\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}{Le car de ramassage}
\newcommand{\soustitre}{Probl\`eme --- Fiche n\textdegree 373}

\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/car_de_ramassage.php?x=373}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0373}
\end{minipage}%
}
}

\begin{document}

\begin{center}
    \small Un car de ramassage doit s'arrêter à chaque carrefour d'un quartier, une fois et une seule. Selon le plan c'est possible ou non — et un simple coloriage permet souvent de le dire d'avance, sans essayer.
\end{center}
\vspace{0.5em}
\sloppy
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6038, line join=round, line cap=round]
\draw[line width=0.5pt] (1.631,10.844) -- (1.631,10.614) -- (1.637,10.385) -- (1.648,10.155) -- (1.665,9.926) -- (1.687,9.697) -- (1.715,9.469) -- (1.748,9.241) -- (1.786,9.014) -- (1.829,8.789) -- (1.878,8.564) -- (1.932,8.34) -- (1.991,8.118) -- (2.056,7.898) -- (2.126,7.679);
\draw[line width=0.5pt] (7.508,8.038) -- (7.539,7.837) -- (7.566,7.635) -- (7.589,7.432) -- (7.606,7.228) -- (7.619,7.025) -- (7.627,6.821) -- (7.63,6.616) -- (7.628,6.412) -- (7.621,6.208) -- (7.61,6.004) -- (7.594,5.801) -- (7.573,5.598) -- (7.547,5.395) -- (7.517,5.194);
\draw[line width=0.5pt] (4.986,7.981) -- (5.166,7.957) -- (5.345,7.938) -- (5.526,7.923) -- (5.706,7.912) -- (5.887,7.906) -- (6.068,7.904) -- (6.249,7.906) -- (6.43,7.912) -- (6.611,7.922) -- (6.791,7.937) -- (6.971,7.956) -- (7.151,7.979) -- (7.33,8.007) -- (7.508,8.038);
\draw[line width=0.5pt] (4.986,7.981) -- (5.026,7.76) -- (5.06,7.538) -- (5.09,7.316) -- (5.114,7.092) -- (5.132,6.869) -- (5.146,6.644) -- (5.154,6.42) -- (5.157,6.195) -- (5.154,5.971) -- (5.147,5.746) -- (5.134,5.522) -- (5.116,5.298) -- (5.092,5.075) -- (5.063,4.853);
\draw[line width=0.5pt] (2.126,7.679) -- (2.332,7.669) -- (2.538,7.664) -- (2.744,7.664) -- (2.951,7.669) -- (3.157,7.678) -- (3.363,7.693) -- (3.568,7.712) -- (3.774,7.736) -- (3.978,7.765) -- (4.182,7.799) -- (4.384,7.837) -- (4.586,7.88) -- (4.787,7.928) -- (4.986,7.981);
\draw[line width=0.5pt] (2.126,7.679) -- (2.105,7.464) -- (2.078,7.25) -- (2.047,7.037) -- (2.011,6.824) -- (1.97,6.612) -- (1.924,6.401) -- (1.872,6.192) -- (1.816,5.983) -- (1.755,5.776) -- (1.69,5.571) -- (1.619,5.367) -- (1.544,5.165) -- (1.464,4.964) -- (1.379,4.766);
\draw[line width=0.5pt] (7.517,5.194) -- (7.76,5.155) -- (8.004,5.123) -- (8.249,5.096) -- (8.495,5.075) -- (8.741,5.06) -- (8.987,5.05) -- (9.233,5.047) -- (9.48,5.049) -- (9.726,5.057) -- (9.973,5.07) -- (10.218,5.09) -- (10.463,5.115) -- (10.708,5.146) -- (10.951,5.183);
\draw[line width=0.5pt] (7.517,5.194) -- (7.551,4.945) -- (7.58,4.695) -- (7.603,4.444) -- (7.62,4.193) -- (7.631,3.942) -- (7.637,3.691) -- (7.636,3.439) -- (7.629,3.187) -- (7.617,2.936) -- (7.598,2.685) -- (7.574,2.435) -- (7.544,2.185) -- (7.508,1.936) -- (7.466,1.688);
\draw[line width=0.5pt] (10.951,5.183) -- (10.944,4.934) -- (10.942,4.686) -- (10.947,4.437) -- (10.957,4.189) -- (10.973,3.941) -- (10.995,3.693) -- (11.023,3.446) -- (11.057,3.199) -- (11.096,2.954) -- (11.141,2.709) -- (11.192,2.466) -- (11.249,2.224) -- (11.311,1.983) -- (11.379,1.744);
\draw[line width=0.5pt] (5.063,4.853) -- (5.233,4.904) -- (5.405,4.951) -- (5.577,4.994) -- (5.751,5.032) -- (5.925,5.067) -- (6.1,5.098) -- (6.276,5.124) -- (6.452,5.147) -- (6.629,5.165) -- (6.806,5.179) -- (6.984,5.189) -- (7.161,5.195) -- (7.339,5.196) -- (7.517,5.194);
\draw[line width=0.5pt] (5.063,4.853) -- (5.0,4.622) -- (4.943,4.39) -- (4.891,4.157) -- (4.845,3.922) -- (4.804,3.687) -- (4.769,3.45) -- (4.739,3.213) -- (4.715,2.975) -- (4.696,2.737) -- (4.683,2.498) -- (4.676,2.259) -- (4.674,2.02) -- (4.678,1.781) -- (4.687,1.542);
\draw[line width=0.5pt] (1.379,4.766) -- (1.639,4.813) -- (1.9,4.853) -- (2.162,4.887) -- (2.425,4.915) -- (2.689,4.937) -- (2.953,4.952) -- (3.217,4.961) -- (3.482,4.964) -- (3.747,4.961) -- (4.011,4.952) -- (4.275,4.937) -- (4.538,4.915) -- (4.801,4.887) -- (5.063,4.853);
\draw[line width=0.5pt] (7.466,1.688) -- (7.744,1.649) -- (8.023,1.617) -- (8.302,1.591) -- (8.582,1.572) -- (8.863,1.56) -- (9.144,1.554) -- (9.425,1.555) -- (9.705,1.562) -- (9.986,1.576) -- (10.266,1.597) -- (10.546,1.624) -- (10.824,1.657) -- (11.102,1.698) -- (11.379,1.744);
\draw[line width=0.5pt] (4.687,1.542) -- (4.883,1.583) -- (5.079,1.619) -- (5.276,1.651) -- (5.474,1.678) -- (5.672,1.7) -- (5.871,1.717) -- (6.071,1.73) -- (6.27,1.738) -- (6.47,1.741) -- (6.67,1.74) -- (6.869,1.734) -- (7.068,1.724) -- (7.267,1.708) -- (7.466,1.688);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.791,10.844) -- (1.769,10.924) -- (1.711,10.983) -- (1.631,11.004) -- (1.551,10.983) -- (1.492,10.924) -- (1.471,10.844) -- (1.492,10.764) -- (1.551,10.706) -- (1.631,10.684) -- (1.711,10.706) -- (1.769,10.764) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.668,8.038) -- (7.646,8.118) -- (7.588,8.177) -- (7.508,8.198) -- (7.428,8.177) -- (7.369,8.118) -- (7.348,8.038) -- (7.369,7.958) -- (7.428,7.9) -- (7.508,7.878) -- (7.588,7.9) -- (7.646,7.958) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.146,7.981) -- (5.125,8.061) -- (5.066,8.119) -- (4.986,8.141) -- (4.906,8.119) -- (4.848,8.061) -- (4.826,7.981) -- (4.848,7.901) -- (4.906,7.842) -- (4.986,7.821) -- (5.066,7.842) -- (5.125,7.901) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.286,7.679) -- (2.264,7.759) -- (2.206,7.818) -- (2.126,7.839) -- (2.046,7.818) -- (1.987,7.759) -- (1.966,7.679) -- (1.987,7.599) -- (2.046,7.54) -- (2.126,7.519) -- (2.206,7.54) -- (2.264,7.599) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.677,5.194) -- (7.655,5.274) -- (7.597,5.332) -- (7.517,5.354) -- (7.437,5.332) -- (7.378,5.274) -- (7.357,5.194) -- (7.378,5.114) -- (7.437,5.055) -- (7.517,5.034) -- (7.597,5.055) -- (7.655,5.114) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.111,5.183) -- (11.09,5.263) -- (11.031,5.321) -- (10.951,5.343) -- (10.871,5.321) -- (10.813,5.263) -- (10.791,5.183) -- (10.813,5.103) -- (10.871,5.044) -- (10.951,5.023) -- (11.031,5.044) -- (11.09,5.103) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.223,4.853) -- (5.202,4.933) -- (5.143,4.991) -- (5.063,5.013) -- (4.983,4.991) -- (4.925,4.933) -- (4.903,4.853) -- (4.925,4.773) -- (4.983,4.714) -- (5.063,4.693) -- (5.143,4.714) -- (5.202,4.773) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.539,4.766) -- (1.517,4.846) -- (1.459,4.905) -- (1.379,4.926) -- (1.299,4.905) -- (1.24,4.846) -- (1.219,4.766) -- (1.24,4.686) -- (1.299,4.627) -- (1.379,4.606) -- (1.459,4.627) -- (1.517,4.686) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.539,1.744) -- (11.517,1.824) -- (11.459,1.883) -- (11.379,1.904) -- (11.299,1.883) -- (11.24,1.824) -- (11.219,1.744) -- (11.24,1.664) -- (11.299,1.606) -- (11.379,1.584) -- (11.459,1.606) -- (11.517,1.664) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (7.626,1.688) -- (7.604,1.768) -- (7.546,1.827) -- (7.466,1.848) -- (7.386,1.827) -- (7.327,1.768) -- (7.306,1.688) -- (7.327,1.608) -- (7.386,1.55) -- (7.466,1.528) -- (7.546,1.55) -- (7.604,1.608) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.847,1.542) -- (4.826,1.622) -- (4.767,1.681) -- (4.687,1.702) -- (4.607,1.681) -- (4.549,1.622) -- (4.527,1.542) -- (4.549,1.462) -- (4.607,1.404) -- (4.687,1.382) -- (4.767,1.404) -- (4.826,1.462) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.137,8.908) -- (2.258,8.927) -- (2.368,8.983) -- (2.455,9.071) -- (2.511,9.18) -- (2.531,9.302) -- (2.511,9.424) -- (2.455,9.534) -- (2.368,9.621) -- (2.258,9.677) -- (2.137,9.696) -- (2.137,9.696) -- (2.015,9.677) -- (1.905,9.621) -- (1.818,9.534) -- (1.762,9.424) -- (1.743,9.302) -- (1.762,9.18) -- (1.818,9.071) -- (1.905,8.983) -- (2.015,8.927) -- (2.137,8.908) -- cycle;
\node at (2.137,9.302) {\scalebox{1.656}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.023,6.224) -- (8.145,6.243) -- (8.255,6.299) -- (8.342,6.386) -- (8.398,6.496) -- (8.417,6.618) -- (8.398,6.739) -- (8.342,6.849) -- (8.255,6.936) -- (8.145,6.992) -- (8.023,7.012) -- (8.023,7.012) -- (7.902,6.992) -- (7.792,6.936) -- (7.705,6.849) -- (7.649,6.739) -- (7.63,6.618) -- (7.649,6.496) -- (7.705,6.386) -- (7.792,6.299) -- (7.902,6.243) -- (8.023,6.224) -- cycle;
\node at (8.023,6.618) {\scalebox{1.656}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.24,7.906) -- (6.362,7.925) -- (6.472,7.981) -- (6.559,8.068) -- (6.615,8.178) -- (6.634,8.299) -- (6.615,8.421) -- (6.559,8.531) -- (6.472,8.618) -- (6.362,8.674) -- (6.24,8.693) -- (6.24,8.693) -- (6.119,8.674) -- (6.009,8.618) -- (5.922,8.531) -- (5.866,8.421) -- (5.847,8.299) -- (5.866,8.178) -- (5.922,8.068) -- (6.009,7.981) -- (6.119,7.925) -- (6.24,7.906) -- cycle;
\node at (6.24,8.299) {\scalebox{1.656}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.548,6.036) -- (5.669,6.055) -- (5.779,6.111) -- (5.866,6.198) -- (5.922,6.308) -- (5.942,6.43) -- (5.922,6.551) -- (5.866,6.661) -- (5.779,6.748) -- (5.669,6.804) -- (5.548,6.824) -- (5.548,6.824) -- (5.426,6.804) -- (5.316,6.748) -- (5.229,6.661) -- (5.173,6.551) -- (5.154,6.43) -- (5.173,6.308) -- (5.229,6.198) -- (5.316,6.111) -- (5.426,6.055) -- (5.548,6.036) -- cycle;
\node at (5.548,6.43) {\scalebox{1.656}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.527,7.71) -- (3.649,7.729) -- (3.759,7.785) -- (3.846,7.872) -- (3.902,7.982) -- (3.921,8.104) -- (3.902,8.225) -- (3.846,8.335) -- (3.759,8.422) -- (3.649,8.478) -- (3.527,8.498) -- (3.527,8.498) -- (3.405,8.478) -- (3.296,8.422) -- (3.208,8.335) -- (3.152,8.225) -- (3.133,8.104) -- (3.152,7.982) -- (3.208,7.872) -- (3.296,7.785) -- (3.405,7.729) -- (3.527,7.71) -- cycle;
\node at (3.527,8.104) {\scalebox{1.656}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.254,5.7) -- (2.376,5.719) -- (2.485,5.775) -- (2.573,5.862) -- (2.629,5.972) -- (2.648,6.094) -- (2.629,6.216) -- (2.573,6.325) -- (2.485,6.413) -- (2.376,6.469) -- (2.254,6.488) -- (2.254,6.488) -- (2.132,6.469) -- (2.022,6.413) -- (1.935,6.325) -- (1.879,6.216) -- (1.86,6.094) -- (1.879,5.972) -- (1.935,5.862) -- (2.022,5.775) -- (2.132,5.719) -- (2.254,5.7) -- cycle;
\node at (2.254,6.094) {\scalebox{1.656}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.235,5.047) -- (9.356,5.066) -- (9.466,5.122) -- (9.553,5.209) -- (9.609,5.319) -- (9.629,5.44) -- (9.609,5.562) -- (9.553,5.672) -- (9.466,5.759) -- (9.356,5.815) -- (9.235,5.834) -- (9.235,5.834) -- (9.113,5.815) -- (9.003,5.759) -- (8.916,5.672) -- (8.86,5.562) -- (8.841,5.44) -- (8.86,5.319) -- (8.916,5.209) -- (9.003,5.122) -- (9.113,5.066) -- (9.235,5.047) -- cycle;
\node at (9.235,5.44) {\scalebox{1.656}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.03,3.039) -- (8.151,3.059) -- (8.261,3.114) -- (8.348,3.202) -- (8.404,3.311) -- (8.424,3.433) -- (8.404,3.555) -- (8.348,3.665) -- (8.261,3.752) -- (8.151,3.808) -- (8.03,3.827) -- (8.03,3.827) -- (7.908,3.808) -- (7.798,3.752) -- (7.711,3.665) -- (7.655,3.555) -- (7.636,3.433) -- (7.655,3.311) -- (7.711,3.202) -- (7.798,3.114) -- (7.908,3.059) -- (8.03,3.039) -- cycle;
\node at (8.03,3.433) {\scalebox{1.656}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.414,3.101) -- (11.536,3.12) -- (11.646,3.176) -- (11.733,3.263) -- (11.789,3.373) -- (11.808,3.495) -- (11.789,3.616) -- (11.733,3.726) -- (11.646,3.813) -- (11.536,3.869) -- (11.414,3.888) -- (11.414,3.888) -- (11.292,3.869) -- (11.182,3.813) -- (11.095,3.726) -- (11.039,3.616) -- (11.02,3.495) -- (11.039,3.373) -- (11.095,3.263) -- (11.182,3.176) -- (11.292,3.12) -- (11.414,3.101) -- cycle;
\node at (11.414,3.495) {\scalebox{1.656}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.407,5.121) -- (6.529,5.14) -- (6.639,5.196) -- (6.726,5.283) -- (6.782,5.393) -- (6.801,5.515) -- (6.782,5.636) -- (6.726,5.746) -- (6.639,5.833) -- (6.529,5.889) -- (6.407,5.908) -- (6.036,5.908) -- (5.915,5.889) -- (5.805,5.833) -- (5.718,5.746) -- (5.662,5.636) -- (5.642,5.515) -- (5.662,5.393) -- (5.718,5.283) -- (5.805,5.196) -- (5.915,5.14) -- (6.036,5.121) -- cycle;
\node at (6.222,5.515) {\scalebox{1.656}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.315,2.775) -- (5.437,2.794) -- (5.547,2.85) -- (5.634,2.937) -- (5.69,3.047) -- (5.709,3.169) -- (5.69,3.29) -- (5.634,3.4) -- (5.547,3.487) -- (5.437,3.543) -- (5.315,3.563) -- (4.945,3.563) -- (4.823,3.543) -- (4.713,3.487) -- (4.626,3.4) -- (4.57,3.29) -- (4.551,3.169) -- (4.57,3.047) -- (4.626,2.937) -- (4.713,2.85) -- (4.823,2.794) -- (4.945,2.775) -- cycle;
\node at (5.13,3.169) {\scalebox{1.656}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.394,4.961) -- (3.515,4.981) -- (3.625,5.036) -- (3.712,5.124) -- (3.768,5.233) -- (3.787,5.355) -- (3.768,5.477) -- (3.712,5.587) -- (3.625,5.674) -- (3.515,5.73) -- (3.394,5.749) -- (3.023,5.749) -- (2.901,5.73) -- (2.791,5.674) -- (2.704,5.587) -- (2.648,5.477) -- (2.629,5.355) -- (2.648,5.233) -- (2.704,5.124) -- (2.791,5.036) -- (2.901,4.981) -- (3.023,4.961) -- cycle;
\node at (3.208,5.355) {\scalebox{1.656}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.604,1.555) -- (9.726,1.574) -- (9.836,1.63) -- (9.923,1.717) -- (9.979,1.827) -- (9.998,1.949) -- (9.979,2.07) -- (9.923,2.18) -- (9.836,2.267) -- (9.726,2.323) -- (9.604,2.343) -- (9.234,2.343) -- (9.112,2.323) -- (9.002,2.267) -- (8.915,2.18) -- (8.859,2.07) -- (8.84,1.949) -- (8.859,1.827) -- (8.915,1.717) -- (9.002,1.63) -- (9.112,1.574) -- (9.234,1.555) -- cycle;
\node at (9.419,1.949) {\scalebox{1.656}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.235,1.729) -- (6.357,1.749) -- (6.467,1.805) -- (6.554,1.892) -- (6.61,2.002) -- (6.629,2.123) -- (6.61,2.245) -- (6.554,2.355) -- (6.467,2.442) -- (6.357,2.498) -- (6.235,2.517) -- (5.865,2.517) -- (5.743,2.498) -- (5.633,2.442) -- (5.546,2.355) -- (5.49,2.245) -- (5.471,2.123) -- (5.49,2.002) -- (5.546,1.892) -- (5.633,1.805) -- (5.743,1.749) -- (5.865,1.729) -- cycle;
\node at (6.05,2.123) {\scalebox{1.656}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.079,4.466) -- (1.679,4.466) -- (1.679,5.066) -- (1.079,5.066) -- cycle;
\node at (1.379,4.766) {\scalebox{1.656}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (0.651,5.225) -- (1.251,5.225) -- (1.251,5.825) -- (0.651,5.825) -- cycle;
\node at (0.951,5.525) {\scalebox{1.656}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.162,6.539) -- (1.762,6.539) -- (1.762,7.139) -- (1.162,7.139) -- cycle;
\node at (1.462,6.839) {\scalebox{1.656}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.226,0.851) -- (5.826,0.851) -- (5.826,1.451) -- (5.226,1.451) -- cycle;
\node at (5.526,1.151) {\scalebox{1.656}{\small d}};
\node at (1.249,11.332) {\scalebox{1.656}{\small A}};
\node at (7.834,8.566) {\scalebox{1.656}{\small B}};
\node at (4.783,8.566) {\scalebox{1.656}{\small C}};
\node at (1.595,8.0) {\scalebox{1.656}{\small D}};
\node at (8.131,5.112) {\scalebox{1.656}{\small E}};
\node at (11.571,5.155) {\scalebox{1.656}{\small F}};
\node at (4.551,4.503) {\scalebox{1.656}{\small G}};
\node at (0.765,4.678) {\scalebox{1.656}{\small H}};
\node at (11.895,1.4) {\scalebox{1.656}{\small I}};
\node at (7.709,1.118) {\scalebox{1.656}{\small J}};
\node at (4.504,0.95) {\scalebox{1.656}{\small K}};
\draw[line width=0.5pt] (12.679,9.894) -- (12.679,10.994);
\draw[line width=0.5pt] (12.448,10.609) -- (12.679,10.994) -- (12.91,10.609);
\draw[line width=0.5pt] (12.376,10.444) -- (12.981,10.444);
\node at (12.679,11.379) {\scalebox{1.656}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour H, et doit s'arrêter à chaque carrefour exactement une fois.} Il emprunte les rues qu'il veut et n'a pas à toutes les parcourir.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|c|}
\hline
\cellcolor{black!32}\bfseries Les rues & ⑧ venelle du Chat & Ⓐ Fontaine Notre-Dame & Ⓘ Place de la Fontaine \\ \hline
① impasse des Cerisiers & ⑨ rue des Lilas & Ⓑ Place du Marché & Ⓙ Carrefour du Lavoir \\ \hline
② rue Verte & ⑩ chemin Creux & Ⓒ Carrefour de la Mairie & Ⓚ Place aux Herbes \\ \hline
③ sentier des Prés & ⑪ rue du Four & Ⓓ Carrefour des Tilleuls & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ avenue de la Gare & ⑫ passage du Marché & Ⓔ Carrefour Saint-Roch & \textbf{a} l'école \\ \hline
⑤ rue de la Poste & ⑬ rue des Jardins & Ⓕ Coin du Lavoir & \textbf{b} la pharmacie \\ \hline
⑥ boulevard du Parc & ⑭ rue Basse & Ⓖ Rond-point du Stade & \textbf{c} le lavoir \\ \hline
⑦ rue de la Mairie & \cellcolor{black!32}\bfseries Les carrefours & Ⓗ Place de l'Église & \textbf{d} le stade \\ \hline
\end{tabular}
\end{center}
\noindent{\small\itshape Légende du plan}\par\addvspace{0.4em}
\begin{enumerate}
  \item Colorier les carrefours du plan avec \textbf{deux couleurs}, de sorte que deux carrefours \textbf{reliés par une rue} ne soient jamais de la même couleur. Combien y en a-t-il de chaque couleur ?
  \item Le car roule d'un carrefour au suivant. Que peut-on dire de la \textbf{couleur} des arrêts successifs ? Qu'en déduire sur les couleurs des arrêts d'une tournée qui les dessert tous ?
  \item \textbf{Le car peut-il desservir tous les carrefours, une fois chacun, et revenir à l'école ?} Si oui, décrire un trajet ; sinon, démontrer que c'est impossible.
  \item \textbf{Et s'il n'est pas obligé de revenir à l'école ?} Même consigne.
\end{enumerate}
\newpage
\section*{\textcolor{red}{Correction}}
\needspace{5\baselineskip}
\subsection*{\textcolor{red}{Le car de ramassage — Correction}}
\needspace{3\baselineskip}
\begin{center}
\begin{tikzpicture}[scale=0.6185, line join=round, line cap=round]
\draw[line width=0.5pt] (1.631,10.844) -- (1.631,10.614) -- (1.637,10.385) -- (1.648,10.155) -- (1.665,9.926) -- (1.687,9.697) -- (1.715,9.469) -- (1.748,9.241) -- (1.786,9.014) -- (1.829,8.789) -- (1.878,8.564) -- (1.932,8.34) -- (1.991,8.118) -- (2.056,7.898) -- (2.126,7.679);
\draw[line width=0.5pt] (7.508,8.038) -- (7.539,7.837) -- (7.566,7.635) -- (7.589,7.432) -- (7.606,7.228) -- (7.619,7.025) -- (7.627,6.821) -- (7.63,6.616) -- (7.628,6.412) -- (7.621,6.208) -- (7.61,6.004) -- (7.594,5.801) -- (7.573,5.598) -- (7.547,5.395) -- (7.517,5.194);
\draw[line width=0.5pt] (4.986,7.981) -- (5.166,7.957) -- (5.345,7.938) -- (5.526,7.923) -- (5.706,7.912) -- (5.887,7.906) -- (6.068,7.904) -- (6.249,7.906) -- (6.43,7.912) -- (6.611,7.922) -- (6.791,7.937) -- (6.971,7.956) -- (7.151,7.979) -- (7.33,8.007) -- (7.508,8.038);
\draw[line width=0.5pt] (4.986,7.981) -- (5.026,7.76) -- (5.06,7.538) -- (5.09,7.316) -- (5.114,7.092) -- (5.132,6.869) -- (5.146,6.644) -- (5.154,6.42) -- (5.157,6.195) -- (5.154,5.971) -- (5.147,5.746) -- (5.134,5.522) -- (5.116,5.298) -- (5.092,5.075) -- (5.063,4.853);
\draw[line width=0.5pt] (2.126,7.679) -- (2.332,7.669) -- (2.538,7.664) -- (2.744,7.664) -- (2.951,7.669) -- (3.157,7.678) -- (3.363,7.693) -- (3.568,7.712) -- (3.774,7.736) -- (3.978,7.765) -- (4.182,7.799) -- (4.384,7.837) -- (4.586,7.88) -- (4.787,7.928) -- (4.986,7.981);
\draw[line width=0.5pt] (2.126,7.679) -- (2.105,7.464) -- (2.078,7.25) -- (2.047,7.037) -- (2.011,6.824) -- (1.97,6.612) -- (1.924,6.401) -- (1.872,6.192) -- (1.816,5.983) -- (1.755,5.776) -- (1.69,5.571) -- (1.619,5.367) -- (1.544,5.165) -- (1.464,4.964) -- (1.379,4.766);
\draw[line width=0.5pt] (7.517,5.194) -- (7.76,5.155) -- (8.004,5.123) -- (8.249,5.096) -- (8.495,5.075) -- (8.741,5.06) -- (8.987,5.05) -- (9.233,5.047) -- (9.48,5.049) -- (9.726,5.057) -- (9.973,5.07) -- (10.218,5.09) -- (10.463,5.115) -- (10.708,5.146) -- (10.951,5.183);
\draw[line width=0.5pt] (7.517,5.194) -- (7.551,4.945) -- (7.58,4.695) -- (7.603,4.444) -- (7.62,4.193) -- (7.631,3.942) -- (7.637,3.691) -- (7.636,3.439) -- (7.629,3.187) -- (7.617,2.936) -- (7.598,2.685) -- (7.574,2.435) -- (7.544,2.185) -- (7.508,1.936) -- (7.466,1.688);
\draw[line width=0.5pt] (10.951,5.183) -- (10.944,4.934) -- (10.942,4.686) -- (10.947,4.437) -- (10.957,4.189) -- (10.973,3.941) -- (10.995,3.693) -- (11.023,3.446) -- (11.057,3.199) -- (11.096,2.954) -- (11.141,2.709) -- (11.192,2.466) -- (11.249,2.224) -- (11.311,1.983) -- (11.379,1.744);
\draw[line width=0.5pt] (5.063,4.853) -- (5.233,4.904) -- (5.405,4.951) -- (5.577,4.994) -- (5.751,5.032) -- (5.925,5.067) -- (6.1,5.098) -- (6.276,5.124) -- (6.452,5.147) -- (6.629,5.165) -- (6.806,5.179) -- (6.984,5.189) -- (7.161,5.195) -- (7.339,5.196) -- (7.517,5.194);
\draw[line width=0.5pt] (5.063,4.853) -- (5.0,4.622) -- (4.943,4.39) -- (4.891,4.157) -- (4.845,3.922) -- (4.804,3.687) -- (4.769,3.45) -- (4.739,3.213) -- (4.715,2.975) -- (4.696,2.737) -- (4.683,2.498) -- (4.676,2.259) -- (4.674,2.02) -- (4.678,1.781) -- (4.687,1.542);
\draw[line width=0.5pt] (1.379,4.766) -- (1.639,4.813) -- (1.9,4.853) -- (2.162,4.887) -- (2.425,4.915) -- (2.689,4.937) -- (2.953,4.952) -- (3.217,4.961) -- (3.482,4.964) -- (3.747,4.961) -- (4.011,4.952) -- (4.275,4.937) -- (4.538,4.915) -- (4.801,4.887) -- (5.063,4.853);
\draw[line width=0.5pt] (7.466,1.688) -- (7.744,1.649) -- (8.023,1.617) -- (8.302,1.591) -- (8.582,1.572) -- (8.863,1.56) -- (9.144,1.554) -- (9.425,1.555) -- (9.705,1.562) -- (9.986,1.576) -- (10.266,1.597) -- (10.546,1.624) -- (10.824,1.657) -- (11.102,1.698) -- (11.379,1.744);
\draw[line width=0.5pt] (4.687,1.542) -- (4.883,1.583) -- (5.079,1.619) -- (5.276,1.651) -- (5.474,1.678) -- (5.672,1.7) -- (5.871,1.717) -- (6.071,1.73) -- (6.27,1.738) -- (6.47,1.741) -- (6.67,1.74) -- (6.869,1.734) -- (7.068,1.724) -- (7.267,1.708) -- (7.466,1.688);
\filldraw[fill=white, draw=black, line width=0.5pt] (2.004,10.844) -- (1.975,10.987) -- (1.894,11.108) -- (1.773,11.189) -- (1.631,11.217) -- (1.488,11.189) -- (1.367,11.108) -- (1.286,10.987) -- (1.257,10.844) -- (1.286,10.701) -- (1.367,10.58) -- (1.488,10.499) -- (1.631,10.471) -- (1.773,10.499) -- (1.894,10.58) -- (1.975,10.701) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.881,8.038) -- (7.853,8.181) -- (7.772,8.302) -- (7.651,8.383) -- (7.508,8.412) -- (7.365,8.383) -- (7.244,8.302) -- (7.163,8.181) -- (7.135,8.038) -- (7.163,7.896) -- (7.244,7.775) -- (7.365,7.694) -- (7.508,7.665) -- (7.651,7.694) -- (7.772,7.775) -- (7.853,7.896) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.36,7.981) -- (5.331,8.124) -- (5.25,8.245) -- (5.129,8.326) -- (4.986,8.354) -- (4.844,8.326) -- (4.722,8.245) -- (4.642,8.124) -- (4.613,7.981) -- (4.642,7.838) -- (4.722,7.717) -- (4.844,7.636) -- (4.986,7.608) -- (5.129,7.636) -- (5.25,7.717) -- (5.331,7.838) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.499,7.679) -- (2.47,7.822) -- (2.39,7.943) -- (2.268,8.024) -- (2.126,8.052) -- (1.983,8.024) -- (1.862,7.943) -- (1.781,7.822) -- (1.752,7.679) -- (1.781,7.536) -- (1.862,7.415) -- (1.983,7.334) -- (2.126,7.306) -- (2.268,7.334) -- (2.39,7.415) -- (2.47,7.536) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.89,5.194) -- (7.861,5.337) -- (7.781,5.458) -- (7.659,5.538) -- (7.517,5.567) -- (7.374,5.538) -- (7.253,5.458) -- (7.172,5.337) -- (7.143,5.194) -- (7.172,5.051) -- (7.253,4.93) -- (7.374,4.849) -- (7.517,4.82) -- (7.659,4.849) -- (7.781,4.93) -- (7.861,5.051) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.324,5.183) -- (11.296,5.326) -- (11.215,5.447) -- (11.094,5.528) -- (10.951,5.556) -- (10.808,5.528) -- (10.687,5.447) -- (10.606,5.326) -- (10.578,5.183) -- (10.606,5.04) -- (10.687,4.919) -- (10.808,4.838) -- (10.951,4.81) -- (11.094,4.838) -- (11.215,4.919) -- (11.296,5.04) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.437,4.853) -- (5.408,4.995) -- (5.327,5.117) -- (5.206,5.197) -- (5.063,5.226) -- (4.92,5.197) -- (4.799,5.117) -- (4.719,4.995) -- (4.69,4.853) -- (4.719,4.71) -- (4.799,4.589) -- (4.92,4.508) -- (5.063,4.479) -- (5.206,4.508) -- (5.327,4.589) -- (5.408,4.71) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.752,4.766) -- (1.724,4.909) -- (1.643,5.03) -- (1.522,5.111) -- (1.379,5.139) -- (1.236,5.111) -- (1.115,5.03) -- (1.034,4.909) -- (1.006,4.766) -- (1.034,4.623) -- (1.115,4.502) -- (1.236,4.421) -- (1.379,4.393) -- (1.522,4.421) -- (1.643,4.502) -- (1.724,4.623) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.752,1.744) -- (11.724,1.887) -- (11.643,2.008) -- (11.522,2.089) -- (11.379,2.118) -- (11.236,2.089) -- (11.115,2.008) -- (11.034,1.887) -- (11.006,1.744) -- (11.034,1.602) -- (11.115,1.48) -- (11.236,1.4) -- (11.379,1.371) -- (11.522,1.4) -- (11.643,1.48) -- (11.724,1.602) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.839,1.688) -- (7.811,1.831) -- (7.73,1.952) -- (7.609,2.033) -- (7.466,2.061) -- (7.323,2.033) -- (7.202,1.952) -- (7.121,1.831) -- (7.093,1.688) -- (7.121,1.545) -- (7.202,1.424) -- (7.323,1.343) -- (7.466,1.315) -- (7.609,1.343) -- (7.73,1.424) -- (7.811,1.545) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.06,1.542) -- (5.032,1.685) -- (4.951,1.806) -- (4.83,1.887) -- (4.687,1.916) -- (4.544,1.887) -- (4.423,1.806) -- (4.342,1.685) -- (4.314,1.542) -- (4.342,1.4) -- (4.423,1.279) -- (4.544,1.198) -- (4.687,1.169) -- (4.83,1.198) -- (4.951,1.279) -- (5.032,1.4) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.127,8.916) -- (2.246,8.935) -- (2.353,8.99) -- (2.439,9.075) -- (2.493,9.182) -- (2.512,9.301) -- (2.493,9.419) -- (2.439,9.527) -- (2.353,9.612) -- (2.246,9.666) -- (2.127,9.685) -- (2.127,9.685) -- (2.009,9.666) -- (1.901,9.612) -- (1.816,9.527) -- (1.762,9.419) -- (1.743,9.301) -- (1.762,9.182) -- (1.816,9.075) -- (1.901,8.99) -- (2.009,8.935) -- (2.127,8.916) -- cycle;
\node at (2.127,9.301) {\scalebox{1.617}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.014,6.233) -- (8.133,6.252) -- (8.24,6.307) -- (8.325,6.392) -- (8.38,6.499) -- (8.399,6.618) -- (8.38,6.736) -- (8.325,6.844) -- (8.24,6.929) -- (8.133,6.983) -- (8.014,7.002) -- (8.014,7.002) -- (7.895,6.983) -- (7.788,6.929) -- (7.703,6.844) -- (7.648,6.736) -- (7.63,6.618) -- (7.648,6.499) -- (7.703,6.392) -- (7.788,6.307) -- (7.895,6.252) -- (8.014,6.233) -- cycle;
\node at (8.014,6.618) {\scalebox{1.617}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.241,7.906) -- (6.359,7.924) -- (6.467,7.979) -- (6.552,8.064) -- (6.606,8.171) -- (6.625,8.29) -- (6.606,8.409) -- (6.552,8.516) -- (6.467,8.601) -- (6.359,8.656) -- (6.241,8.675) -- (6.241,8.675) -- (6.122,8.656) -- (6.015,8.601) -- (5.93,8.516) -- (5.875,8.409) -- (5.856,8.29) -- (5.875,8.171) -- (5.93,8.064) -- (6.015,7.979) -- (6.122,7.924) -- (6.241,7.906) -- cycle;
\node at (6.241,8.29) {\scalebox{1.617}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.538,6.045) -- (5.657,6.064) -- (5.764,6.118) -- (5.849,6.203) -- (5.904,6.311) -- (5.923,6.429) -- (5.904,6.548) -- (5.849,6.655) -- (5.764,6.74) -- (5.657,6.795) -- (5.538,6.814) -- (5.538,6.814) -- (5.419,6.795) -- (5.312,6.74) -- (5.227,6.655) -- (5.173,6.548) -- (5.154,6.429) -- (5.173,6.311) -- (5.227,6.203) -- (5.312,6.118) -- (5.419,6.064) -- (5.538,6.045) -- cycle;
\node at (5.538,6.429) {\scalebox{1.617}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.528,7.71) -- (3.647,7.729) -- (3.754,7.783) -- (3.839,7.868) -- (3.894,7.975) -- (3.913,8.094) -- (3.894,8.213) -- (3.839,8.32) -- (3.754,8.405) -- (3.647,8.46) -- (3.528,8.479) -- (3.528,8.479) -- (3.409,8.46) -- (3.302,8.405) -- (3.217,8.32) -- (3.162,8.213) -- (3.144,8.094) -- (3.162,7.975) -- (3.217,7.868) -- (3.302,7.783) -- (3.409,7.729) -- (3.528,7.71) -- cycle;
\node at (3.528,8.094) {\scalebox{1.617}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.245,5.712) -- (2.364,5.731) -- (2.471,5.785) -- (2.556,5.87) -- (2.611,5.977) -- (2.629,6.096) -- (2.611,6.215) -- (2.556,6.322) -- (2.471,6.407) -- (2.364,6.462) -- (2.245,6.481) -- (2.245,6.481) -- (2.126,6.462) -- (2.019,6.407) -- (1.934,6.322) -- (1.879,6.215) -- (1.86,6.096) -- (1.879,5.977) -- (1.934,5.87) -- (2.019,5.785) -- (2.126,5.731) -- (2.245,5.712) -- cycle;
\node at (2.245,6.096) {\scalebox{1.617}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.235,5.047) -- (9.354,5.065) -- (9.461,5.12) -- (9.546,5.205) -- (9.6,5.312) -- (9.619,5.431) -- (9.6,5.55) -- (9.546,5.657) -- (9.461,5.742) -- (9.354,5.797) -- (9.235,5.816) -- (9.235,5.816) -- (9.116,5.797) -- (9.009,5.742) -- (8.924,5.657) -- (8.869,5.55) -- (8.85,5.431) -- (8.869,5.312) -- (8.924,5.205) -- (9.009,5.12) -- (9.116,5.065) -- (9.235,5.047) -- cycle;
\node at (9.235,5.431) {\scalebox{1.617}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.02,3.049) -- (8.139,3.068) -- (8.246,3.122) -- (8.331,3.207) -- (8.386,3.314) -- (8.405,3.433) -- (8.386,3.552) -- (8.331,3.659) -- (8.246,3.744) -- (8.139,3.799) -- (8.02,3.818) -- (8.02,3.818) -- (7.902,3.799) -- (7.794,3.744) -- (7.709,3.659) -- (7.655,3.552) -- (7.636,3.433) -- (7.655,3.314) -- (7.709,3.207) -- (7.794,3.122) -- (7.902,3.068) -- (8.02,3.049) -- cycle;
\node at (8.02,3.433) {\scalebox{1.617}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.405,3.109) -- (11.524,3.128) -- (11.631,3.182) -- (11.716,3.267) -- (11.77,3.375) -- (11.789,3.493) -- (11.77,3.612) -- (11.716,3.719) -- (11.631,3.804) -- (11.524,3.859) -- (11.405,3.878) -- (11.405,3.878) -- (11.286,3.859) -- (11.179,3.804) -- (11.094,3.719) -- (11.039,3.612) -- (11.02,3.493) -- (11.039,3.375) -- (11.094,3.267) -- (11.179,3.182) -- (11.286,3.128) -- (11.405,3.109) -- cycle;
\node at (11.405,3.493) {\scalebox{1.617}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.404,5.121) -- (6.523,5.14) -- (6.63,5.194) -- (6.715,5.279) -- (6.77,5.386) -- (6.788,5.505) -- (6.77,5.624) -- (6.715,5.731) -- (6.63,5.816) -- (6.523,5.871) -- (6.404,5.89) -- (6.042,5.89) -- (5.923,5.871) -- (5.816,5.816) -- (5.731,5.731) -- (5.676,5.624) -- (5.658,5.505) -- (5.676,5.386) -- (5.731,5.279) -- (5.816,5.194) -- (5.923,5.14) -- (6.042,5.121) -- cycle;
\node at (6.223,5.505) {\scalebox{1.617}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.302,2.785) -- (5.421,2.804) -- (5.528,2.859) -- (5.613,2.944) -- (5.667,3.051) -- (5.686,3.17) -- (5.667,3.288) -- (5.613,3.396) -- (5.528,3.481) -- (5.421,3.535) -- (5.302,3.554) -- (4.94,3.554) -- (4.821,3.535) -- (4.714,3.481) -- (4.629,3.396) -- (4.574,3.288) -- (4.555,3.17) -- (4.574,3.051) -- (4.629,2.944) -- (4.714,2.859) -- (4.821,2.804) -- (4.94,2.785) -- cycle;
\node at (5.121,3.17) {\scalebox{1.617}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.389,4.961) -- (3.508,4.98) -- (3.615,5.035) -- (3.7,5.12) -- (3.755,5.227) -- (3.774,5.346) -- (3.755,5.465) -- (3.7,5.572) -- (3.615,5.657) -- (3.508,5.711) -- (3.389,5.73) -- (3.027,5.73) -- (2.909,5.711) -- (2.801,5.657) -- (2.716,5.572) -- (2.662,5.465) -- (2.643,5.346) -- (2.662,5.227) -- (2.716,5.12) -- (2.801,5.035) -- (2.909,4.98) -- (3.027,4.961) -- cycle;
\node at (3.208,5.346) {\scalebox{1.617}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.6,1.555) -- (9.719,1.574) -- (9.826,1.628) -- (9.911,1.713) -- (9.966,1.821) -- (9.985,1.939) -- (9.966,2.058) -- (9.911,2.165) -- (9.826,2.25) -- (9.719,2.305) -- (9.6,2.324) -- (9.238,2.324) -- (9.119,2.305) -- (9.012,2.25) -- (8.927,2.165) -- (8.872,2.058) -- (8.854,1.939) -- (8.872,1.821) -- (8.927,1.713) -- (9.012,1.628) -- (9.119,1.574) -- (9.238,1.555) -- cycle;
\node at (9.419,1.939) {\scalebox{1.617}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.231,1.729) -- (6.35,1.748) -- (6.457,1.803) -- (6.542,1.888) -- (6.597,1.995) -- (6.616,2.114) -- (6.597,2.233) -- (6.542,2.34) -- (6.457,2.425) -- (6.35,2.48) -- (6.231,2.498) -- (5.869,2.498) -- (5.751,2.48) -- (5.643,2.425) -- (5.558,2.34) -- (5.504,2.233) -- (5.485,2.114) -- (5.504,1.995) -- (5.558,1.888) -- (5.643,1.803) -- (5.751,1.748) -- (5.869,1.729) -- cycle;
\node at (6.05,2.114) {\scalebox{1.617}{\small 14}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (1.079,4.466) -- (1.679,4.466) -- (1.679,5.066) -- (1.079,5.066) -- cycle;
\node at (1.379,4.766) {\scalebox{1.617}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (0.651,5.225) -- (1.251,5.225) -- (1.251,5.825) -- (0.651,5.825) -- cycle;
\node at (0.951,5.525) {\scalebox{1.617}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.162,6.539) -- (1.762,6.539) -- (1.762,7.139) -- (1.162,7.139) -- cycle;
\node at (1.462,6.839) {\scalebox{1.617}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (5.226,0.851) -- (5.826,0.851) -- (5.826,1.451) -- (5.226,1.451) -- cycle;
\node at (5.526,1.151) {\scalebox{1.617}{\small d}};
\node at (1.249,11.332) {\scalebox{1.617}{\small A}};
\node at (7.834,8.566) {\scalebox{1.617}{\small B}};
\node at (4.783,8.566) {\scalebox{1.617}{\small C}};
\node at (1.595,8.0) {\scalebox{1.617}{\small D}};
\node at (8.131,5.112) {\scalebox{1.617}{\small E}};
\node at (11.571,5.155) {\scalebox{1.617}{\small F}};
\node at (4.551,4.503) {\scalebox{1.617}{\small G}};
\node at (0.765,4.678) {\scalebox{1.617}{\small H}};
\node at (11.895,1.4) {\scalebox{1.617}{\small I}};
\node at (7.709,1.118) {\scalebox{1.617}{\small J}};
\node at (4.504,0.95) {\scalebox{1.617}{\small K}};
\draw[line width=0.5pt] (12.679,9.894) -- (12.679,10.994);
\draw[line width=0.5pt] (12.448,10.609) -- (12.679,10.994) -- (12.91,10.609);
\draw[line width=0.5pt] (12.376,10.444) -- (12.981,10.444);
\node at (12.679,11.379) {\scalebox{1.617}{\small N}};
\end{tikzpicture}
\end{center}
\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & B, D, F, G, J & 5 \\ \hline
clairs & A, C, E, H, I, K & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{6} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 11 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 5 carrefours sombres et 6 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{H} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend le \textbf{passage du Marché ⑫} vers l'est jusqu'à Rond-point du Stade \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Four ⑪}, vers le sud, jusqu'à Place aux Herbes \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ⑭}, vers l'est, jusqu'à Carrefour du Lavoir \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Jardins ⑬}, vers l'est, jusqu'à Place de la Fontaine \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ⑨}, vers le nord, jusqu'à Coin du Lavoir \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de la Mairie ⑦}, vers l'ouest, jusqu'à Carrefour Saint-Roch \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Verte ②}, vers le nord, jusqu'à Place du Marché \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{sentier des Prés ③}, vers l'ouest, jusqu'à Carrefour de la Mairie \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de la Poste ⑤}, vers l'ouest, jusqu'à Carrefour des Tilleuls \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{impasse des Cerisiers ①}, vers le nord, jusqu'à Fontaine Notre-Dame \textbf{(A)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 11 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
