -- (9.102,9.754) -- (8.833,9.759) -- (8.565,9.757) -- (8.297,9.749) -- (8.029,9.735) -- (7.761,9.714) -- (7.494,9.687) -- (7.228,9.654) -- (6.963,9.615) -- (6.698,9.569) -- (6.435,9.517);
\draw[line width=0.5pt] (10.171,9.673) -- (10.295,9.44) -- (10.413,9.205) -- (10.526,8.966) -- (10.633,8.725) -- (10.735,8.482) -- (10.83,8.236) -- (10.92,7.988) -- (11.004,7.738) -- (11.083,7.486) -- (11.155,7.233) -- (11.221,6.978) -- (11.281,6.721) -- (11.335,6.463) -- (11.383,6.204);
\draw[line width=0.5pt] (6.435,9.517) -- (6.296,9.462) -- (6.158,9.404) -- (6.022,9.343) -- (5.886,9.278) -- (5.753,9.21) -- (5.621,9.139) -- (5.491,9.065) -- (5.362,8.988) -- (5.236,8.908) -- (5.111,8.825) -- (4.988,8.739) -- (4.867,8.65) -- (4.749,8.559) -- (4.633,8.464);
\draw[line width=0.5pt] (6.435,9.517) -- (6.479,9.308) -- (6.518,9.097) -- (6.552,8.885) -- (6.581,8.673) -- (6.605,8.46) -- (6.624,8.247) -- (6.638,8.033) -- (6.647,7.819) -- (6.651,7.604) -- (6.649,7.39) -- (6.643,7.176) -- (6.632,6.962) -- (6.616,6.749) -- (6.594,6.536);
\draw[line width=0.5pt] (6.435,9.517) -- (6.603,9.418) -- (6.773,9.323) -- (6.945,9.231) -- (7.12,9.144) -- (7.296,9.061) -- (7.475,8.982) -- (7.655,8.908) -- (7.837,8.837) -- (8.02,8.771) -- (8.205,8.71) -- (8.392,8.652) -- (8.579,8.599) -- (8.768,8.551) -- (8.958,8.507) -- (9.155,8.37) -- (9.349,8.229) -- (9.539,8.083) -- (9.726,7.933) -- (9.91,7.778) -- (10.09,7.619) -- (10.266,7.455) -- (10.438,7.288) -- (10.606,7.117) -- (10.77,6.942) -- (10.93,6.763) -- (11.085,6.58) -- (11.237,6.394) -- (11.383,6.204);
\draw[line width=0.5pt] (4.633,8.464) -- (4.793,8.349) -- (4.95,8.23) -- (5.105,8.107) -- (5.256,7.981) -- (5.405,7.851) -- (5.551,7.718) -- (5.693,7.581) -- (5.832,7.441) -- (5.968,7.297) -- (6.1,7.151) -- (6.229,7.002) -- (6.355,6.849) -- (6.476,6.694) -- (6.594,6.536);
\draw[line width=0.5pt] (4.633,8.464) -- (4.378,8.339) -- (4.126,8.208) -- (3.877,8.071) -- (3.631,7.927) -- (3.388,7.779) -- (3.15,7.624) -- (2.914,7.464) -- (2.683,7.298) -- (2.456,7.127) -- (2.233,6.951) -- (2.013,6.77) -- (1.799,6.583) -- (1.589,6.392) -- (1.383,6.195);
\draw[line width=0.5pt] (6.594,6.536) -- (6.937,6.565) -- (7.281,6.585) -- (7.625,6.598) -- (7.969,6.602) -- (8.314,6.599) -- (8.658,6.587) -- (9.002,6.567) -- (9.346,6.54) -- (9.689,6.504) -- (10.03,6.46) -- (10.371,6.408) -- (10.71,6.348) -- (11.048,6.28) -- (11.383,6.204);
\draw[line width=0.5pt] (6.594,6.536) -- (6.221,6.569) -- (5.847,6.593) -- (5.473,6.608) -- (5.098,6.614) -- (4.724,6.612) -- (4.349,6.6) -- (3.975,6.58) -- (3.601,6.552) -- (3.228,6.514) -- (2.856,6.468) -- (2.485,6.413) -- (2.116,6.349) -- (1.749,6.276) -- (1.383,6.195);
\draw[line width=0.5pt] (6.594,6.536) -- (6.759,6.351) -- (6.92,6.162) -- (7.076,5.97) -- (7.228,5.774) -- (7.375,5.574) -- (7.518,5.371) -- (7.655,5.164) -- (7.788,4.955) -- (7.916,4.742) -- (8.038,4.526) -- (8.156,4.308) -- (8.268,4.087) -- (8.375,3.864) -- (8.477,3.638);
\draw[line width=0.5pt] (6.594,6.536) -- (6.491,6.19) -- (6.396,5.841) -- (6.31,5.49) -- (6.232,5.138) -- (6.162,4.783) -- (6.1,4.427) -- (6.047,4.07) -- (6.002,3.711) -- (5.966,3.351) -- (5.938,2.991) -- (5.918,2.63) -- (5.907,2.269) -- (5.905,1.908) -- (5.911,1.547);
\draw[line width=0.5pt] (11.383,6.204) -- (11.403,5.865) -- (11.415,5.525) -- (11.419,5.185) -- (11.415,4.845) -- (11.403,4.505) -- (11.383,4.165) -- (11.355,3.826) -- (11.319,3.487) -- (11.275,3.15) -- (11.224,2.814) -- (11.164,2.479) -- (11.096,2.146) -- (11.021,1.814) -- (10.938,1.484);
\draw[line width=0.5pt] (1.383,6.195) -- (1.617,6.034) -- (1.854,5.879) -- (2.095,5.729) -- (2.34,5.585) -- (2.588,5.446) -- (2.839,5.314) -- (3.094,5.187) -- (3.351,5.066) -- (3.611,4.952) -- (3.873,4.843) -- (4.138,4.741) -- (4.405,4.645) -- (4.675,4.556) -- (4.946,4.473);
\draw[line width=0.5pt] (4.946,4.473) -- (5.047,4.276) -- (5.142,4.077) -- (5.233,3.876) -- (5.32,3.672) -- (5.401,3.466) -- (5.478,3.259) -- (5.549,3.05) -- (5.616,2.839) -- (5.678,2.627) -- (5.735,2.413) -- (5.786,2.198) -- (5.833,1.982) -- (5.875,1.765) -- (5.911,1.547);
\draw[line width=0.5pt] (8.477,3.638) -- (8.272,3.518) -- (8.071,3.393) -- (7.872,3.263) -- (7.676,3.129) -- (7.483,2.99) -- (7.294,2.846) -- (7.108,2.698) -- (6.925,2.546) -- (6.747,2.39) -- (6.572,2.229) -- (6.4,2.064) -- (6.233,1.896) -- (6.07,1.723) -- (5.911,1.547);
\draw[line width=0.5pt] (8.477,3.638) -- (8.675,3.512) -- (8.87,3.382) -- (9.062,3.247) -- (9.251,3.107) -- (9.436,2.963) -- (9.618,2.815) -- (9.796,2.662) -- (9.971,2.506) -- (10.142,2.345) -- (10.309,2.181) -- (10.473,2.012) -- (10.632,1.84) -- (10.787,1.664) -- (10.938,1.484);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.331,9.673) -- (10.31,9.753) -- (10.251,9.811) -- (10.171,9.833) -- (10.091,9.811) -- (10.032,9.753) -- (10.011,9.673) -- (10.032,9.593) -- (10.091,9.534) -- (10.171,9.513) -- (10.251,9.534) -- (10.31,9.593) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.595,9.517) -- (6.574,9.597) -- (6.515,9.656) -- (6.435,9.677) -- (6.355,9.656) -- (6.297,9.597) -- (6.275,9.517) -- (6.297,9.437) -- (6.355,9.378) -- (6.435,9.357) -- (6.515,9.378) -- (6.574,9.437) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.793,8.464) -- (4.771,8.544) -- (4.713,8.603) -- (4.633,8.624) -- (4.553,8.603) -- (4.494,8.544) -- (4.473,8.464) -- (4.494,8.384) -- (4.553,8.326) -- (4.633,8.304) -- (4.713,8.326) -- (4.771,8.384) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.754,6.536) -- (6.733,6.616) -- (6.674,6.674) -- (6.594,6.696) -- (6.514,6.674) -- (6.456,6.616) -- (6.434,6.536) -- (6.456,6.456) -- (6.514,6.397) -- (6.594,6.376) -- (6.674,6.397) -- (6.733,6.456) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.543,6.204) -- (11.522,6.284) -- (11.463,6.343) -- (11.383,6.364) -- (11.303,6.343) -- (11.245,6.284) -- (11.223,6.204) -- (11.245,6.124) -- (11.303,6.065) -- (11.383,6.044) -- (11.463,6.065) -- (11.522,6.124) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.543,6.195) -- (1.522,6.275) -- (1.463,6.334) -- (1.383,6.355) -- (1.303,6.334) -- (1.245,6.275) -- (1.223,6.195) -- (1.245,6.115) -- (1.303,6.057) -- (1.383,6.035) -- (1.463,6.057) -- (1.522,6.115) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.106,4.473) -- (5.085,4.553) -- (5.026,4.611) -- (4.946,4.633) -- (4.866,4.611) -- (4.808,4.553) -- (4.786,4.473) -- (4.808,4.393) -- (4.866,4.334) -- (4.946,4.313) -- (5.026,4.334) -- (5.085,4.393) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.637,3.638) -- (8.616,3.718) -- (8.557,3.776) -- (8.477,3.798) -- (8.397,3.776) -- (8.339,3.718) -- (8.317,3.638) -- (8.339,3.558) -- (8.397,3.499) -- (8.477,3.478) -- (8.557,3.499) -- (8.616,3.558) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.071,1.547) -- (6.05,1.627) -- (5.991,1.686) -- (5.911,1.707) -- (5.831,1.686) -- (5.772,1.627) -- (5.751,1.547) -- (5.772,1.467) -- (5.831,1.409) -- (5.911,1.387) -- (5.991,1.409) -- (6.05,1.467) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.098,1.484) -- (11.076,1.564) -- (11.018,1.623) -- (10.938,1.644) -- (10.858,1.623) -- (10.799,1.564) -- (10.778,1.484) -- (10.799,1.404) -- (10.858,1.346) -- (10.938,1.324) -- (11.018,1.346) -- (11.076,1.404) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.65,9.429) -- (8.604,9.599) -- (8.48,9.724) -- (8.31,9.769) -- (8.14,9.724) -- (8.016,9.599) -- (7.97,9.429) -- (8.016,9.259) -- (8.14,9.135) -- (8.31,9.089) -- (8.48,9.135) -- (8.604,9.259) -- cycle;
\node at (8.31,9.429) {\scalebox{1.598}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.562,8.094) -- (11.517,8.264) -- (11.392,8.388) -- (11.222,8.434) -- (11.052,8.388) -- (10.928,8.264) -- (10.882,8.094) -- (10.928,7.924) -- (11.052,7.799) -- (11.222,7.754) -- (11.392,7.799) -- (11.517,7.924) -- cycle;
\node at (11.222,8.094) {\scalebox{1.598}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.992,8.789) -- (5.946,8.959) -- (5.822,9.083) -- (5.652,9.129) -- (5.482,9.083) -- (5.358,8.959) -- (5.312,8.789) -- (5.358,8.619) -- (5.482,8.494) -- (5.652,8.449) -- (5.822,8.494) -- (5.946,8.619) -- cycle;
\node at (5.652,8.789) {\scalebox{1.598}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.297,8.05) -- (7.252,8.22) -- (7.127,8.344) -- (6.957,8.39) -- (6.787,8.344) -- (6.663,8.22) -- (6.617,8.05) -- (6.663,7.88) -- (6.787,7.755) -- (6.957,7.71) -- (7.127,7.755) -- (7.252,7.88) -- cycle;
\node at (6.957,8.05) {\scalebox{1.598}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.433,8.797) -- (9.388,8.967) -- (9.263,9.091) -- (9.093,9.137) -- (8.923,9.091) -- (8.799,8.967) -- (8.753,8.797) -- (8.799,8.627) -- (8.923,8.502) -- (9.093,8.457) -- (9.263,8.502) -- (9.388,8.627) -- cycle;
\node at (9.093,8.797) {\scalebox{1.598}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.257,7.809) -- (6.212,7.979) -- (6.087,8.103) -- (5.917,8.149) -- (5.747,8.103) -- (5.623,7.979) -- (5.577,7.809) -- (5.623,7.639) -- (5.747,7.515) -- (5.917,7.469) -- (6.087,7.515) -- (6.212,7.639) -- cycle;
\node at (5.917,7.809) {\scalebox{1.598}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.438,7.201) -- (3.392,7.371) -- (3.268,7.496) -- (3.098,7.541) -- (2.928,7.496) -- (2.803,7.371) -- (2.758,7.201) -- (2.803,7.031) -- (2.928,6.907) -- (3.098,6.861) -- (3.268,6.907) -- (3.392,7.031) -- cycle;
\node at (3.098,7.201) {\scalebox{1.598}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.365,6.887) -- (9.319,7.057) -- (9.195,7.181) -- (9.025,7.227) -- (8.855,7.181) -- (8.73,7.057) -- (8.685,6.887) -- (8.73,6.717) -- (8.855,6.592) -- (9.025,6.547) -- (9.195,6.592) -- (9.319,6.717) -- cycle;
\node at (9.025,6.887) {\scalebox{1.598}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.336,6.261) -- (4.29,6.431) -- (4.166,6.556) -- (3.996,6.601) -- (3.826,6.556) -- (3.701,6.431) -- (3.656,6.261) -- (3.701,6.091) -- (3.826,5.967) -- (3.996,5.921) -- (4.166,5.967) -- (4.29,6.091) -- cycle;
\node at (3.996,6.261) {\scalebox{1.598}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.264,5.339) -- (8.218,5.509) -- (8.094,5.633) -- (7.924,5.679) -- (7.754,5.633) -- (7.629,5.509) -- (7.584,5.339) -- (7.629,5.169) -- (7.754,5.044) -- (7.924,4.999) -- (8.094,5.044) -- (8.218,5.169) -- cycle;
\node at (7.924,5.339) {\scalebox{1.598}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.704,4.026) -- (6.658,4.196) -- (6.534,4.321) -- (6.364,4.366) -- (6.194,4.321) -- (6.069,4.196) -- (6.024,4.026) -- (6.069,3.856) -- (6.194,3.732) -- (6.364,3.686) -- (6.534,3.732) -- (6.658,3.856) -- cycle;
\node at (6.364,4.026) {\scalebox{1.598}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (12.014,3.796) -- (11.968,3.966) -- (11.844,4.09) -- (11.674,4.136) -- (11.504,4.09) -- (11.379,3.966) -- (11.334,3.796) -- (11.379,3.626) -- (11.504,3.501) -- (11.674,3.456) -- (11.844,3.501) -- (11.968,3.626) -- cycle;
\node at (11.674,3.796) {\scalebox{1.598}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.573,5.475) -- (3.527,5.645) -- (3.403,5.77) -- (3.233,5.815) -- (3.063,5.77) -- (2.939,5.645) -- (2.893,5.475) -- (2.939,5.305) -- (3.063,5.181) -- (3.233,5.135) -- (3.403,5.181) -- (3.527,5.305) -- cycle;
\node at (3.233,5.475) {\scalebox{1.598}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.193,3.15) -- (6.148,3.32) -- (6.023,3.445) -- (5.853,3.49) -- (5.683,3.445) -- (5.559,3.32) -- (5.513,3.15) -- (5.559,2.98) -- (5.683,2.856) -- (5.853,2.81) -- (6.023,2.856) -- (6.148,2.98) -- cycle;
\node at (5.853,3.15) {\scalebox{1.598}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.65,2.45) -- (7.604,2.62) -- (7.48,2.745) -- (7.31,2.79) -- (7.14,2.745) -- (7.016,2.62) -- (6.97,2.45) -- (7.016,2.28) -- (7.14,2.156) -- (7.31,2.11) -- (7.48,2.156) -- (7.604,2.28) -- cycle;
\node at (7.31,2.45) {\scalebox{1.598}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.347,2.903) -- (10.301,3.073) -- (10.177,3.198) -- (10.007,3.243) -- (9.837,3.198) -- (9.713,3.073) -- (9.667,2.903) -- (9.713,2.733) -- (9.837,2.609) -- (10.007,2.563) -- (10.177,2.609) -- (10.301,2.733) -- cycle;
\node at (10.007,2.903) {\scalebox{1.598}{\small 16}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (6.294,6.236) -- (6.894,6.236) -- (6.894,6.836) -- (6.294,6.836) -- cycle;
\node at (6.594,6.536) {\scalebox{1.598}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (9.647,7.439) -- (10.247,7.439) -- (10.247,8.039) -- (9.647,8.039) -- cycle;
\node at (9.947,7.739) {\scalebox{1.598}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (7.27,0.878) -- (7.87,0.878) -- (7.87,1.478) -- (7.27,1.478) -- cycle;
\node at (7.57,1.178) {\scalebox{1.598}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.392,9.934) -- (8.992,9.934) -- (8.992,10.534) -- (8.392,10.534) -- cycle;
\node at (8.692,10.234) {\scalebox{1.598}{\small d}};
\node at (10.556,10.159) {\scalebox{1.598}{\small A}};
\node at (6.329,10.128) {\scalebox{1.598}{\small B}};
\node at (4.215,8.922) {\scalebox{1.598}{\small C}};
\node at (6.258,7.056) {\scalebox{1.598}{\small D}};
\node at (12.0,6.266) {\scalebox{1.598}{\small E}};
\node at (0.765,6.241) {\scalebox{1.598}{\small F}};
\node at (4.416,4.151) {\scalebox{1.598}{\small G}};
\node at (8.815,3.118) {\scalebox{1.598}{\small H}};
\node at (5.745,0.95) {\scalebox{1.598}{\small I}};
\node at (11.352,1.023) {\scalebox{1.598}{\small J}};
\draw[line width=0.5pt] (12.683,8.723) -- (12.683,9.823);
\draw[line width=0.5pt] (12.452,9.438) -- (12.683,9.823) -- (12.914,9.438);
\draw[line width=0.5pt] (12.381,9.273) -- (12.986,9.273);
\node at (12.683,10.208) {\scalebox{1.598}{\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 & 2 & 4 & 3 & 6 & 4 & 3 & 2 & 3 & 3 & 2 \\ \hline
Parité & pair & pair & \textbf{impair} & pair & pair & \textbf{impair} & pair & \textbf{impair} & \textbf{impair} & pair \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 32 = 2 \(\times\) 16, 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{C}, \textbf{F}, \textbf{H}, \textbf{I} — il y en a 4.}\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{D}, 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{Non.} Le quartier compte \textbf{4 carrefours impairs} (C, F, H, I).}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Chacun d'eux exigerait d'être un départ ou une arrivée. Une tournée n'a qu'un départ et qu'une arrivée : au moins 2 d'entre eux resteraient sans solution.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non plus.} Où que l'on installe la poste, il reste 4 carrefours impairs, et une tournée n'en tolère que deux : un pour partir, un pour arriver.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}C'est la différence avec un quartier à deux carrefours impairs, où déplacer la poste suffisait. Ici, le problème ne vient pas du point de départ mais du plan lui-même.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}\textbf{Combien de rues en double, au minimum ?} Il faut apparier les carrefours impairs deux à deux, ce qui demande 1 passage(s) supplémentaire(s) au mieux pour une tournée ouverte, un de plus pour revenir à la poste. C'est le « problème du postier chinois », étudié bien après Euler.}\par\addvspace{0.4em}
\end{document}
