\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 17}

\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=17}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0017}
\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=0.6917, line join=round, line cap=round]
\draw[line width=0.5pt] (4.771,11.261) -- (4.573,11.268) -- (4.374,11.27) -- (4.175,11.267) -- (3.976,11.26) -- (3.777,11.248) -- (3.579,11.231) -- (3.381,11.209) -- (3.184,11.183) -- (2.987,11.153) -- (2.791,11.118) -- (2.596,11.078) -- (2.402,11.033) -- (2.209,10.984) -- (2.018,10.931);
\draw[line width=0.5pt] (4.771,11.261) -- (5.05,11.226) -- (5.329,11.184) -- (5.606,11.135) -- (5.882,11.08) -- (6.157,11.018) -- (6.43,10.95) -- (6.702,10.876) -- (6.972,10.795) -- (7.24,10.708) -- (7.505,10.614) -- (7.769,10.515) -- (8.03,10.409) -- (8.288,10.297) -- (8.544,10.179);
\draw[line width=0.5pt] (4.771,11.261) -- (4.748,10.963) -- (4.731,10.664) -- (4.722,10.365) -- (4.719,10.065) -- (4.724,9.766) -- (4.736,9.467) -- (4.755,9.168) -- (4.78,8.869) -- (4.813,8.572) -- (4.853,8.275) -- (4.9,7.979) -- (4.953,7.685) -- (5.014,7.392) -- (5.081,7.1);
\draw[line width=0.5pt] (2.018,10.931) -- (2.277,10.693) -- (2.53,10.449) -- (2.778,10.199) -- (3.019,9.943) -- (3.255,9.681) -- (3.484,9.414) -- (3.708,9.142) -- (3.924,8.864) -- (4.134,8.582) -- (4.338,8.294) -- (4.534,8.002) -- (4.724,7.706) -- (4.906,7.405) -- (5.081,7.1);
\draw[line width=0.5pt] (5.127,1.56) -- (5.3,1.555) -- (5.474,1.555) -- (5.648,1.558) -- (5.821,1.566) -- (5.995,1.578) -- (6.168,1.594) -- (6.34,1.614) -- (6.513,1.638) -- (6.684,1.666) -- (6.855,1.698) -- (7.025,1.734) -- (7.194,1.774) -- (7.362,1.818) -- (7.529,1.866) -- (7.816,1.886) -- (8.104,1.9) -- (8.392,1.907) -- (8.68,1.906) -- (8.968,1.9) -- (9.256,1.886) -- (9.543,1.866) -- (9.83,1.838) -- (10.117,1.805) -- (10.402,1.764) -- (10.686,1.717) -- (10.969,1.663) -- (11.251,1.602) -- (11.531,1.535);
\draw[line width=0.5pt] (8.544,10.179) -- (8.534,10.014) -- (8.521,9.848) -- (8.504,9.683) -- (8.483,9.518) -- (8.458,9.354) -- (8.429,9.19) -- (8.396,9.027) -- (8.36,8.865) -- (8.32,8.704) -- (8.276,8.544) -- (8.228,8.385) -- (8.177,8.227) -- (8.121,8.07) -- (8.062,7.915);
\draw[line width=0.5pt] (8.544,10.179) -- (8.634,9.951) -- (8.73,9.726) -- (8.831,9.502) -- (8.937,9.281) -- (9.049,9.062) -- (9.165,8.847) -- (9.287,8.633) -- (9.414,8.423) -- (9.545,8.216) -- (9.682,8.012) -- (9.823,7.811) -- (9.969,7.614) -- (10.119,7.42) -- (10.274,7.23);
\draw[line width=0.5pt] (8.544,10.179) -- (8.332,9.923) -- (8.114,9.672) -- (7.891,9.426) -- (7.661,9.186) -- (7.426,8.951) -- (7.185,8.721) -- (6.939,8.497) -- (6.688,8.279) -- (6.432,8.067) -- (6.171,7.861) -- (5.905,7.661) -- (5.635,7.467) -- (5.36,7.281) -- (5.081,7.1);
\draw[line width=0.5pt] (8.062,7.915) -- (7.842,7.89) -- (7.623,7.86) -- (7.404,7.824) -- (7.186,7.783) -- (6.969,7.738) -- (6.753,7.687) -- (6.538,7.631) -- (6.325,7.57) -- (6.113,7.504) -- (5.903,7.433) -- (5.695,7.357) -- (5.488,7.276) -- (5.284,7.191) -- (5.081,7.1);
\draw[line width=0.5pt] (10.274,7.23) -- (10.101,7.042) -- (9.924,6.857) -- (9.743,6.677) -- (9.557,6.501) -- (9.367,6.33) -- (9.173,6.163) -- (8.975,6.0) -- (8.773,5.843) -- (8.568,5.69) -- (8.359,5.542) -- (8.147,5.398) -- (7.932,5.26) -- (7.713,5.128) -- (7.492,5.0);
\draw[line width=0.5pt] (5.081,7.1) -- (5.229,6.925) -- (5.381,6.753) -- (5.537,6.585) -- (5.697,6.421) -- (5.861,6.26) -- (6.029,6.103) -- (6.2,5.951) -- (6.375,5.802) -- (6.553,5.657) -- (6.735,5.517) -- (6.919,5.381) -- (7.107,5.249) -- (7.298,5.122) -- (7.492,5.0);
\draw[line width=0.5pt] (5.081,7.1) -- (5.111,6.942) -- (5.137,6.783) -- (5.159,6.623) -- (5.178,6.463) -- (5.192,6.302) -- (5.203,6.141) -- (5.21,5.98) -- (5.214,5.818) -- (5.213,5.657) -- (5.209,5.496) -- (5.201,5.334) -- (5.189,5.174) -- (5.174,5.013) -- (5.154,4.853);
\draw[line width=0.5pt] (1.531,5.069) -- (1.747,4.781) -- (1.971,4.498) -- (2.2,4.221) -- (2.437,3.949) -- (2.68,3.682) -- (2.929,3.421) -- (3.184,3.166) -- (3.445,2.917) -- (3.712,2.675) -- (3.984,2.438) -- (4.262,2.208) -- (4.545,1.985) -- (4.834,1.769) -- (5.127,1.56);
\draw[line width=0.5pt] (1.531,5.069) -- (1.615,4.844) -- (1.694,4.618) -- (1.768,4.389) -- (1.836,4.159) -- (1.899,3.927) -- (1.956,3.693) -- (2.008,3.458) -- (2.054,3.222) -- (2.095,2.986) -- (2.13,2.748) -- (2.16,2.509) -- (2.184,2.27) -- (2.202,2.031) -- (2.215,1.791);
\draw[line width=0.5pt] (7.492,5.0) -- (7.287,4.782) -- (7.087,4.559) -- (6.892,4.332) -- (6.702,4.1) -- (6.519,3.863) -- (6.34,3.622) -- (6.168,3.377) -- (6.001,3.128) -- (5.84,2.876) -- (5.685,2.619) -- (5.536,2.359) -- (5.393,2.096) -- (5.257,1.829) -- (5.127,1.56);
\draw[line width=0.5pt] (7.492,5.0) -- (7.328,4.964) -- (7.163,4.932) -- (6.997,4.904) -- (6.831,4.879) -- (6.664,4.859) -- (6.497,4.843) -- (6.329,4.83) -- (6.161,4.821) -- (5.993,4.817) -- (5.825,4.816) -- (5.657,4.819) -- (5.489,4.827) -- (5.321,4.838) -- (5.154,4.853);
\draw[line width=0.5pt] (2.215,1.791) -- (2.419,1.743) -- (2.624,1.699) -- (2.83,1.661) -- (3.036,1.627) -- (3.244,1.598) -- (3.452,1.574) -- (3.661,1.555) -- (3.871,1.541) -- (4.08,1.532) -- (4.29,1.528) -- (4.499,1.528) -- (4.709,1.534) -- (4.918,1.544) -- (5.127,1.56);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.931,11.261) -- (4.91,11.341) -- (4.851,11.4) -- (4.771,11.421) -- (4.691,11.4) -- (4.633,11.341) -- (4.611,11.261) -- (4.633,11.181) -- (4.691,11.123) -- (4.771,11.101) -- (4.851,11.123) -- (4.91,11.181) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.178,10.931) -- (2.156,11.011) -- (2.098,11.069) -- (2.018,11.091) -- (1.938,11.069) -- (1.879,11.011) -- (1.858,10.931) -- (1.879,10.851) -- (1.938,10.792) -- (2.018,10.771) -- (2.098,10.792) -- (2.156,10.851) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.704,10.179) -- (8.682,10.259) -- (8.624,10.318) -- (8.544,10.339) -- (8.464,10.318) -- (8.405,10.259) -- (8.384,10.179) -- (8.405,10.099) -- (8.464,10.041) -- (8.544,10.019) -- (8.624,10.041) -- (8.682,10.099) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.222,7.915) -- (8.201,7.995) -- (8.142,8.054) -- (8.062,8.075) -- (7.982,8.054) -- (7.924,7.995) -- (7.902,7.915) -- (7.924,7.835) -- (7.982,7.777) -- (8.062,7.755) -- (8.142,7.777) -- (8.201,7.835) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.434,7.23) -- (10.413,7.31) -- (10.354,7.369) -- (10.274,7.39) -- (10.194,7.369) -- (10.135,7.31) -- (10.114,7.23) -- (10.135,7.15) -- (10.194,7.092) -- (10.274,7.07) -- (10.354,7.092) -- (10.413,7.15) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.241,7.1) -- (5.22,7.18) -- (5.161,7.239) -- (5.081,7.26) -- (5.001,7.239) -- (4.943,7.18) -- (4.921,7.1) -- (4.943,7.02) -- (5.001,6.962) -- (5.081,6.94) -- (5.161,6.962) -- (5.22,7.02) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.691,5.069) -- (1.669,5.149) -- (1.611,5.208) -- (1.531,5.229) -- (1.451,5.208) -- (1.392,5.149) -- (1.371,5.069) -- (1.392,4.989) -- (1.451,4.931) -- (1.531,4.909) -- (1.611,4.931) -- (1.669,4.989) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.652,5.0) -- (7.631,5.08) -- (7.572,5.138) -- (7.492,5.16) -- (7.412,5.138) -- (7.353,5.08) -- (7.332,5.0) -- (7.353,4.92) -- (7.412,4.861) -- (7.492,4.84) -- (7.572,4.861) -- (7.631,4.92) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.314,4.853) -- (5.293,4.933) -- (5.234,4.992) -- (5.154,5.013) -- (5.074,4.992) -- (5.016,4.933) -- (4.994,4.853) -- (5.016,4.773) -- (5.074,4.714) -- (5.154,4.693) -- (5.234,4.714) -- (5.293,4.773) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.375,1.791) -- (2.353,1.871) -- (2.295,1.929) -- (2.215,1.951) -- (2.135,1.929) -- (2.076,1.871) -- (2.055,1.791) -- (2.076,1.711) -- (2.135,1.652) -- (2.215,1.631) -- (2.295,1.652) -- (2.353,1.711) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.287,1.56) -- (5.266,1.64) -- (5.207,1.698) -- (5.127,1.72) -- (5.047,1.698) -- (4.988,1.64) -- (4.967,1.56) -- (4.988,1.48) -- (5.047,1.421) -- (5.127,1.4) -- (5.207,1.421) -- (5.266,1.48) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.691,1.535) -- (11.669,1.615) -- (11.611,1.674) -- (11.531,1.695) -- (11.451,1.674) -- (11.392,1.615) -- (11.371,1.535) -- (11.392,1.455) -- (11.451,1.397) -- (11.531,1.375) -- (11.611,1.397) -- (11.669,1.455) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.759,10.892) -- (3.714,11.062) -- (3.589,11.186) -- (3.419,11.232) -- (3.249,11.186) -- (3.125,11.062) -- (3.079,10.892) -- (3.125,10.722) -- (3.249,10.597) -- (3.419,10.552) -- (3.589,10.597) -- (3.714,10.722) -- cycle;
\node at (3.419,10.892) {\scalebox{1.446}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.13,11.183) -- (7.085,11.353) -- (6.96,11.478) -- (6.79,11.523) -- (6.62,11.478) -- (6.496,11.353) -- (6.45,11.183) -- (6.496,11.013) -- (6.62,10.889) -- (6.79,10.843) -- (6.96,10.889) -- (7.085,11.013) -- cycle;
\node at (6.79,11.183) {\scalebox{1.446}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.414,9.192) -- (5.368,9.362) -- (5.244,9.486) -- (5.074,9.532) -- (4.904,9.486) -- (4.779,9.362) -- (4.734,9.192) -- (4.779,9.022) -- (4.904,8.897) -- (5.074,8.852) -- (5.244,8.897) -- (5.368,9.022) -- cycle;
\node at (5.074,9.192) {\scalebox{1.446}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.297,9.342) -- (4.252,9.512) -- (4.127,9.636) -- (3.957,9.682) -- (3.787,9.636) -- (3.663,9.512) -- (3.617,9.342) -- (3.663,9.172) -- (3.787,9.047) -- (3.957,9.002) -- (4.127,9.047) -- (4.252,9.172) -- cycle;
\node at (3.957,9.342) {\scalebox{1.446}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.821,2.182) -- (7.776,2.352) -- (7.651,2.477) -- (7.481,2.522) -- (7.311,2.477) -- (7.187,2.352) -- (7.141,2.182) -- (7.187,2.012) -- (7.311,1.888) -- (7.481,1.842) -- (7.651,1.888) -- (7.776,2.012) -- cycle;
\node at (7.481,2.182) {\scalebox{1.446}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.049,8.961) -- (9.004,9.131) -- (8.879,9.255) -- (8.709,9.301) -- (8.539,9.255) -- (8.415,9.131) -- (8.369,8.961) -- (8.415,8.791) -- (8.539,8.666) -- (8.709,8.621) -- (8.879,8.666) -- (9.004,8.791) -- cycle;
\node at (8.709,8.961) {\scalebox{1.446}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.903,8.795) -- (9.858,8.965) -- (9.733,9.09) -- (9.563,9.135) -- (9.393,9.09) -- (9.269,8.965) -- (9.223,8.795) -- (9.269,8.625) -- (9.393,8.501) -- (9.563,8.455) -- (9.733,8.501) -- (9.858,8.625) -- cycle;
\node at (9.563,8.795) {\scalebox{1.446}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.492,8.258) -- (7.447,8.428) -- (7.322,8.552) -- (7.152,8.598) -- (6.982,8.552) -- (6.858,8.428) -- (6.812,8.258) -- (6.858,8.088) -- (6.982,7.963) -- (7.152,7.918) -- (7.322,7.963) -- (7.447,8.088) -- cycle;
\node at (7.152,8.258) {\scalebox{1.446}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.963,7.322) -- (6.917,7.492) -- (6.793,7.616) -- (6.623,7.662) -- (6.453,7.616) -- (6.328,7.492) -- (6.283,7.322) -- (6.328,7.152) -- (6.453,7.028) -- (6.623,6.982) -- (6.793,7.028) -- (6.917,7.152) -- cycle;
\node at (6.623,7.322) {\scalebox{1.446}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.515,5.751) -- (9.47,5.921) -- (9.345,6.045) -- (9.175,6.091) -- (9.005,6.045) -- (8.881,5.921) -- (8.835,5.751) -- (8.881,5.581) -- (9.005,5.456) -- (9.175,5.411) -- (9.345,5.456) -- (9.47,5.581) -- cycle;
\node at (9.175,5.751) {\scalebox{1.446}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.75,6.192) -- (6.705,6.362) -- (6.58,6.486) -- (6.41,6.532) -- (6.24,6.486) -- (6.116,6.362) -- (6.07,6.192) -- (6.116,6.022) -- (6.24,5.897) -- (6.41,5.852) -- (6.58,5.897) -- (6.705,6.022) -- cycle;
\node at (6.41,6.192) {\scalebox{1.446}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.87,5.99) -- (5.825,6.16) -- (5.7,6.284) -- (5.53,6.33) -- (5.36,6.284) -- (5.236,6.16) -- (5.19,5.99) -- (5.236,5.82) -- (5.36,5.696) -- (5.53,5.65) -- (5.7,5.696) -- (5.825,5.82) -- cycle;
\node at (5.53,5.99) {\scalebox{1.446}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.748,3.395) -- (3.702,3.565) -- (3.578,3.69) -- (3.408,3.735) -- (3.238,3.69) -- (3.113,3.565) -- (3.068,3.395) -- (3.113,3.225) -- (3.238,3.101) -- (3.408,3.055) -- (3.578,3.101) -- (3.702,3.225) -- cycle;
\node at (3.408,3.395) {\scalebox{1.446}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.661,3.524) -- (2.616,3.694) -- (2.491,3.818) -- (2.321,3.864) -- (2.151,3.818) -- (2.027,3.694) -- (1.981,3.524) -- (2.027,3.354) -- (2.151,3.229) -- (2.321,3.184) -- (2.491,3.229) -- (2.616,3.354) -- cycle;
\node at (2.321,3.524) {\scalebox{1.446}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.771,3.196) -- (6.726,3.366) -- (6.601,3.491) -- (6.431,3.536) -- (6.261,3.491) -- (6.137,3.366) -- (6.091,3.196) -- (6.137,3.026) -- (6.261,2.902) -- (6.431,2.856) -- (6.601,2.902) -- (6.726,3.026) -- cycle;
\node at (6.431,3.196) {\scalebox{1.446}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.689,4.511) -- (6.644,4.681) -- (6.519,4.805) -- (6.349,4.851) -- (6.179,4.805) -- (6.055,4.681) -- (6.009,4.511) -- (6.055,4.341) -- (6.179,4.216) -- (6.349,4.171) -- (6.519,4.216) -- (6.644,4.341) -- cycle;
\node at (6.349,4.511) {\scalebox{1.446}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.027,1.874) -- (3.981,2.044) -- (3.857,2.169) -- (3.687,2.214) -- (3.517,2.169) -- (3.392,2.044) -- (3.347,1.874) -- (3.392,1.704) -- (3.517,1.58) -- (3.687,1.534) -- (3.857,1.58) -- (3.981,1.704) -- cycle;
\node at (3.687,1.874) {\scalebox{1.446}{\small 17}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.471,10.961) -- (5.071,10.961) -- (5.071,11.561) -- (4.471,11.561) -- cycle;
\node at (4.771,11.261) {\scalebox{1.446}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (10.355,9.253) -- (10.955,9.253) -- (10.955,9.853) -- (10.355,9.853) -- cycle;
\node at (10.655,9.553) {\scalebox{1.446}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.603,11.32) -- (3.203,11.32) -- (3.203,11.92) -- (2.603,11.92) -- cycle;
\node at (2.903,11.62) {\scalebox{1.446}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (7.32,6.364) -- (7.92,6.364) -- (7.92,6.964) -- (7.32,6.964) -- cycle;
\node at (7.62,6.664) {\scalebox{1.446}{\small d}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (6.659,1.1) -- (7.259,1.1) -- (7.259,1.7) -- (6.659,1.7) -- cycle;
\node at (6.959,1.4) {\scalebox{1.446}{\small e}};
\node at (4.627,11.864) {\scalebox{1.446}{\small S0}};
\node at (1.619,11.406) {\scalebox{1.446}{\small S1}};
\node at (8.879,10.701) {\scalebox{1.446}{\small S2}};
\node at (8.541,8.309) {\scalebox{1.446}{\small S3}};
\node at (10.877,7.375) {\scalebox{1.446}{\small S4}};
\node at (4.642,7.538) {\scalebox{1.446}{\small S5}};
\node at (0.93,4.916) {\scalebox{1.446}{\small S6}};
\node at (7.977,4.614) {\scalebox{1.446}{\small S7}};
\node at (4.83,4.325) {\scalebox{1.446}{\small S8}};
\node at (1.812,1.32) {\scalebox{1.446}{\small S9}};
\node at (5.014,0.95) {\scalebox{1.446}{\small S10}};
\node at (12.005,1.136) {\scalebox{1.446}{\small S11}};
\draw[line width=0.5pt] (12.831,10.311) -- (12.831,11.411);
\draw[line width=0.5pt] (12.6,11.026) -- (12.831,11.411) -- (13.062,11.026);
\draw[line width=0.5pt] (12.528,10.861) -- (13.133,10.861);
\node at (12.831,11.796) {\scalebox{1.446}{\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 rue Neuve & ⑩ rue du Four & S0 Carrefour du Lavoir & \textbf{a} la poste \\ \hline
② venelle du Chat & ⑪ rue du Puits & S1 Coin du Lavoir & \textbf{b} le gymnase \\ \hline
③ rue Verte & ⑫ rue du Pont & S2 Rond-point du Stade & \textbf{c} le marché couvert \\ \hline
④ rue des Jardins & ⑬ rue de l'Église & S3 Fontaine Notre-Dame & \textbf{d} la gendarmerie \\ \hline
⑤ impasse des Roses & ⑭ chemin Creux & S4 Place de l'Église & \textbf{e} l'église \\ \hline
⑥ chemin des Vignes & ⑮ rue Haute & S5 Carrefour des Quatre-Chemins &  \\ \hline
⑦ chemin du Lavoir & ⑯ rue du Stade & S6 Place Sainte-Croix &  \\ \hline
⑧ rue de la Mairie & ⑰ rue des Écoles & S7 Rond-point de la Gare &  \\ \hline
⑨ rue de la Poste &  & S8 Place aux Herbes &  \\ \hline
 &  & S9 Place du Marché &  \\ \hline
 &  & S10 Carrefour de la Mairie &  \\ \hline
 &  & S11 Place du Vieux-Puits &  \\ \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=0.6189, line join=round, line cap=round]
\draw[line width=0.5pt] (4.771,11.261) -- (4.573,11.268) -- (4.374,11.27) -- (4.175,11.267) -- (3.976,11.26) -- (3.777,11.248) -- (3.579,11.231) -- (3.381,11.209) -- (3.184,11.183) -- (2.987,11.153) -- (2.791,11.118) -- (2.596,11.078) -- (2.402,11.033) -- (2.209,10.984) -- (2.018,10.931);
\draw[line width=0.5pt] (4.771,11.261) -- (5.05,11.226) -- (5.329,11.184) -- (5.606,11.135) -- (5.882,11.08) -- (6.157,11.018) -- (6.43,10.95) -- (6.702,10.876) -- (6.972,10.795) -- (7.24,10.708) -- (7.505,10.614) -- (7.769,10.515) -- (8.03,10.409) -- (8.288,10.297) -- (8.544,10.179);
\draw[line width=0.5pt] (4.771,11.261) -- (4.748,10.963) -- (4.731,10.664) -- (4.722,10.365) -- (4.719,10.065) -- (4.724,9.766) -- (4.736,9.467) -- (4.755,9.168) -- (4.78,8.869) -- (4.813,8.572) -- (4.853,8.275) -- (4.9,7.979) -- (4.953,7.685) -- (5.014,7.392) -- (5.081,7.1);
\draw[line width=0.5pt] (2.018,10.931) -- (2.277,10.693) -- (2.53,10.449) -- (2.778,10.199) -- (3.019,9.943) -- (3.255,9.681) -- (3.484,9.414) -- (3.708,9.142) -- (3.924,8.864) -- (4.134,8.582) -- (4.338,8.294) -- (4.534,8.002) -- (4.724,7.706) -- (4.906,7.405) -- (5.081,7.1);
\draw[line width=0.5pt] (5.127,1.56) -- (5.3,1.555) -- (5.474,1.555) -- (5.648,1.558) -- (5.821,1.566) -- (5.995,1.578) -- (6.168,1.594) -- (6.34,1.614) -- (6.513,1.638) -- (6.684,1.666) -- (6.855,1.698) -- (7.025,1.734) -- (7.194,1.774) -- (7.362,1.818) -- (7.529,1.866) -- (7.816,1.886) -- (8.104,1.9) -- (8.392,1.907) -- (8.68,1.906) -- (8.968,1.9) -- (9.256,1.886) -- (9.543,1.866) -- (9.83,1.838) -- (10.117,1.805) -- (10.402,1.764) -- (10.686,1.717) -- (10.969,1.663) -- (11.251,1.602) -- (11.531,1.535);
\draw[line width=0.5pt] (8.544,10.179) -- (8.534,10.014) -- (8.521,9.848) -- (8.504,9.683) -- (8.483,9.518) -- (8.458,9.354) -- (8.429,9.19) -- (8.396,9.027) -- (8.36,8.865) -- (8.32,8.704) -- (8.276,8.544) -- (8.228,8.385) -- (8.177,8.227) -- (8.121,8.07) -- (8.062,7.915);
\draw[line width=0.5pt] (8.544,10.179) -- (8.634,9.951) -- (8.73,9.726) -- (8.831,9.502) -- (8.937,9.281) -- (9.049,9.062) -- (9.165,8.847) -- (9.287,8.633) -- (9.414,8.423) -- (9.545,8.216) -- (9.682,8.012) -- (9.823,7.811) -- (9.969,7.614) -- (10.119,7.42) -- (10.274,7.23);
\draw[line width=0.5pt] (8.544,10.179) -- (8.332,9.923) -- (8.114,9.672) -- (7.891,9.426) -- (7.661,9.186) -- (7.426,8.951) -- (7.185,8.721) -- (6.939,8.497) -- (6.688,8.279) -- (6.432,8.067) -- (6.171,7.861) -- (5.905,7.661) -- (5.635,7.467) -- (5.36,7.281) -- (5.081,7.1);
\draw[line width=0.5pt] (8.062,7.915) -- (7.842,7.89) -- (7.623,7.86) -- (7.404,7.824) -- (7.186,7.783) -- (6.969,7.738) -- (6.753,7.687) -- (6.538,7.631) -- (6.325,7.57) -- (6.113,7.504) -- (5.903,7.433) -- (5.695,7.357) -- (5.488,7.276) -- (5.284,7.191) -- (5.081,7.1);
\draw[line width=0.5pt] (10.274,7.23) -- (10.101,7.042) -- (9.924,6.857) -- (9.743,6.677) -- (9.557,6.501) -- (9.367,6.33) -- (9.173,6.163) -- (8.975,6.0) -- (8.773,5.843) -- (8.568,5.69) -- (8.359,5.542) -- (8.147,5.398) -- (7.932,5.26) -- (7.713,5.128) -- (7.492,5.0);
\draw[line width=0.5pt] (5.081,7.1) -- (5.229,6.925) -- (5.381,6.753) -- (5.537,6.585) -- (5.697,6.421) -- (5.861,6.26) -- (6.029,6.103) -- (6.2,5.951) -- (6.375,5.802) -- (6.553,5.657) -- (6.735,5.517) -- (6.919,5.381) -- (7.107,5.249) -- (7.298,5.122) -- (7.492,5.0);
\draw[line width=0.5pt] (5.081,7.1) -- (5.111,6.942) -- (5.137,6.783) -- (5.159,6.623) -- (5.178,6.463) -- (5.192,6.302) -- (5.203,6.141) -- (5.21,5.98) -- (5.214,5.818) -- (5.213,5.657) -- (5.209,5.496) -- (5.201,5.334) -- (5.189,5.174) -- (5.174,5.013) -- (5.154,4.853);
\draw[line width=0.5pt] (1.531,5.069) -- (1.747,4.781) -- (1.971,4.498) -- (2.2,4.221) -- (2.437,3.949) -- (2.68,3.682) -- (2.929,3.421) -- (3.184,3.166) -- (3.445,2.917) -- (3.712,2.675) -- (3.984,2.438) -- (4.262,2.208) -- (4.545,1.985) -- (4.834,1.769) -- (5.127,1.56);
\draw[line width=0.5pt] (1.531,5.069) -- (1.615,4.844) -- (1.694,4.618) -- (1.768,4.389) -- (1.836,4.159) -- (1.899,3.927) -- (1.956,3.693) -- (2.008,3.458) -- (2.054,3.222) -- (2.095,2.986) -- (2.13,2.748) -- (2.16,2.509) -- (2.184,2.27) -- (2.202,2.031) -- (2.215,1.791);
\draw[line width=0.5pt] (7.492,5.0) -- (7.287,4.782) -- (7.087,4.559) -- (6.892,4.332) -- (6.702,4.1) -- (6.519,3.863) -- (6.34,3.622) -- (6.168,3.377) -- (6.001,3.128) -- (5.84,2.876) -- (5.685,2.619) -- (5.536,2.359) -- (5.393,2.096) -- (5.257,1.829) -- (5.127,1.56);
\draw[line width=0.5pt] (7.492,5.0) -- (7.328,4.964) -- (7.163,4.932) -- (6.997,4.904) -- (6.831,4.879) -- (6.664,4.859) -- (6.497,4.843) -- (6.329,4.83) -- (6.161,4.821) -- (5.993,4.817) -- (5.825,4.816) -- (5.657,4.819) -- (5.489,4.827) -- (5.321,4.838) -- (5.154,4.853);
\draw[line width=0.5pt] (2.215,1.791) -- (2.419,1.743) -- (2.624,1.699) -- (2.83,1.661) -- (3.036,1.627) -- (3.244,1.598) -- (3.452,1.574) -- (3.661,1.555) -- (3.871,1.541) -- (4.08,1.532) -- (4.29,1.528) -- (4.499,1.528) -- (4.709,1.534) -- (4.918,1.544) -- (5.127,1.56);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.931,11.261) -- (4.91,11.341) -- (4.851,11.4) -- (4.771,11.421) -- (4.691,11.4) -- (4.633,11.341) -- (4.611,11.261) -- (4.633,11.181) -- (4.691,11.123) -- (4.771,11.101) -- (4.851,11.123) -- (4.91,11.181) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.178,10.931) -- (2.156,11.011) -- (2.098,11.069) -- (2.018,11.091) -- (1.938,11.069) -- (1.879,11.011) -- (1.858,10.931) -- (1.879,10.851) -- (1.938,10.792) -- (2.018,10.771) -- (2.098,10.792) -- (2.156,10.851) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.704,10.179) -- (8.682,10.259) -- (8.624,10.318) -- (8.544,10.339) -- (8.464,10.318) -- (8.405,10.259) -- (8.384,10.179) -- (8.405,10.099) -- (8.464,10.041) -- (8.544,10.019) -- (8.624,10.041) -- (8.682,10.099) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.222,7.915) -- (8.201,7.995) -- (8.142,8.054) -- (8.062,8.075) -- (7.982,8.054) -- (7.924,7.995) -- (7.902,7.915) -- (7.924,7.835) -- (7.982,7.777) -- (8.062,7.755) -- (8.142,7.777) -- (8.201,7.835) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.434,7.23) -- (10.413,7.31) -- (10.354,7.369) -- (10.274,7.39) -- (10.194,7.369) -- (10.135,7.31) -- (10.114,7.23) -- (10.135,7.15) -- (10.194,7.092) -- (10.274,7.07) -- (10.354,7.092) -- (10.413,7.15) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.241,7.1) -- (5.22,7.18) -- (5.161,7.239) -- (5.081,7.26) -- (5.001,7.239) -- (4.943,7.18) -- (4.921,7.1) -- (4.943,7.02) -- (5.001,6.962) -- (5.081,6.94) -- (5.161,6.962) -- (5.22,7.02) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.691,5.069) -- (1.669,5.149) -- (1.611,5.208) -- (1.531,5.229) -- (1.451,5.208) -- (1.392,5.149) -- (1.371,5.069) -- (1.392,4.989) -- (1.451,4.931) -- (1.531,4.909) -- (1.611,4.931) -- (1.669,4.989) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.652,5.0) -- (7.631,5.08) -- (7.572,5.138) -- (7.492,5.16) -- (7.412,5.138) -- (7.353,5.08) -- (7.332,5.0) -- (7.353,4.92) -- (7.412,4.861) -- (7.492,4.84) -- (7.572,4.861) -- (7.631,4.92) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.314,4.853) -- (5.293,4.933) -- (5.234,4.992) -- (5.154,5.013) -- (5.074,4.992) -- (5.016,4.933) -- (4.994,4.853) -- (5.016,4.773) -- (5.074,4.714) -- (5.154,4.693) -- (5.234,4.714) -- (5.293,4.773) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.375,1.791) -- (2.353,1.871) -- (2.295,1.929) -- (2.215,1.951) -- (2.135,1.929) -- (2.076,1.871) -- (2.055,1.791) -- (2.076,1.711) -- (2.135,1.652) -- (2.215,1.631) -- (2.295,1.652) -- (2.353,1.711) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.287,1.56) -- (5.266,1.64) -- (5.207,1.698) -- (5.127,1.72) -- (5.047,1.698) -- (4.988,1.64) -- (4.967,1.56) -- (4.988,1.48) -- (5.047,1.421) -- (5.127,1.4) -- (5.207,1.421) -- (5.266,1.48) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.691,1.535) -- (11.669,1.615) -- (11.611,1.674) -- (11.531,1.695) -- (11.451,1.674) -- (11.392,1.615) -- (11.371,1.535) -- (11.392,1.455) -- (11.451,1.397) -- (11.531,1.375) -- (11.611,1.397) -- (11.669,1.455) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.759,10.892) -- (3.714,11.062) -- (3.589,11.186) -- (3.419,11.232) -- (3.249,11.186) -- (3.125,11.062) -- (3.079,10.892) -- (3.125,10.722) -- (3.249,10.597) -- (3.419,10.552) -- (3.589,10.597) -- (3.714,10.722) -- cycle;
\node at (3.419,10.892) {\scalebox{1.616}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.13,11.183) -- (7.085,11.353) -- (6.96,11.478) -- (6.79,11.523) -- (6.62,11.478) -- (6.496,11.353) -- (6.45,11.183) -- (6.496,11.013) -- (6.62,10.889) -- (6.79,10.843) -- (6.96,10.889) -- (7.085,11.013) -- cycle;
\node at (6.79,11.183) {\scalebox{1.616}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.414,9.192) -- (5.368,9.362) -- (5.244,9.486) -- (5.074,9.532) -- (4.904,9.486) -- (4.779,9.362) -- (4.734,9.192) -- (4.779,9.022) -- (4.904,8.897) -- (5.074,8.852) -- (5.244,8.897) -- (5.368,9.022) -- cycle;
\node at (5.074,9.192) {\scalebox{1.616}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.297,9.342) -- (4.252,9.512) -- (4.127,9.636) -- (3.957,9.682) -- (3.787,9.636) -- (3.663,9.512) -- (3.617,9.342) -- (3.663,9.172) -- (3.787,9.047) -- (3.957,9.002) -- (4.127,9.047) -- (4.252,9.172) -- cycle;
\node at (3.957,9.342) {\scalebox{1.616}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.821,2.182) -- (7.776,2.352) -- (7.651,2.477) -- (7.481,2.522) -- (7.311,2.477) -- (7.187,2.352) -- (7.141,2.182) -- (7.187,2.012) -- (7.311,1.888) -- (7.481,1.842) -- (7.651,1.888) -- (7.776,2.012) -- cycle;
\node at (7.481,2.182) {\scalebox{1.616}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.049,8.961) -- (9.004,9.131) -- (8.879,9.255) -- (8.709,9.301) -- (8.539,9.255) -- (8.415,9.131) -- (8.369,8.961) -- (8.415,8.791) -- (8.539,8.666) -- (8.709,8.621) -- (8.879,8.666) -- (9.004,8.791) -- cycle;
\node at (8.709,8.961) {\scalebox{1.616}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.903,8.795) -- (9.858,8.965) -- (9.733,9.09) -- (9.563,9.135) -- (9.393,9.09) -- (9.269,8.965) -- (9.223,8.795) -- (9.269,8.625) -- (9.393,8.501) -- (9.563,8.455) -- (9.733,8.501) -- (9.858,8.625) -- cycle;
\node at (9.563,8.795) {\scalebox{1.616}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.492,8.258) -- (7.447,8.428) -- (7.322,8.552) -- (7.152,8.598) -- (6.982,8.552) -- (6.858,8.428) -- (6.812,8.258) -- (6.858,8.088) -- (6.982,7.963) -- (7.152,7.918) -- (7.322,7.963) -- (7.447,8.088) -- cycle;
\node at (7.152,8.258) {\scalebox{1.616}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.963,7.322) -- (6.917,7.492) -- (6.793,7.616) -- (6.623,7.662) -- (6.453,7.616) -- (6.328,7.492) -- (6.283,7.322) -- (6.328,7.152) -- (6.453,7.028) -- (6.623,6.982) -- (6.793,7.028) -- (6.917,7.152) -- cycle;
\node at (6.623,7.322) {\scalebox{1.616}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.515,5.751) -- (9.47,5.921) -- (9.345,6.045) -- (9.175,6.091) -- (9.005,6.045) -- (8.881,5.921) -- (8.835,5.751) -- (8.881,5.581) -- (9.005,5.456) -- (9.175,5.411) -- (9.345,5.456) -- (9.47,5.581) -- cycle;
\node at (9.175,5.751) {\scalebox{1.616}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.75,6.192) -- (6.705,6.362) -- (6.58,6.486) -- (6.41,6.532) -- (6.24,6.486) -- (6.116,6.362) -- (6.07,6.192) -- (6.116,6.022) -- (6.24,5.897) -- (6.41,5.852) -- (6.58,5.897) -- (6.705,6.022) -- cycle;
\node at (6.41,6.192) {\scalebox{1.616}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.87,5.99) -- (5.825,6.16) -- (5.7,6.284) -- (5.53,6.33) -- (5.36,6.284) -- (5.236,6.16) -- (5.19,5.99) -- (5.236,5.82) -- (5.36,5.696) -- (5.53,5.65) -- (5.7,5.696) -- (5.825,5.82) -- cycle;
\node at (5.53,5.99) {\scalebox{1.616}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.748,3.395) -- (3.702,3.565) -- (3.578,3.69) -- (3.408,3.735) -- (3.238,3.69) -- (3.113,3.565) -- (3.068,3.395) -- (3.113,3.225) -- (3.238,3.101) -- (3.408,3.055) -- (3.578,3.101) -- (3.702,3.225) -- cycle;
\node at (3.408,3.395) {\scalebox{1.616}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.661,3.524) -- (2.616,3.694) -- (2.491,3.818) -- (2.321,3.864) -- (2.151,3.818) -- (2.027,3.694) -- (1.981,3.524) -- (2.027,3.354) -- (2.151,3.229) -- (2.321,3.184) -- (2.491,3.229) -- (2.616,3.354) -- cycle;
\node at (2.321,3.524) {\scalebox{1.616}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.771,3.196) -- (6.726,3.366) -- (6.601,3.491) -- (6.431,3.536) -- (6.261,3.491) -- (6.137,3.366) -- (6.091,3.196) -- (6.137,3.026) -- (6.261,2.902) -- (6.431,2.856) -- (6.601,2.902) -- (6.726,3.026) -- cycle;
\node at (6.431,3.196) {\scalebox{1.616}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.689,4.511) -- (6.644,4.681) -- (6.519,4.805) -- (6.349,4.851) -- (6.179,4.805) -- (6.055,4.681) -- (6.009,4.511) -- (6.055,4.341) -- (6.179,4.216) -- (6.349,4.171) -- (6.519,4.216) -- (6.644,4.341) -- cycle;
\node at (6.349,4.511) {\scalebox{1.616}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.027,1.874) -- (3.981,2.044) -- (3.857,2.169) -- (3.687,2.214) -- (3.517,2.169) -- (3.392,2.044) -- (3.347,1.874) -- (3.392,1.704) -- (3.517,1.58) -- (3.687,1.534) -- (3.857,1.58) -- (3.981,1.704) -- cycle;
\node at (3.687,1.874) {\scalebox{1.616}{\small 17}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (4.471,10.961) -- (5.071,10.961) -- (5.071,11.561) -- (4.471,11.561) -- cycle;
\node at (4.771,11.261) {\scalebox{1.616}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (10.355,9.253) -- (10.955,9.253) -- (10.955,9.853) -- (10.355,9.853) -- cycle;
\node at (10.655,9.553) {\scalebox{1.616}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.603,11.32) -- (3.203,11.32) -- (3.203,11.92) -- (2.603,11.92) -- cycle;
\node at (2.903,11.62) {\scalebox{1.616}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (7.32,6.364) -- (7.92,6.364) -- (7.92,6.964) -- (7.32,6.964) -- cycle;
\node at (7.62,6.664) {\scalebox{1.616}{\small d}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (6.659,1.1) -- (7.259,1.1) -- (7.259,1.7) -- (6.659,1.7) -- cycle;
\node at (6.959,1.4) {\scalebox{1.616}{\small e}};
\node at (4.627,11.864) {\scalebox{1.616}{\small S0}};
\node at (1.619,11.406) {\scalebox{1.616}{\small S1}};
\node at (8.879,10.701) {\scalebox{1.616}{\small S2}};
\node at (8.541,8.309) {\scalebox{1.616}{\small S3}};
\node at (10.877,7.375) {\scalebox{1.616}{\small S4}};
\node at (4.642,7.538) {\scalebox{1.616}{\small S5}};
\node at (0.93,4.916) {\scalebox{1.616}{\small S6}};
\node at (7.977,4.614) {\scalebox{1.616}{\small S7}};
\node at (4.83,4.325) {\scalebox{1.616}{\small S8}};
\node at (1.812,1.32) {\scalebox{1.616}{\small S9}};
\node at (5.014,0.95) {\scalebox{1.616}{\small S10}};
\node at (12.005,1.136) {\scalebox{1.616}{\small S11}};
\draw[line width=0.5pt] (12.831,10.311) -- (12.831,11.411);
\draw[line width=0.5pt] (12.6,11.026) -- (12.831,11.411) -- (13.062,11.026);
\draw[line width=0.5pt] (12.528,10.861) -- (13.133,10.861);
\node at (12.831,11.796) {\scalebox{1.616}{\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|c|c|}
\hline
\cellcolor{blue!7}\bfseries Carrefour & \cellcolor{blue!7}\bfseries S0 & \cellcolor{blue!7}\bfseries S1 & \cellcolor{blue!7}\bfseries S2 & \cellcolor{blue!7}\bfseries S3 & \cellcolor{blue!7}\bfseries S4 & \cellcolor{blue!7}\bfseries S5 & \cellcolor{blue!7}\bfseries S6 & \cellcolor{blue!7}\bfseries S7 & \cellcolor{blue!7}\bfseries S8 & \cellcolor{blue!7}\bfseries S9 & \cellcolor{blue!7}\bfseries S10 & \cellcolor{blue!7}\bfseries S11 \\ \hline
Nombre de rues & 3 & 2 & 4 & 2 & 2 & 6 & 2 & 4 & 2 & 2 & 4 & 1 \\ \hline
Parité & \textbf{impair} & pair & pair & pair & pair & pair & pair & pair & pair & pair & pair & \textbf{impair} \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 34 = 2 \(\times\) 17, 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{S0}, \textbf{S11} — il y en a 2.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Quand le facteur \textbf{traverse} un carrefour, il y arrive par une rue et en repart par une autre : il consomme les rues \textbf{deux par deux}. Un carrefour traversé 3 fois voit donc 6 de ses rues utilisées — toujours un nombre \textbf{pair}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Un carrefour où aboutit un nombre \textbf{impair} de rues ne peut donc pas être seulement traversé : il doit être le \textbf{départ} ou l'\textbf{arrivée} de la tournée. Là, une rue reste « non appariée ».}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Ici la poste est \textbf{sur un carrefour}, en \textbf{S0}, et le facteur n'a pas à y revenir. Ce carrefour est donc le \textbf{départ} — il a droit d'être impair. Mais il n'y a qu'\textbf{une seule arrivée} : tous les autres carrefours doivent être pairs, sauf un au plus.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui.} Deux carrefours sont impairs, \textbf{S0} et \textbf{S11}, et la poste est justement sur l'un des deux (\textbf{S0}). Il part donc de la poste et \textbf{termine en S11}, l'autre carrefour impair.}\par\addvspace{0.4em}
\newpage
\noindent Voici une tournée possible :\par\addvspace{0.35em}
\begin{enumerate}
  \item Il part de Carrefour du Lavoir \textbf{(S0)}, prend la \textbf{rue Neuve ①} vers l'ouest et atteint Coin du Lavoir \textbf{(S1)}.
  \item Il tourne à gauche dans la \textbf{rue des Jardins ④}, vers le sud-est, jusqu'à Carrefour des Quatre-Chemins \textbf{(S5)}.
  \item Il prend dans la \textbf{rue Verte ③}, vers le nord, jusqu'à Carrefour du Lavoir \textbf{(S0)}.
  \item Il tourne à droite dans la \textbf{venelle du Chat ②}, vers l'est, jusqu'à Rond-point du Stade \textbf{(S2)}.
  \item Il tourne à droite dans le \textbf{chemin des Vignes ⑥}, vers le sud, jusqu'à Fontaine Notre-Dame \textbf{(S3)}.
  \item Il tourne à droite dans la \textbf{rue de la Poste ⑨}, vers l'ouest, jusqu'à Carrefour des Quatre-Chemins \textbf{(S5)}.
  \item Il prend dans la \textbf{rue de la Mairie ⑧}, vers le nord-est, jusqu'à Rond-point du Stade \textbf{(S2)}.
  \item Il tourne à droite dans le \textbf{chemin du Lavoir ⑦}, vers le sud-est, jusqu'à Place de l'Église \textbf{(S4)}.
  \item Il tourne à droite dans la \textbf{rue du Four ⑩}, vers le sud-ouest, jusqu'à Rond-point de la Gare \textbf{(S7)}.
  \item Il tourne à droite dans la \textbf{rue du Puits ⑪}, vers le nord-ouest, jusqu'à Carrefour des Quatre-Chemins \textbf{(S5)}.
  \item Il prend dans la \textbf{rue du Pont ⑫}, vers le sud, jusqu'à Place aux Herbes \textbf{(S8)}.
  \item Il tourne à gauche dans la \textbf{rue du Stade ⑯}, vers l'est, jusqu'à Rond-point de la Gare \textbf{(S7)}.
  \item Il tourne à droite dans la \textbf{rue Haute ⑮}, vers le sud-ouest, jusqu'à Carrefour de la Mairie \textbf{(S10)}.
  \item Il tourne à droite dans la \textbf{rue de l'Église ⑬}, vers le nord-ouest, jusqu'à Place Sainte-Croix \textbf{(S6)}.
  \item Il prend dans le \textbf{chemin Creux ⑭}, vers le sud, jusqu'à Place du Marché \textbf{(S9)}.
  \item Il tourne à gauche dans la \textbf{rue des Écoles ⑰}, vers l'est, jusqu'à Carrefour de la Mairie \textbf{(S10)}.
  \item Il continue tout droit dans l'\textbf{impasse des Roses ⑤}, vers l'est, jusqu'à Place du Vieux-Puits \textbf{(S11)}.
\end{enumerate}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries C'était déjà possible. Notons seulement que la poste devait être en \textbf{S0} ou en \textbf{S11} : ailleurs, la tournée n'aurait pas pu commencer.}\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}
