\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 24}

\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=24}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0024}
\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=0.7384, line join=round, line cap=round]
\draw[line width=0.5pt] (8.649,11.48) -- (8.78,11.4) -- (8.912,11.324) -- (9.046,11.251) -- (9.181,11.181) -- (9.319,11.114) -- (9.458,11.051) -- (9.598,10.99) -- (9.739,10.933) -- (9.882,10.88) -- (10.027,10.83) -- (10.172,10.783) -- (10.318,10.739) -- (10.465,10.699) -- (10.614,10.663);
\draw[line width=0.5pt] (8.649,11.48) -- (8.83,11.259) -- (9.005,11.034) -- (9.174,10.805) -- (9.338,10.572) -- (9.497,10.335) -- (9.65,10.094) -- (9.798,9.85) -- (9.939,9.603) -- (10.075,9.352) -- (10.205,9.098) -- (10.329,8.841) -- (10.446,8.581) -- (10.558,8.319) -- (10.663,8.054);
\draw[line width=0.5pt] (10.614,10.663) -- (10.589,10.478) -- (10.568,10.292) -- (10.552,10.105) -- (10.54,9.918) -- (10.532,9.731) -- (10.529,9.544) -- (10.531,9.357) -- (10.536,9.169) -- (10.547,8.982) -- (10.561,8.796) -- (10.58,8.609) -- (10.603,8.424) -- (10.631,8.239) -- (10.663,8.054);
\draw[line width=0.5pt] (1.465,8.448) -- (1.721,8.478) -- (1.978,8.503) -- (2.236,8.521) -- (2.494,8.533) -- (2.752,8.539) -- (3.01,8.539) -- (3.269,8.533) -- (3.527,8.521) -- (3.785,8.503) -- (4.042,8.479) -- (4.298,8.449) -- (4.554,8.412) -- (4.808,8.37) -- (5.062,8.322);
\draw[line width=0.5pt] (1.465,8.448) -- (1.446,8.216) -- (1.433,7.984) -- (1.426,7.752) -- (1.423,7.52) -- (1.427,7.287) -- (1.435,7.055) -- (1.449,6.822) -- (1.469,6.591) -- (1.494,6.36) -- (1.525,6.129) -- (1.56,5.899) -- (1.602,5.671) -- (1.648,5.443) -- (1.7,5.217);
\draw[line width=0.5pt] (5.062,8.322) -- (5.274,8.349) -- (5.487,8.371) -- (5.7,8.389) -- (5.914,8.401) -- (6.127,8.408) -- (6.341,8.41) -- (6.555,8.407) -- (6.769,8.4) -- (6.983,8.387) -- (7.196,8.369) -- (7.409,8.346) -- (7.621,8.318) -- (7.832,8.285) -- (8.043,8.247) -- (8.226,8.205) -- (8.411,8.167) -- (8.596,8.133) -- (8.783,8.104) -- (8.97,8.079) -- (9.157,8.059) -- (9.345,8.043) -- (9.533,8.031) -- (9.722,8.024) -- (9.91,8.021) -- (10.099,8.023) -- (10.287,8.029) -- (10.475,8.039) -- (10.663,8.054);
\draw[line width=0.5pt] (5.062,8.322) -- (5.24,8.101) -- (5.423,7.885) -- (5.612,7.674) -- (5.805,7.467) -- (6.004,7.264) -- (6.207,7.066) -- (6.414,6.873) -- (6.626,6.685) -- (6.843,6.501) -- (7.063,6.323) -- (7.288,6.151) -- (7.517,5.983) -- (7.749,5.822) -- (7.986,5.665);
\draw[line width=0.5pt] (10.663,8.054) -- (10.499,7.856) -- (10.331,7.661) -- (10.158,7.47) -- (9.981,7.284) -- (9.799,7.101) -- (9.613,6.923) -- (9.423,6.749) -- (9.229,6.58) -- (9.031,6.415) -- (8.829,6.256) -- (8.623,6.101) -- (8.414,5.95) -- (8.201,5.805) -- (7.986,5.665);
\draw[line width=0.5pt] (7.986,5.665) -- (7.809,5.677) -- (7.633,5.685) -- (7.456,5.689) -- (7.279,5.688) -- (7.102,5.683) -- (6.925,5.675) -- (6.749,5.662) -- (6.573,5.644) -- (6.397,5.623) -- (6.222,5.598) -- (6.048,5.568) -- (5.874,5.535) -- (5.701,5.497) -- (5.53,5.455);
\draw[line width=0.5pt] (7.986,5.665) -- (7.809,5.34) -- (7.624,5.019) -- (7.432,4.702) -- (7.232,4.39) -- (7.025,4.083) -- (6.811,3.78) -- (6.589,3.483) -- (6.361,3.191) -- (6.126,2.904) -- (5.885,2.623) -- (5.636,2.348) -- (5.382,2.079) -- (5.121,1.816) -- (4.854,1.559);
\draw[line width=0.5pt] (5.53,5.455) -- (5.256,5.48) -- (4.981,5.499) -- (4.706,5.511) -- (4.431,5.516) -- (4.155,5.515) -- (3.88,5.508) -- (3.605,5.494) -- (3.33,5.474) -- (3.056,5.447) -- (2.783,5.413) -- (2.511,5.374) -- (2.239,5.328) -- (1.969,5.275) -- (1.7,5.217);
\draw[line width=0.5pt] (5.53,5.455) -- (5.524,5.172) -- (5.512,4.889) -- (5.494,4.605) -- (5.468,4.323) -- (5.436,4.041) -- (5.398,3.76) -- (5.353,3.479) -- (5.301,3.2) -- (5.243,2.923) -- (5.178,2.646) -- (5.107,2.372) -- (5.029,2.099) -- (4.945,1.828) -- (4.854,1.559);
\draw[line width=0.5pt] (4.854,1.559) -- (4.62,1.59) -- (4.386,1.614) -- (4.151,1.634) -- (3.915,1.647) -- (3.679,1.656) -- (3.443,1.658) -- (3.206,1.655) -- (2.97,1.647) -- (2.735,1.633) -- (2.499,1.613) -- (2.265,1.588) -- (2.03,1.558) -- (1.797,1.521) -- (1.565,1.48);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.809,11.48) -- (8.788,11.56) -- (8.729,11.618) -- (8.649,11.64) -- (8.569,11.618) -- (8.511,11.56) -- (8.489,11.48) -- (8.511,11.4) -- (8.569,11.341) -- (8.649,11.32) -- (8.729,11.341) -- (8.788,11.4) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.774,10.663) -- (10.752,10.743) -- (10.694,10.802) -- (10.614,10.823) -- (10.534,10.802) -- (10.475,10.743) -- (10.454,10.663) -- (10.475,10.583) -- (10.534,10.525) -- (10.614,10.503) -- (10.694,10.525) -- (10.752,10.583) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.625,8.448) -- (1.604,8.528) -- (1.545,8.586) -- (1.465,8.608) -- (1.385,8.586) -- (1.327,8.528) -- (1.305,8.448) -- (1.327,8.368) -- (1.385,8.309) -- (1.465,8.288) -- (1.545,8.309) -- (1.604,8.368) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.222,8.322) -- (5.2,8.402) -- (5.142,8.46) -- (5.062,8.482) -- (4.982,8.46) -- (4.923,8.402) -- (4.902,8.322) -- (4.923,8.242) -- (4.982,8.183) -- (5.062,8.162) -- (5.142,8.183) -- (5.2,8.242) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.823,8.054) -- (10.802,8.134) -- (10.743,8.193) -- (10.663,8.214) -- (10.583,8.193) -- (10.524,8.134) -- (10.503,8.054) -- (10.524,7.974) -- (10.583,7.916) -- (10.663,7.894) -- (10.743,7.916) -- (10.802,7.974) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.146,5.665) -- (8.124,5.745) -- (8.066,5.804) -- (7.986,5.825) -- (7.906,5.804) -- (7.847,5.745) -- (7.826,5.665) -- (7.847,5.585) -- (7.906,5.527) -- (7.986,5.505) -- (8.066,5.527) -- (8.124,5.585) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.69,5.455) -- (5.668,5.535) -- (5.61,5.594) -- (5.53,5.615) -- (5.45,5.594) -- (5.391,5.535) -- (5.37,5.455) -- (5.391,5.375) -- (5.45,5.317) -- (5.53,5.295) -- (5.61,5.317) -- (5.668,5.375) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.86,5.217) -- (1.839,5.297) -- (1.78,5.355) -- (1.7,5.377) -- (1.62,5.355) -- (1.562,5.297) -- (1.54,5.217) -- (1.562,5.137) -- (1.62,5.078) -- (1.7,5.057) -- (1.78,5.078) -- (1.839,5.137) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.014,1.559) -- (4.993,1.639) -- (4.934,1.698) -- (4.854,1.719) -- (4.774,1.698) -- (4.716,1.639) -- (4.694,1.559) -- (4.716,1.479) -- (4.774,1.421) -- (4.854,1.399) -- (4.934,1.421) -- (4.993,1.479) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.725,1.48) -- (1.704,1.56) -- (1.645,1.618) -- (1.565,1.64) -- (1.485,1.618) -- (1.426,1.56) -- (1.405,1.48) -- (1.426,1.4) -- (1.485,1.341) -- (1.565,1.32) -- (1.645,1.341) -- (1.704,1.4) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.061,11.286) -- (10.015,11.456) -- (9.891,11.58) -- (9.721,11.626) -- (9.551,11.58) -- (9.426,11.456) -- (9.381,11.286) -- (9.426,11.116) -- (9.551,10.991) -- (9.721,10.946) -- (9.891,10.991) -- (10.015,11.116) -- cycle;
\node at (9.721,11.286) {\scalebox{1.354}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.413,10.012) -- (10.368,10.182) -- (10.243,10.307) -- (10.073,10.352) -- (9.903,10.307) -- (9.779,10.182) -- (9.733,10.012) -- (9.779,9.842) -- (9.903,9.718) -- (10.073,9.672) -- (10.243,9.718) -- (10.368,9.842) -- cycle;
\node at (10.073,10.012) {\scalebox{1.354}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.191,9.363) -- (11.145,9.533) -- (11.021,9.657) -- (10.851,9.703) -- (10.681,9.657) -- (10.556,9.533) -- (10.511,9.363) -- (10.556,9.193) -- (10.681,9.068) -- (10.851,9.023) -- (11.021,9.068) -- (11.145,9.193) -- cycle;
\node at (10.851,9.363) {\scalebox{1.354}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.62,8.853) -- (3.574,9.023) -- (3.45,9.147) -- (3.28,9.193) -- (3.11,9.147) -- (2.986,9.023) -- (2.94,8.853) -- (2.986,8.683) -- (3.11,8.558) -- (3.28,8.513) -- (3.45,8.558) -- (3.574,8.683) -- cycle;
\node at (3.28,8.853) {\scalebox{1.354}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.109,6.846) -- (2.063,7.016) -- (1.939,7.14) -- (1.769,7.186) -- (1.599,7.14) -- (1.474,7.016) -- (1.429,6.846) -- (1.474,6.676) -- (1.599,6.551) -- (1.769,6.506) -- (1.939,6.551) -- (2.063,6.676) -- cycle;
\node at (1.769,6.846) {\scalebox{1.354}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.447,8.561) -- (8.401,8.731) -- (8.277,8.855) -- (8.107,8.901) -- (7.937,8.855) -- (7.812,8.731) -- (7.767,8.561) -- (7.812,8.391) -- (7.937,8.266) -- (8.107,8.221) -- (8.277,8.266) -- (8.401,8.391) -- cycle;
\node at (8.107,8.561) {\scalebox{1.354}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.969,7.11) -- (6.924,7.28) -- (6.799,7.404) -- (6.629,7.45) -- (6.459,7.404) -- (6.335,7.28) -- (6.289,7.11) -- (6.335,6.94) -- (6.459,6.815) -- (6.629,6.77) -- (6.799,6.815) -- (6.924,6.94) -- cycle;
\node at (6.629,7.11) {\scalebox{1.354}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.976,6.511) -- (9.93,6.681) -- (9.806,6.805) -- (9.636,6.851) -- (9.466,6.805) -- (9.341,6.681) -- (9.296,6.511) -- (9.341,6.341) -- (9.466,6.216) -- (9.636,6.171) -- (9.806,6.216) -- (9.93,6.341) -- cycle;
\node at (9.636,6.511) {\scalebox{1.354}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.116,5.343) -- (7.071,5.513) -- (6.946,5.637) -- (6.776,5.683) -- (6.606,5.637) -- (6.482,5.513) -- (6.436,5.343) -- (6.482,5.173) -- (6.606,5.048) -- (6.776,5.003) -- (6.946,5.048) -- (7.071,5.173) -- cycle;
\node at (6.776,5.343) {\scalebox{1.354}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.184,3.289) -- (7.138,3.459) -- (7.014,3.584) -- (6.844,3.629) -- (6.674,3.584) -- (6.549,3.459) -- (6.504,3.289) -- (6.549,3.119) -- (6.674,2.995) -- (6.844,2.949) -- (7.014,2.995) -- (7.138,3.119) -- cycle;
\node at (6.844,3.289) {\scalebox{1.354}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.965,5.175) -- (3.919,5.345) -- (3.795,5.469) -- (3.625,5.515) -- (3.455,5.469) -- (3.331,5.345) -- (3.285,5.175) -- (3.331,5.005) -- (3.455,4.88) -- (3.625,4.835) -- (3.795,4.88) -- (3.919,5.005) -- cycle;
\node at (3.625,5.175) {\scalebox{1.354}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.008,3.425) -- (5.962,3.595) -- (5.838,3.719) -- (5.668,3.765) -- (5.498,3.719) -- (5.374,3.595) -- (5.328,3.425) -- (5.374,3.255) -- (5.498,3.13) -- (5.668,3.085) -- (5.838,3.13) -- (5.962,3.255) -- cycle;
\node at (5.668,3.425) {\scalebox{1.354}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.554,1.335) -- (3.509,1.505) -- (3.384,1.63) -- (3.214,1.675) -- (3.044,1.63) -- (2.92,1.505) -- (2.874,1.335) -- (2.92,1.165) -- (3.044,1.041) -- (3.214,0.995) -- (3.384,1.041) -- (3.509,1.165) -- cycle;
\node at (3.214,1.335) {\scalebox{1.354}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (5.23,5.155) -- (5.83,5.155) -- (5.83,5.755) -- (5.23,5.755) -- cycle;
\node at (5.53,5.455) {\scalebox{1.354}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.471,3.969) -- (3.071,3.969) -- (3.071,4.569) -- (2.471,4.569) -- cycle;
\node at (2.771,4.269) {\scalebox{1.354}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.09,1.811) -- (7.69,1.811) -- (7.69,2.411) -- (7.09,2.411) -- cycle;
\node at (7.39,2.111) {\scalebox{1.354}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (0.6,6.628) -- (1.2,6.628) -- (1.2,7.228) -- (0.6,7.228) -- cycle;
\node at (0.9,6.928) {\scalebox{1.354}{\small d}};
\node at (8.963,12.015) {\scalebox{1.354}{\small A}};
\node at (11.089,11.061) {\scalebox{1.354}{\small B}};
\node at (0.893,8.687) {\scalebox{1.354}{\small C}};
\node at (4.811,8.889) {\scalebox{1.354}{\small D}};
\node at (11.258,8.228) {\scalebox{1.354}{\small E}};
\node at (8.552,5.413) {\scalebox{1.354}{\small F}};
\node at (5.387,4.852) {\scalebox{1.354}{\small G}};
\node at (1.114,5.014) {\scalebox{1.354}{\small H}};
\node at (4.74,0.95) {\scalebox{1.354}{\small I}};
\node at (1.171,1.001) {\scalebox{1.354}{\small J}};
\draw[line width=0.5pt] (11.963,10.53) -- (11.963,11.63);
\draw[line width=0.5pt] (11.732,11.245) -- (11.963,11.63) -- (12.194,11.245);
\draw[line width=0.5pt] (11.661,11.08) -- (12.266,11.08);
\node at (11.963,12.015) {\scalebox{1.354}{\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 de la Poste & ⑧ rue Verte & Ⓐ Place du Marché & \textbf{a} la poste \\ \hline
② rue du Puits & ⑨ rue de la Mairie & Ⓑ Place de l'Église & \textbf{b} la boulangerie \\ \hline
③ rue du Four & ⑩ allée des Tilleuls & Ⓒ Place de la Fontaine & \textbf{c} le lavoir \\ \hline
④ rue des Lilas & ⑪ chemin du Lavoir & Ⓓ Rond-point de la Gare & \textbf{d} la bibliothèque \\ \hline
⑤ chemin de Ronde & ⑫ boulevard du Parc & Ⓔ Carrefour des Quatre-Chemins &  \\ \hline
⑥ rue Basse & ⑬ impasse Sainte-Anne & Ⓕ Carrefour des Tilleuls &  \\ \hline
⑦ rue des Écoles &  & Ⓖ Carrefour Saint-Roch &  \\ \hline
 &  & Ⓗ Coin du Lavoir &  \\ \hline
 &  & Ⓘ Rond-point des Vaches &  \\ \hline
 &  & Ⓙ 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=0.6607, line join=round, line cap=round]
\draw[line width=0.5pt] (8.649,11.48) -- (8.78,11.4) -- (8.912,11.324) -- (9.046,11.251) -- (9.181,11.181) -- (9.319,11.114) -- (9.458,11.051) -- (9.598,10.99) -- (9.739,10.933) -- (9.882,10.88) -- (10.027,10.83) -- (10.172,10.783) -- (10.318,10.739) -- (10.465,10.699) -- (10.614,10.663);
\draw[line width=0.5pt] (8.649,11.48) -- (8.83,11.259) -- (9.005,11.034) -- (9.174,10.805) -- (9.338,10.572) -- (9.497,10.335) -- (9.65,10.094) -- (9.798,9.85) -- (9.939,9.603) -- (10.075,9.352) -- (10.205,9.098) -- (10.329,8.841) -- (10.446,8.581) -- (10.558,8.319) -- (10.663,8.054);
\draw[line width=0.5pt] (10.614,10.663) -- (10.589,10.478) -- (10.568,10.292) -- (10.552,10.105) -- (10.54,9.918) -- (10.532,9.731) -- (10.529,9.544) -- (10.531,9.357) -- (10.536,9.169) -- (10.547,8.982) -- (10.561,8.796) -- (10.58,8.609) -- (10.603,8.424) -- (10.631,8.239) -- (10.663,8.054);
\draw[line width=0.5pt] (1.465,8.448) -- (1.721,8.478) -- (1.978,8.503) -- (2.236,8.521) -- (2.494,8.533) -- (2.752,8.539) -- (3.01,8.539) -- (3.269,8.533) -- (3.527,8.521) -- (3.785,8.503) -- (4.042,8.479) -- (4.298,8.449) -- (4.554,8.412) -- (4.808,8.37) -- (5.062,8.322);
\draw[line width=0.5pt] (1.465,8.448) -- (1.446,8.216) -- (1.433,7.984) -- (1.426,7.752) -- (1.423,7.52) -- (1.427,7.287) -- (1.435,7.055) -- (1.449,6.822) -- (1.469,6.591) -- (1.494,6.36) -- (1.525,6.129) -- (1.56,5.899) -- (1.602,5.671) -- (1.648,5.443) -- (1.7,5.217);
\draw[line width=0.5pt] (5.062,8.322) -- (5.274,8.349) -- (5.487,8.371) -- (5.7,8.389) -- (5.914,8.401) -- (6.127,8.408) -- (6.341,8.41) -- (6.555,8.407) -- (6.769,8.4) -- (6.983,8.387) -- (7.196,8.369) -- (7.409,8.346) -- (7.621,8.318) -- (7.832,8.285) -- (8.043,8.247) -- (8.226,8.205) -- (8.411,8.167) -- (8.596,8.133) -- (8.783,8.104) -- (8.97,8.079) -- (9.157,8.059) -- (9.345,8.043) -- (9.533,8.031) -- (9.722,8.024) -- (9.91,8.021) -- (10.099,8.023) -- (10.287,8.029) -- (10.475,8.039) -- (10.663,8.054);
\draw[line width=0.5pt] (5.062,8.322) -- (5.24,8.101) -- (5.423,7.885) -- (5.612,7.674) -- (5.805,7.467) -- (6.004,7.264) -- (6.207,7.066) -- (6.414,6.873) -- (6.626,6.685) -- (6.843,6.501) -- (7.063,6.323) -- (7.288,6.151) -- (7.517,5.983) -- (7.749,5.822) -- (7.986,5.665);
\draw[line width=0.5pt] (10.663,8.054) -- (10.499,7.856) -- (10.331,7.661) -- (10.158,7.47) -- (9.981,7.284) -- (9.799,7.101) -- (9.613,6.923) -- (9.423,6.749) -- (9.229,6.58) -- (9.031,6.415) -- (8.829,6.256) -- (8.623,6.101) -- (8.414,5.95) -- (8.201,5.805) -- (7.986,5.665);
\draw[line width=0.5pt] (7.986,5.665) -- (7.809,5.677) -- (7.633,5.685) -- (7.456,5.689) -- (7.279,5.688) -- (7.102,5.683) -- (6.925,5.675) -- (6.749,5.662) -- (6.573,5.644) -- (6.397,5.623) -- (6.222,5.598) -- (6.048,5.568) -- (5.874,5.535) -- (5.701,5.497) -- (5.53,5.455);
\draw[line width=0.5pt] (7.986,5.665) -- (7.809,5.34) -- (7.624,5.019) -- (7.432,4.702) -- (7.232,4.39) -- (7.025,4.083) -- (6.811,3.78) -- (6.589,3.483) -- (6.361,3.191) -- (6.126,2.904) -- (5.885,2.623) -- (5.636,2.348) -- (5.382,2.079) -- (5.121,1.816) -- (4.854,1.559);
\draw[line width=0.5pt] (5.53,5.455) -- (5.256,5.48) -- (4.981,5.499) -- (4.706,5.511) -- (4.431,5.516) -- (4.155,5.515) -- (3.88,5.508) -- (3.605,5.494) -- (3.33,5.474) -- (3.056,5.447) -- (2.783,5.413) -- (2.511,5.374) -- (2.239,5.328) -- (1.969,5.275) -- (1.7,5.217);
\draw[line width=0.5pt] (5.53,5.455) -- (5.524,5.172) -- (5.512,4.889) -- (5.494,4.605) -- (5.468,4.323) -- (5.436,4.041) -- (5.398,3.76) -- (5.353,3.479) -- (5.301,3.2) -- (5.243,2.923) -- (5.178,2.646) -- (5.107,2.372) -- (5.029,2.099) -- (4.945,1.828) -- (4.854,1.559);
\draw[line width=0.5pt] (4.854,1.559) -- (4.62,1.59) -- (4.386,1.614) -- (4.151,1.634) -- (3.915,1.647) -- (3.679,1.656) -- (3.443,1.658) -- (3.206,1.655) -- (2.97,1.647) -- (2.735,1.633) -- (2.499,1.613) -- (2.265,1.588) -- (2.03,1.558) -- (1.797,1.521) -- (1.565,1.48);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.809,11.48) -- (8.788,11.56) -- (8.729,11.618) -- (8.649,11.64) -- (8.569,11.618) -- (8.511,11.56) -- (8.489,11.48) -- (8.511,11.4) -- (8.569,11.341) -- (8.649,11.32) -- (8.729,11.341) -- (8.788,11.4) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.774,10.663) -- (10.752,10.743) -- (10.694,10.802) -- (10.614,10.823) -- (10.534,10.802) -- (10.475,10.743) -- (10.454,10.663) -- (10.475,10.583) -- (10.534,10.525) -- (10.614,10.503) -- (10.694,10.525) -- (10.752,10.583) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.625,8.448) -- (1.604,8.528) -- (1.545,8.586) -- (1.465,8.608) -- (1.385,8.586) -- (1.327,8.528) -- (1.305,8.448) -- (1.327,8.368) -- (1.385,8.309) -- (1.465,8.288) -- (1.545,8.309) -- (1.604,8.368) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.222,8.322) -- (5.2,8.402) -- (5.142,8.46) -- (5.062,8.482) -- (4.982,8.46) -- (4.923,8.402) -- (4.902,8.322) -- (4.923,8.242) -- (4.982,8.183) -- (5.062,8.162) -- (5.142,8.183) -- (5.2,8.242) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.823,8.054) -- (10.802,8.134) -- (10.743,8.193) -- (10.663,8.214) -- (10.583,8.193) -- (10.524,8.134) -- (10.503,8.054) -- (10.524,7.974) -- (10.583,7.916) -- (10.663,7.894) -- (10.743,7.916) -- (10.802,7.974) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.146,5.665) -- (8.124,5.745) -- (8.066,5.804) -- (7.986,5.825) -- (7.906,5.804) -- (7.847,5.745) -- (7.826,5.665) -- (7.847,5.585) -- (7.906,5.527) -- (7.986,5.505) -- (8.066,5.527) -- (8.124,5.585) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.69,5.455) -- (5.668,5.535) -- (5.61,5.594) -- (5.53,5.615) -- (5.45,5.594) -- (5.391,5.535) -- (5.37,5.455) -- (5.391,5.375) -- (5.45,5.317) -- (5.53,5.295) -- (5.61,5.317) -- (5.668,5.375) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.86,5.217) -- (1.839,5.297) -- (1.78,5.355) -- (1.7,5.377) -- (1.62,5.355) -- (1.562,5.297) -- (1.54,5.217) -- (1.562,5.137) -- (1.62,5.078) -- (1.7,5.057) -- (1.78,5.078) -- (1.839,5.137) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.014,1.559) -- (4.993,1.639) -- (4.934,1.698) -- (4.854,1.719) -- (4.774,1.698) -- (4.716,1.639) -- (4.694,1.559) -- (4.716,1.479) -- (4.774,1.421) -- (4.854,1.399) -- (4.934,1.421) -- (4.993,1.479) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.725,1.48) -- (1.704,1.56) -- (1.645,1.618) -- (1.565,1.64) -- (1.485,1.618) -- (1.426,1.56) -- (1.405,1.48) -- (1.426,1.4) -- (1.485,1.341) -- (1.565,1.32) -- (1.645,1.341) -- (1.704,1.4) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.061,11.286) -- (10.015,11.456) -- (9.891,11.58) -- (9.721,11.626) -- (9.551,11.58) -- (9.426,11.456) -- (9.381,11.286) -- (9.426,11.116) -- (9.551,10.991) -- (9.721,10.946) -- (9.891,10.991) -- (10.015,11.116) -- cycle;
\node at (9.721,11.286) {\scalebox{1.514}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.413,10.012) -- (10.368,10.182) -- (10.243,10.307) -- (10.073,10.352) -- (9.903,10.307) -- (9.779,10.182) -- (9.733,10.012) -- (9.779,9.842) -- (9.903,9.718) -- (10.073,9.672) -- (10.243,9.718) -- (10.368,9.842) -- cycle;
\node at (10.073,10.012) {\scalebox{1.514}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.191,9.363) -- (11.145,9.533) -- (11.021,9.657) -- (10.851,9.703) -- (10.681,9.657) -- (10.556,9.533) -- (10.511,9.363) -- (10.556,9.193) -- (10.681,9.068) -- (10.851,9.023) -- (11.021,9.068) -- (11.145,9.193) -- cycle;
\node at (10.851,9.363) {\scalebox{1.514}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.62,8.853) -- (3.574,9.023) -- (3.45,9.147) -- (3.28,9.193) -- (3.11,9.147) -- (2.986,9.023) -- (2.94,8.853) -- (2.986,8.683) -- (3.11,8.558) -- (3.28,8.513) -- (3.45,8.558) -- (3.574,8.683) -- cycle;
\node at (3.28,8.853) {\scalebox{1.514}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.109,6.846) -- (2.063,7.016) -- (1.939,7.14) -- (1.769,7.186) -- (1.599,7.14) -- (1.474,7.016) -- (1.429,6.846) -- (1.474,6.676) -- (1.599,6.551) -- (1.769,6.506) -- (1.939,6.551) -- (2.063,6.676) -- cycle;
\node at (1.769,6.846) {\scalebox{1.514}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.447,8.561) -- (8.401,8.731) -- (8.277,8.855) -- (8.107,8.901) -- (7.937,8.855) -- (7.812,8.731) -- (7.767,8.561) -- (7.812,8.391) -- (7.937,8.266) -- (8.107,8.221) -- (8.277,8.266) -- (8.401,8.391) -- cycle;
\node at (8.107,8.561) {\scalebox{1.514}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.969,7.11) -- (6.924,7.28) -- (6.799,7.404) -- (6.629,7.45) -- (6.459,7.404) -- (6.335,7.28) -- (6.289,7.11) -- (6.335,6.94) -- (6.459,6.815) -- (6.629,6.77) -- (6.799,6.815) -- (6.924,6.94) -- cycle;
\node at (6.629,7.11) {\scalebox{1.514}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.976,6.511) -- (9.93,6.681) -- (9.806,6.805) -- (9.636,6.851) -- (9.466,6.805) -- (9.341,6.681) -- (9.296,6.511) -- (9.341,6.341) -- (9.466,6.216) -- (9.636,6.171) -- (9.806,6.216) -- (9.93,6.341) -- cycle;
\node at (9.636,6.511) {\scalebox{1.514}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.116,5.343) -- (7.071,5.513) -- (6.946,5.637) -- (6.776,5.683) -- (6.606,5.637) -- (6.482,5.513) -- (6.436,5.343) -- (6.482,5.173) -- (6.606,5.048) -- (6.776,5.003) -- (6.946,5.048) -- (7.071,5.173) -- cycle;
\node at (6.776,5.343) {\scalebox{1.514}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.184,3.289) -- (7.138,3.459) -- (7.014,3.584) -- (6.844,3.629) -- (6.674,3.584) -- (6.549,3.459) -- (6.504,3.289) -- (6.549,3.119) -- (6.674,2.995) -- (6.844,2.949) -- (7.014,2.995) -- (7.138,3.119) -- cycle;
\node at (6.844,3.289) {\scalebox{1.514}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.965,5.175) -- (3.919,5.345) -- (3.795,5.469) -- (3.625,5.515) -- (3.455,5.469) -- (3.331,5.345) -- (3.285,5.175) -- (3.331,5.005) -- (3.455,4.88) -- (3.625,4.835) -- (3.795,4.88) -- (3.919,5.005) -- cycle;
\node at (3.625,5.175) {\scalebox{1.514}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.008,3.425) -- (5.962,3.595) -- (5.838,3.719) -- (5.668,3.765) -- (5.498,3.719) -- (5.374,3.595) -- (5.328,3.425) -- (5.374,3.255) -- (5.498,3.13) -- (5.668,3.085) -- (5.838,3.13) -- (5.962,3.255) -- cycle;
\node at (5.668,3.425) {\scalebox{1.514}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.554,1.335) -- (3.509,1.505) -- (3.384,1.63) -- (3.214,1.675) -- (3.044,1.63) -- (2.92,1.505) -- (2.874,1.335) -- (2.92,1.165) -- (3.044,1.041) -- (3.214,0.995) -- (3.384,1.041) -- (3.509,1.165) -- cycle;
\node at (3.214,1.335) {\scalebox{1.514}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (5.23,5.155) -- (5.83,5.155) -- (5.83,5.755) -- (5.23,5.755) -- cycle;
\node at (5.53,5.455) {\scalebox{1.514}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.471,3.969) -- (3.071,3.969) -- (3.071,4.569) -- (2.471,4.569) -- cycle;
\node at (2.771,4.269) {\scalebox{1.514}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.09,1.811) -- (7.69,1.811) -- (7.69,2.411) -- (7.09,2.411) -- cycle;
\node at (7.39,2.111) {\scalebox{1.514}{\small c}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (0.6,6.628) -- (1.2,6.628) -- (1.2,7.228) -- (0.6,7.228) -- cycle;
\node at (0.9,6.928) {\scalebox{1.514}{\small d}};
\node at (8.963,12.015) {\scalebox{1.514}{\small A}};
\node at (11.089,11.061) {\scalebox{1.514}{\small B}};
\node at (0.893,8.687) {\scalebox{1.514}{\small C}};
\node at (4.811,8.889) {\scalebox{1.514}{\small D}};
\node at (11.258,8.228) {\scalebox{1.514}{\small E}};
\node at (8.552,5.413) {\scalebox{1.514}{\small F}};
\node at (5.387,4.852) {\scalebox{1.514}{\small G}};
\node at (1.114,5.014) {\scalebox{1.514}{\small H}};
\node at (4.74,0.95) {\scalebox{1.514}{\small I}};
\node at (1.171,1.001) {\scalebox{1.514}{\small J}};
\draw[line width=0.5pt] (11.963,10.53) -- (11.963,11.63);
\draw[line width=0.5pt] (11.732,11.245) -- (11.963,11.63) -- (12.194,11.245);
\draw[line width=0.5pt] (11.661,11.08) -- (12.266,11.08);
\node at (11.963,12.015) {\scalebox{1.514}{\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 & 2 & 2 & 3 & 4 & 4 & 3 & 2 & 3 & 1 \\ \hline
Parité & pair & pair & pair & \textbf{impair} & pair & pair & \textbf{impair} & pair & \textbf{impair} & \textbf{impair} \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 26 = 2 \(\times\) 13, 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{D}, \textbf{G}, \textbf{I}, \textbf{J} — 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{G}, 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} (D, G, I, J).}\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}
