\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 324}

\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=324}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0324}
\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=0.7051, line join=round, line cap=round]
\draw[line width=0.5pt] (5.557,11.412) -- (5.249,11.43) -- (4.941,11.441) -- (4.632,11.444) -- (4.323,11.44) -- (4.014,11.429) -- (3.705,11.411) -- (3.397,11.385) -- (3.09,11.352) -- (2.783,11.312) -- (2.478,11.265) -- (2.174,11.211) -- (1.871,11.149) -- (1.57,11.081) -- (1.271,11.005);
\draw[line width=0.5pt] (5.557,11.412) -- (5.98,11.362) -- (6.401,11.301) -- (6.82,11.231) -- (7.238,11.151) -- (7.654,11.06) -- (8.068,10.961) -- (8.48,10.851) -- (8.888,10.732) -- (9.294,10.603) -- (9.697,10.465) -- (10.096,10.318) -- (10.492,10.161) -- (10.883,9.994) -- (11.271,9.819);
\draw[line width=0.5pt] (5.557,11.412) -- (5.613,11.087) -- (5.676,10.763) -- (5.747,10.44) -- (5.825,10.119) -- (5.911,9.8) -- (6.004,9.484) -- (6.105,9.169) -- (6.213,8.857) -- (6.329,8.547) -- (6.452,8.241) -- (6.581,7.937) -- (6.718,7.637) -- (6.862,7.34) -- (7.013,7.046);
\draw[line width=0.5pt] (1.271,11.005) -- (1.721,10.787) -- (2.166,10.559) -- (2.606,10.32) -- (3.04,10.071) -- (3.468,9.811) -- (3.89,9.542) -- (4.305,9.263) -- (4.714,8.974) -- (5.116,8.675) -- (5.511,8.367) -- (5.898,8.05) -- (6.278,7.724) -- (6.65,7.39) -- (7.013,7.046);
\draw[line width=0.5pt] (1.271,11.005) -- (1.276,10.612) -- (1.291,10.219) -- (1.316,9.827) -- (1.349,9.435) -- (1.392,9.043) -- (1.444,8.653) -- (1.505,8.265) -- (1.575,7.877) -- (1.654,7.492) -- (1.743,7.108) -- (1.84,6.727) -- (1.946,6.348) -- (2.061,5.973) -- (2.185,5.599) -- (2.227,5.303) -- (2.262,5.005) -- (2.291,4.706) -- (2.311,4.407) -- (2.325,4.107) -- (2.332,3.807) -- (2.332,3.507) -- (2.325,3.207) -- (2.311,2.907) -- (2.289,2.608) -- (2.261,2.31) -- (2.226,2.012) -- (2.183,1.715) -- (2.134,1.419);
\draw[line width=0.5pt] (11.271,9.819) -- (11.261,9.59) -- (11.245,9.362) -- (11.224,9.134) -- (11.198,8.907) -- (11.167,8.68) -- (11.13,8.454) -- (11.087,8.229) -- (11.04,8.005) -- (10.987,7.782) -- (10.929,7.56) -- (10.866,7.34) -- (10.798,7.122) -- (10.725,6.905) -- (10.646,6.69);
\draw[line width=0.5pt] (7.013,7.046) -- (7.267,6.981) -- (7.522,6.922) -- (7.778,6.869) -- (8.036,6.822) -- (8.295,6.782) -- (8.555,6.747) -- (8.815,6.718) -- (9.076,6.696) -- (9.338,6.679) -- (9.599,6.669) -- (9.861,6.665) -- (10.123,6.667) -- (10.385,6.675) -- (10.646,6.69);
\draw[line width=0.5pt] (7.013,7.046) -- (7.335,6.693) -- (7.647,6.332) -- (7.952,5.964) -- (8.247,5.589) -- (8.534,5.207) -- (8.811,4.818) -- (9.08,4.423) -- (9.339,4.021) -- (9.589,3.614) -- (9.828,3.201) -- (10.059,2.782) -- (10.279,2.358) -- (10.489,1.93) -- (10.688,1.496);
\draw[line width=0.5pt] (7.013,7.046) -- (6.606,6.701) -- (6.207,6.346) -- (5.816,5.981) -- (5.434,5.608) -- (5.061,5.225) -- (4.697,4.834) -- (4.342,4.434) -- (3.996,4.026) -- (3.66,3.61) -- (3.334,3.186) -- (3.019,2.755) -- (2.713,2.317) -- (2.418,1.871) -- (2.134,1.419);
\draw[line width=0.5pt] (10.646,6.69) -- (10.706,6.322) -- (10.757,5.953) -- (10.799,5.583) -- (10.833,5.212) -- (10.858,4.84) -- (10.874,4.468) -- (10.881,4.095) -- (10.88,3.722) -- (10.87,3.349) -- (10.851,2.977) -- (10.824,2.605) -- (10.787,2.234) -- (10.742,1.865) -- (10.688,1.496);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.717,11.412) -- (5.696,11.492) -- (5.637,11.55) -- (5.557,11.572) -- (5.477,11.55) -- (5.419,11.492) -- (5.397,11.412) -- (5.419,11.332) -- (5.477,11.273) -- (5.557,11.252) -- (5.637,11.273) -- (5.696,11.332) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.431,11.005) -- (1.409,11.085) -- (1.351,11.144) -- (1.271,11.165) -- (1.191,11.144) -- (1.132,11.085) -- (1.111,11.005) -- (1.132,10.925) -- (1.191,10.867) -- (1.271,10.845) -- (1.351,10.867) -- (1.409,10.925) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.431,9.819) -- (11.409,9.899) -- (11.351,9.958) -- (11.271,9.979) -- (11.191,9.958) -- (11.132,9.899) -- (11.111,9.819) -- (11.132,9.739) -- (11.191,9.68) -- (11.271,9.659) -- (11.351,9.68) -- (11.409,9.739) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.173,7.046) -- (7.152,7.126) -- (7.093,7.185) -- (7.013,7.206) -- (6.933,7.185) -- (6.875,7.126) -- (6.853,7.046) -- (6.875,6.966) -- (6.933,6.908) -- (7.013,6.886) -- (7.093,6.908) -- (7.152,6.966) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.806,6.69) -- (10.785,6.77) -- (10.726,6.829) -- (10.646,6.85) -- (10.566,6.829) -- (10.508,6.77) -- (10.486,6.69) -- (10.508,6.61) -- (10.566,6.551) -- (10.646,6.53) -- (10.726,6.551) -- (10.785,6.61) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.848,1.496) -- (10.827,1.576) -- (10.768,1.635) -- (10.688,1.656) -- (10.608,1.635) -- (10.55,1.576) -- (10.528,1.496) -- (10.55,1.416) -- (10.608,1.358) -- (10.688,1.336) -- (10.768,1.358) -- (10.827,1.416) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.294,1.419) -- (2.273,1.499) -- (2.214,1.558) -- (2.134,1.579) -- (2.054,1.558) -- (1.996,1.499) -- (1.974,1.419) -- (1.996,1.339) -- (2.054,1.281) -- (2.134,1.259) -- (2.214,1.281) -- (2.273,1.339) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.767,11.067) -- (3.722,11.237) -- (3.597,11.361) -- (3.427,11.407) -- (3.257,11.361) -- (3.133,11.237) -- (3.087,11.067) -- (3.133,10.897) -- (3.257,10.772) -- (3.427,10.727) -- (3.597,10.772) -- (3.722,10.897) -- cycle;
\node at (3.427,11.067) {\scalebox{1.418}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.906,11.159) -- (8.86,11.329) -- (8.736,11.454) -- (8.566,11.499) -- (8.396,11.454) -- (8.271,11.329) -- (8.226,11.159) -- (8.271,10.989) -- (8.396,10.865) -- (8.566,10.819) -- (8.736,10.865) -- (8.86,10.989) -- cycle;
\node at (8.566,11.159) {\scalebox{1.418}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.749,9.27) -- (6.703,9.44) -- (6.579,9.565) -- (6.409,9.61) -- (6.239,9.565) -- (6.114,9.44) -- (6.069,9.27) -- (6.114,9.1) -- (6.239,8.976) -- (6.409,8.93) -- (6.579,8.976) -- (6.703,9.1) -- cycle;
\node at (6.409,9.27) {\scalebox{1.418}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.827,9.526) -- (4.781,9.696) -- (4.657,9.82) -- (4.487,9.866) -- (4.317,9.82) -- (4.193,9.696) -- (4.147,9.526) -- (4.193,9.356) -- (4.317,9.232) -- (4.487,9.186) -- (4.657,9.232) -- (4.781,9.356) -- cycle;
\node at (4.487,9.526) {\scalebox{1.418}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.836,5.676) -- (2.79,5.846) -- (2.666,5.971) -- (2.496,6.016) -- (2.326,5.971) -- (2.201,5.846) -- (2.156,5.676) -- (2.201,5.506) -- (2.326,5.382) -- (2.496,5.336) -- (2.666,5.382) -- (2.79,5.506) -- cycle;
\node at (2.496,5.676) {\scalebox{1.418}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.741,8.166) -- (11.696,8.336) -- (11.571,8.461) -- (11.401,8.506) -- (11.231,8.461) -- (11.107,8.336) -- (11.061,8.166) -- (11.107,7.996) -- (11.231,7.872) -- (11.401,7.826) -- (11.571,7.872) -- (11.696,7.996) -- cycle;
\node at (11.401,8.166) {\scalebox{1.418}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.186,7.037) -- (9.141,7.207) -- (9.016,7.331) -- (8.846,7.377) -- (8.676,7.331) -- (8.552,7.207) -- (8.506,7.037) -- (8.552,6.867) -- (8.676,6.742) -- (8.846,6.697) -- (9.016,6.742) -- (9.141,6.867) -- cycle;
\node at (8.846,7.037) {\scalebox{1.418}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.687,4.599) -- (9.641,4.769) -- (9.517,4.894) -- (9.347,4.939) -- (9.177,4.894) -- (9.052,4.769) -- (9.007,4.599) -- (9.052,4.429) -- (9.177,4.305) -- (9.347,4.259) -- (9.517,4.305) -- (9.641,4.429) -- cycle;
\node at (9.347,4.599) {\scalebox{1.418}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.923,4.224) -- (4.878,4.394) -- (4.753,4.519) -- (4.583,4.564) -- (4.413,4.519) -- (4.289,4.394) -- (4.243,4.224) -- (4.289,4.054) -- (4.413,3.93) -- (4.583,3.884) -- (4.753,3.93) -- (4.878,4.054) -- cycle;
\node at (4.583,4.224) {\scalebox{1.418}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.541,4.097) -- (11.496,4.267) -- (11.371,4.392) -- (11.201,4.437) -- (11.031,4.392) -- (10.907,4.267) -- (10.861,4.097) -- (10.907,3.927) -- (11.031,3.803) -- (11.201,3.757) -- (11.371,3.803) -- (11.496,3.927) -- cycle;
\node at (11.201,4.097) {\scalebox{1.418}{\small 10}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (5.257,11.112) -- (5.857,11.112) -- (5.857,11.712) -- (5.257,11.712) -- cycle;
\node at (5.557,11.412) {\scalebox{1.418}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (6.182,10.133) -- (6.782,10.133) -- (6.782,10.733) -- (6.182,10.733) -- cycle;
\node at (6.482,10.433) {\scalebox{1.418}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (3.797,2.894) -- (4.397,2.894) -- (4.397,3.494) -- (3.797,3.494) -- cycle;
\node at (4.097,3.194) {\scalebox{1.418}{\small c}};
\node at (5.373,12.004) {\scalebox{1.418}{\small A}};
\node at (0.765,11.364) {\scalebox{1.418}{\small B}};
\node at (11.789,10.159) {\scalebox{1.418}{\small C}};
\node at (7.486,7.448) {\scalebox{1.418}{\small D}};
\node at (11.264,6.641) {\scalebox{1.418}{\small E}};
\node at (11.038,0.984) {\scalebox{1.418}{\small F}};
\node at (1.729,0.95) {\scalebox{1.418}{\small G}};
\draw[line width=0.5pt] (12.571,10.462) -- (12.571,11.562);
\draw[line width=0.5pt] (12.34,11.177) -- (12.571,11.562) -- (12.802,11.177);
\draw[line width=0.5pt] (12.268,11.012) -- (12.873,11.012);
\node at (12.571,11.947) {\scalebox{1.418}{\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 & ⑥ chemin de Ronde & Ⓐ Rond-point du Stade & \textbf{a} la poste \\ \hline
② rue de la Poste & ⑦ rue du Puits & Ⓑ Carrefour du Moulin & \textbf{b} le gymnase \\ \hline
③ chemin des Vignes & ⑧ rue du Four & Ⓒ Place du Marché & \textbf{c} la gendarmerie \\ \hline
④ avenue de la Gare & ⑨ rue Pasteur & Ⓓ Carrefour Saint-Roch &  \\ \hline
⑤ rue Verte & ⑩ allée des Peupliers & Ⓔ Rond-point de la Poste &  \\ \hline
 &  & Ⓕ Place aux Herbes &  \\ \hline
 &  & Ⓖ Place des Tanneurs &  \\ \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=0.6309, line join=round, line cap=round]
\draw[line width=0.5pt] (5.557,11.412) -- (5.249,11.43) -- (4.941,11.441) -- (4.632,11.444) -- (4.323,11.44) -- (4.014,11.429) -- (3.705,11.411) -- (3.397,11.385) -- (3.09,11.352) -- (2.783,11.312) -- (2.478,11.265) -- (2.174,11.211) -- (1.871,11.149) -- (1.57,11.081) -- (1.271,11.005);
\draw[line width=0.5pt] (5.557,11.412) -- (5.98,11.362) -- (6.401,11.301) -- (6.82,11.231) -- (7.238,11.151) -- (7.654,11.06) -- (8.068,10.961) -- (8.48,10.851) -- (8.888,10.732) -- (9.294,10.603) -- (9.697,10.465) -- (10.096,10.318) -- (10.492,10.161) -- (10.883,9.994) -- (11.271,9.819);
\draw[line width=0.5pt] (5.557,11.412) -- (5.613,11.087) -- (5.676,10.763) -- (5.747,10.44) -- (5.825,10.119) -- (5.911,9.8) -- (6.004,9.484) -- (6.105,9.169) -- (6.213,8.857) -- (6.329,8.547) -- (6.452,8.241) -- (6.581,7.937) -- (6.718,7.637) -- (6.862,7.34) -- (7.013,7.046);
\draw[line width=0.5pt] (1.271,11.005) -- (1.721,10.787) -- (2.166,10.559) -- (2.606,10.32) -- (3.04,10.071) -- (3.468,9.811) -- (3.89,9.542) -- (4.305,9.263) -- (4.714,8.974) -- (5.116,8.675) -- (5.511,8.367) -- (5.898,8.05) -- (6.278,7.724) -- (6.65,7.39) -- (7.013,7.046);
\draw[line width=0.5pt] (1.271,11.005) -- (1.276,10.612) -- (1.291,10.219) -- (1.316,9.827) -- (1.349,9.435) -- (1.392,9.043) -- (1.444,8.653) -- (1.505,8.265) -- (1.575,7.877) -- (1.654,7.492) -- (1.743,7.108) -- (1.84,6.727) -- (1.946,6.348) -- (2.061,5.973) -- (2.185,5.599) -- (2.227,5.303) -- (2.262,5.005) -- (2.291,4.706) -- (2.311,4.407) -- (2.325,4.107) -- (2.332,3.807) -- (2.332,3.507) -- (2.325,3.207) -- (2.311,2.907) -- (2.289,2.608) -- (2.261,2.31) -- (2.226,2.012) -- (2.183,1.715) -- (2.134,1.419);
\draw[line width=0.5pt] (11.271,9.819) -- (11.261,9.59) -- (11.245,9.362) -- (11.224,9.134) -- (11.198,8.907) -- (11.167,8.68) -- (11.13,8.454) -- (11.087,8.229) -- (11.04,8.005) -- (10.987,7.782) -- (10.929,7.56) -- (10.866,7.34) -- (10.798,7.122) -- (10.725,6.905) -- (10.646,6.69);
\draw[line width=0.5pt] (7.013,7.046) -- (7.267,6.981) -- (7.522,6.922) -- (7.778,6.869) -- (8.036,6.822) -- (8.295,6.782) -- (8.555,6.747) -- (8.815,6.718) -- (9.076,6.696) -- (9.338,6.679) -- (9.599,6.669) -- (9.861,6.665) -- (10.123,6.667) -- (10.385,6.675) -- (10.646,6.69);
\draw[line width=0.5pt] (7.013,7.046) -- (7.335,6.693) -- (7.647,6.332) -- (7.952,5.964) -- (8.247,5.589) -- (8.534,5.207) -- (8.811,4.818) -- (9.08,4.423) -- (9.339,4.021) -- (9.589,3.614) -- (9.828,3.201) -- (10.059,2.782) -- (10.279,2.358) -- (10.489,1.93) -- (10.688,1.496);
\draw[line width=0.5pt] (7.013,7.046) -- (6.606,6.701) -- (6.207,6.346) -- (5.816,5.981) -- (5.434,5.608) -- (5.061,5.225) -- (4.697,4.834) -- (4.342,4.434) -- (3.996,4.026) -- (3.66,3.61) -- (3.334,3.186) -- (3.019,2.755) -- (2.713,2.317) -- (2.418,1.871) -- (2.134,1.419);
\draw[line width=0.5pt] (10.646,6.69) -- (10.706,6.322) -- (10.757,5.953) -- (10.799,5.583) -- (10.833,5.212) -- (10.858,4.84) -- (10.874,4.468) -- (10.881,4.095) -- (10.88,3.722) -- (10.87,3.349) -- (10.851,2.977) -- (10.824,2.605) -- (10.787,2.234) -- (10.742,1.865) -- (10.688,1.496);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.717,11.412) -- (5.696,11.492) -- (5.637,11.55) -- (5.557,11.572) -- (5.477,11.55) -- (5.419,11.492) -- (5.397,11.412) -- (5.419,11.332) -- (5.477,11.273) -- (5.557,11.252) -- (5.637,11.273) -- (5.696,11.332) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.431,11.005) -- (1.409,11.085) -- (1.351,11.144) -- (1.271,11.165) -- (1.191,11.144) -- (1.132,11.085) -- (1.111,11.005) -- (1.132,10.925) -- (1.191,10.867) -- (1.271,10.845) -- (1.351,10.867) -- (1.409,10.925) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.431,9.819) -- (11.409,9.899) -- (11.351,9.958) -- (11.271,9.979) -- (11.191,9.958) -- (11.132,9.899) -- (11.111,9.819) -- (11.132,9.739) -- (11.191,9.68) -- (11.271,9.659) -- (11.351,9.68) -- (11.409,9.739) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.173,7.046) -- (7.152,7.126) -- (7.093,7.185) -- (7.013,7.206) -- (6.933,7.185) -- (6.875,7.126) -- (6.853,7.046) -- (6.875,6.966) -- (6.933,6.908) -- (7.013,6.886) -- (7.093,6.908) -- (7.152,6.966) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.806,6.69) -- (10.785,6.77) -- (10.726,6.829) -- (10.646,6.85) -- (10.566,6.829) -- (10.508,6.77) -- (10.486,6.69) -- (10.508,6.61) -- (10.566,6.551) -- (10.646,6.53) -- (10.726,6.551) -- (10.785,6.61) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.848,1.496) -- (10.827,1.576) -- (10.768,1.635) -- (10.688,1.656) -- (10.608,1.635) -- (10.55,1.576) -- (10.528,1.496) -- (10.55,1.416) -- (10.608,1.358) -- (10.688,1.336) -- (10.768,1.358) -- (10.827,1.416) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.294,1.419) -- (2.273,1.499) -- (2.214,1.558) -- (2.134,1.579) -- (2.054,1.558) -- (1.996,1.499) -- (1.974,1.419) -- (1.996,1.339) -- (2.054,1.281) -- (2.134,1.259) -- (2.214,1.281) -- (2.273,1.339) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.767,11.067) -- (3.722,11.237) -- (3.597,11.361) -- (3.427,11.407) -- (3.257,11.361) -- (3.133,11.237) -- (3.087,11.067) -- (3.133,10.897) -- (3.257,10.772) -- (3.427,10.727) -- (3.597,10.772) -- (3.722,10.897) -- cycle;
\node at (3.427,11.067) {\scalebox{1.585}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.906,11.159) -- (8.86,11.329) -- (8.736,11.454) -- (8.566,11.499) -- (8.396,11.454) -- (8.271,11.329) -- (8.226,11.159) -- (8.271,10.989) -- (8.396,10.865) -- (8.566,10.819) -- (8.736,10.865) -- (8.86,10.989) -- cycle;
\node at (8.566,11.159) {\scalebox{1.585}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.749,9.27) -- (6.703,9.44) -- (6.579,9.565) -- (6.409,9.61) -- (6.239,9.565) -- (6.114,9.44) -- (6.069,9.27) -- (6.114,9.1) -- (6.239,8.976) -- (6.409,8.93) -- (6.579,8.976) -- (6.703,9.1) -- cycle;
\node at (6.409,9.27) {\scalebox{1.585}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.827,9.526) -- (4.781,9.696) -- (4.657,9.82) -- (4.487,9.866) -- (4.317,9.82) -- (4.193,9.696) -- (4.147,9.526) -- (4.193,9.356) -- (4.317,9.232) -- (4.487,9.186) -- (4.657,9.232) -- (4.781,9.356) -- cycle;
\node at (4.487,9.526) {\scalebox{1.585}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.836,5.676) -- (2.79,5.846) -- (2.666,5.971) -- (2.496,6.016) -- (2.326,5.971) -- (2.201,5.846) -- (2.156,5.676) -- (2.201,5.506) -- (2.326,5.382) -- (2.496,5.336) -- (2.666,5.382) -- (2.79,5.506) -- cycle;
\node at (2.496,5.676) {\scalebox{1.585}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.741,8.166) -- (11.696,8.336) -- (11.571,8.461) -- (11.401,8.506) -- (11.231,8.461) -- (11.107,8.336) -- (11.061,8.166) -- (11.107,7.996) -- (11.231,7.872) -- (11.401,7.826) -- (11.571,7.872) -- (11.696,7.996) -- cycle;
\node at (11.401,8.166) {\scalebox{1.585}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.186,7.037) -- (9.141,7.207) -- (9.016,7.331) -- (8.846,7.377) -- (8.676,7.331) -- (8.552,7.207) -- (8.506,7.037) -- (8.552,6.867) -- (8.676,6.742) -- (8.846,6.697) -- (9.016,6.742) -- (9.141,6.867) -- cycle;
\node at (8.846,7.037) {\scalebox{1.585}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.687,4.599) -- (9.641,4.769) -- (9.517,4.894) -- (9.347,4.939) -- (9.177,4.894) -- (9.052,4.769) -- (9.007,4.599) -- (9.052,4.429) -- (9.177,4.305) -- (9.347,4.259) -- (9.517,4.305) -- (9.641,4.429) -- cycle;
\node at (9.347,4.599) {\scalebox{1.585}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.923,4.224) -- (4.878,4.394) -- (4.753,4.519) -- (4.583,4.564) -- (4.413,4.519) -- (4.289,4.394) -- (4.243,4.224) -- (4.289,4.054) -- (4.413,3.93) -- (4.583,3.884) -- (4.753,3.93) -- (4.878,4.054) -- cycle;
\node at (4.583,4.224) {\scalebox{1.585}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.541,4.097) -- (11.496,4.267) -- (11.371,4.392) -- (11.201,4.437) -- (11.031,4.392) -- (10.907,4.267) -- (10.861,4.097) -- (10.907,3.927) -- (11.031,3.803) -- (11.201,3.757) -- (11.371,3.803) -- (11.496,3.927) -- cycle;
\node at (11.201,4.097) {\scalebox{1.585}{\small 10}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (5.257,11.112) -- (5.857,11.112) -- (5.857,11.712) -- (5.257,11.712) -- cycle;
\node at (5.557,11.412) {\scalebox{1.585}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (6.182,10.133) -- (6.782,10.133) -- (6.782,10.733) -- (6.182,10.733) -- cycle;
\node at (6.482,10.433) {\scalebox{1.585}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (3.797,2.894) -- (4.397,2.894) -- (4.397,3.494) -- (3.797,3.494) -- cycle;
\node at (4.097,3.194) {\scalebox{1.585}{\small c}};
\node at (5.373,12.004) {\scalebox{1.585}{\small A}};
\node at (0.765,11.364) {\scalebox{1.585}{\small B}};
\node at (11.789,10.159) {\scalebox{1.585}{\small C}};
\node at (7.486,7.448) {\scalebox{1.585}{\small D}};
\node at (11.264,6.641) {\scalebox{1.585}{\small E}};
\node at (11.038,0.984) {\scalebox{1.585}{\small F}};
\node at (1.729,0.95) {\scalebox{1.585}{\small G}};
\draw[line width=0.5pt] (12.571,10.462) -- (12.571,11.562);
\draw[line width=0.5pt] (12.34,11.177) -- (12.571,11.562) -- (12.802,11.177);
\draw[line width=0.5pt] (12.268,11.012) -- (12.873,11.012);
\node at (12.571,11.947) {\scalebox{1.585}{\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|}
\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 \\ \hline
Nombre de rues & 3 & 3 & 2 & 5 & 3 & 2 & 2 \\ \hline
Parité & \textbf{impair} & \textbf{impair} & pair & \textbf{impair} & \textbf{impair} & pair & pair \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 20 = 2 \(\times\) 10, 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{B}, \textbf{D}, \textbf{E} — 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{A}, 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} (A, B, D, E).}\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}
