\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 216}

\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=216}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0216}
\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=0.737, line join=round, line cap=round]
\draw[line width=0.5pt] (7.959,11.57) -- (8.131,11.481) -- (8.305,11.397) -- (8.481,11.317) -- (8.658,11.241) -- (8.838,11.169) -- (9.019,11.101) -- (9.202,11.038) -- (9.387,10.978) -- (9.572,10.924) -- (9.759,10.873) -- (9.947,10.827) -- (10.136,10.786) -- (10.326,10.749) -- (10.517,10.716);
\draw[line width=0.5pt] (7.959,11.57) -- (8.15,11.344) -- (8.337,11.113) -- (8.518,10.878) -- (8.694,10.639) -- (8.863,10.396) -- (9.028,10.149) -- (9.186,9.897) -- (9.338,9.643) -- (9.485,9.385) -- (9.625,9.123) -- (9.759,8.858) -- (9.886,8.59) -- (10.008,8.32) -- (10.123,8.046);
\draw[line width=0.5pt] (7.959,11.57) -- (7.674,11.388) -- (7.394,11.199) -- (7.118,11.003) -- (6.846,10.801) -- (6.58,10.593) -- (6.318,10.378) -- (6.062,10.157) -- (5.811,9.93) -- (5.565,9.698) -- (5.325,9.459) -- (5.09,9.216) -- (4.861,8.966) -- (4.639,8.712) -- (4.422,8.452) -- (4.243,8.187) -- (4.057,7.926) -- (3.865,7.669) -- (3.667,7.417) -- (3.464,7.169) -- (3.254,6.927) -- (3.039,6.689) -- (2.818,6.456) -- (2.592,6.229) -- (2.36,6.007) -- (2.124,5.791) -- (1.882,5.58) -- (1.636,5.375) -- (1.385,5.176);
\draw[line width=0.5pt] (7.959,11.57) -- (7.989,11.374) -- (8.015,11.177) -- (8.036,10.98) -- (8.053,10.782) -- (8.065,10.584) -- (8.072,10.385) -- (8.075,10.187) -- (8.073,9.988) -- (8.066,9.79) -- (8.055,9.591) -- (8.039,9.393) -- (8.018,9.196) -- (7.993,8.999) -- (7.963,8.803) -- (7.853,8.52) -- (7.75,8.235) -- (7.653,7.947) -- (7.563,7.657) -- (7.481,7.365) -- (7.405,7.07) -- (7.336,6.775) -- (7.273,6.477) -- (7.218,6.179) -- (7.17,5.879) -- (7.129,5.578) -- (7.095,5.276) -- (7.068,4.974) -- (7.049,4.671);
\draw[line width=0.5pt] (10.517,10.716) -- (10.459,10.531) -- (10.407,10.345) -- (10.358,10.158) -- (10.314,9.969) -- (10.275,9.779) -- (10.24,9.589) -- (10.209,9.398) -- (10.183,9.206) -- (10.162,9.013) -- (10.145,8.82) -- (10.132,8.627) -- (10.125,8.433) -- (10.121,8.24) -- (10.123,8.046);
\draw[line width=0.5pt] (10.123,8.046) -- (10.182,7.859) -- (10.236,7.67) -- (10.287,7.48) -- (10.332,7.289) -- (10.374,7.097) -- (10.41,6.903) -- (10.443,6.709) -- (10.47,6.514) -- (10.493,6.319) -- (10.512,6.123) -- (10.525,5.927) -- (10.535,5.73) -- (10.539,5.534) -- (10.539,5.337);
\draw[line width=0.5pt] (10.123,8.046) -- (9.868,7.841) -- (9.618,7.629) -- (9.373,7.412) -- (9.133,7.188) -- (8.899,6.96) -- (8.67,6.725) -- (8.446,6.485) -- (8.229,6.241) -- (8.017,5.991) -- (7.811,5.736) -- (7.611,5.476) -- (7.417,5.212) -- (7.23,4.944) -- (7.049,4.671);
\draw[line width=0.5pt] (10.539,5.337) -- (10.299,5.252) -- (10.057,5.172) -- (9.813,5.098) -- (9.567,5.03) -- (9.32,4.968) -- (9.071,4.911) -- (8.821,4.86) -- (8.57,4.815) -- (8.318,4.776) -- (8.065,4.743) -- (7.812,4.716) -- (7.558,4.695) -- (7.304,4.68) -- (7.049,4.671);
\draw[line width=0.5pt] (1.385,5.176) -- (1.614,5.146) -- (1.843,5.121) -- (2.073,5.102) -- (2.304,5.089) -- (2.535,5.08) -- (2.766,5.078) -- (2.997,5.08) -- (3.228,5.088) -- (3.459,5.102) -- (3.689,5.12) -- (3.919,5.145) -- (4.148,5.174) -- (4.376,5.209) -- (4.604,5.25) -- (4.783,5.235) -- (4.963,5.217) -- (5.141,5.194) -- (5.32,5.167) -- (5.497,5.136) -- (5.674,5.101) -- (5.85,5.061) -- (6.025,5.018) -- (6.199,4.97) -- (6.372,4.918) -- (6.543,4.862) -- (6.713,4.802) -- (6.882,4.739) -- (7.049,4.671);
\draw[line width=0.5pt] (1.385,5.176) -- (1.443,4.978) -- (1.497,4.778) -- (1.547,4.577) -- (1.591,4.375) -- (1.631,4.172) -- (1.666,3.968) -- (1.696,3.763) -- (1.722,3.558) -- (1.742,3.352) -- (1.758,3.146) -- (1.769,2.939) -- (1.775,2.732) -- (1.776,2.526) -- (1.772,2.319);
\draw[line width=0.5pt] (1.385,5.176) -- (1.601,4.895) -- (1.824,4.619) -- (2.054,4.348) -- (2.29,4.082) -- (2.532,3.823) -- (2.78,3.569) -- (3.035,3.32) -- (3.295,3.078) -- (3.56,2.842) -- (3.831,2.613) -- (4.108,2.389) -- (4.389,2.173) -- (4.675,1.963) -- (4.967,1.76);
\draw[line width=0.5pt] (7.049,4.671) -- (7.331,4.554) -- (7.61,4.43) -- (7.887,4.3) -- (8.16,4.163) -- (8.43,4.019) -- (8.696,3.87) -- (8.959,3.714) -- (9.219,3.552) -- (9.474,3.384) -- (9.726,3.21) -- (9.973,3.03) -- (10.215,2.845) -- (10.454,2.654) -- (10.687,2.457);
\draw[line width=0.5pt] (7.049,4.671) -- (6.87,4.487) -- (6.694,4.3) -- (6.524,4.108) -- (6.358,3.912) -- (6.196,3.712) -- (6.04,3.509) -- (5.888,3.301) -- (5.741,3.091) -- (5.599,2.877) -- (5.462,2.659) -- (5.33,2.439) -- (5.203,2.216) -- (5.082,1.989) -- (4.967,1.76);
\draw[line width=0.5pt] (7.049,4.671) -- (7.098,4.453) -- (7.142,4.235) -- (7.18,4.015) -- (7.214,3.795) -- (7.242,3.573) -- (7.265,3.352) -- (7.283,3.129) -- (7.295,2.906) -- (7.302,2.683) -- (7.305,2.46) -- (7.301,2.237) -- (7.293,2.015) -- (7.279,1.792) -- (7.26,1.57);
\draw[line width=0.5pt] (10.687,2.457) -- (10.435,2.431) -- (10.183,2.4) -- (9.932,2.363) -- (9.681,2.32) -- (9.432,2.27) -- (9.184,2.216) -- (8.937,2.155) -- (8.692,2.088) -- (8.448,2.016) -- (8.207,1.938) -- (7.967,1.854) -- (7.729,1.765) -- (7.494,1.67) -- (7.26,1.57);
\draw[line width=0.5pt] (1.772,2.319) -- (2.005,2.314) -- (2.237,2.304) -- (2.469,2.289) -- (2.701,2.267) -- (2.932,2.241) -- (3.163,2.209) -- (3.392,2.171) -- (3.621,2.129) -- (3.849,2.08) -- (4.075,2.027) -- (4.301,1.968) -- (4.524,1.904) -- (4.746,1.835) -- (4.967,1.76);
\draw[line width=0.5pt] (4.967,1.76) -- (5.127,1.722) -- (5.288,1.687) -- (5.451,1.656) -- (5.614,1.629) -- (5.777,1.606) -- (5.941,1.586) -- (6.106,1.571) -- (6.27,1.559) -- (6.435,1.551) -- (6.601,1.547) -- (6.766,1.547) -- (6.931,1.551) -- (7.096,1.558) -- (7.26,1.57);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.119,11.57) -- (8.097,11.65) -- (8.039,11.709) -- (7.959,11.73) -- (7.879,11.709) -- (7.82,11.65) -- (7.799,11.57) -- (7.82,11.49) -- (7.879,11.431) -- (7.959,11.41) -- (8.039,11.431) -- (8.097,11.49) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.677,10.716) -- (10.655,10.796) -- (10.597,10.855) -- (10.517,10.876) -- (10.437,10.855) -- (10.378,10.796) -- (10.357,10.716) -- (10.378,10.636) -- (10.437,10.578) -- (10.517,10.556) -- (10.597,10.578) -- (10.655,10.636) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.283,8.046) -- (10.261,8.126) -- (10.203,8.185) -- (10.123,8.206) -- (10.043,8.185) -- (9.984,8.126) -- (9.963,8.046) -- (9.984,7.966) -- (10.043,7.908) -- (10.123,7.886) -- (10.203,7.908) -- (10.261,7.966) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.699,5.337) -- (10.678,5.417) -- (10.619,5.476) -- (10.539,5.497) -- (10.459,5.476) -- (10.4,5.417) -- (10.379,5.337) -- (10.4,5.257) -- (10.459,5.199) -- (10.539,5.177) -- (10.619,5.199) -- (10.678,5.257) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.545,5.176) -- (1.523,5.256) -- (1.465,5.315) -- (1.385,5.336) -- (1.305,5.315) -- (1.246,5.256) -- (1.225,5.176) -- (1.246,5.096) -- (1.305,5.037) -- (1.385,5.016) -- (1.465,5.037) -- (1.523,5.096) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.209,4.671) -- (7.187,4.751) -- (7.129,4.809) -- (7.049,4.831) -- (6.969,4.809) -- (6.91,4.751) -- (6.889,4.671) -- (6.91,4.591) -- (6.969,4.532) -- (7.049,4.511) -- (7.129,4.532) -- (7.187,4.591) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.847,2.457) -- (10.826,2.537) -- (10.767,2.595) -- (10.687,2.617) -- (10.607,2.595) -- (10.549,2.537) -- (10.527,2.457) -- (10.549,2.377) -- (10.607,2.318) -- (10.687,2.297) -- (10.767,2.318) -- (10.826,2.377) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.932,2.319) -- (1.911,2.399) -- (1.852,2.457) -- (1.772,2.479) -- (1.692,2.457) -- (1.634,2.399) -- (1.612,2.319) -- (1.634,2.239) -- (1.692,2.18) -- (1.772,2.159) -- (1.852,2.18) -- (1.911,2.239) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.127,1.76) -- (5.105,1.84) -- (5.047,1.899) -- (4.967,1.92) -- (4.887,1.899) -- (4.828,1.84) -- (4.807,1.76) -- (4.828,1.68) -- (4.887,1.622) -- (4.967,1.6) -- (5.047,1.622) -- (5.105,1.68) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.42,1.57) -- (7.399,1.65) -- (7.34,1.709) -- (7.26,1.73) -- (7.18,1.709) -- (7.122,1.65) -- (7.1,1.57) -- (7.122,1.49) -- (7.18,1.431) -- (7.26,1.41) -- (7.34,1.431) -- (7.399,1.49) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.644,11.341) -- (9.598,11.511) -- (9.474,11.636) -- (9.304,11.681) -- (9.134,11.636) -- (9.009,11.511) -- (8.964,11.341) -- (9.009,11.171) -- (9.134,11.047) -- (9.304,11.001) -- (9.474,11.047) -- (9.598,11.171) -- cycle;
\node at (9.304,11.341) {\scalebox{1.357}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.799,10.065) -- (9.753,10.235) -- (9.629,10.359) -- (9.459,10.405) -- (9.289,10.359) -- (9.164,10.235) -- (9.119,10.065) -- (9.164,9.895) -- (9.289,9.771) -- (9.459,9.725) -- (9.629,9.771) -- (9.753,9.895) -- cycle;
\node at (9.459,10.065) {\scalebox{1.357}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.018,8.26) -- (4.972,8.43) -- (4.848,8.554) -- (4.678,8.6) -- (4.508,8.554) -- (4.383,8.43) -- (4.338,8.26) -- (4.383,8.09) -- (4.508,7.965) -- (4.678,7.92) -- (4.848,7.965) -- (4.972,8.09) -- cycle;
\node at (4.678,8.26) {\scalebox{1.357}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.61,8.713) -- (8.565,8.883) -- (8.44,9.008) -- (8.27,9.053) -- (8.1,9.008) -- (7.976,8.883) -- (7.93,8.713) -- (7.976,8.543) -- (8.1,8.419) -- (8.27,8.373) -- (8.44,8.419) -- (8.565,8.543) -- cycle;
\node at (8.27,8.713) {\scalebox{1.357}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.866,9.351) -- (10.82,9.521) -- (10.696,9.645) -- (10.526,9.691) -- (10.356,9.645) -- (10.232,9.521) -- (10.186,9.351) -- (10.232,9.181) -- (10.356,9.056) -- (10.526,9.011) -- (10.696,9.056) -- (10.82,9.181) -- cycle;
\node at (10.526,9.351) {\scalebox{1.357}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.099,6.758) -- (11.053,6.928) -- (10.929,7.052) -- (10.759,7.098) -- (10.589,7.052) -- (10.464,6.928) -- (10.419,6.758) -- (10.464,6.588) -- (10.589,6.463) -- (10.759,6.418) -- (10.929,6.463) -- (11.053,6.588) -- cycle;
\node at (10.759,6.758) {\scalebox{1.357}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.023,6.27) -- (8.978,6.44) -- (8.853,6.564) -- (8.683,6.61) -- (8.513,6.564) -- (8.389,6.44) -- (8.343,6.27) -- (8.389,6.1) -- (8.513,5.976) -- (8.683,5.93) -- (8.853,5.976) -- (8.978,6.1) -- cycle;
\node at (8.683,6.27) {\scalebox{1.357}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.222,4.546) -- (9.176,4.716) -- (9.052,4.84) -- (8.882,4.886) -- (8.712,4.84) -- (8.587,4.716) -- (8.542,4.546) -- (8.587,4.376) -- (8.712,4.251) -- (8.882,4.206) -- (9.052,4.251) -- (9.176,4.376) -- cycle;
\node at (8.882,4.546) {\scalebox{1.357}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.923,5.569) -- (4.878,5.739) -- (4.753,5.864) -- (4.583,5.909) -- (4.413,5.864) -- (4.289,5.739) -- (4.243,5.569) -- (4.289,5.399) -- (4.413,5.275) -- (4.583,5.229) -- (4.753,5.275) -- (4.878,5.399) -- cycle;
\node at (4.583,5.569) {\scalebox{1.357}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.353,3.806) -- (2.308,3.976) -- (2.183,4.101) -- (2.013,4.146) -- (1.843,4.101) -- (1.719,3.976) -- (1.673,3.806) -- (1.719,3.636) -- (1.843,3.512) -- (2.013,3.466) -- (2.183,3.512) -- (2.308,3.636) -- cycle;
\node at (2.013,3.806) {\scalebox{1.357}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.596,3.552) -- (3.55,3.722) -- (3.426,3.846) -- (3.256,3.892) -- (3.086,3.846) -- (2.961,3.722) -- (2.916,3.552) -- (2.961,3.382) -- (3.086,3.258) -- (3.256,3.212) -- (3.426,3.258) -- (3.55,3.382) -- cycle;
\node at (3.256,3.552) {\scalebox{1.357}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.133,3.441) -- (9.088,3.611) -- (8.963,3.735) -- (8.793,3.781) -- (8.623,3.735) -- (8.499,3.611) -- (8.453,3.441) -- (8.499,3.271) -- (8.623,3.146) -- (8.793,3.101) -- (8.963,3.146) -- (9.088,3.271) -- cycle;
\node at (8.793,3.441) {\scalebox{1.357}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.488,3.115) -- (6.442,3.285) -- (6.318,3.41) -- (6.148,3.455) -- (5.978,3.41) -- (5.853,3.285) -- (5.808,3.115) -- (5.853,2.945) -- (5.978,2.821) -- (6.148,2.775) -- (6.318,2.821) -- (6.442,2.945) -- cycle;
\node at (6.148,3.115) {\scalebox{1.357}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.942,3.151) -- (7.896,3.321) -- (7.772,3.445) -- (7.602,3.491) -- (7.432,3.445) -- (7.307,3.321) -- (7.262,3.151) -- (7.307,2.981) -- (7.432,2.856) -- (7.602,2.811) -- (7.772,2.856) -- (7.896,2.981) -- cycle;
\node at (7.602,3.151) {\scalebox{1.357}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.357,1.845) -- (9.312,2.015) -- (9.187,2.139) -- (9.017,2.185) -- (8.847,2.139) -- (8.723,2.015) -- (8.677,1.845) -- (8.723,1.675) -- (8.847,1.551) -- (9.017,1.505) -- (9.187,1.551) -- (9.312,1.675) -- cycle;
\node at (9.017,1.845) {\scalebox{1.357}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.788,2.487) -- (3.742,2.657) -- (3.618,2.781) -- (3.448,2.827) -- (3.278,2.781) -- (3.153,2.657) -- (3.108,2.487) -- (3.153,2.317) -- (3.278,2.192) -- (3.448,2.147) -- (3.618,2.192) -- (3.742,2.317) -- cycle;
\node at (3.448,2.487) {\scalebox{1.357}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.472,1.889) -- (6.427,2.059) -- (6.302,2.184) -- (6.132,2.229) -- (5.962,2.184) -- (5.838,2.059) -- (5.792,1.889) -- (5.838,1.719) -- (5.962,1.595) -- (6.132,1.549) -- (6.302,1.595) -- (6.427,1.719) -- cycle;
\node at (6.132,1.889) {\scalebox{1.357}{\small 17}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (6.96,1.27) -- (7.56,1.27) -- (7.56,1.87) -- (6.96,1.87) -- cycle;
\node at (7.26,1.57) {\scalebox{1.357}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.035,2.681) -- (6.635,2.681) -- (6.635,3.281) -- (6.035,3.281) -- cycle;
\node at (6.335,2.981) {\scalebox{1.357}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (10.119,11.404) -- (10.719,11.404) -- (10.719,12.004) -- (10.119,12.004) -- cycle;
\node at (10.419,11.704) {\scalebox{1.357}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.191,8.349) -- (8.791,8.349) -- (8.791,8.949) -- (8.191,8.949) -- cycle;
\node at (8.491,8.649) {\scalebox{1.357}{\small d}};
\node at (8.031,12.186) {\scalebox{1.357}{\small A}};
\node at (10.841,11.244) {\scalebox{1.357}{\small B}};
\node at (10.577,8.468) {\scalebox{1.357}{\small C}};
\node at (11.159,5.333) {\scalebox{1.357}{\small D}};
\node at (0.765,5.156) {\scalebox{1.357}{\small E}};
\node at (6.895,4.07) {\scalebox{1.357}{\small F}};
\node at (11.162,2.058) {\scalebox{1.357}{\small G}};
\node at (1.231,2.017) {\scalebox{1.357}{\small H}};
\node at (4.637,1.235) {\scalebox{1.357}{\small I}};
\node at (7.266,0.95) {\scalebox{1.357}{\small J}};
\draw[line width=0.5pt] (11.987,10.62) -- (11.987,11.72);
\draw[line width=0.5pt] (11.756,11.335) -- (11.987,11.72) -- (12.218,11.335);
\draw[line width=0.5pt] (11.685,11.17) -- (12.29,11.17);
\node at (11.987,12.105) {\scalebox{1.357}{\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 boulevard du Parc & Ⓐ Carrefour du Moulin & \textbf{a} la poste \\ \hline
② rue Verte & ⑪ passage du Marché & Ⓑ Place aux Herbes & \textbf{b} la pharmacie \\ \hline
③ quai aux Fleurs & ⑫ rue Pasteur & Ⓒ Carrefour Saint-Roch & \textbf{c} la salle des fêtes \\ \hline
④ rue de l'Église & ⑬ rue du Puits & Ⓓ Place des Tanneurs & \textbf{d} la gendarmerie \\ \hline
⑤ rue du Four & ⑭ allée des Peupliers & Ⓔ Carrefour des Quatre-Chemins &  \\ \hline
⑥ rue du Stade & ⑮ rue Haute & Ⓕ Rond-point des Vaches &  \\ \hline
⑦ rue de la Poste & ⑯ rue de la Mairie & Ⓖ Place Sainte-Croix &  \\ \hline
⑧ chemin du Lavoir & ⑰ rue du Pont & Ⓗ Coin du Lavoir &  \\ \hline
⑨ rue des Jardins &  & Ⓘ Carrefour de l'Hôpital &  \\ \hline
 &  & Ⓙ Carrefour 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=0.6594, line join=round, line cap=round]
\draw[line width=0.5pt] (7.959,11.57) -- (8.131,11.481) -- (8.305,11.397) -- (8.481,11.317) -- (8.658,11.241) -- (8.838,11.169) -- (9.019,11.101) -- (9.202,11.038) -- (9.387,10.978) -- (9.572,10.924) -- (9.759,10.873) -- (9.947,10.827) -- (10.136,10.786) -- (10.326,10.749) -- (10.517,10.716);
\draw[line width=0.5pt] (7.959,11.57) -- (8.15,11.344) -- (8.337,11.113) -- (8.518,10.878) -- (8.694,10.639) -- (8.863,10.396) -- (9.028,10.149) -- (9.186,9.897) -- (9.338,9.643) -- (9.485,9.385) -- (9.625,9.123) -- (9.759,8.858) -- (9.886,8.59) -- (10.008,8.32) -- (10.123,8.046);
\draw[line width=0.5pt] (7.959,11.57) -- (7.674,11.388) -- (7.394,11.199) -- (7.118,11.003) -- (6.846,10.801) -- (6.58,10.593) -- (6.318,10.378) -- (6.062,10.157) -- (5.811,9.93) -- (5.565,9.698) -- (5.325,9.459) -- (5.09,9.216) -- (4.861,8.966) -- (4.639,8.712) -- (4.422,8.452) -- (4.243,8.187) -- (4.057,7.926) -- (3.865,7.669) -- (3.667,7.417) -- (3.464,7.169) -- (3.254,6.927) -- (3.039,6.689) -- (2.818,6.456) -- (2.592,6.229) -- (2.36,6.007) -- (2.124,5.791) -- (1.882,5.58) -- (1.636,5.375) -- (1.385,5.176);
\draw[line width=0.5pt] (7.959,11.57) -- (7.989,11.374) -- (8.015,11.177) -- (8.036,10.98) -- (8.053,10.782) -- (8.065,10.584) -- (8.072,10.385) -- (8.075,10.187) -- (8.073,9.988) -- (8.066,9.79) -- (8.055,9.591) -- (8.039,9.393) -- (8.018,9.196) -- (7.993,8.999) -- (7.963,8.803) -- (7.853,8.52) -- (7.75,8.235) -- (7.653,7.947) -- (7.563,7.657) -- (7.481,7.365) -- (7.405,7.07) -- (7.336,6.775) -- (7.273,6.477) -- (7.218,6.179) -- (7.17,5.879) -- (7.129,5.578) -- (7.095,5.276) -- (7.068,4.974) -- (7.049,4.671);
\draw[line width=0.5pt] (10.517,10.716) -- (10.459,10.531) -- (10.407,10.345) -- (10.358,10.158) -- (10.314,9.969) -- (10.275,9.779) -- (10.24,9.589) -- (10.209,9.398) -- (10.183,9.206) -- (10.162,9.013) -- (10.145,8.82) -- (10.132,8.627) -- (10.125,8.433) -- (10.121,8.24) -- (10.123,8.046);
\draw[line width=0.5pt] (10.123,8.046) -- (10.182,7.859) -- (10.236,7.67) -- (10.287,7.48) -- (10.332,7.289) -- (10.374,7.097) -- (10.41,6.903) -- (10.443,6.709) -- (10.47,6.514) -- (10.493,6.319) -- (10.512,6.123) -- (10.525,5.927) -- (10.535,5.73) -- (10.539,5.534) -- (10.539,5.337);
\draw[line width=0.5pt] (10.123,8.046) -- (9.868,7.841) -- (9.618,7.629) -- (9.373,7.412) -- (9.133,7.188) -- (8.899,6.96) -- (8.67,6.725) -- (8.446,6.485) -- (8.229,6.241) -- (8.017,5.991) -- (7.811,5.736) -- (7.611,5.476) -- (7.417,5.212) -- (7.23,4.944) -- (7.049,4.671);
\draw[line width=0.5pt] (10.539,5.337) -- (10.299,5.252) -- (10.057,5.172) -- (9.813,5.098) -- (9.567,5.03) -- (9.32,4.968) -- (9.071,4.911) -- (8.821,4.86) -- (8.57,4.815) -- (8.318,4.776) -- (8.065,4.743) -- (7.812,4.716) -- (7.558,4.695) -- (7.304,4.68) -- (7.049,4.671);
\draw[line width=0.5pt] (1.385,5.176) -- (1.614,5.146) -- (1.843,5.121) -- (2.073,5.102) -- (2.304,5.089) -- (2.535,5.08) -- (2.766,5.078) -- (2.997,5.08) -- (3.228,5.088) -- (3.459,5.102) -- (3.689,5.12) -- (3.919,5.145) -- (4.148,5.174) -- (4.376,5.209) -- (4.604,5.25) -- (4.783,5.235) -- (4.963,5.217) -- (5.141,5.194) -- (5.32,5.167) -- (5.497,5.136) -- (5.674,5.101) -- (5.85,5.061) -- (6.025,5.018) -- (6.199,4.97) -- (6.372,4.918) -- (6.543,4.862) -- (6.713,4.802) -- (6.882,4.739) -- (7.049,4.671);
\draw[line width=0.5pt] (1.385,5.176) -- (1.443,4.978) -- (1.497,4.778) -- (1.547,4.577) -- (1.591,4.375) -- (1.631,4.172) -- (1.666,3.968) -- (1.696,3.763) -- (1.722,3.558) -- (1.742,3.352) -- (1.758,3.146) -- (1.769,2.939) -- (1.775,2.732) -- (1.776,2.526) -- (1.772,2.319);
\draw[line width=0.5pt] (1.385,5.176) -- (1.601,4.895) -- (1.824,4.619) -- (2.054,4.348) -- (2.29,4.082) -- (2.532,3.823) -- (2.78,3.569) -- (3.035,3.32) -- (3.295,3.078) -- (3.56,2.842) -- (3.831,2.613) -- (4.108,2.389) -- (4.389,2.173) -- (4.675,1.963) -- (4.967,1.76);
\draw[line width=0.5pt] (7.049,4.671) -- (7.331,4.554) -- (7.61,4.43) -- (7.887,4.3) -- (8.16,4.163) -- (8.43,4.019) -- (8.696,3.87) -- (8.959,3.714) -- (9.219,3.552) -- (9.474,3.384) -- (9.726,3.21) -- (9.973,3.03) -- (10.215,2.845) -- (10.454,2.654) -- (10.687,2.457);
\draw[line width=0.5pt] (7.049,4.671) -- (6.87,4.487) -- (6.694,4.3) -- (6.524,4.108) -- (6.358,3.912) -- (6.196,3.712) -- (6.04,3.509) -- (5.888,3.301) -- (5.741,3.091) -- (5.599,2.877) -- (5.462,2.659) -- (5.33,2.439) -- (5.203,2.216) -- (5.082,1.989) -- (4.967,1.76);
\draw[line width=0.5pt] (7.049,4.671) -- (7.098,4.453) -- (7.142,4.235) -- (7.18,4.015) -- (7.214,3.795) -- (7.242,3.573) -- (7.265,3.352) -- (7.283,3.129) -- (7.295,2.906) -- (7.302,2.683) -- (7.305,2.46) -- (7.301,2.237) -- (7.293,2.015) -- (7.279,1.792) -- (7.26,1.57);
\draw[line width=0.5pt] (10.687,2.457) -- (10.435,2.431) -- (10.183,2.4) -- (9.932,2.363) -- (9.681,2.32) -- (9.432,2.27) -- (9.184,2.216) -- (8.937,2.155) -- (8.692,2.088) -- (8.448,2.016) -- (8.207,1.938) -- (7.967,1.854) -- (7.729,1.765) -- (7.494,1.67) -- (7.26,1.57);
\draw[line width=0.5pt] (1.772,2.319) -- (2.005,2.314) -- (2.237,2.304) -- (2.469,2.289) -- (2.701,2.267) -- (2.932,2.241) -- (3.163,2.209) -- (3.392,2.171) -- (3.621,2.129) -- (3.849,2.08) -- (4.075,2.027) -- (4.301,1.968) -- (4.524,1.904) -- (4.746,1.835) -- (4.967,1.76);
\draw[line width=0.5pt] (4.967,1.76) -- (5.127,1.722) -- (5.288,1.687) -- (5.451,1.656) -- (5.614,1.629) -- (5.777,1.606) -- (5.941,1.586) -- (6.106,1.571) -- (6.27,1.559) -- (6.435,1.551) -- (6.601,1.547) -- (6.766,1.547) -- (6.931,1.551) -- (7.096,1.558) -- (7.26,1.57);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.119,11.57) -- (8.097,11.65) -- (8.039,11.709) -- (7.959,11.73) -- (7.879,11.709) -- (7.82,11.65) -- (7.799,11.57) -- (7.82,11.49) -- (7.879,11.431) -- (7.959,11.41) -- (8.039,11.431) -- (8.097,11.49) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.677,10.716) -- (10.655,10.796) -- (10.597,10.855) -- (10.517,10.876) -- (10.437,10.855) -- (10.378,10.796) -- (10.357,10.716) -- (10.378,10.636) -- (10.437,10.578) -- (10.517,10.556) -- (10.597,10.578) -- (10.655,10.636) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.283,8.046) -- (10.261,8.126) -- (10.203,8.185) -- (10.123,8.206) -- (10.043,8.185) -- (9.984,8.126) -- (9.963,8.046) -- (9.984,7.966) -- (10.043,7.908) -- (10.123,7.886) -- (10.203,7.908) -- (10.261,7.966) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.699,5.337) -- (10.678,5.417) -- (10.619,5.476) -- (10.539,5.497) -- (10.459,5.476) -- (10.4,5.417) -- (10.379,5.337) -- (10.4,5.257) -- (10.459,5.199) -- (10.539,5.177) -- (10.619,5.199) -- (10.678,5.257) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.545,5.176) -- (1.523,5.256) -- (1.465,5.315) -- (1.385,5.336) -- (1.305,5.315) -- (1.246,5.256) -- (1.225,5.176) -- (1.246,5.096) -- (1.305,5.037) -- (1.385,5.016) -- (1.465,5.037) -- (1.523,5.096) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.209,4.671) -- (7.187,4.751) -- (7.129,4.809) -- (7.049,4.831) -- (6.969,4.809) -- (6.91,4.751) -- (6.889,4.671) -- (6.91,4.591) -- (6.969,4.532) -- (7.049,4.511) -- (7.129,4.532) -- (7.187,4.591) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.847,2.457) -- (10.826,2.537) -- (10.767,2.595) -- (10.687,2.617) -- (10.607,2.595) -- (10.549,2.537) -- (10.527,2.457) -- (10.549,2.377) -- (10.607,2.318) -- (10.687,2.297) -- (10.767,2.318) -- (10.826,2.377) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.932,2.319) -- (1.911,2.399) -- (1.852,2.457) -- (1.772,2.479) -- (1.692,2.457) -- (1.634,2.399) -- (1.612,2.319) -- (1.634,2.239) -- (1.692,2.18) -- (1.772,2.159) -- (1.852,2.18) -- (1.911,2.239) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.127,1.76) -- (5.105,1.84) -- (5.047,1.899) -- (4.967,1.92) -- (4.887,1.899) -- (4.828,1.84) -- (4.807,1.76) -- (4.828,1.68) -- (4.887,1.622) -- (4.967,1.6) -- (5.047,1.622) -- (5.105,1.68) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.42,1.57) -- (7.399,1.65) -- (7.34,1.709) -- (7.26,1.73) -- (7.18,1.709) -- (7.122,1.65) -- (7.1,1.57) -- (7.122,1.49) -- (7.18,1.431) -- (7.26,1.41) -- (7.34,1.431) -- (7.399,1.49) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.644,11.341) -- (9.598,11.511) -- (9.474,11.636) -- (9.304,11.681) -- (9.134,11.636) -- (9.009,11.511) -- (8.964,11.341) -- (9.009,11.171) -- (9.134,11.047) -- (9.304,11.001) -- (9.474,11.047) -- (9.598,11.171) -- cycle;
\node at (9.304,11.341) {\scalebox{1.517}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.799,10.065) -- (9.753,10.235) -- (9.629,10.359) -- (9.459,10.405) -- (9.289,10.359) -- (9.164,10.235) -- (9.119,10.065) -- (9.164,9.895) -- (9.289,9.771) -- (9.459,9.725) -- (9.629,9.771) -- (9.753,9.895) -- cycle;
\node at (9.459,10.065) {\scalebox{1.517}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.018,8.26) -- (4.972,8.43) -- (4.848,8.554) -- (4.678,8.6) -- (4.508,8.554) -- (4.383,8.43) -- (4.338,8.26) -- (4.383,8.09) -- (4.508,7.965) -- (4.678,7.92) -- (4.848,7.965) -- (4.972,8.09) -- cycle;
\node at (4.678,8.26) {\scalebox{1.517}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.61,8.713) -- (8.565,8.883) -- (8.44,9.008) -- (8.27,9.053) -- (8.1,9.008) -- (7.976,8.883) -- (7.93,8.713) -- (7.976,8.543) -- (8.1,8.419) -- (8.27,8.373) -- (8.44,8.419) -- (8.565,8.543) -- cycle;
\node at (8.27,8.713) {\scalebox{1.517}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.866,9.351) -- (10.82,9.521) -- (10.696,9.645) -- (10.526,9.691) -- (10.356,9.645) -- (10.232,9.521) -- (10.186,9.351) -- (10.232,9.181) -- (10.356,9.056) -- (10.526,9.011) -- (10.696,9.056) -- (10.82,9.181) -- cycle;
\node at (10.526,9.351) {\scalebox{1.517}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.099,6.758) -- (11.053,6.928) -- (10.929,7.052) -- (10.759,7.098) -- (10.589,7.052) -- (10.464,6.928) -- (10.419,6.758) -- (10.464,6.588) -- (10.589,6.463) -- (10.759,6.418) -- (10.929,6.463) -- (11.053,6.588) -- cycle;
\node at (10.759,6.758) {\scalebox{1.517}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.023,6.27) -- (8.978,6.44) -- (8.853,6.564) -- (8.683,6.61) -- (8.513,6.564) -- (8.389,6.44) -- (8.343,6.27) -- (8.389,6.1) -- (8.513,5.976) -- (8.683,5.93) -- (8.853,5.976) -- (8.978,6.1) -- cycle;
\node at (8.683,6.27) {\scalebox{1.517}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.222,4.546) -- (9.176,4.716) -- (9.052,4.84) -- (8.882,4.886) -- (8.712,4.84) -- (8.587,4.716) -- (8.542,4.546) -- (8.587,4.376) -- (8.712,4.251) -- (8.882,4.206) -- (9.052,4.251) -- (9.176,4.376) -- cycle;
\node at (8.882,4.546) {\scalebox{1.517}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.923,5.569) -- (4.878,5.739) -- (4.753,5.864) -- (4.583,5.909) -- (4.413,5.864) -- (4.289,5.739) -- (4.243,5.569) -- (4.289,5.399) -- (4.413,5.275) -- (4.583,5.229) -- (4.753,5.275) -- (4.878,5.399) -- cycle;
\node at (4.583,5.569) {\scalebox{1.517}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.353,3.806) -- (2.308,3.976) -- (2.183,4.101) -- (2.013,4.146) -- (1.843,4.101) -- (1.719,3.976) -- (1.673,3.806) -- (1.719,3.636) -- (1.843,3.512) -- (2.013,3.466) -- (2.183,3.512) -- (2.308,3.636) -- cycle;
\node at (2.013,3.806) {\scalebox{1.517}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.596,3.552) -- (3.55,3.722) -- (3.426,3.846) -- (3.256,3.892) -- (3.086,3.846) -- (2.961,3.722) -- (2.916,3.552) -- (2.961,3.382) -- (3.086,3.258) -- (3.256,3.212) -- (3.426,3.258) -- (3.55,3.382) -- cycle;
\node at (3.256,3.552) {\scalebox{1.517}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.133,3.441) -- (9.088,3.611) -- (8.963,3.735) -- (8.793,3.781) -- (8.623,3.735) -- (8.499,3.611) -- (8.453,3.441) -- (8.499,3.271) -- (8.623,3.146) -- (8.793,3.101) -- (8.963,3.146) -- (9.088,3.271) -- cycle;
\node at (8.793,3.441) {\scalebox{1.517}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.488,3.115) -- (6.442,3.285) -- (6.318,3.41) -- (6.148,3.455) -- (5.978,3.41) -- (5.853,3.285) -- (5.808,3.115) -- (5.853,2.945) -- (5.978,2.821) -- (6.148,2.775) -- (6.318,2.821) -- (6.442,2.945) -- cycle;
\node at (6.148,3.115) {\scalebox{1.517}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.942,3.151) -- (7.896,3.321) -- (7.772,3.445) -- (7.602,3.491) -- (7.432,3.445) -- (7.307,3.321) -- (7.262,3.151) -- (7.307,2.981) -- (7.432,2.856) -- (7.602,2.811) -- (7.772,2.856) -- (7.896,2.981) -- cycle;
\node at (7.602,3.151) {\scalebox{1.517}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.357,1.845) -- (9.312,2.015) -- (9.187,2.139) -- (9.017,2.185) -- (8.847,2.139) -- (8.723,2.015) -- (8.677,1.845) -- (8.723,1.675) -- (8.847,1.551) -- (9.017,1.505) -- (9.187,1.551) -- (9.312,1.675) -- cycle;
\node at (9.017,1.845) {\scalebox{1.517}{\small 15}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.788,2.487) -- (3.742,2.657) -- (3.618,2.781) -- (3.448,2.827) -- (3.278,2.781) -- (3.153,2.657) -- (3.108,2.487) -- (3.153,2.317) -- (3.278,2.192) -- (3.448,2.147) -- (3.618,2.192) -- (3.742,2.317) -- cycle;
\node at (3.448,2.487) {\scalebox{1.517}{\small 16}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.472,1.889) -- (6.427,2.059) -- (6.302,2.184) -- (6.132,2.229) -- (5.962,2.184) -- (5.838,2.059) -- (5.792,1.889) -- (5.838,1.719) -- (5.962,1.595) -- (6.132,1.549) -- (6.302,1.595) -- (6.427,1.719) -- cycle;
\node at (6.132,1.889) {\scalebox{1.517}{\small 17}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (6.96,1.27) -- (7.56,1.27) -- (7.56,1.87) -- (6.96,1.87) -- cycle;
\node at (7.26,1.57) {\scalebox{1.517}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.035,2.681) -- (6.635,2.681) -- (6.635,3.281) -- (6.035,3.281) -- cycle;
\node at (6.335,2.981) {\scalebox{1.517}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (10.119,11.404) -- (10.719,11.404) -- (10.719,12.004) -- (10.119,12.004) -- cycle;
\node at (10.419,11.704) {\scalebox{1.517}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (8.191,8.349) -- (8.791,8.349) -- (8.791,8.949) -- (8.191,8.949) -- cycle;
\node at (8.491,8.649) {\scalebox{1.517}{\small d}};
\node at (8.031,12.186) {\scalebox{1.517}{\small A}};
\node at (10.841,11.244) {\scalebox{1.517}{\small B}};
\node at (10.577,8.468) {\scalebox{1.517}{\small C}};
\node at (11.159,5.333) {\scalebox{1.517}{\small D}};
\node at (0.765,5.156) {\scalebox{1.517}{\small E}};
\node at (6.895,4.07) {\scalebox{1.517}{\small F}};
\node at (11.162,2.058) {\scalebox{1.517}{\small G}};
\node at (1.231,2.017) {\scalebox{1.517}{\small H}};
\node at (4.637,1.235) {\scalebox{1.517}{\small I}};
\node at (7.266,0.95) {\scalebox{1.517}{\small J}};
\draw[line width=0.5pt] (11.987,10.62) -- (11.987,11.72);
\draw[line width=0.5pt] (11.756,11.335) -- (11.987,11.72) -- (12.218,11.335);
\draw[line width=0.5pt] (11.685,11.17) -- (12.29,11.17);
\node at (11.987,12.105) {\scalebox{1.517}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{1.} On compte les rues aboutissant à chaque carrefour :\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Carrefour & \cellcolor{blue!7}\bfseries A & \cellcolor{blue!7}\bfseries B & \cellcolor{blue!7}\bfseries C & \cellcolor{blue!7}\bfseries D & \cellcolor{blue!7}\bfseries E & \cellcolor{blue!7}\bfseries F & \cellcolor{blue!7}\bfseries G & \cellcolor{blue!7}\bfseries H & \cellcolor{blue!7}\bfseries I & \cellcolor{blue!7}\bfseries J \\ \hline
Nombre de rues & 4 & 2 & 4 & 2 & 4 & 7 & 2 & 2 & 4 & 3 \\ \hline
Parité & pair & pair & pair & pair & pair & \textbf{impair} & 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{F}, \textbf{J} — 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{J}, 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{F} et \textbf{J}, et la poste est justement sur l'un des deux (\textbf{J}). Il part donc de la poste et \textbf{termine en F}, 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{(J)}, prend l'\textbf{allée des Peupliers ⑭} vers le nord et atteint Rond-point des Vaches \textbf{(F)}.
  \item Il continue tout droit dans la \textbf{rue de l'Église ④}, vers le nord, jusqu'à Carrefour du Moulin \textbf{(A)}.
  \item Il tourne à droite dans la \textbf{rue Neuve ①}, vers l'est, jusqu'à Place aux Herbes \textbf{(B)}.
  \item Il tourne à droite dans la \textbf{rue du Four ⑤}, vers le sud, jusqu'à Carrefour Saint-Roch \textbf{(C)}.
  \item Il prend dans la \textbf{rue Verte ②}, vers le nord-ouest, jusqu'à Carrefour du Moulin \textbf{(A)}.
  \item Il tourne à gauche dans le \textbf{quai aux Fleurs ③}, vers le sud-ouest, jusqu'à Carrefour des Quatre-Chemins \textbf{(E)}.
  \item Il tourne à gauche dans la \textbf{rue des Jardins ⑨}, vers l'est, jusqu'à Rond-point des Vaches \textbf{(F)}.
  \item Il tourne à gauche dans la \textbf{rue de la Poste ⑦}, vers le nord-est, jusqu'à Carrefour Saint-Roch \textbf{(C)}.
  \item Il tourne à droite dans la \textbf{rue du Stade ⑥}, vers le sud, jusqu'à Place des Tanneurs \textbf{(D)}.
  \item Il tourne à droite dans le \textbf{chemin du Lavoir ⑧}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(F)}.
  \item Il prend dans la \textbf{rue Pasteur ⑫}, vers le sud-est, jusqu'à Place Sainte-Croix \textbf{(G)}.
  \item Il tourne à droite dans la \textbf{rue Haute ⑮}, vers l'ouest, jusqu'à Carrefour du Lavoir \textbf{(J)}.
  \item Il continue tout droit dans la \textbf{rue du Pont ⑰}, vers l'ouest, jusqu'à Carrefour de l'Hôpital \textbf{(I)}.
  \item Il continue tout droit dans le \textbf{passage du Marché ⑪}, vers le nord-ouest, jusqu'à Carrefour des Quatre-Chemins \textbf{(E)}.
  \item Il prend dans le \textbf{boulevard du Parc ⑩}, vers le sud, jusqu'à Coin du Lavoir \textbf{(H)}.
  \item Il tourne à gauche dans la \textbf{rue de la Mairie ⑯}, vers l'est, jusqu'à Carrefour de l'Hôpital \textbf{(I)}.
  \item Il tourne à gauche dans la \textbf{rue du Puits ⑬}, vers le nord-est, jusqu'à Rond-point des Vaches \textbf{(F)}.
\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{F} ou en \textbf{J} : ailleurs, la tournée n'aurait pas pu commencer.}\par\addvspace{0.4em}
\end{document}
