\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 264}

\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=264}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0264}
\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=0.6992, line join=round, line cap=round]
\draw[line width=0.5pt] (7.767,11.317) -- (7.582,11.146) -- (7.402,10.97) -- (7.226,10.791) -- (7.054,10.607) -- (6.887,10.419) -- (6.724,10.227) -- (6.566,10.032) -- (6.412,9.832) -- (6.263,9.63) -- (6.118,9.423) -- (5.979,9.214) -- (5.845,9.001) -- (5.715,8.785) -- (5.591,8.567);
\draw[line width=0.5pt] (7.767,11.317) -- (8.063,11.097) -- (8.354,10.87) -- (8.64,10.637) -- (8.92,10.396) -- (9.195,10.149) -- (9.464,9.896) -- (9.727,9.636) -- (9.983,9.37) -- (10.234,9.099) -- (10.477,8.821) -- (10.715,8.538) -- (10.945,8.249) -- (11.168,7.956) -- (11.385,7.657);
\draw[line width=0.5pt] (5.591,8.567) -- (5.821,8.54) -- (6.052,8.518) -- (6.283,8.502) -- (6.514,8.491) -- (6.746,8.486) -- (6.978,8.486) -- (7.209,8.492) -- (7.441,8.503) -- (7.672,8.52) -- (7.903,8.542) -- (8.133,8.569) -- (8.362,8.602) -- (8.59,8.64) -- (8.818,8.683) -- (9.011,8.639) -- (9.203,8.59) -- (9.394,8.536) -- (9.584,8.477) -- (9.772,8.415) -- (9.959,8.348) -- (10.144,8.276) -- (10.327,8.2) -- (10.509,8.12) -- (10.688,8.036) -- (10.866,7.947) -- (11.041,7.854) -- (11.214,7.758) -- (11.385,7.657);
\draw[line width=0.5pt] (5.591,8.567) -- (5.328,8.299) -- (5.057,8.037) -- (4.781,7.781) -- (4.499,7.532) -- (4.211,7.29) -- (3.917,7.054) -- (3.617,6.826) -- (3.313,6.604) -- (3.003,6.39) -- (2.689,6.183) -- (2.369,5.983) -- (2.045,5.792) -- (1.717,5.607) -- (1.385,5.431);
\draw[line width=0.5pt] (5.591,8.567) -- (5.531,8.307) -- (5.477,8.046) -- (5.429,7.783) -- (5.387,7.52) -- (5.352,7.255) -- (5.322,6.99) -- (5.299,6.723) -- (5.282,6.457) -- (5.272,6.19) -- (5.267,5.923) -- (5.269,5.656) -- (5.278,5.39) -- (5.292,5.123) -- (5.313,4.857);
\draw[line width=0.5pt] (11.385,7.657) -- (11.384,7.469) -- (11.379,7.281) -- (11.37,7.094) -- (11.356,6.907) -- (11.338,6.72) -- (11.315,6.533) -- (11.288,6.347) -- (11.257,6.162) -- (11.221,5.978) -- (11.181,5.794) -- (11.137,5.612) -- (11.088,5.43) -- (11.035,5.25) -- (10.978,5.072);
\draw[line width=0.5pt] (1.385,5.431) -- (1.575,5.164) -- (1.772,4.901) -- (1.974,4.643) -- (2.183,4.389) -- (2.398,4.141) -- (2.619,3.897) -- (2.845,3.659) -- (3.077,3.426) -- (3.314,3.199) -- (3.557,2.977) -- (3.804,2.762) -- (4.057,2.552) -- (4.314,2.348) -- (4.576,2.15);
\draw[line width=0.5pt] (1.385,5.431) -- (1.433,5.168) -- (1.475,4.904) -- (1.511,4.639) -- (1.54,4.373) -- (1.564,4.107) -- (1.581,3.84) -- (1.592,3.572) -- (1.596,3.305) -- (1.594,3.037) -- (1.586,2.77) -- (1.572,2.502) -- (1.551,2.236) -- (1.524,1.97) -- (1.491,1.704);
\draw[line width=0.5pt] (10.978,5.072) -- (10.736,5.095) -- (10.494,5.113) -- (10.251,5.125) -- (10.008,5.131) -- (9.765,5.131) -- (9.522,5.126) -- (9.279,5.115) -- (9.037,5.098) -- (8.795,5.076) -- (8.553,5.048) -- (8.313,5.014) -- (8.073,4.975) -- (7.834,4.93) -- (7.597,4.879);
\draw[line width=0.5pt] (7.597,4.879) -- (7.435,4.853) -- (7.273,4.83) -- (7.11,4.811) -- (6.947,4.796) -- (6.783,4.785) -- (6.62,4.778) -- (6.456,4.774) -- (6.292,4.774) -- (6.128,4.779) -- (5.964,4.787) -- (5.801,4.799) -- (5.638,4.814) -- (5.475,4.834) -- (5.313,4.857);
\draw[line width=0.5pt] (7.597,4.879) -- (7.353,4.719) -- (7.113,4.553) -- (6.876,4.381) -- (6.644,4.204) -- (6.416,4.021) -- (6.193,3.833) -- (5.974,3.64) -- (5.759,3.441) -- (5.55,3.238) -- (5.345,3.029) -- (5.145,2.816) -- (4.95,2.599) -- (4.76,2.377) -- (4.576,2.15);
\draw[line width=0.5pt] (7.597,4.879) -- (7.718,4.652) -- (7.834,4.421) -- (7.944,4.188) -- (8.049,3.952) -- (8.148,3.713) -- (8.242,3.472) -- (8.33,3.23) -- (8.412,2.985) -- (8.489,2.738) -- (8.559,2.489) -- (8.624,2.24) -- (8.683,1.988) -- (8.736,1.735) -- (8.783,1.482);
\draw[line width=0.5pt] (5.313,4.857) -- (5.231,4.673) -- (5.154,4.488) -- (5.081,4.3) -- (5.012,4.111) -- (4.948,3.92) -- (4.888,3.728) -- (4.833,3.534) -- (4.782,3.339) -- (4.736,3.143) -- (4.695,2.946) -- (4.658,2.748) -- (4.626,2.55) -- (4.599,2.35) -- (4.576,2.15);
\draw[line width=0.5pt] (4.576,2.15) -- (4.882,2.149) -- (5.187,2.14) -- (5.492,2.125) -- (5.797,2.102) -- (6.101,2.071) -- (6.405,2.034) -- (6.707,1.99) -- (7.008,1.938) -- (7.308,1.879) -- (7.607,1.814) -- (7.904,1.741) -- (8.199,1.662) -- (8.492,1.575) -- (8.783,1.482);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.927,11.317) -- (7.905,11.397) -- (7.847,11.455) -- (7.767,11.477) -- (7.687,11.455) -- (7.628,11.397) -- (7.607,11.317) -- (7.628,11.237) -- (7.687,11.178) -- (7.767,11.157) -- (7.847,11.178) -- (7.905,11.237) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.751,8.567) -- (5.73,8.647) -- (5.671,8.705) -- (5.591,8.727) -- (5.511,8.705) -- (5.453,8.647) -- (5.431,8.567) -- (5.453,8.487) -- (5.511,8.428) -- (5.591,8.407) -- (5.671,8.428) -- (5.73,8.487) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.545,7.657) -- (11.523,7.737) -- (11.465,7.795) -- (11.385,7.817) -- (11.305,7.795) -- (11.246,7.737) -- (11.225,7.657) -- (11.246,7.577) -- (11.305,7.518) -- (11.385,7.497) -- (11.465,7.518) -- (11.523,7.577) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.545,5.431) -- (1.523,5.511) -- (1.465,5.57) -- (1.385,5.591) -- (1.305,5.57) -- (1.246,5.511) -- (1.225,5.431) -- (1.246,5.351) -- (1.305,5.293) -- (1.385,5.271) -- (1.465,5.293) -- (1.523,5.351) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.138,5.072) -- (11.117,5.152) -- (11.058,5.21) -- (10.978,5.232) -- (10.898,5.21) -- (10.839,5.152) -- (10.818,5.072) -- (10.839,4.992) -- (10.898,4.933) -- (10.978,4.912) -- (11.058,4.933) -- (11.117,4.992) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.757,4.879) -- (7.735,4.959) -- (7.677,5.018) -- (7.597,5.039) -- (7.517,5.018) -- (7.458,4.959) -- (7.437,4.879) -- (7.458,4.799) -- (7.517,4.741) -- (7.597,4.719) -- (7.677,4.741) -- (7.735,4.799) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.473,4.857) -- (5.451,4.937) -- (5.393,4.996) -- (5.313,5.017) -- (5.233,4.996) -- (5.174,4.937) -- (5.153,4.857) -- (5.174,4.777) -- (5.233,4.719) -- (5.313,4.697) -- (5.393,4.719) -- (5.451,4.777) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.736,2.15) -- (4.715,2.23) -- (4.656,2.289) -- (4.576,2.31) -- (4.496,2.289) -- (4.438,2.23) -- (4.416,2.15) -- (4.438,2.07) -- (4.496,2.012) -- (4.576,1.99) -- (4.656,2.012) -- (4.715,2.07) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.651,1.704) -- (1.629,1.784) -- (1.571,1.843) -- (1.491,1.864) -- (1.411,1.843) -- (1.352,1.784) -- (1.331,1.704) -- (1.352,1.624) -- (1.411,1.566) -- (1.491,1.544) -- (1.571,1.566) -- (1.629,1.624) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.943,1.482) -- (8.921,1.562) -- (8.863,1.62) -- (8.783,1.642) -- (8.703,1.62) -- (8.644,1.562) -- (8.623,1.482) -- (8.644,1.402) -- (8.703,1.343) -- (8.783,1.322) -- (8.863,1.343) -- (8.921,1.402) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.156,9.833) -- (7.111,10.003) -- (6.986,10.127) -- (6.816,10.173) -- (6.646,10.127) -- (6.522,10.003) -- (6.476,9.833) -- (6.522,9.663) -- (6.646,9.539) -- (6.816,9.493) -- (6.986,9.539) -- (7.111,9.663) -- cycle;
\node at (6.816,9.833) {\scalebox{1.43}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.294,9.861) -- (10.249,10.031) -- (10.124,10.155) -- (9.954,10.201) -- (9.784,10.155) -- (9.66,10.031) -- (9.614,9.861) -- (9.66,9.691) -- (9.784,9.567) -- (9.954,9.521) -- (10.124,9.567) -- (10.249,9.691) -- cycle;
\node at (9.954,9.861) {\scalebox{1.43}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.159,9.003) -- (9.113,9.173) -- (8.989,9.298) -- (8.819,9.343) -- (8.649,9.298) -- (8.524,9.173) -- (8.479,9.003) -- (8.524,8.833) -- (8.649,8.709) -- (8.819,8.663) -- (8.989,8.709) -- (9.113,8.833) -- cycle;
\node at (8.819,9.003) {\scalebox{1.43}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.149,6.569) -- (4.103,6.739) -- (3.979,6.863) -- (3.809,6.909) -- (3.639,6.863) -- (3.514,6.739) -- (3.469,6.569) -- (3.514,6.399) -- (3.639,6.275) -- (3.809,6.229) -- (3.979,6.275) -- (4.103,6.399) -- cycle;
\node at (3.809,6.569) {\scalebox{1.43}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.958,6.7) -- (5.913,6.87) -- (5.788,6.994) -- (5.618,7.04) -- (5.448,6.994) -- (5.324,6.87) -- (5.278,6.7) -- (5.324,6.53) -- (5.448,6.405) -- (5.618,6.36) -- (5.788,6.405) -- (5.913,6.53) -- cycle;
\node at (5.618,6.7) {\scalebox{1.43}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.944,6.298) -- (11.899,6.468) -- (11.774,6.592) -- (11.604,6.638) -- (11.434,6.592) -- (11.31,6.468) -- (11.264,6.298) -- (11.31,6.128) -- (11.434,6.003) -- (11.604,5.958) -- (11.774,6.003) -- (11.899,6.128) -- cycle;
\node at (11.604,6.298) {\scalebox{1.43}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.415,3.882) -- (3.369,4.052) -- (3.245,4.177) -- (3.075,4.222) -- (2.905,4.177) -- (2.78,4.052) -- (2.735,3.882) -- (2.78,3.712) -- (2.905,3.588) -- (3.075,3.542) -- (3.245,3.588) -- (3.369,3.712) -- cycle;
\node at (3.075,3.882) {\scalebox{1.43}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.251,3.581) -- (2.206,3.751) -- (2.081,3.876) -- (1.911,3.921) -- (1.741,3.876) -- (1.617,3.751) -- (1.571,3.581) -- (1.617,3.411) -- (1.741,3.287) -- (1.911,3.241) -- (2.081,3.287) -- (2.206,3.411) -- cycle;
\node at (1.911,3.581) {\scalebox{1.43}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.637,4.796) -- (9.592,4.966) -- (9.467,5.09) -- (9.297,5.136) -- (9.127,5.09) -- (9.003,4.966) -- (8.957,4.796) -- (9.003,4.626) -- (9.127,4.501) -- (9.297,4.456) -- (9.467,4.501) -- (9.592,4.626) -- cycle;
\node at (9.297,4.796) {\scalebox{1.43}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.799,4.454) -- (6.753,4.624) -- (6.629,4.749) -- (6.459,4.794) -- (6.289,4.749) -- (6.164,4.624) -- (6.119,4.454) -- (6.164,4.284) -- (6.289,4.16) -- (6.459,4.114) -- (6.629,4.16) -- (6.753,4.284) -- cycle;
\node at (6.459,4.454) {\scalebox{1.43}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.528,3.402) -- (6.483,3.572) -- (6.358,3.697) -- (6.188,3.742) -- (6.018,3.697) -- (5.894,3.572) -- (5.848,3.402) -- (5.894,3.232) -- (6.018,3.108) -- (6.188,3.062) -- (6.358,3.108) -- (6.483,3.232) -- cycle;
\node at (6.188,3.402) {\scalebox{1.43}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.972,3.335) -- (8.926,3.505) -- (8.802,3.629) -- (8.632,3.675) -- (8.462,3.629) -- (8.337,3.505) -- (8.292,3.335) -- (8.337,3.165) -- (8.462,3.041) -- (8.632,2.995) -- (8.802,3.041) -- (8.926,3.165) -- cycle;
\node at (8.632,3.335) {\scalebox{1.43}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.482,3.45) -- (5.436,3.62) -- (5.312,3.745) -- (5.142,3.79) -- (4.972,3.745) -- (4.847,3.62) -- (4.802,3.45) -- (4.847,3.28) -- (4.972,3.156) -- (5.142,3.11) -- (5.312,3.156) -- (5.436,3.28) -- cycle;
\node at (5.142,3.45) {\scalebox{1.43}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.097,2.306) -- (7.052,2.476) -- (6.927,2.6) -- (6.757,2.646) -- (6.587,2.6) -- (6.463,2.476) -- (6.417,2.306) -- (6.463,2.136) -- (6.587,2.011) -- (6.757,1.966) -- (6.927,2.011) -- (7.052,2.136) -- cycle;
\node at (6.757,2.306) {\scalebox{1.43}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.085,5.131) -- (1.685,5.131) -- (1.685,5.731) -- (1.085,5.731) -- cycle;
\node at (1.385,5.431) {\scalebox{1.43}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (2.939,3.743) -- (3.539,3.743) -- (3.539,4.343) -- (2.939,4.343) -- cycle;
\node at (3.239,4.043) {\scalebox{1.43}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.406,2.421) -- (10.006,2.421) -- (10.006,3.021) -- (9.406,3.021) -- cycle;
\node at (9.706,2.721) {\scalebox{1.43}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.925,1.235) -- (6.525,1.235) -- (6.525,1.835) -- (5.925,1.835) -- cycle;
\node at (6.225,1.535) {\scalebox{1.43}{\small d}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (10.074,3.985) -- (10.674,3.985) -- (10.674,4.585) -- (10.074,4.585) -- cycle;
\node at (10.374,4.285) {\scalebox{1.43}{\small e}};
\node at (7.896,11.923) {\scalebox{1.43}{\small A}};
\node at (5.427,9.165) {\scalebox{1.43}{\small B}};
\node at (11.944,7.924) {\scalebox{1.43}{\small C}};
\node at (0.765,5.446) {\scalebox{1.43}{\small D}};
\node at (11.597,5.039) {\scalebox{1.43}{\small E}};
\node at (8.174,4.655) {\scalebox{1.43}{\small F}};
\node at (4.735,4.633) {\scalebox{1.43}{\small G}};
\node at (4.256,1.62) {\scalebox{1.43}{\small H}};
\node at (0.988,1.341) {\scalebox{1.43}{\small I}};
\node at (9.102,0.95) {\scalebox{1.43}{\small J}};
\draw[line width=0.5pt] (12.685,10.367) -- (12.685,11.467);
\draw[line width=0.5pt] (12.454,11.082) -- (12.685,11.467) -- (12.916,11.082);
\draw[line width=0.5pt] (12.382,10.917) -- (12.987,10.917);
\node at (12.685,11.852) {\scalebox{1.43}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le facteur part de la poste et doit emprunter chaque rue exactement une fois. Il n'est pas obligé d'y revenir : il termine sa tournée où il veut.}\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Les rues & \cellcolor{blue!7}\bfseries  & \cellcolor{blue!7}\bfseries Les carrefours & \cellcolor{blue!7}\bfseries Les bâtiments \\ \hline
① allée des Peupliers & ⑧ impasse du Puits & Ⓐ Place du Marché & \textbf{a} la poste \\ \hline
② rue des Lilas & ⑨ boulevard du Parc & Ⓑ Carrefour des Tilleuls & \textbf{b} le gymnase \\ \hline
③ rue de la Poste & ⑩ rue de l'Église & Ⓒ Rond-point du Pont & \textbf{c} la pharmacie \\ \hline
④ rue des Jardins & ⑪ chemin Creux & Ⓓ Rond-point du Stade & \textbf{d} le cimetière \\ \hline
⑤ avenue de la Gare & ⑫ chemin de Ronde & Ⓔ Placette des Ormes & \textbf{e} le stade \\ \hline
⑥ rue du Pont & ⑬ chemin du Lavoir & Ⓕ Rond-point de la Poste &  \\ \hline
⑦ rue Basse & ⑭ rue de la Mairie & Ⓖ Carrefour des Quatre-Chemins &  \\ \hline
 &  & Ⓗ Rond-point des Vaches &  \\ \hline
 &  & Ⓘ Place des Tanneurs &  \\ \hline
 &  & Ⓙ Place Sainte-Croix &  \\ \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=0.6256, line join=round, line cap=round]
\draw[line width=0.5pt] (7.767,11.317) -- (7.582,11.146) -- (7.402,10.97) -- (7.226,10.791) -- (7.054,10.607) -- (6.887,10.419) -- (6.724,10.227) -- (6.566,10.032) -- (6.412,9.832) -- (6.263,9.63) -- (6.118,9.423) -- (5.979,9.214) -- (5.845,9.001) -- (5.715,8.785) -- (5.591,8.567);
\draw[line width=0.5pt] (7.767,11.317) -- (8.063,11.097) -- (8.354,10.87) -- (8.64,10.637) -- (8.92,10.396) -- (9.195,10.149) -- (9.464,9.896) -- (9.727,9.636) -- (9.983,9.37) -- (10.234,9.099) -- (10.477,8.821) -- (10.715,8.538) -- (10.945,8.249) -- (11.168,7.956) -- (11.385,7.657);
\draw[line width=0.5pt] (5.591,8.567) -- (5.821,8.54) -- (6.052,8.518) -- (6.283,8.502) -- (6.514,8.491) -- (6.746,8.486) -- (6.978,8.486) -- (7.209,8.492) -- (7.441,8.503) -- (7.672,8.52) -- (7.903,8.542) -- (8.133,8.569) -- (8.362,8.602) -- (8.59,8.64) -- (8.818,8.683) -- (9.011,8.639) -- (9.203,8.59) -- (9.394,8.536) -- (9.584,8.477) -- (9.772,8.415) -- (9.959,8.348) -- (10.144,8.276) -- (10.327,8.2) -- (10.509,8.12) -- (10.688,8.036) -- (10.866,7.947) -- (11.041,7.854) -- (11.214,7.758) -- (11.385,7.657);
\draw[line width=0.5pt] (5.591,8.567) -- (5.328,8.299) -- (5.057,8.037) -- (4.781,7.781) -- (4.499,7.532) -- (4.211,7.29) -- (3.917,7.054) -- (3.617,6.826) -- (3.313,6.604) -- (3.003,6.39) -- (2.689,6.183) -- (2.369,5.983) -- (2.045,5.792) -- (1.717,5.607) -- (1.385,5.431);
\draw[line width=0.5pt] (5.591,8.567) -- (5.531,8.307) -- (5.477,8.046) -- (5.429,7.783) -- (5.387,7.52) -- (5.352,7.255) -- (5.322,6.99) -- (5.299,6.723) -- (5.282,6.457) -- (5.272,6.19) -- (5.267,5.923) -- (5.269,5.656) -- (5.278,5.39) -- (5.292,5.123) -- (5.313,4.857);
\draw[line width=0.5pt] (11.385,7.657) -- (11.384,7.469) -- (11.379,7.281) -- (11.37,7.094) -- (11.356,6.907) -- (11.338,6.72) -- (11.315,6.533) -- (11.288,6.347) -- (11.257,6.162) -- (11.221,5.978) -- (11.181,5.794) -- (11.137,5.612) -- (11.088,5.43) -- (11.035,5.25) -- (10.978,5.072);
\draw[line width=0.5pt] (1.385,5.431) -- (1.575,5.164) -- (1.772,4.901) -- (1.974,4.643) -- (2.183,4.389) -- (2.398,4.141) -- (2.619,3.897) -- (2.845,3.659) -- (3.077,3.426) -- (3.314,3.199) -- (3.557,2.977) -- (3.804,2.762) -- (4.057,2.552) -- (4.314,2.348) -- (4.576,2.15);
\draw[line width=0.5pt] (1.385,5.431) -- (1.433,5.168) -- (1.475,4.904) -- (1.511,4.639) -- (1.54,4.373) -- (1.564,4.107) -- (1.581,3.84) -- (1.592,3.572) -- (1.596,3.305) -- (1.594,3.037) -- (1.586,2.77) -- (1.572,2.502) -- (1.551,2.236) -- (1.524,1.97) -- (1.491,1.704);
\draw[line width=0.5pt] (10.978,5.072) -- (10.736,5.095) -- (10.494,5.113) -- (10.251,5.125) -- (10.008,5.131) -- (9.765,5.131) -- (9.522,5.126) -- (9.279,5.115) -- (9.037,5.098) -- (8.795,5.076) -- (8.553,5.048) -- (8.313,5.014) -- (8.073,4.975) -- (7.834,4.93) -- (7.597,4.879);
\draw[line width=0.5pt] (7.597,4.879) -- (7.435,4.853) -- (7.273,4.83) -- (7.11,4.811) -- (6.947,4.796) -- (6.783,4.785) -- (6.62,4.778) -- (6.456,4.774) -- (6.292,4.774) -- (6.128,4.779) -- (5.964,4.787) -- (5.801,4.799) -- (5.638,4.814) -- (5.475,4.834) -- (5.313,4.857);
\draw[line width=0.5pt] (7.597,4.879) -- (7.353,4.719) -- (7.113,4.553) -- (6.876,4.381) -- (6.644,4.204) -- (6.416,4.021) -- (6.193,3.833) -- (5.974,3.64) -- (5.759,3.441) -- (5.55,3.238) -- (5.345,3.029) -- (5.145,2.816) -- (4.95,2.599) -- (4.76,2.377) -- (4.576,2.15);
\draw[line width=0.5pt] (7.597,4.879) -- (7.718,4.652) -- (7.834,4.421) -- (7.944,4.188) -- (8.049,3.952) -- (8.148,3.713) -- (8.242,3.472) -- (8.33,3.23) -- (8.412,2.985) -- (8.489,2.738) -- (8.559,2.489) -- (8.624,2.24) -- (8.683,1.988) -- (8.736,1.735) -- (8.783,1.482);
\draw[line width=0.5pt] (5.313,4.857) -- (5.231,4.673) -- (5.154,4.488) -- (5.081,4.3) -- (5.012,4.111) -- (4.948,3.92) -- (4.888,3.728) -- (4.833,3.534) -- (4.782,3.339) -- (4.736,3.143) -- (4.695,2.946) -- (4.658,2.748) -- (4.626,2.55) -- (4.599,2.35) -- (4.576,2.15);
\draw[line width=0.5pt] (4.576,2.15) -- (4.882,2.149) -- (5.187,2.14) -- (5.492,2.125) -- (5.797,2.102) -- (6.101,2.071) -- (6.405,2.034) -- (6.707,1.99) -- (7.008,1.938) -- (7.308,1.879) -- (7.607,1.814) -- (7.904,1.741) -- (8.199,1.662) -- (8.492,1.575) -- (8.783,1.482);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.927,11.317) -- (7.905,11.397) -- (7.847,11.455) -- (7.767,11.477) -- (7.687,11.455) -- (7.628,11.397) -- (7.607,11.317) -- (7.628,11.237) -- (7.687,11.178) -- (7.767,11.157) -- (7.847,11.178) -- (7.905,11.237) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.751,8.567) -- (5.73,8.647) -- (5.671,8.705) -- (5.591,8.727) -- (5.511,8.705) -- (5.453,8.647) -- (5.431,8.567) -- (5.453,8.487) -- (5.511,8.428) -- (5.591,8.407) -- (5.671,8.428) -- (5.73,8.487) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.545,7.657) -- (11.523,7.737) -- (11.465,7.795) -- (11.385,7.817) -- (11.305,7.795) -- (11.246,7.737) -- (11.225,7.657) -- (11.246,7.577) -- (11.305,7.518) -- (11.385,7.497) -- (11.465,7.518) -- (11.523,7.577) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.545,5.431) -- (1.523,5.511) -- (1.465,5.57) -- (1.385,5.591) -- (1.305,5.57) -- (1.246,5.511) -- (1.225,5.431) -- (1.246,5.351) -- (1.305,5.293) -- (1.385,5.271) -- (1.465,5.293) -- (1.523,5.351) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.138,5.072) -- (11.117,5.152) -- (11.058,5.21) -- (10.978,5.232) -- (10.898,5.21) -- (10.839,5.152) -- (10.818,5.072) -- (10.839,4.992) -- (10.898,4.933) -- (10.978,4.912) -- (11.058,4.933) -- (11.117,4.992) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.757,4.879) -- (7.735,4.959) -- (7.677,5.018) -- (7.597,5.039) -- (7.517,5.018) -- (7.458,4.959) -- (7.437,4.879) -- (7.458,4.799) -- (7.517,4.741) -- (7.597,4.719) -- (7.677,4.741) -- (7.735,4.799) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.473,4.857) -- (5.451,4.937) -- (5.393,4.996) -- (5.313,5.017) -- (5.233,4.996) -- (5.174,4.937) -- (5.153,4.857) -- (5.174,4.777) -- (5.233,4.719) -- (5.313,4.697) -- (5.393,4.719) -- (5.451,4.777) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.736,2.15) -- (4.715,2.23) -- (4.656,2.289) -- (4.576,2.31) -- (4.496,2.289) -- (4.438,2.23) -- (4.416,2.15) -- (4.438,2.07) -- (4.496,2.012) -- (4.576,1.99) -- (4.656,2.012) -- (4.715,2.07) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.651,1.704) -- (1.629,1.784) -- (1.571,1.843) -- (1.491,1.864) -- (1.411,1.843) -- (1.352,1.784) -- (1.331,1.704) -- (1.352,1.624) -- (1.411,1.566) -- (1.491,1.544) -- (1.571,1.566) -- (1.629,1.624) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.943,1.482) -- (8.921,1.562) -- (8.863,1.62) -- (8.783,1.642) -- (8.703,1.62) -- (8.644,1.562) -- (8.623,1.482) -- (8.644,1.402) -- (8.703,1.343) -- (8.783,1.322) -- (8.863,1.343) -- (8.921,1.402) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.156,9.833) -- (7.111,10.003) -- (6.986,10.127) -- (6.816,10.173) -- (6.646,10.127) -- (6.522,10.003) -- (6.476,9.833) -- (6.522,9.663) -- (6.646,9.539) -- (6.816,9.493) -- (6.986,9.539) -- (7.111,9.663) -- cycle;
\node at (6.816,9.833) {\scalebox{1.598}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.294,9.861) -- (10.249,10.031) -- (10.124,10.155) -- (9.954,10.201) -- (9.784,10.155) -- (9.66,10.031) -- (9.614,9.861) -- (9.66,9.691) -- (9.784,9.567) -- (9.954,9.521) -- (10.124,9.567) -- (10.249,9.691) -- cycle;
\node at (9.954,9.861) {\scalebox{1.598}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.159,9.003) -- (9.113,9.173) -- (8.989,9.298) -- (8.819,9.343) -- (8.649,9.298) -- (8.524,9.173) -- (8.479,9.003) -- (8.524,8.833) -- (8.649,8.709) -- (8.819,8.663) -- (8.989,8.709) -- (9.113,8.833) -- cycle;
\node at (8.819,9.003) {\scalebox{1.598}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.149,6.569) -- (4.103,6.739) -- (3.979,6.863) -- (3.809,6.909) -- (3.639,6.863) -- (3.514,6.739) -- (3.469,6.569) -- (3.514,6.399) -- (3.639,6.275) -- (3.809,6.229) -- (3.979,6.275) -- (4.103,6.399) -- cycle;
\node at (3.809,6.569) {\scalebox{1.598}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.958,6.7) -- (5.913,6.87) -- (5.788,6.994) -- (5.618,7.04) -- (5.448,6.994) -- (5.324,6.87) -- (5.278,6.7) -- (5.324,6.53) -- (5.448,6.405) -- (5.618,6.36) -- (5.788,6.405) -- (5.913,6.53) -- cycle;
\node at (5.618,6.7) {\scalebox{1.598}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.944,6.298) -- (11.899,6.468) -- (11.774,6.592) -- (11.604,6.638) -- (11.434,6.592) -- (11.31,6.468) -- (11.264,6.298) -- (11.31,6.128) -- (11.434,6.003) -- (11.604,5.958) -- (11.774,6.003) -- (11.899,6.128) -- cycle;
\node at (11.604,6.298) {\scalebox{1.598}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.415,3.882) -- (3.369,4.052) -- (3.245,4.177) -- (3.075,4.222) -- (2.905,4.177) -- (2.78,4.052) -- (2.735,3.882) -- (2.78,3.712) -- (2.905,3.588) -- (3.075,3.542) -- (3.245,3.588) -- (3.369,3.712) -- cycle;
\node at (3.075,3.882) {\scalebox{1.598}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.251,3.581) -- (2.206,3.751) -- (2.081,3.876) -- (1.911,3.921) -- (1.741,3.876) -- (1.617,3.751) -- (1.571,3.581) -- (1.617,3.411) -- (1.741,3.287) -- (1.911,3.241) -- (2.081,3.287) -- (2.206,3.411) -- cycle;
\node at (1.911,3.581) {\scalebox{1.598}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.637,4.796) -- (9.592,4.966) -- (9.467,5.09) -- (9.297,5.136) -- (9.127,5.09) -- (9.003,4.966) -- (8.957,4.796) -- (9.003,4.626) -- (9.127,4.501) -- (9.297,4.456) -- (9.467,4.501) -- (9.592,4.626) -- cycle;
\node at (9.297,4.796) {\scalebox{1.598}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.799,4.454) -- (6.753,4.624) -- (6.629,4.749) -- (6.459,4.794) -- (6.289,4.749) -- (6.164,4.624) -- (6.119,4.454) -- (6.164,4.284) -- (6.289,4.16) -- (6.459,4.114) -- (6.629,4.16) -- (6.753,4.284) -- cycle;
\node at (6.459,4.454) {\scalebox{1.598}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.528,3.402) -- (6.483,3.572) -- (6.358,3.697) -- (6.188,3.742) -- (6.018,3.697) -- (5.894,3.572) -- (5.848,3.402) -- (5.894,3.232) -- (6.018,3.108) -- (6.188,3.062) -- (6.358,3.108) -- (6.483,3.232) -- cycle;
\node at (6.188,3.402) {\scalebox{1.598}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.972,3.335) -- (8.926,3.505) -- (8.802,3.629) -- (8.632,3.675) -- (8.462,3.629) -- (8.337,3.505) -- (8.292,3.335) -- (8.337,3.165) -- (8.462,3.041) -- (8.632,2.995) -- (8.802,3.041) -- (8.926,3.165) -- cycle;
\node at (8.632,3.335) {\scalebox{1.598}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.482,3.45) -- (5.436,3.62) -- (5.312,3.745) -- (5.142,3.79) -- (4.972,3.745) -- (4.847,3.62) -- (4.802,3.45) -- (4.847,3.28) -- (4.972,3.156) -- (5.142,3.11) -- (5.312,3.156) -- (5.436,3.28) -- cycle;
\node at (5.142,3.45) {\scalebox{1.598}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.097,2.306) -- (7.052,2.476) -- (6.927,2.6) -- (6.757,2.646) -- (6.587,2.6) -- (6.463,2.476) -- (6.417,2.306) -- (6.463,2.136) -- (6.587,2.011) -- (6.757,1.966) -- (6.927,2.011) -- (7.052,2.136) -- cycle;
\node at (6.757,2.306) {\scalebox{1.598}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.085,5.131) -- (1.685,5.131) -- (1.685,5.731) -- (1.085,5.731) -- cycle;
\node at (1.385,5.431) {\scalebox{1.598}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (2.939,3.743) -- (3.539,3.743) -- (3.539,4.343) -- (2.939,4.343) -- cycle;
\node at (3.239,4.043) {\scalebox{1.598}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (9.406,2.421) -- (10.006,2.421) -- (10.006,3.021) -- (9.406,3.021) -- cycle;
\node at (9.706,2.721) {\scalebox{1.598}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.925,1.235) -- (6.525,1.235) -- (6.525,1.835) -- (5.925,1.835) -- cycle;
\node at (6.225,1.535) {\scalebox{1.598}{\small d}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (10.074,3.985) -- (10.674,3.985) -- (10.674,4.585) -- (10.074,4.585) -- cycle;
\node at (10.374,4.285) {\scalebox{1.598}{\small e}};
\node at (7.896,11.923) {\scalebox{1.598}{\small A}};
\node at (5.427,9.165) {\scalebox{1.598}{\small B}};
\node at (11.944,7.924) {\scalebox{1.598}{\small C}};
\node at (0.765,5.446) {\scalebox{1.598}{\small D}};
\node at (11.597,5.039) {\scalebox{1.598}{\small E}};
\node at (8.174,4.655) {\scalebox{1.598}{\small F}};
\node at (4.735,4.633) {\scalebox{1.598}{\small G}};
\node at (4.256,1.62) {\scalebox{1.598}{\small H}};
\node at (0.988,1.341) {\scalebox{1.598}{\small I}};
\node at (9.102,0.95) {\scalebox{1.598}{\small J}};
\draw[line width=0.5pt] (12.685,10.367) -- (12.685,11.467);
\draw[line width=0.5pt] (12.454,11.082) -- (12.685,11.467) -- (12.916,11.082);
\draw[line width=0.5pt] (12.382,10.917) -- (12.987,10.917);
\node at (12.685,11.852) {\scalebox{1.598}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{1.} On compte les rues aboutissant à chaque carrefour :\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Carrefour & \cellcolor{blue!7}\bfseries A & \cellcolor{blue!7}\bfseries B & \cellcolor{blue!7}\bfseries C & \cellcolor{blue!7}\bfseries D & \cellcolor{blue!7}\bfseries E & \cellcolor{blue!7}\bfseries F & \cellcolor{blue!7}\bfseries G & \cellcolor{blue!7}\bfseries H & \cellcolor{blue!7}\bfseries I & \cellcolor{blue!7}\bfseries J \\ \hline
Nombre de rues & 2 & 4 & 3 & 3 & 2 & 4 & 3 & 4 & 1 & 2 \\ \hline
Parité & pair & pair & \textbf{impair} & \textbf{impair} & pair & pair & \textbf{impair} & pair & \textbf{impair} & pair \\ \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{C}, \textbf{D}, \textbf{G}, \textbf{I} — il y en a 4.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Quand le facteur \textbf{traverse} un carrefour, il y arrive par une rue et en repart par une autre : il consomme les rues \textbf{deux par deux}. Un carrefour traversé 3 fois voit donc 6 de ses rues utilisées — toujours un nombre \textbf{pair}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Un carrefour où aboutit un nombre \textbf{impair} de rues ne peut donc pas être seulement traversé : il doit être le \textbf{départ} ou l'\textbf{arrivée} de la tournée. Là, une rue reste « non appariée ».}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Ici la poste est \textbf{sur un carrefour}, en \textbf{D}, et le facteur n'a pas à y revenir. Ce carrefour est donc le \textbf{départ} — il a droit d'être impair. Mais il n'y a qu'\textbf{une seule arrivée} : tous les autres carrefours doivent être pairs, sauf un au plus.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Le quartier compte \textbf{4 carrefours impairs} (C, D, G, 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}
