\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 340}

\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=340}\\[2pt]
    {\small\ttfamily\bfseries\color{red} EULC0340}
\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=0.7337, line join=round, line cap=round]
\draw[line width=0.5pt] (7.868,11.548) -- (8.039,11.307) -- (8.217,11.071) -- (8.4,10.838) -- (8.588,10.61) -- (8.782,10.387) -- (8.981,10.168) -- (9.185,9.954) -- (9.394,9.744) -- (9.608,9.54) -- (9.827,9.34) -- (10.05,9.146) -- (10.278,8.958) -- (10.51,8.774) -- (10.746,8.597);
\draw[line width=0.5pt] (7.868,11.548) -- (7.93,11.267) -- (7.986,10.984) -- (8.035,10.701) -- (8.078,10.416) -- (8.114,10.13) -- (8.143,9.843) -- (8.166,9.556) -- (8.181,9.268) -- (8.19,8.98) -- (8.192,8.692) -- (8.188,8.404) -- (8.176,8.117) -- (8.158,7.829) -- (8.133,7.543);
\draw[line width=0.5pt] (1.32,10.798) -- (1.557,10.572) -- (1.799,10.352) -- (2.047,10.138) -- (2.299,9.929) -- (2.556,9.727) -- (2.818,9.53) -- (3.085,9.34) -- (3.356,9.156) -- (3.631,8.979) -- (3.91,8.808) -- (4.193,8.643) -- (4.48,8.486) -- (4.771,8.335) -- (5.065,8.191);
\draw[line width=0.5pt] (1.32,10.798) -- (1.359,10.57) -- (1.391,10.341) -- (1.419,10.111) -- (1.441,9.881) -- (1.458,9.65) -- (1.469,9.419) -- (1.474,9.187) -- (1.475,8.956) -- (1.47,8.724) -- (1.459,8.493) -- (1.443,8.262) -- (1.422,8.032) -- (1.395,7.802) -- (1.363,7.573) -- (1.331,7.378) -- (1.303,7.182) -- (1.281,6.985) -- (1.263,6.788) -- (1.25,6.59) -- (1.241,6.392) -- (1.237,6.194) -- (1.237,5.996) -- (1.243,5.798) -- (1.253,5.6) -- (1.267,5.403) -- (1.286,5.206) -- (1.31,5.009) -- (1.339,4.813);
\draw[line width=0.5pt] (10.746,8.597) -- (10.482,8.347) -- (10.223,8.09) -- (9.97,7.827) -- (9.724,7.559) -- (9.484,7.284) -- (9.25,7.004) -- (9.023,6.719) -- (8.803,6.428) -- (8.59,6.132) -- (8.384,5.831) -- (8.184,5.526) -- (7.993,5.216) -- (7.808,4.902) -- (7.631,4.583);
\draw[line width=0.5pt] (5.065,8.191) -- (5.289,8.179) -- (5.513,8.161) -- (5.737,8.138) -- (5.96,8.11) -- (6.183,8.076) -- (6.405,8.037) -- (6.626,7.993) -- (6.845,7.944) -- (7.064,7.89) -- (7.281,7.831) -- (7.497,7.766) -- (7.711,7.697) -- (7.923,7.622) -- (8.133,7.543);
\draw[line width=0.5pt] (5.065,8.191) -- (4.972,7.953) -- (4.884,7.714) -- (4.802,7.472) -- (4.726,7.228) -- (4.656,6.982) -- (4.592,6.735) -- (4.533,6.486) -- (4.48,6.236) -- (4.433,5.985) -- (4.392,5.733) -- (4.357,5.48) -- (4.328,5.226) -- (4.305,4.972) -- (4.288,4.717);
\draw[line width=0.5pt] (5.065,8.191) -- (5.286,7.964) -- (5.502,7.731) -- (5.712,7.493) -- (5.917,7.25) -- (6.116,7.002) -- (6.31,6.75) -- (6.497,6.493) -- (6.678,6.232) -- (6.853,5.967) -- (7.022,5.697) -- (7.184,5.424) -- (7.34,5.147) -- (7.489,4.867) -- (7.631,4.583);
\draw[line width=0.5pt] (1.339,4.813) -- (1.547,4.774) -- (1.756,4.74) -- (1.965,4.711) -- (2.176,4.687) -- (2.386,4.667) -- (2.598,4.653) -- (2.809,4.644) -- (3.021,4.639) -- (3.233,4.64) -- (3.445,4.645) -- (3.656,4.656) -- (3.867,4.671) -- (4.078,4.692) -- (4.288,4.717);
\draw[line width=0.5pt] (1.339,4.813) -- (1.409,4.595) -- (1.474,4.375) -- (1.534,4.153) -- (1.589,3.93) -- (1.639,3.705) -- (1.683,3.48) -- (1.722,3.253) -- (1.755,3.026) -- (1.784,2.798) -- (1.807,2.569) -- (1.824,2.34) -- (1.836,2.111) -- (1.843,1.881) -- (1.844,1.652);
\draw[line width=0.5pt] (1.339,4.813) -- (1.483,4.554) -- (1.633,4.298) -- (1.789,4.046) -- (1.951,3.798) -- (2.119,3.553) -- (2.293,3.312) -- (2.472,3.076) -- (2.657,2.844) -- (2.848,2.616) -- (3.043,2.393) -- (3.244,2.174) -- (3.449,1.96) -- (3.66,1.751) -- (3.876,1.548);
\draw[line width=0.5pt] (1.844,1.652) -- (1.989,1.667) -- (2.135,1.678) -- (2.28,1.686) -- (2.426,1.691) -- (2.572,1.692) -- (2.718,1.689) -- (2.864,1.684) -- (3.01,1.674) -- (3.155,1.662) -- (3.3,1.646) -- (3.445,1.626) -- (3.589,1.604) -- (3.733,1.577) -- (3.876,1.548);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.028,11.548) -- (8.006,11.628) -- (7.948,11.686) -- (7.868,11.708) -- (7.788,11.686) -- (7.729,11.628) -- (7.708,11.548) -- (7.729,11.468) -- (7.788,11.409) -- (7.868,11.388) -- (7.948,11.409) -- (8.006,11.468) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.48,10.798) -- (1.459,10.878) -- (1.4,10.937) -- (1.32,10.958) -- (1.24,10.937) -- (1.182,10.878) -- (1.16,10.798) -- (1.182,10.718) -- (1.24,10.66) -- (1.32,10.638) -- (1.4,10.66) -- (1.459,10.718) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.906,8.597) -- (10.885,8.677) -- (10.826,8.735) -- (10.746,8.757) -- (10.666,8.735) -- (10.608,8.677) -- (10.586,8.597) -- (10.608,8.517) -- (10.666,8.458) -- (10.746,8.437) -- (10.826,8.458) -- (10.885,8.517) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.225,8.191) -- (5.203,8.271) -- (5.145,8.33) -- (5.065,8.351) -- (4.985,8.33) -- (4.926,8.271) -- (4.905,8.191) -- (4.926,8.111) -- (4.985,8.053) -- (5.065,8.031) -- (5.145,8.053) -- (5.203,8.111) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.293,7.543) -- (8.272,7.623) -- (8.213,7.681) -- (8.133,7.703) -- (8.053,7.681) -- (7.995,7.623) -- (7.973,7.543) -- (7.995,7.463) -- (8.053,7.404) -- (8.133,7.383) -- (8.213,7.404) -- (8.272,7.463) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.499,4.813) -- (1.477,4.893) -- (1.419,4.952) -- (1.339,4.973) -- (1.259,4.952) -- (1.2,4.893) -- (1.179,4.813) -- (1.2,4.733) -- (1.259,4.675) -- (1.339,4.653) -- (1.419,4.675) -- (1.477,4.733) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.448,4.717) -- (4.426,4.797) -- (4.368,4.856) -- (4.288,4.877) -- (4.208,4.856) -- (4.149,4.797) -- (4.128,4.717) -- (4.149,4.637) -- (4.208,4.579) -- (4.288,4.557) -- (4.368,4.579) -- (4.426,4.637) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.791,4.583) -- (7.77,4.663) -- (7.711,4.722) -- (7.631,4.743) -- (7.551,4.722) -- (7.493,4.663) -- (7.471,4.583) -- (7.493,4.503) -- (7.551,4.445) -- (7.631,4.423) -- (7.711,4.445) -- (7.77,4.503) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.004,1.652) -- (1.983,1.732) -- (1.924,1.79) -- (1.844,1.812) -- (1.764,1.79) -- (1.706,1.732) -- (1.684,1.652) -- (1.706,1.572) -- (1.764,1.513) -- (1.844,1.492) -- (1.924,1.513) -- (1.983,1.572) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.036,1.548) -- (4.014,1.628) -- (3.956,1.686) -- (3.876,1.708) -- (3.796,1.686) -- (3.737,1.628) -- (3.716,1.548) -- (3.737,1.468) -- (3.796,1.409) -- (3.876,1.388) -- (3.956,1.409) -- (4.014,1.468) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.754,10.177) -- (9.709,10.347) -- (9.584,10.471) -- (9.414,10.517) -- (9.244,10.471) -- (9.12,10.347) -- (9.074,10.177) -- (9.12,10.007) -- (9.244,9.883) -- (9.414,9.837) -- (9.584,9.883) -- (9.709,10.007) -- cycle;
\node at (9.414,10.177) {\scalebox{1.363}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.825,9.577) -- (8.779,9.747) -- (8.655,9.872) -- (8.485,9.917) -- (8.315,9.872) -- (8.19,9.747) -- (8.145,9.577) -- (8.19,9.407) -- (8.315,9.283) -- (8.485,9.237) -- (8.655,9.283) -- (8.779,9.407) -- cycle;
\node at (8.485,9.577) {\scalebox{1.363}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.608,9.603) -- (3.562,9.773) -- (3.438,9.897) -- (3.268,9.943) -- (3.098,9.897) -- (2.973,9.773) -- (2.928,9.603) -- (2.973,9.433) -- (3.098,9.308) -- (3.268,9.263) -- (3.438,9.308) -- (3.562,9.433) -- cycle;
\node at (3.268,9.603) {\scalebox{1.363}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.019,7.525) -- (1.973,7.695) -- (1.849,7.82) -- (1.679,7.865) -- (1.509,7.82) -- (1.385,7.695) -- (1.339,7.525) -- (1.385,7.355) -- (1.509,7.231) -- (1.679,7.185) -- (1.849,7.231) -- (1.973,7.355) -- cycle;
\node at (1.679,7.525) {\scalebox{1.363}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.616,6.522) -- (9.571,6.692) -- (9.446,6.817) -- (9.276,6.862) -- (9.106,6.817) -- (8.982,6.692) -- (8.936,6.522) -- (8.982,6.352) -- (9.106,6.228) -- (9.276,6.182) -- (9.446,6.228) -- (9.571,6.352) -- cycle;
\node at (9.276,6.522) {\scalebox{1.363}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.032,8.307) -- (6.986,8.477) -- (6.862,8.601) -- (6.692,8.647) -- (6.522,8.601) -- (6.397,8.477) -- (6.352,8.307) -- (6.397,8.137) -- (6.522,8.012) -- (6.692,7.967) -- (6.862,8.012) -- (6.986,8.137) -- cycle;
\node at (6.692,8.307) {\scalebox{1.363}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.185,6.416) -- (5.14,6.586) -- (5.015,6.711) -- (4.845,6.756) -- (4.675,6.711) -- (4.551,6.586) -- (4.505,6.416) -- (4.551,6.246) -- (4.675,6.122) -- (4.845,6.076) -- (5.015,6.122) -- (5.14,6.246) -- cycle;
\node at (4.845,6.416) {\scalebox{1.363}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.098,6.679) -- (7.052,6.849) -- (6.928,6.973) -- (6.758,7.019) -- (6.588,6.973) -- (6.463,6.849) -- (6.418,6.679) -- (6.463,6.509) -- (6.588,6.384) -- (6.758,6.339) -- (6.928,6.384) -- (7.052,6.509) -- cycle;
\node at (6.758,6.679) {\scalebox{1.363}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.16,4.964) -- (3.114,5.134) -- (2.99,5.258) -- (2.82,5.304) -- (2.65,5.258) -- (2.525,5.134) -- (2.48,4.964) -- (2.525,4.794) -- (2.65,4.669) -- (2.82,4.624) -- (2.99,4.669) -- (3.114,4.794) -- cycle;
\node at (2.82,4.964) {\scalebox{1.363}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.378,3.304) -- (2.332,3.474) -- (2.208,3.598) -- (2.038,3.644) -- (1.868,3.598) -- (1.743,3.474) -- (1.698,3.304) -- (1.743,3.134) -- (1.868,3.009) -- (2.038,2.964) -- (2.208,3.009) -- (2.332,3.134) -- cycle;
\node at (2.038,3.304) {\scalebox{1.363}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.431,2.824) -- (3.385,2.994) -- (3.261,3.118) -- (3.091,3.164) -- (2.921,3.118) -- (2.796,2.994) -- (2.751,2.824) -- (2.796,2.654) -- (2.921,2.529) -- (3.091,2.484) -- (3.261,2.529) -- (3.385,2.654) -- cycle;
\node at (3.091,2.824) {\scalebox{1.363}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.221,2.003) -- (3.175,2.173) -- (3.051,2.298) -- (2.881,2.343) -- (2.711,2.298) -- (2.586,2.173) -- (2.541,2.003) -- (2.586,1.833) -- (2.711,1.709) -- (2.881,1.663) -- (3.051,1.709) -- (3.175,1.833) -- cycle;
\node at (2.881,2.003) {\scalebox{1.363}{\small 12}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.039,4.513) -- (1.639,4.513) -- (1.639,5.113) -- (1.039,5.113) -- cycle;
\node at (1.339,4.813) {\scalebox{1.363}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (3.698,5.818) -- (4.298,5.818) -- (4.298,6.418) -- (3.698,6.418) -- cycle;
\node at (3.998,6.118) {\scalebox{1.363}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.414,9.293) -- (9.014,9.293) -- (9.014,9.893) -- (8.414,9.893) -- cycle;
\node at (8.714,9.593) {\scalebox{1.363}{\small c}};
\filldraw[fill=blue!12, draw=black, line width=0.5pt] (10.096,10.774) -- (10.696,10.774) -- (10.696,11.374) -- (10.096,11.374) -- cycle;
\node at (10.396,11.074) {\scalebox{1.363}{\small d}};
\node at (8.152,12.099) {\scalebox{1.363}{\small A}};
\node at (0.91,11.263) {\scalebox{1.363}{\small B}};
\node at (11.323,8.826) {\scalebox{1.363}{\small C}};
\node at (5.014,8.809) {\scalebox{1.363}{\small D}};
\node at (8.711,7.769) {\scalebox{1.363}{\small E}};
\node at (0.765,4.578) {\scalebox{1.363}{\small F}};
\node at (3.989,4.174) {\scalebox{1.363}{\small G}};
\node at (8.127,4.211) {\scalebox{1.363}{\small H}};
\node at (1.485,1.146) {\scalebox{1.363}{\small I}};
\node at (3.711,0.95) {\scalebox{1.363}{\small J}};
\draw[line width=0.5pt] (12.046,10.598) -- (12.046,11.698);
\draw[line width=0.5pt] (11.815,11.313) -- (12.046,11.698) -- (12.277,11.313);
\draw[line width=0.5pt] (11.744,11.148) -- (12.349,11.148);
\node at (12.046,12.083) {\scalebox{1.363}{\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 & ⑦ sentier des Prés & Ⓐ Place du Vieux-Puits & \textbf{a} la poste \\ \hline
② rue des Écoles & ⑧ chemin des Vignes & Ⓑ Place du Marché & \textbf{b} la gendarmerie \\ \hline
③ rue du Pont & ⑨ rue de la Mairie & Ⓒ Place de l'Église & \textbf{c} le gymnase \\ \hline
④ rue du Puits & ⑩ allée des Peupliers & Ⓓ Rond-point de la Gare & \textbf{d} l'école \\ \hline
⑤ rue de l'Église & ⑪ rue des Lilas & Ⓔ Fontaine Notre-Dame &  \\ \hline
⑥ chemin de Ronde & ⑫ rue Basse & Ⓕ Rond-point du Stade &  \\ \hline
 &  & Ⓖ Coin du Lavoir &  \\ \hline
 &  & Ⓗ Place des Tanneurs &  \\ \hline
 &  & Ⓘ Place aux Herbes &  \\ \hline
 &  & Ⓙ Placette des Ormes &  \\ \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=0.6564, line join=round, line cap=round]
\draw[line width=0.5pt] (7.868,11.548) -- (8.039,11.307) -- (8.217,11.071) -- (8.4,10.838) -- (8.588,10.61) -- (8.782,10.387) -- (8.981,10.168) -- (9.185,9.954) -- (9.394,9.744) -- (9.608,9.54) -- (9.827,9.34) -- (10.05,9.146) -- (10.278,8.958) -- (10.51,8.774) -- (10.746,8.597);
\draw[line width=0.5pt] (7.868,11.548) -- (7.93,11.267) -- (7.986,10.984) -- (8.035,10.701) -- (8.078,10.416) -- (8.114,10.13) -- (8.143,9.843) -- (8.166,9.556) -- (8.181,9.268) -- (8.19,8.98) -- (8.192,8.692) -- (8.188,8.404) -- (8.176,8.117) -- (8.158,7.829) -- (8.133,7.543);
\draw[line width=0.5pt] (1.32,10.798) -- (1.557,10.572) -- (1.799,10.352) -- (2.047,10.138) -- (2.299,9.929) -- (2.556,9.727) -- (2.818,9.53) -- (3.085,9.34) -- (3.356,9.156) -- (3.631,8.979) -- (3.91,8.808) -- (4.193,8.643) -- (4.48,8.486) -- (4.771,8.335) -- (5.065,8.191);
\draw[line width=0.5pt] (1.32,10.798) -- (1.359,10.57) -- (1.391,10.341) -- (1.419,10.111) -- (1.441,9.881) -- (1.458,9.65) -- (1.469,9.419) -- (1.474,9.187) -- (1.475,8.956) -- (1.47,8.724) -- (1.459,8.493) -- (1.443,8.262) -- (1.422,8.032) -- (1.395,7.802) -- (1.363,7.573) -- (1.331,7.378) -- (1.303,7.182) -- (1.281,6.985) -- (1.263,6.788) -- (1.25,6.59) -- (1.241,6.392) -- (1.237,6.194) -- (1.237,5.996) -- (1.243,5.798) -- (1.253,5.6) -- (1.267,5.403) -- (1.286,5.206) -- (1.31,5.009) -- (1.339,4.813);
\draw[line width=0.5pt] (10.746,8.597) -- (10.482,8.347) -- (10.223,8.09) -- (9.97,7.827) -- (9.724,7.559) -- (9.484,7.284) -- (9.25,7.004) -- (9.023,6.719) -- (8.803,6.428) -- (8.59,6.132) -- (8.384,5.831) -- (8.184,5.526) -- (7.993,5.216) -- (7.808,4.902) -- (7.631,4.583);
\draw[line width=0.5pt] (5.065,8.191) -- (5.289,8.179) -- (5.513,8.161) -- (5.737,8.138) -- (5.96,8.11) -- (6.183,8.076) -- (6.405,8.037) -- (6.626,7.993) -- (6.845,7.944) -- (7.064,7.89) -- (7.281,7.831) -- (7.497,7.766) -- (7.711,7.697) -- (7.923,7.622) -- (8.133,7.543);
\draw[line width=0.5pt] (5.065,8.191) -- (4.972,7.953) -- (4.884,7.714) -- (4.802,7.472) -- (4.726,7.228) -- (4.656,6.982) -- (4.592,6.735) -- (4.533,6.486) -- (4.48,6.236) -- (4.433,5.985) -- (4.392,5.733) -- (4.357,5.48) -- (4.328,5.226) -- (4.305,4.972) -- (4.288,4.717);
\draw[line width=0.5pt] (5.065,8.191) -- (5.286,7.964) -- (5.502,7.731) -- (5.712,7.493) -- (5.917,7.25) -- (6.116,7.002) -- (6.31,6.75) -- (6.497,6.493) -- (6.678,6.232) -- (6.853,5.967) -- (7.022,5.697) -- (7.184,5.424) -- (7.34,5.147) -- (7.489,4.867) -- (7.631,4.583);
\draw[line width=0.5pt] (1.339,4.813) -- (1.547,4.774) -- (1.756,4.74) -- (1.965,4.711) -- (2.176,4.687) -- (2.386,4.667) -- (2.598,4.653) -- (2.809,4.644) -- (3.021,4.639) -- (3.233,4.64) -- (3.445,4.645) -- (3.656,4.656) -- (3.867,4.671) -- (4.078,4.692) -- (4.288,4.717);
\draw[line width=0.5pt] (1.339,4.813) -- (1.409,4.595) -- (1.474,4.375) -- (1.534,4.153) -- (1.589,3.93) -- (1.639,3.705) -- (1.683,3.48) -- (1.722,3.253) -- (1.755,3.026) -- (1.784,2.798) -- (1.807,2.569) -- (1.824,2.34) -- (1.836,2.111) -- (1.843,1.881) -- (1.844,1.652);
\draw[line width=0.5pt] (1.339,4.813) -- (1.483,4.554) -- (1.633,4.298) -- (1.789,4.046) -- (1.951,3.798) -- (2.119,3.553) -- (2.293,3.312) -- (2.472,3.076) -- (2.657,2.844) -- (2.848,2.616) -- (3.043,2.393) -- (3.244,2.174) -- (3.449,1.96) -- (3.66,1.751) -- (3.876,1.548);
\draw[line width=0.5pt] (1.844,1.652) -- (1.989,1.667) -- (2.135,1.678) -- (2.28,1.686) -- (2.426,1.691) -- (2.572,1.692) -- (2.718,1.689) -- (2.864,1.684) -- (3.01,1.674) -- (3.155,1.662) -- (3.3,1.646) -- (3.445,1.626) -- (3.589,1.604) -- (3.733,1.577) -- (3.876,1.548);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.028,11.548) -- (8.006,11.628) -- (7.948,11.686) -- (7.868,11.708) -- (7.788,11.686) -- (7.729,11.628) -- (7.708,11.548) -- (7.729,11.468) -- (7.788,11.409) -- (7.868,11.388) -- (7.948,11.409) -- (8.006,11.468) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.48,10.798) -- (1.459,10.878) -- (1.4,10.937) -- (1.32,10.958) -- (1.24,10.937) -- (1.182,10.878) -- (1.16,10.798) -- (1.182,10.718) -- (1.24,10.66) -- (1.32,10.638) -- (1.4,10.66) -- (1.459,10.718) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (10.906,8.597) -- (10.885,8.677) -- (10.826,8.735) -- (10.746,8.757) -- (10.666,8.735) -- (10.608,8.677) -- (10.586,8.597) -- (10.608,8.517) -- (10.666,8.458) -- (10.746,8.437) -- (10.826,8.458) -- (10.885,8.517) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.225,8.191) -- (5.203,8.271) -- (5.145,8.33) -- (5.065,8.351) -- (4.985,8.33) -- (4.926,8.271) -- (4.905,8.191) -- (4.926,8.111) -- (4.985,8.053) -- (5.065,8.031) -- (5.145,8.053) -- (5.203,8.111) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.293,7.543) -- (8.272,7.623) -- (8.213,7.681) -- (8.133,7.703) -- (8.053,7.681) -- (7.995,7.623) -- (7.973,7.543) -- (7.995,7.463) -- (8.053,7.404) -- (8.133,7.383) -- (8.213,7.404) -- (8.272,7.463) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.499,4.813) -- (1.477,4.893) -- (1.419,4.952) -- (1.339,4.973) -- (1.259,4.952) -- (1.2,4.893) -- (1.179,4.813) -- (1.2,4.733) -- (1.259,4.675) -- (1.339,4.653) -- (1.419,4.675) -- (1.477,4.733) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.448,4.717) -- (4.426,4.797) -- (4.368,4.856) -- (4.288,4.877) -- (4.208,4.856) -- (4.149,4.797) -- (4.128,4.717) -- (4.149,4.637) -- (4.208,4.579) -- (4.288,4.557) -- (4.368,4.579) -- (4.426,4.637) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.791,4.583) -- (7.77,4.663) -- (7.711,4.722) -- (7.631,4.743) -- (7.551,4.722) -- (7.493,4.663) -- (7.471,4.583) -- (7.493,4.503) -- (7.551,4.445) -- (7.631,4.423) -- (7.711,4.445) -- (7.77,4.503) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.004,1.652) -- (1.983,1.732) -- (1.924,1.79) -- (1.844,1.812) -- (1.764,1.79) -- (1.706,1.732) -- (1.684,1.652) -- (1.706,1.572) -- (1.764,1.513) -- (1.844,1.492) -- (1.924,1.513) -- (1.983,1.572) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.036,1.548) -- (4.014,1.628) -- (3.956,1.686) -- (3.876,1.708) -- (3.796,1.686) -- (3.737,1.628) -- (3.716,1.548) -- (3.737,1.468) -- (3.796,1.409) -- (3.876,1.388) -- (3.956,1.409) -- (4.014,1.468) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.754,10.177) -- (9.709,10.347) -- (9.584,10.471) -- (9.414,10.517) -- (9.244,10.471) -- (9.12,10.347) -- (9.074,10.177) -- (9.12,10.007) -- (9.244,9.883) -- (9.414,9.837) -- (9.584,9.883) -- (9.709,10.007) -- cycle;
\node at (9.414,10.177) {\scalebox{1.523}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.825,9.577) -- (8.779,9.747) -- (8.655,9.872) -- (8.485,9.917) -- (8.315,9.872) -- (8.19,9.747) -- (8.145,9.577) -- (8.19,9.407) -- (8.315,9.283) -- (8.485,9.237) -- (8.655,9.283) -- (8.779,9.407) -- cycle;
\node at (8.485,9.577) {\scalebox{1.523}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.608,9.603) -- (3.562,9.773) -- (3.438,9.897) -- (3.268,9.943) -- (3.098,9.897) -- (2.973,9.773) -- (2.928,9.603) -- (2.973,9.433) -- (3.098,9.308) -- (3.268,9.263) -- (3.438,9.308) -- (3.562,9.433) -- cycle;
\node at (3.268,9.603) {\scalebox{1.523}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.019,7.525) -- (1.973,7.695) -- (1.849,7.82) -- (1.679,7.865) -- (1.509,7.82) -- (1.385,7.695) -- (1.339,7.525) -- (1.385,7.355) -- (1.509,7.231) -- (1.679,7.185) -- (1.849,7.231) -- (1.973,7.355) -- cycle;
\node at (1.679,7.525) {\scalebox{1.523}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.616,6.522) -- (9.571,6.692) -- (9.446,6.817) -- (9.276,6.862) -- (9.106,6.817) -- (8.982,6.692) -- (8.936,6.522) -- (8.982,6.352) -- (9.106,6.228) -- (9.276,6.182) -- (9.446,6.228) -- (9.571,6.352) -- cycle;
\node at (9.276,6.522) {\scalebox{1.523}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.032,8.307) -- (6.986,8.477) -- (6.862,8.601) -- (6.692,8.647) -- (6.522,8.601) -- (6.397,8.477) -- (6.352,8.307) -- (6.397,8.137) -- (6.522,8.012) -- (6.692,7.967) -- (6.862,8.012) -- (6.986,8.137) -- cycle;
\node at (6.692,8.307) {\scalebox{1.523}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.185,6.416) -- (5.14,6.586) -- (5.015,6.711) -- (4.845,6.756) -- (4.675,6.711) -- (4.551,6.586) -- (4.505,6.416) -- (4.551,6.246) -- (4.675,6.122) -- (4.845,6.076) -- (5.015,6.122) -- (5.14,6.246) -- cycle;
\node at (4.845,6.416) {\scalebox{1.523}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.098,6.679) -- (7.052,6.849) -- (6.928,6.973) -- (6.758,7.019) -- (6.588,6.973) -- (6.463,6.849) -- (6.418,6.679) -- (6.463,6.509) -- (6.588,6.384) -- (6.758,6.339) -- (6.928,6.384) -- (7.052,6.509) -- cycle;
\node at (6.758,6.679) {\scalebox{1.523}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.16,4.964) -- (3.114,5.134) -- (2.99,5.258) -- (2.82,5.304) -- (2.65,5.258) -- (2.525,5.134) -- (2.48,4.964) -- (2.525,4.794) -- (2.65,4.669) -- (2.82,4.624) -- (2.99,4.669) -- (3.114,4.794) -- cycle;
\node at (2.82,4.964) {\scalebox{1.523}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.378,3.304) -- (2.332,3.474) -- (2.208,3.598) -- (2.038,3.644) -- (1.868,3.598) -- (1.743,3.474) -- (1.698,3.304) -- (1.743,3.134) -- (1.868,3.009) -- (2.038,2.964) -- (2.208,3.009) -- (2.332,3.134) -- cycle;
\node at (2.038,3.304) {\scalebox{1.523}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.431,2.824) -- (3.385,2.994) -- (3.261,3.118) -- (3.091,3.164) -- (2.921,3.118) -- (2.796,2.994) -- (2.751,2.824) -- (2.796,2.654) -- (2.921,2.529) -- (3.091,2.484) -- (3.261,2.529) -- (3.385,2.654) -- cycle;
\node at (3.091,2.824) {\scalebox{1.523}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.221,2.003) -- (3.175,2.173) -- (3.051,2.298) -- (2.881,2.343) -- (2.711,2.298) -- (2.586,2.173) -- (2.541,2.003) -- (2.586,1.833) -- (2.711,1.709) -- (2.881,1.663) -- (3.051,1.709) -- (3.175,1.833) -- cycle;
\node at (2.881,2.003) {\scalebox{1.523}{\small 12}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.039,4.513) -- (1.639,4.513) -- (1.639,5.113) -- (1.039,5.113) -- cycle;
\node at (1.339,4.813) {\scalebox{1.523}{\small a}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (3.698,5.818) -- (4.298,5.818) -- (4.298,6.418) -- (3.698,6.418) -- cycle;
\node at (3.998,6.118) {\scalebox{1.523}{\small b}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (8.414,9.293) -- (9.014,9.293) -- (9.014,9.893) -- (8.414,9.893) -- cycle;
\node at (8.714,9.593) {\scalebox{1.523}{\small c}};
\filldraw[fill=blue!12, draw=black, line width=0.5pt] (10.096,10.774) -- (10.696,10.774) -- (10.696,11.374) -- (10.096,11.374) -- cycle;
\node at (10.396,11.074) {\scalebox{1.523}{\small d}};
\node at (8.152,12.099) {\scalebox{1.523}{\small A}};
\node at (0.91,11.263) {\scalebox{1.523}{\small B}};
\node at (11.323,8.826) {\scalebox{1.523}{\small C}};
\node at (5.014,8.809) {\scalebox{1.523}{\small D}};
\node at (8.711,7.769) {\scalebox{1.523}{\small E}};
\node at (0.765,4.578) {\scalebox{1.523}{\small F}};
\node at (3.989,4.174) {\scalebox{1.523}{\small G}};
\node at (8.127,4.211) {\scalebox{1.523}{\small H}};
\node at (1.485,1.146) {\scalebox{1.523}{\small I}};
\node at (3.711,0.95) {\scalebox{1.523}{\small J}};
\draw[line width=0.5pt] (12.046,10.598) -- (12.046,11.698);
\draw[line width=0.5pt] (11.815,11.313) -- (12.046,11.698) -- (12.277,11.313);
\draw[line width=0.5pt] (11.744,11.148) -- (12.349,11.148);
\node at (12.046,12.083) {\scalebox{1.523}{\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 & 4 & 2 & 4 & 2 & 2 & 2 & 2 \\ \hline
Parité & pair & pair & pair & pair & pair & pair & pair & pair & pair & pair \\ \hline
\end{tabular}
\end{center}
\noindent\hspace{1.2em}Vérification : la somme de ces nombres vaut 24 = 2 \(\times\) 12, car chaque rue est comptée à ses \textbf{deux} extrémités.\par\addvspace{0.3em}
\noindent{\color{red!80!black}\bfseries \textbf{Aucun} carrefour n'a un nombre impair de rues : ils sont tous pairs.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Quand le facteur \textbf{traverse} un carrefour, il y arrive par une rue et en repart par une autre : il consomme les rues \textbf{deux par deux}. Un carrefour traversé 3 fois voit donc 6 de ses rues utilisées — toujours un nombre \textbf{pair}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Un carrefour où aboutit un nombre \textbf{impair} de rues ne peut donc pas être seulement traversé : il doit être le \textbf{départ} ou l'\textbf{arrivée} de la tournée. Là, une rue reste « non appariée ».}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Ici la poste est \textbf{sur un carrefour}, en \textbf{F}, et le facteur n'a pas à y revenir. Ce carrefour est donc le \textbf{départ} — il a droit d'être impair. Mais il n'y a qu'\textbf{une seule arrivée} : tous les autres carrefours doivent être pairs, sauf un au plus.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui.} Aucun carrefour n'est impair : chacun peut être traversé de part en part. Le facteur revient même à la poste sans l'avoir cherché.}\par\addvspace{0.4em}
\newpage
\noindent Voici une tournée possible — ce n'est pas la seule :\par\addvspace{0.35em}
\begin{enumerate}
  \item Il part de Rond-point du Stade \textbf{(F)}, prend la \textbf{rue du Puits ④} vers le nord et atteint Place du Marché \textbf{(B)}.
  \item Il tourne à droite dans la \textbf{rue du Pont ③}, vers le sud-est, jusqu'à Rond-point de la Gare \textbf{(D)}.
  \item Il continue tout droit dans le \textbf{chemin de Ronde ⑥}, vers l'est, jusqu'à Fontaine Notre-Dame \textbf{(E)}.
  \item Il tourne à gauche dans la \textbf{rue des Écoles ②}, vers le nord, jusqu'à Place du Vieux-Puits \textbf{(A)}.
  \item Il prend dans la \textbf{rue Neuve ①}, vers le sud-est, jusqu'à Place de l'Église \textbf{(C)}.
  \item Il tourne à droite dans la \textbf{rue de l'Église ⑤}, vers le sud-ouest, jusqu'à Place des Tanneurs \textbf{(H)}.
  \item Il tourne à droite dans le \textbf{chemin des Vignes ⑧}, vers le nord-ouest, jusqu'à Rond-point de la Gare \textbf{(D)}.
  \item Il tourne à gauche dans le \textbf{sentier des Prés ⑦}, vers le sud, jusqu'à Coin du Lavoir \textbf{(G)}.
  \item Il tourne à droite dans la \textbf{rue de la Mairie ⑨}, vers l'ouest, jusqu'à Rond-point du Stade \textbf{(F)}.
  \item Il tourne à gauche dans l'\textbf{allée des Peupliers ⑩}, vers le sud, jusqu'à Place aux Herbes \textbf{(I)}.
  \item Il tourne à gauche dans la \textbf{rue Basse ⑫}, vers l'est, jusqu'à Placette des Ormes \textbf{(J)}.
  \item Il tourne à gauche dans la \textbf{rue des Lilas ⑪}, vers le nord-ouest, jusqu'à Rond-point du Stade \textbf{(F)}.
\end{enumerate}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries La question ne se pose pas : la tournée est possible au départ de la poste, et elle le resterait où qu'on installe celle-ci — aucun carrefour n'est impair.}\par\addvspace{0.4em}
\end{document}
