\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 289}

\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=289}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0289}
\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=0.7032, line join=round, line cap=round]
\draw[line width=0.5pt] (2.495,9.693) -- (2.671,9.677) -- (2.848,9.664) -- (3.025,9.656) -- (3.202,9.653) -- (3.379,9.653) -- (3.556,9.657) -- (3.734,9.666) -- (3.91,9.679) -- (4.087,9.696) -- (4.263,9.717) -- (4.438,9.742) -- (4.613,9.771) -- (4.787,9.805) -- (4.96,9.842) -- (5.159,9.838) -- (5.359,9.829) -- (5.558,9.815) -- (5.756,9.797) -- (5.955,9.774) -- (6.152,9.746) -- (6.349,9.714) -- (6.545,9.677) -- (6.741,9.635) -- (6.935,9.589) -- (7.128,9.538) -- (7.32,9.483) -- (7.51,9.423) -- (7.699,9.359);
\draw[line width=0.5pt] (2.495,9.693) -- (2.461,9.392) -- (2.419,9.093) -- (2.371,8.794) -- (2.316,8.496) -- (2.253,8.2) -- (2.184,7.905) -- (2.108,7.612) -- (2.025,7.321) -- (1.935,7.032) -- (1.838,6.745) -- (1.735,6.46) -- (1.625,6.178) -- (1.508,5.899) -- (1.385,5.623);
\draw[line width=0.5pt] (2.495,9.693) -- (2.584,9.382) -- (2.682,9.073) -- (2.786,8.766) -- (2.898,8.462) -- (3.017,8.161) -- (3.143,7.862) -- (3.275,7.566) -- (3.415,7.274) -- (3.562,6.985) -- (3.715,6.699) -- (3.875,6.417) -- (4.042,6.139) -- (4.215,5.866) -- (4.394,5.596);
\draw[line width=0.5pt] (7.699,9.359) -- (7.986,9.195) -- (8.269,9.025) -- (8.548,8.848) -- (8.823,8.664) -- (9.094,8.474) -- (9.36,8.277) -- (9.622,8.074) -- (9.878,7.865) -- (10.13,7.651) -- (10.376,7.43) -- (10.618,7.204) -- (10.853,6.972) -- (11.083,6.734) -- (11.308,6.492);
\draw[line width=0.5pt] (7.699,9.359) -- (7.663,9.101) -- (7.633,8.842) -- (7.609,8.582) -- (7.592,8.321) -- (7.58,8.06) -- (7.575,7.799) -- (7.576,7.538) -- (7.582,7.277) -- (7.596,7.016) -- (7.615,6.756) -- (7.64,6.496) -- (7.671,6.237) -- (7.709,5.979) -- (7.753,5.721);
\draw[line width=0.5pt] (7.699,9.359) -- (7.506,9.056) -- (7.306,8.758) -- (7.098,8.465) -- (6.884,8.176) -- (6.663,7.892) -- (6.436,7.614) -- (6.202,7.341) -- (5.961,7.074) -- (5.715,6.812) -- (5.462,6.557) -- (5.204,6.307) -- (4.939,6.064) -- (4.669,5.827) -- (4.394,5.596);
\draw[line width=0.5pt] (11.308,6.492) -- (11.08,6.259) -- (10.858,6.02) -- (10.642,5.777) -- (10.431,5.528) -- (10.227,5.274) -- (10.028,5.016) -- (9.835,4.753) -- (9.649,4.486) -- (9.469,4.214) -- (9.295,3.938) -- (9.128,3.658) -- (8.968,3.375) -- (8.814,3.087) -- (8.668,2.797);
\draw[line width=0.5pt] (7.753,5.721) -- (7.516,5.676) -- (7.278,5.636) -- (7.04,5.601) -- (6.8,5.573) -- (6.56,5.549) -- (6.319,5.532) -- (6.079,5.52) -- (5.837,5.514) -- (5.596,5.514) -- (5.355,5.519) -- (5.114,5.529) -- (4.873,5.546) -- (4.633,5.568) -- (4.394,5.596);
\draw[line width=0.5pt] (7.753,5.721) -- (7.785,5.504) -- (7.823,5.288) -- (7.867,5.072) -- (7.915,4.858) -- (7.968,4.644) -- (8.026,4.432) -- (8.089,4.221) -- (8.158,4.012) -- (8.231,3.805) -- (8.309,3.599) -- (8.391,3.395) -- (8.479,3.194) -- (8.571,2.994) -- (8.668,2.797);
\draw[line width=0.5pt] (1.385,5.623) -- (1.598,5.654) -- (1.813,5.68) -- (2.028,5.701) -- (2.243,5.717) -- (2.459,5.727) -- (2.675,5.733) -- (2.891,5.734) -- (3.107,5.729) -- (3.322,5.72) -- (3.538,5.705) -- (3.753,5.685) -- (3.967,5.661) -- (4.181,5.631) -- (4.394,5.596);
\draw[line width=0.5pt] (1.385,5.623) -- (1.344,5.395) -- (1.309,5.166) -- (1.279,4.936) -- (1.254,4.705) -- (1.235,4.474) -- (1.221,4.242) -- (1.213,4.01) -- (1.21,3.778) -- (1.213,3.546) -- (1.221,3.315) -- (1.235,3.083) -- (1.253,2.852) -- (1.278,2.621) -- (1.308,2.392);
\draw[line width=0.5pt] (4.394,5.596) -- (4.728,5.444) -- (5.057,5.285) -- (5.384,5.118) -- (5.706,4.943) -- (6.024,4.76) -- (6.337,4.57) -- (6.646,4.373) -- (6.951,4.168) -- (7.25,3.957) -- (7.545,3.738) -- (7.834,3.513) -- (8.117,3.281) -- (8.396,3.042) -- (8.668,2.797);
\draw[line width=0.5pt] (4.394,5.596) -- (4.392,5.379) -- (4.396,5.161) -- (4.404,4.944) -- (4.418,4.727) -- (4.436,4.51) -- (4.46,4.294) -- (4.489,4.078) -- (4.523,3.863) -- (4.562,3.649) -- (4.606,3.436) -- (4.655,3.224) -- (4.709,3.014) -- (4.768,2.804) -- (4.832,2.596);
\draw[line width=0.5pt] (8.668,2.797) -- (8.398,2.741) -- (8.127,2.691) -- (7.855,2.647) -- (7.582,2.611) -- (7.308,2.58) -- (7.033,2.556) -- (6.758,2.538) -- (6.482,2.527) -- (6.207,2.523) -- (5.931,2.524) -- (5.656,2.533) -- (5.38,2.547) -- (5.106,2.569) -- (4.832,2.596);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.655,9.693) -- (2.633,9.773) -- (2.575,9.832) -- (2.495,9.853) -- (2.415,9.832) -- (2.356,9.773) -- (2.335,9.693) -- (2.356,9.613) -- (2.415,9.554) -- (2.495,9.533) -- (2.575,9.554) -- (2.633,9.613) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.859,9.359) -- (7.837,9.439) -- (7.779,9.498) -- (7.699,9.519) -- (7.619,9.498) -- (7.56,9.439) -- (7.539,9.359) -- (7.56,9.279) -- (7.619,9.221) -- (7.699,9.199) -- (7.779,9.221) -- (7.837,9.279) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.468,6.492) -- (11.446,6.572) -- (11.388,6.63) -- (11.308,6.652) -- (11.228,6.63) -- (11.169,6.572) -- (11.148,6.492) -- (11.169,6.412) -- (11.228,6.353) -- (11.308,6.332) -- (11.388,6.353) -- (11.446,6.412) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.913,5.721) -- (7.891,5.801) -- (7.833,5.86) -- (7.753,5.881) -- (7.673,5.86) -- (7.614,5.801) -- (7.593,5.721) -- (7.614,5.641) -- (7.673,5.583) -- (7.753,5.561) -- (7.833,5.583) -- (7.891,5.641) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.545,5.623) -- (1.524,5.703) -- (1.465,5.762) -- (1.385,5.783) -- (1.305,5.762) -- (1.246,5.703) -- (1.225,5.623) -- (1.246,5.543) -- (1.305,5.484) -- (1.385,5.463) -- (1.465,5.484) -- (1.524,5.543) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.554,5.596) -- (4.533,5.676) -- (4.474,5.735) -- (4.394,5.756) -- (4.314,5.735) -- (4.256,5.676) -- (4.234,5.596) -- (4.256,5.516) -- (4.314,5.457) -- (4.394,5.436) -- (4.474,5.457) -- (4.533,5.516) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.828,2.797) -- (8.806,2.877) -- (8.748,2.935) -- (8.668,2.957) -- (8.588,2.935) -- (8.529,2.877) -- (8.508,2.797) -- (8.529,2.717) -- (8.588,2.658) -- (8.668,2.637) -- (8.748,2.658) -- (8.806,2.717) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.992,2.596) -- (4.97,2.676) -- (4.912,2.735) -- (4.832,2.756) -- (4.752,2.735) -- (4.693,2.676) -- (4.672,2.596) -- (4.693,2.516) -- (4.752,2.458) -- (4.832,2.436) -- (4.912,2.458) -- (4.97,2.516) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.468,2.392) -- (1.446,2.472) -- (1.388,2.53) -- (1.308,2.552) -- (1.228,2.53) -- (1.169,2.472) -- (1.148,2.392) -- (1.169,2.312) -- (1.228,2.253) -- (1.308,2.232) -- (1.388,2.253) -- (1.446,2.312) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.272,10.161) -- (5.226,10.331) -- (5.102,10.456) -- (4.932,10.501) -- (4.762,10.456) -- (4.637,10.331) -- (4.592,10.161) -- (4.637,9.991) -- (4.762,9.867) -- (4.932,9.821) -- (5.102,9.867) -- (5.226,9.991) -- cycle;
\node at (4.932,10.161) {\scalebox{1.422}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.756,7.528) -- (2.711,7.698) -- (2.586,7.822) -- (2.416,7.868) -- (2.246,7.822) -- (2.122,7.698) -- (2.076,7.528) -- (2.122,7.358) -- (2.246,7.234) -- (2.416,7.188) -- (2.586,7.234) -- (2.711,7.358) -- cycle;
\node at (2.416,7.528) {\scalebox{1.422}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.906,7.701) -- (3.86,7.871) -- (3.736,7.995) -- (3.566,8.041) -- (3.396,7.995) -- (3.271,7.871) -- (3.226,7.701) -- (3.271,7.531) -- (3.396,7.406) -- (3.566,7.361) -- (3.736,7.406) -- (3.86,7.531) -- cycle;
\node at (3.566,7.701) {\scalebox{1.422}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.161,8.325) -- (10.115,8.495) -- (9.991,8.619) -- (9.821,8.665) -- (9.651,8.619) -- (9.526,8.495) -- (9.481,8.325) -- (9.526,8.155) -- (9.651,8.03) -- (9.821,7.985) -- (9.991,8.03) -- (10.115,8.155) -- cycle;
\node at (9.821,8.325) {\scalebox{1.422}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.236,7.543) -- (8.19,7.713) -- (8.066,7.837) -- (7.896,7.883) -- (7.726,7.837) -- (7.601,7.713) -- (7.556,7.543) -- (7.601,7.373) -- (7.726,7.248) -- (7.896,7.203) -- (8.066,7.248) -- (8.19,7.373) -- cycle;
\node at (7.896,7.543) {\scalebox{1.422}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.782,7.13) -- (6.737,7.3) -- (6.612,7.425) -- (6.442,7.47) -- (6.272,7.425) -- (6.148,7.3) -- (6.102,7.13) -- (6.148,6.96) -- (6.272,6.836) -- (6.442,6.79) -- (6.612,6.836) -- (6.737,6.96) -- cycle;
\node at (6.442,7.13) {\scalebox{1.422}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.436,4.567) -- (10.39,4.737) -- (10.266,4.861) -- (10.096,4.907) -- (9.926,4.861) -- (9.801,4.737) -- (9.756,4.567) -- (9.801,4.397) -- (9.926,4.273) -- (10.096,4.227) -- (10.266,4.273) -- (10.39,4.397) -- cycle;
\node at (10.096,4.567) {\scalebox{1.422}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.43,5.2) -- (6.385,5.37) -- (6.26,5.495) -- (6.09,5.54) -- (5.92,5.495) -- (5.796,5.37) -- (5.75,5.2) -- (5.796,5.03) -- (5.92,4.906) -- (6.09,4.86) -- (6.26,4.906) -- (6.385,5.03) -- cycle;
\node at (6.09,5.2) {\scalebox{1.422}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.735,4.317) -- (8.689,4.487) -- (8.565,4.611) -- (8.395,4.657) -- (8.225,4.611) -- (8.1,4.487) -- (8.055,4.317) -- (8.1,4.147) -- (8.225,4.022) -- (8.395,3.977) -- (8.565,4.022) -- (8.689,4.147) -- cycle;
\node at (8.395,4.317) {\scalebox{1.422}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.234,6.054) -- (3.188,6.224) -- (3.064,6.348) -- (2.894,6.394) -- (2.724,6.348) -- (2.599,6.224) -- (2.554,6.054) -- (2.599,5.884) -- (2.724,5.759) -- (2.894,5.714) -- (3.064,5.759) -- (3.188,5.884) -- cycle;
\node at (2.894,6.054) {\scalebox{1.422}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.873,4.003) -- (1.827,4.173) -- (1.703,4.297) -- (1.533,4.343) -- (1.363,4.297) -- (1.239,4.173) -- (1.193,4.003) -- (1.239,3.833) -- (1.363,3.708) -- (1.533,3.663) -- (1.703,3.708) -- (1.827,3.833) -- cycle;
\node at (1.533,4.003) {\scalebox{1.422}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.162,4.64) -- (7.116,4.81) -- (6.992,4.935) -- (6.822,4.98) -- (6.652,4.935) -- (6.527,4.81) -- (6.482,4.64) -- (6.527,4.47) -- (6.652,4.346) -- (6.822,4.3) -- (6.992,4.346) -- (7.116,4.47) -- cycle;
\node at (6.822,4.64) {\scalebox{1.422}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.146,4.124) -- (5.1,4.294) -- (4.976,4.419) -- (4.806,4.464) -- (4.636,4.419) -- (4.511,4.294) -- (4.466,4.124) -- (4.511,3.954) -- (4.636,3.83) -- (4.806,3.784) -- (4.976,3.83) -- (5.1,3.954) -- cycle;
\node at (4.806,4.124) {\scalebox{1.422}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.115,2.219) -- (7.069,2.389) -- (6.945,2.513) -- (6.775,2.559) -- (6.605,2.513) -- (6.48,2.389) -- (6.435,2.219) -- (6.48,2.049) -- (6.605,1.924) -- (6.775,1.879) -- (6.945,1.924) -- (7.069,2.049) -- cycle;
\node at (6.775,2.219) {\scalebox{1.422}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.094,5.296) -- (4.694,5.296) -- (4.694,5.896) -- (4.094,5.896) -- cycle;
\node at (4.394,5.596) {\scalebox{1.422}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.052,3.874) -- (11.652,3.874) -- (11.652,4.474) -- (11.052,4.474) -- cycle;
\node at (11.352,4.174) {\scalebox{1.422}{\small b}};
\filldraw[fill=blue!12, draw=black, line width=0.5pt] (6.847,0.65) -- (7.447,0.65) -- (7.447,1.25) -- (6.847,1.25) -- cycle;
\node at (7.147,0.95) {\scalebox{1.422}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.532,6.265) -- (2.132,6.265) -- (2.132,6.865) -- (1.532,6.865) -- cycle;
\node at (1.832,6.565) {\scalebox{1.422}{\small d}};
\node at (2.126,10.191) {\scalebox{1.422}{\small A}};
\node at (8.007,9.897) {\scalebox{1.422}{\small B}};
\node at (11.92,6.588) {\scalebox{1.422}{\small C}};
\node at (8.371,5.759) {\scalebox{1.422}{\small D}};
\node at (0.765,5.629) {\scalebox{1.422}{\small E}};
\node at (3.774,5.602) {\scalebox{1.422}{\small F}};
\node at (9.131,2.384) {\scalebox{1.422}{\small G}};
\node at (4.689,1.993) {\scalebox{1.422}{\small H}};
\node at (0.813,2.018) {\scalebox{1.422}{\small I}};
\draw[line width=0.5pt] (12.608,8.743) -- (12.608,9.843);
\draw[line width=0.5pt] (12.377,9.458) -- (12.608,9.843) -- (12.839,9.458);
\draw[line width=0.5pt] (12.305,9.293) -- (12.91,9.293);
\node at (12.608,10.228) {\scalebox{1.422}{\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 avenue de la Gare & ⑧ rue de l'Église & Ⓐ Carrefour des Quatre-Chemins & \textbf{a} la poste \\ \hline
② boulevard du Parc & ⑨ quai aux Fleurs & Ⓑ Rond-point de la Poste & \textbf{b} la pharmacie \\ \hline
③ rue de la Poste & ⑩ chemin de Ronde & Ⓒ Place de l'Église & \textbf{c} l'école \\ \hline
④ chemin du Lavoir & ⑪ impasse Verte & Ⓓ Carrefour des Tilleuls & \textbf{d} la gendarmerie \\ \hline
⑤ rue de la Mairie & ⑫ rue Verte & Ⓔ Rond-point du Stade &  \\ \hline
⑥ rue Haute & ⑬ rue du Stade & Ⓕ Rond-point des Vaches &  \\ \hline
⑦ chemin Creux & ⑭ allée des Tilleuls & Ⓖ Carrefour de la Mairie &  \\ \hline
 &  & Ⓗ Place du Marché &  \\ \hline
 &  & Ⓘ Coin du Lavoir &  \\ \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=0.6291, line join=round, line cap=round]
\draw[line width=0.5pt] (2.495,9.693) -- (2.671,9.677) -- (2.848,9.664) -- (3.025,9.656) -- (3.202,9.653) -- (3.379,9.653) -- (3.556,9.657) -- (3.734,9.666) -- (3.91,9.679) -- (4.087,9.696) -- (4.263,9.717) -- (4.438,9.742) -- (4.613,9.771) -- (4.787,9.805) -- (4.96,9.842) -- (5.159,9.838) -- (5.359,9.829) -- (5.558,9.815) -- (5.756,9.797) -- (5.955,9.774) -- (6.152,9.746) -- (6.349,9.714) -- (6.545,9.677) -- (6.741,9.635) -- (6.935,9.589) -- (7.128,9.538) -- (7.32,9.483) -- (7.51,9.423) -- (7.699,9.359);
\draw[line width=0.5pt] (2.495,9.693) -- (2.461,9.392) -- (2.419,9.093) -- (2.371,8.794) -- (2.316,8.496) -- (2.253,8.2) -- (2.184,7.905) -- (2.108,7.612) -- (2.025,7.321) -- (1.935,7.032) -- (1.838,6.745) -- (1.735,6.46) -- (1.625,6.178) -- (1.508,5.899) -- (1.385,5.623);
\draw[line width=0.5pt] (2.495,9.693) -- (2.584,9.382) -- (2.682,9.073) -- (2.786,8.766) -- (2.898,8.462) -- (3.017,8.161) -- (3.143,7.862) -- (3.275,7.566) -- (3.415,7.274) -- (3.562,6.985) -- (3.715,6.699) -- (3.875,6.417) -- (4.042,6.139) -- (4.215,5.866) -- (4.394,5.596);
\draw[line width=0.5pt] (7.699,9.359) -- (7.986,9.195) -- (8.269,9.025) -- (8.548,8.848) -- (8.823,8.664) -- (9.094,8.474) -- (9.36,8.277) -- (9.622,8.074) -- (9.878,7.865) -- (10.13,7.651) -- (10.376,7.43) -- (10.618,7.204) -- (10.853,6.972) -- (11.083,6.734) -- (11.308,6.492);
\draw[line width=0.5pt] (7.699,9.359) -- (7.663,9.101) -- (7.633,8.842) -- (7.609,8.582) -- (7.592,8.321) -- (7.58,8.06) -- (7.575,7.799) -- (7.576,7.538) -- (7.582,7.277) -- (7.596,7.016) -- (7.615,6.756) -- (7.64,6.496) -- (7.671,6.237) -- (7.709,5.979) -- (7.753,5.721);
\draw[line width=0.5pt] (7.699,9.359) -- (7.506,9.056) -- (7.306,8.758) -- (7.098,8.465) -- (6.884,8.176) -- (6.663,7.892) -- (6.436,7.614) -- (6.202,7.341) -- (5.961,7.074) -- (5.715,6.812) -- (5.462,6.557) -- (5.204,6.307) -- (4.939,6.064) -- (4.669,5.827) -- (4.394,5.596);
\draw[line width=0.5pt] (11.308,6.492) -- (11.08,6.259) -- (10.858,6.02) -- (10.642,5.777) -- (10.431,5.528) -- (10.227,5.274) -- (10.028,5.016) -- (9.835,4.753) -- (9.649,4.486) -- (9.469,4.214) -- (9.295,3.938) -- (9.128,3.658) -- (8.968,3.375) -- (8.814,3.087) -- (8.668,2.797);
\draw[line width=0.5pt] (7.753,5.721) -- (7.516,5.676) -- (7.278,5.636) -- (7.04,5.601) -- (6.8,5.573) -- (6.56,5.549) -- (6.319,5.532) -- (6.079,5.52) -- (5.837,5.514) -- (5.596,5.514) -- (5.355,5.519) -- (5.114,5.529) -- (4.873,5.546) -- (4.633,5.568) -- (4.394,5.596);
\draw[line width=0.5pt] (7.753,5.721) -- (7.785,5.504) -- (7.823,5.288) -- (7.867,5.072) -- (7.915,4.858) -- (7.968,4.644) -- (8.026,4.432) -- (8.089,4.221) -- (8.158,4.012) -- (8.231,3.805) -- (8.309,3.599) -- (8.391,3.395) -- (8.479,3.194) -- (8.571,2.994) -- (8.668,2.797);
\draw[line width=0.5pt] (1.385,5.623) -- (1.598,5.654) -- (1.813,5.68) -- (2.028,5.701) -- (2.243,5.717) -- (2.459,5.727) -- (2.675,5.733) -- (2.891,5.734) -- (3.107,5.729) -- (3.322,5.72) -- (3.538,5.705) -- (3.753,5.685) -- (3.967,5.661) -- (4.181,5.631) -- (4.394,5.596);
\draw[line width=0.5pt] (1.385,5.623) -- (1.344,5.395) -- (1.309,5.166) -- (1.279,4.936) -- (1.254,4.705) -- (1.235,4.474) -- (1.221,4.242) -- (1.213,4.01) -- (1.21,3.778) -- (1.213,3.546) -- (1.221,3.315) -- (1.235,3.083) -- (1.253,2.852) -- (1.278,2.621) -- (1.308,2.392);
\draw[line width=0.5pt] (4.394,5.596) -- (4.728,5.444) -- (5.057,5.285) -- (5.384,5.118) -- (5.706,4.943) -- (6.024,4.76) -- (6.337,4.57) -- (6.646,4.373) -- (6.951,4.168) -- (7.25,3.957) -- (7.545,3.738) -- (7.834,3.513) -- (8.117,3.281) -- (8.396,3.042) -- (8.668,2.797);
\draw[line width=0.5pt] (4.394,5.596) -- (4.392,5.379) -- (4.396,5.161) -- (4.404,4.944) -- (4.418,4.727) -- (4.436,4.51) -- (4.46,4.294) -- (4.489,4.078) -- (4.523,3.863) -- (4.562,3.649) -- (4.606,3.436) -- (4.655,3.224) -- (4.709,3.014) -- (4.768,2.804) -- (4.832,2.596);
\draw[line width=0.5pt] (8.668,2.797) -- (8.398,2.741) -- (8.127,2.691) -- (7.855,2.647) -- (7.582,2.611) -- (7.308,2.58) -- (7.033,2.556) -- (6.758,2.538) -- (6.482,2.527) -- (6.207,2.523) -- (5.931,2.524) -- (5.656,2.533) -- (5.38,2.547) -- (5.106,2.569) -- (4.832,2.596);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.655,9.693) -- (2.633,9.773) -- (2.575,9.832) -- (2.495,9.853) -- (2.415,9.832) -- (2.356,9.773) -- (2.335,9.693) -- (2.356,9.613) -- (2.415,9.554) -- (2.495,9.533) -- (2.575,9.554) -- (2.633,9.613) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.859,9.359) -- (7.837,9.439) -- (7.779,9.498) -- (7.699,9.519) -- (7.619,9.498) -- (7.56,9.439) -- (7.539,9.359) -- (7.56,9.279) -- (7.619,9.221) -- (7.699,9.199) -- (7.779,9.221) -- (7.837,9.279) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.468,6.492) -- (11.446,6.572) -- (11.388,6.63) -- (11.308,6.652) -- (11.228,6.63) -- (11.169,6.572) -- (11.148,6.492) -- (11.169,6.412) -- (11.228,6.353) -- (11.308,6.332) -- (11.388,6.353) -- (11.446,6.412) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.913,5.721) -- (7.891,5.801) -- (7.833,5.86) -- (7.753,5.881) -- (7.673,5.86) -- (7.614,5.801) -- (7.593,5.721) -- (7.614,5.641) -- (7.673,5.583) -- (7.753,5.561) -- (7.833,5.583) -- (7.891,5.641) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.545,5.623) -- (1.524,5.703) -- (1.465,5.762) -- (1.385,5.783) -- (1.305,5.762) -- (1.246,5.703) -- (1.225,5.623) -- (1.246,5.543) -- (1.305,5.484) -- (1.385,5.463) -- (1.465,5.484) -- (1.524,5.543) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.554,5.596) -- (4.533,5.676) -- (4.474,5.735) -- (4.394,5.756) -- (4.314,5.735) -- (4.256,5.676) -- (4.234,5.596) -- (4.256,5.516) -- (4.314,5.457) -- (4.394,5.436) -- (4.474,5.457) -- (4.533,5.516) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.828,2.797) -- (8.806,2.877) -- (8.748,2.935) -- (8.668,2.957) -- (8.588,2.935) -- (8.529,2.877) -- (8.508,2.797) -- (8.529,2.717) -- (8.588,2.658) -- (8.668,2.637) -- (8.748,2.658) -- (8.806,2.717) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.992,2.596) -- (4.97,2.676) -- (4.912,2.735) -- (4.832,2.756) -- (4.752,2.735) -- (4.693,2.676) -- (4.672,2.596) -- (4.693,2.516) -- (4.752,2.458) -- (4.832,2.436) -- (4.912,2.458) -- (4.97,2.516) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.468,2.392) -- (1.446,2.472) -- (1.388,2.53) -- (1.308,2.552) -- (1.228,2.53) -- (1.169,2.472) -- (1.148,2.392) -- (1.169,2.312) -- (1.228,2.253) -- (1.308,2.232) -- (1.388,2.253) -- (1.446,2.312) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.272,10.161) -- (5.226,10.331) -- (5.102,10.456) -- (4.932,10.501) -- (4.762,10.456) -- (4.637,10.331) -- (4.592,10.161) -- (4.637,9.991) -- (4.762,9.867) -- (4.932,9.821) -- (5.102,9.867) -- (5.226,9.991) -- cycle;
\node at (4.932,10.161) {\scalebox{1.59}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.756,7.528) -- (2.711,7.698) -- (2.586,7.822) -- (2.416,7.868) -- (2.246,7.822) -- (2.122,7.698) -- (2.076,7.528) -- (2.122,7.358) -- (2.246,7.234) -- (2.416,7.188) -- (2.586,7.234) -- (2.711,7.358) -- cycle;
\node at (2.416,7.528) {\scalebox{1.59}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.906,7.701) -- (3.86,7.871) -- (3.736,7.995) -- (3.566,8.041) -- (3.396,7.995) -- (3.271,7.871) -- (3.226,7.701) -- (3.271,7.531) -- (3.396,7.406) -- (3.566,7.361) -- (3.736,7.406) -- (3.86,7.531) -- cycle;
\node at (3.566,7.701) {\scalebox{1.59}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.161,8.325) -- (10.115,8.495) -- (9.991,8.619) -- (9.821,8.665) -- (9.651,8.619) -- (9.526,8.495) -- (9.481,8.325) -- (9.526,8.155) -- (9.651,8.03) -- (9.821,7.985) -- (9.991,8.03) -- (10.115,8.155) -- cycle;
\node at (9.821,8.325) {\scalebox{1.59}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.236,7.543) -- (8.19,7.713) -- (8.066,7.837) -- (7.896,7.883) -- (7.726,7.837) -- (7.601,7.713) -- (7.556,7.543) -- (7.601,7.373) -- (7.726,7.248) -- (7.896,7.203) -- (8.066,7.248) -- (8.19,7.373) -- cycle;
\node at (7.896,7.543) {\scalebox{1.59}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.782,7.13) -- (6.737,7.3) -- (6.612,7.425) -- (6.442,7.47) -- (6.272,7.425) -- (6.148,7.3) -- (6.102,7.13) -- (6.148,6.96) -- (6.272,6.836) -- (6.442,6.79) -- (6.612,6.836) -- (6.737,6.96) -- cycle;
\node at (6.442,7.13) {\scalebox{1.59}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.436,4.567) -- (10.39,4.737) -- (10.266,4.861) -- (10.096,4.907) -- (9.926,4.861) -- (9.801,4.737) -- (9.756,4.567) -- (9.801,4.397) -- (9.926,4.273) -- (10.096,4.227) -- (10.266,4.273) -- (10.39,4.397) -- cycle;
\node at (10.096,4.567) {\scalebox{1.59}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.43,5.2) -- (6.385,5.37) -- (6.26,5.495) -- (6.09,5.54) -- (5.92,5.495) -- (5.796,5.37) -- (5.75,5.2) -- (5.796,5.03) -- (5.92,4.906) -- (6.09,4.86) -- (6.26,4.906) -- (6.385,5.03) -- cycle;
\node at (6.09,5.2) {\scalebox{1.59}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.735,4.317) -- (8.689,4.487) -- (8.565,4.611) -- (8.395,4.657) -- (8.225,4.611) -- (8.1,4.487) -- (8.055,4.317) -- (8.1,4.147) -- (8.225,4.022) -- (8.395,3.977) -- (8.565,4.022) -- (8.689,4.147) -- cycle;
\node at (8.395,4.317) {\scalebox{1.59}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.234,6.054) -- (3.188,6.224) -- (3.064,6.348) -- (2.894,6.394) -- (2.724,6.348) -- (2.599,6.224) -- (2.554,6.054) -- (2.599,5.884) -- (2.724,5.759) -- (2.894,5.714) -- (3.064,5.759) -- (3.188,5.884) -- cycle;
\node at (2.894,6.054) {\scalebox{1.59}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.873,4.003) -- (1.827,4.173) -- (1.703,4.297) -- (1.533,4.343) -- (1.363,4.297) -- (1.239,4.173) -- (1.193,4.003) -- (1.239,3.833) -- (1.363,3.708) -- (1.533,3.663) -- (1.703,3.708) -- (1.827,3.833) -- cycle;
\node at (1.533,4.003) {\scalebox{1.59}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.162,4.64) -- (7.116,4.81) -- (6.992,4.935) -- (6.822,4.98) -- (6.652,4.935) -- (6.527,4.81) -- (6.482,4.64) -- (6.527,4.47) -- (6.652,4.346) -- (6.822,4.3) -- (6.992,4.346) -- (7.116,4.47) -- cycle;
\node at (6.822,4.64) {\scalebox{1.59}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.146,4.124) -- (5.1,4.294) -- (4.976,4.419) -- (4.806,4.464) -- (4.636,4.419) -- (4.511,4.294) -- (4.466,4.124) -- (4.511,3.954) -- (4.636,3.83) -- (4.806,3.784) -- (4.976,3.83) -- (5.1,3.954) -- cycle;
\node at (4.806,4.124) {\scalebox{1.59}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.115,2.219) -- (7.069,2.389) -- (6.945,2.513) -- (6.775,2.559) -- (6.605,2.513) -- (6.48,2.389) -- (6.435,2.219) -- (6.48,2.049) -- (6.605,1.924) -- (6.775,1.879) -- (6.945,1.924) -- (7.069,2.049) -- cycle;
\node at (6.775,2.219) {\scalebox{1.59}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.094,5.296) -- (4.694,5.296) -- (4.694,5.896) -- (4.094,5.896) -- cycle;
\node at (4.394,5.596) {\scalebox{1.59}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.052,3.874) -- (11.652,3.874) -- (11.652,4.474) -- (11.052,4.474) -- cycle;
\node at (11.352,4.174) {\scalebox{1.59}{\small b}};
\filldraw[fill=blue!12, draw=black, line width=0.5pt] (6.847,0.65) -- (7.447,0.65) -- (7.447,1.25) -- (6.847,1.25) -- cycle;
\node at (7.147,0.95) {\scalebox{1.59}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (1.532,6.265) -- (2.132,6.265) -- (2.132,6.865) -- (1.532,6.865) -- cycle;
\node at (1.832,6.565) {\scalebox{1.59}{\small d}};
\node at (2.126,10.191) {\scalebox{1.59}{\small A}};
\node at (8.007,9.897) {\scalebox{1.59}{\small B}};
\node at (11.92,6.588) {\scalebox{1.59}{\small C}};
\node at (8.371,5.759) {\scalebox{1.59}{\small D}};
\node at (0.765,5.629) {\scalebox{1.59}{\small E}};
\node at (3.774,5.602) {\scalebox{1.59}{\small F}};
\node at (9.131,2.384) {\scalebox{1.59}{\small G}};
\node at (4.689,1.993) {\scalebox{1.59}{\small H}};
\node at (0.813,2.018) {\scalebox{1.59}{\small I}};
\draw[line width=0.5pt] (12.608,8.743) -- (12.608,9.843);
\draw[line width=0.5pt] (12.377,9.458) -- (12.608,9.843) -- (12.839,9.458);
\draw[line width=0.5pt] (12.305,9.293) -- (12.91,9.293);
\node at (12.608,10.228) {\scalebox{1.59}{\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|}
\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 \\ \hline
Nombre de rues & 3 & 4 & 2 & 3 & 3 & 6 & 4 & 2 & 1 \\ \hline
Parité & \textbf{impair} & pair & pair & \textbf{impair} & \textbf{impair} & pair & pair & pair & \textbf{impair} \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 28 = 2 \(\times\) 14, 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{D}, \textbf{E}, \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{F}, 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, D, E, 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}
\noindent{\color{green!45!black}\textbf{À remarquer.} Ce quartier a une impasse. On y entre et on en ressort par la même rue : elle serait forcément empruntée deux fois. Une impasse interdit donc à elle seule la tournée qui revient à son point de départ — sans qu'il soit besoin de compter quoi que ce soit.}\par\addvspace{0.4em}
\end{document}
