\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 131}

\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=131}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0131}
\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=0.7543, line join=round, line cap=round]
\draw[line width=0.5pt] (1.331,11.561) -- (1.599,11.49) -- (1.869,11.425) -- (2.141,11.367) -- (2.413,11.315) -- (2.687,11.269) -- (2.963,11.23) -- (3.238,11.198) -- (3.515,11.171) -- (3.792,11.152) -- (4.07,11.139) -- (4.347,11.132) -- (4.625,11.132) -- (4.902,11.138) -- (5.18,11.151);
\draw[line width=0.5pt] (1.331,11.561) -- (1.359,11.363) -- (1.382,11.163) -- (1.4,10.963) -- (1.413,10.763) -- (1.422,10.562) -- (1.426,10.361) -- (1.425,10.16) -- (1.42,9.959) -- (1.41,9.759) -- (1.395,9.559) -- (1.376,9.359) -- (1.351,9.159) -- (1.323,8.961) -- (1.289,8.763);
\draw[line width=0.5pt] (5.18,11.151) -- (5.161,10.928) -- (5.148,10.705) -- (5.14,10.481) -- (5.138,10.256) -- (5.141,10.032) -- (5.149,9.808) -- (5.162,9.584) -- (5.181,9.361) -- (5.205,9.138) -- (5.234,8.916) -- (5.268,8.694) -- (5.307,8.474) -- (5.352,8.254) -- (5.402,8.036);
\draw[line width=0.5pt] (1.289,8.763) -- (1.588,8.756) -- (1.888,8.743) -- (2.186,8.722) -- (2.485,8.694) -- (2.783,8.659) -- (3.079,8.618) -- (3.375,8.569) -- (3.67,8.513) -- (3.963,8.451) -- (4.255,8.381) -- (4.545,8.305) -- (4.832,8.222) -- (5.118,8.132) -- (5.402,8.036);
\draw[line width=0.5pt] (1.289,8.763) -- (1.304,8.511) -- (1.325,8.26) -- (1.352,8.009) -- (1.385,7.758) -- (1.423,7.509) -- (1.468,7.261) -- (1.518,7.013) -- (1.575,6.767) -- (1.637,6.522) -- (1.704,6.279) -- (1.778,6.038) -- (1.857,5.798) -- (1.941,5.56) -- (2.032,5.325);
\draw[line width=0.5pt] (8.039,8.381) -- (8.114,8.19) -- (8.184,7.998) -- (8.251,7.804) -- (8.312,7.608) -- (8.369,7.411) -- (8.421,7.212) -- (8.469,7.013) -- (8.512,6.812) -- (8.55,6.61) -- (8.583,6.408) -- (8.612,6.205) -- (8.636,6.001) -- (8.655,5.797) -- (8.669,5.592);
\draw[line width=0.5pt] (5.402,8.036) -- (5.592,8.031) -- (5.783,8.031) -- (5.974,8.036) -- (6.164,8.045) -- (6.355,8.059) -- (6.545,8.077) -- (6.734,8.1) -- (6.923,8.127) -- (7.112,8.158) -- (7.299,8.194) -- (7.486,8.234) -- (7.671,8.279) -- (7.856,8.328) -- (8.039,8.381);
\draw[line width=0.5pt] (5.402,8.036) -- (5.411,7.815) -- (5.416,7.593) -- (5.415,7.372) -- (5.409,7.151) -- (5.398,6.929) -- (5.382,6.709) -- (5.36,6.488) -- (5.334,6.268) -- (5.302,6.049) -- (5.265,5.831) -- (5.223,5.613) -- (5.175,5.397) -- (5.123,5.182) -- (5.066,4.968);
\draw[line width=0.5pt] (2.032,5.325) -- (2.243,5.266) -- (2.455,5.213) -- (2.669,5.164) -- (2.884,5.121) -- (3.1,5.083) -- (3.317,5.049) -- (3.534,5.021) -- (3.752,4.998) -- (3.971,4.98) -- (4.19,4.968) -- (4.409,4.96) -- (4.628,4.957) -- (4.847,4.96) -- (5.066,4.968);
\draw[line width=0.5pt] (2.032,5.325) -- (2.079,5.108) -- (2.122,4.891) -- (2.16,4.672) -- (2.192,4.453) -- (2.219,4.233) -- (2.241,4.012) -- (2.258,3.791) -- (2.27,3.569) -- (2.276,3.347) -- (2.277,3.126) -- (2.273,2.904) -- (2.264,2.682) -- (2.25,2.461) -- (2.23,2.24);
\draw[line width=0.5pt] (5.066,4.968) -- (5.328,4.973) -- (5.59,4.984) -- (5.852,5.001) -- (6.113,5.025) -- (6.374,5.054) -- (6.634,5.09) -- (6.893,5.132) -- (7.151,5.179) -- (7.408,5.233) -- (7.664,5.293) -- (7.918,5.359) -- (8.17,5.431) -- (8.421,5.509) -- (8.669,5.592);
\draw[line width=0.5pt] (5.066,4.968) -- (5.122,4.73) -- (5.173,4.49) -- (5.218,4.249) -- (5.257,4.007) -- (5.291,3.764) -- (5.319,3.52) -- (5.341,3.276) -- (5.358,3.031) -- (5.368,2.786) -- (5.373,2.541) -- (5.373,2.296) -- (5.366,2.05) -- (5.354,1.806) -- (5.336,1.561);
\draw[line width=0.5pt] (2.23,2.24) -- (2.442,2.158) -- (2.657,2.081) -- (2.873,2.009) -- (3.092,1.942) -- (3.311,1.881) -- (3.532,1.824) -- (3.755,1.773) -- (3.978,1.727) -- (4.203,1.686) -- (4.428,1.65) -- (4.654,1.62) -- (4.881,1.595) -- (5.108,1.575) -- (5.336,1.561);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.491,11.561) -- (1.47,11.641) -- (1.411,11.7) -- (1.331,11.721) -- (1.251,11.7) -- (1.192,11.641) -- (1.171,11.561) -- (1.192,11.481) -- (1.251,11.423) -- (1.331,11.401) -- (1.411,11.423) -- (1.47,11.481) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.34,11.151) -- (5.318,11.231) -- (5.26,11.29) -- (5.18,11.311) -- (5.1,11.29) -- (5.041,11.231) -- (5.02,11.151) -- (5.041,11.071) -- (5.1,11.013) -- (5.18,10.991) -- (5.26,11.013) -- (5.318,11.071) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.449,8.763) -- (1.427,8.843) -- (1.369,8.901) -- (1.289,8.923) -- (1.209,8.901) -- (1.15,8.843) -- (1.129,8.763) -- (1.15,8.683) -- (1.209,8.624) -- (1.289,8.603) -- (1.369,8.624) -- (1.427,8.683) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.199,8.381) -- (8.177,8.461) -- (8.119,8.52) -- (8.039,8.541) -- (7.959,8.52) -- (7.9,8.461) -- (7.879,8.381) -- (7.9,8.301) -- (7.959,8.242) -- (8.039,8.221) -- (8.119,8.242) -- (8.177,8.301) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.562,8.036) -- (5.54,8.116) -- (5.482,8.174) -- (5.402,8.196) -- (5.322,8.174) -- (5.263,8.116) -- (5.242,8.036) -- (5.263,7.956) -- (5.322,7.897) -- (5.402,7.876) -- (5.482,7.897) -- (5.54,7.956) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.829,5.592) -- (8.808,5.672) -- (8.749,5.731) -- (8.669,5.752) -- (8.589,5.731) -- (8.531,5.672) -- (8.509,5.592) -- (8.531,5.512) -- (8.589,5.454) -- (8.669,5.432) -- (8.749,5.454) -- (8.808,5.512) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.192,5.325) -- (2.17,5.405) -- (2.112,5.463) -- (2.032,5.485) -- (1.952,5.463) -- (1.893,5.405) -- (1.872,5.325) -- (1.893,5.245) -- (1.952,5.186) -- (2.032,5.165) -- (2.112,5.186) -- (2.17,5.245) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.226,4.968) -- (5.204,5.048) -- (5.146,5.107) -- (5.066,5.128) -- (4.986,5.107) -- (4.927,5.048) -- (4.906,4.968) -- (4.927,4.888) -- (4.986,4.83) -- (5.066,4.808) -- (5.146,4.83) -- (5.204,4.888) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (2.39,2.24) -- (2.368,2.32) -- (2.31,2.379) -- (2.23,2.4) -- (2.15,2.379) -- (2.091,2.32) -- (2.07,2.24) -- (2.091,2.16) -- (2.15,2.102) -- (2.23,2.08) -- (2.31,2.102) -- (2.368,2.16) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (5.496,1.561) -- (5.474,1.641) -- (5.416,1.7) -- (5.336,1.721) -- (5.256,1.7) -- (5.197,1.641) -- (5.176,1.561) -- (5.197,1.481) -- (5.256,1.423) -- (5.336,1.401) -- (5.416,1.423) -- (5.474,1.481) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.28,11.195) -- (3.402,11.215) -- (3.512,11.271) -- (3.599,11.358) -- (3.655,11.468) -- (3.674,11.589) -- (3.655,11.711) -- (3.599,11.821) -- (3.512,11.908) -- (3.402,11.964) -- (3.28,11.983) -- (3.28,11.983) -- (3.158,11.964) -- (3.049,11.908) -- (2.961,11.821) -- (2.905,11.711) -- (2.886,11.589) -- (2.905,11.468) -- (2.961,11.358) -- (3.049,11.271) -- (3.158,11.215) -- (3.28,11.195) -- cycle;
\node at (3.28,11.589) {\scalebox{1.326}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.819,9.76) -- (1.941,9.78) -- (2.051,9.836) -- (2.138,9.923) -- (2.194,10.033) -- (2.213,10.154) -- (2.194,10.276) -- (2.138,10.386) -- (2.051,10.473) -- (1.941,10.529) -- (1.819,10.548) -- (1.819,10.548) -- (1.698,10.529) -- (1.588,10.473) -- (1.501,10.386) -- (1.445,10.276) -- (1.425,10.154) -- (1.445,10.033) -- (1.501,9.923) -- (1.588,9.836) -- (1.698,9.78) -- (1.819,9.76) -- cycle;
\node at (1.819,10.154) {\scalebox{1.326}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.555,9.219) -- (5.677,9.238) -- (5.787,9.294) -- (5.874,9.381) -- (5.93,9.491) -- (5.949,9.612) -- (5.93,9.734) -- (5.874,9.844) -- (5.787,9.931) -- (5.677,9.987) -- (5.555,10.006) -- (5.555,10.006) -- (5.433,9.987) -- (5.324,9.931) -- (5.236,9.844) -- (5.18,9.734) -- (5.161,9.612) -- (5.18,9.491) -- (5.236,9.381) -- (5.324,9.294) -- (5.433,9.238) -- (5.555,9.219) -- cycle;
\node at (5.555,9.612) {\scalebox{1.326}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.444,8.563) -- (3.566,8.582) -- (3.675,8.638) -- (3.763,8.725) -- (3.818,8.835) -- (3.838,8.957) -- (3.818,9.079) -- (3.763,9.188) -- (3.675,9.275) -- (3.566,9.331) -- (3.444,9.351) -- (3.444,9.351) -- (3.322,9.331) -- (3.212,9.275) -- (3.125,9.188) -- (3.069,9.079) -- (3.05,8.957) -- (3.069,8.835) -- (3.125,8.725) -- (3.212,8.638) -- (3.322,8.582) -- (3.444,8.563) -- cycle;
\node at (3.444,8.957) {\scalebox{1.326}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.904,6.702) -- (2.025,6.722) -- (2.135,6.778) -- (2.222,6.865) -- (2.278,6.975) -- (2.297,7.096) -- (2.278,7.218) -- (2.222,7.328) -- (2.135,7.415) -- (2.025,7.471) -- (1.904,7.49) -- (1.904,7.49) -- (1.782,7.471) -- (1.672,7.415) -- (1.585,7.328) -- (1.529,7.218) -- (1.51,7.096) -- (1.529,6.975) -- (1.585,6.865) -- (1.672,6.778) -- (1.782,6.722) -- (1.904,6.702) -- cycle;
\node at (1.904,7.096) {\scalebox{1.326}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.853,6.706) -- (8.975,6.725) -- (9.085,6.781) -- (9.172,6.868) -- (9.228,6.978) -- (9.247,7.1) -- (9.228,7.221) -- (9.172,7.331) -- (9.085,7.418) -- (8.975,7.474) -- (8.853,7.493) -- (8.853,7.493) -- (8.731,7.474) -- (8.622,7.418) -- (8.534,7.331) -- (8.478,7.221) -- (8.459,7.1) -- (8.478,6.978) -- (8.534,6.868) -- (8.622,6.781) -- (8.731,6.725) -- (8.853,6.706) -- cycle;
\node at (8.853,7.1) {\scalebox{1.326}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.683,8.096) -- (6.805,8.116) -- (6.915,8.171) -- (7.002,8.259) -- (7.058,8.368) -- (7.077,8.49) -- (7.058,8.612) -- (7.002,8.722) -- (6.915,8.809) -- (6.805,8.865) -- (6.683,8.884) -- (6.683,8.884) -- (6.562,8.865) -- (6.452,8.809) -- (6.365,8.722) -- (6.309,8.612) -- (6.289,8.49) -- (6.309,8.368) -- (6.365,8.259) -- (6.452,8.171) -- (6.562,8.116) -- (6.683,8.096) -- cycle;
\node at (6.683,8.49) {\scalebox{1.326}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.752,6.051) -- (5.874,6.071) -- (5.983,6.127) -- (6.071,6.214) -- (6.126,6.323) -- (6.146,6.445) -- (6.126,6.567) -- (6.071,6.677) -- (5.983,6.764) -- (5.874,6.82) -- (5.752,6.839) -- (5.752,6.839) -- (5.63,6.82) -- (5.52,6.764) -- (5.433,6.677) -- (5.377,6.567) -- (5.358,6.445) -- (5.377,6.323) -- (5.433,6.214) -- (5.52,6.127) -- (5.63,6.071) -- (5.752,6.051) -- cycle;
\node at (5.752,6.445) {\scalebox{1.326}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.58,5.019) -- (3.702,5.038) -- (3.812,5.094) -- (3.899,5.181) -- (3.955,5.291) -- (3.974,5.412) -- (3.955,5.534) -- (3.899,5.644) -- (3.812,5.731) -- (3.702,5.787) -- (3.58,5.806) -- (3.58,5.806) -- (3.458,5.787) -- (3.348,5.731) -- (3.261,5.644) -- (3.205,5.534) -- (3.186,5.412) -- (3.205,5.291) -- (3.261,5.181) -- (3.348,5.094) -- (3.458,5.038) -- (3.58,5.019) -- cycle;
\node at (3.58,5.412) {\scalebox{1.326}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.836,3.422) -- (2.958,3.441) -- (3.068,3.497) -- (3.155,3.584) -- (3.211,3.694) -- (3.23,3.816) -- (3.211,3.938) -- (3.155,4.047) -- (3.068,4.135) -- (2.958,4.191) -- (2.836,4.21) -- (2.466,4.21) -- (2.344,4.191) -- (2.234,4.135) -- (2.147,4.047) -- (2.091,3.938) -- (2.072,3.816) -- (2.091,3.694) -- (2.147,3.584) -- (2.234,3.497) -- (2.344,3.441) -- (2.466,3.422) -- cycle;
\node at (2.651,3.816) {\scalebox{1.326}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.011,5.126) -- (7.133,5.145) -- (7.243,5.201) -- (7.33,5.288) -- (7.386,5.398) -- (7.405,5.52) -- (7.386,5.641) -- (7.33,5.751) -- (7.243,5.838) -- (7.133,5.894) -- (7.011,5.914) -- (6.641,5.914) -- (6.519,5.894) -- (6.409,5.838) -- (6.322,5.751) -- (6.266,5.641) -- (6.247,5.52) -- (6.266,5.398) -- (6.322,5.288) -- (6.409,5.201) -- (6.519,5.145) -- (6.641,5.126) -- cycle;
\node at (6.826,5.52) {\scalebox{1.326}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.919,2.913) -- (6.041,2.932) -- (6.151,2.988) -- (6.238,3.075) -- (6.294,3.185) -- (6.313,3.307) -- (6.294,3.429) -- (6.238,3.538) -- (6.151,3.626) -- (6.041,3.682) -- (5.919,3.701) -- (5.549,3.701) -- (5.427,3.682) -- (5.317,3.626) -- (5.23,3.538) -- (5.174,3.429) -- (5.155,3.307) -- (5.174,3.185) -- (5.23,3.075) -- (5.317,2.988) -- (5.427,2.932) -- (5.549,2.913) -- cycle;
\node at (5.734,3.307) {\scalebox{1.326}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.024,1.764) -- (4.146,1.783) -- (4.256,1.839) -- (4.343,1.926) -- (4.399,2.036) -- (4.418,2.157) -- (4.399,2.279) -- (4.343,2.389) -- (4.256,2.476) -- (4.146,2.532) -- (4.024,2.551) -- (3.653,2.551) -- (3.532,2.532) -- (3.422,2.476) -- (3.335,2.389) -- (3.279,2.279) -- (3.26,2.157) -- (3.279,2.036) -- (3.335,1.926) -- (3.422,1.839) -- (3.532,1.783) -- (3.653,1.764) -- cycle;
\node at (3.839,2.157) {\scalebox{1.326}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (8.369,5.292) -- (8.969,5.292) -- (8.969,5.892) -- (8.369,5.892) -- cycle;
\node at (8.669,5.592) {\scalebox{1.326}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.504,2.592) -- (2.104,2.592) -- (2.104,3.192) -- (1.504,3.192) -- cycle;
\node at (1.804,2.892) {\scalebox{1.326}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (7.446,4.397) -- (8.046,4.397) -- (8.046,4.997) -- (7.446,4.997) -- cycle;
\node at (7.746,4.697) {\scalebox{1.326}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (2.069,4.592) -- (2.669,4.592) -- (2.669,5.192) -- (2.069,5.192) -- cycle;
\node at (2.369,4.892) {\scalebox{1.326}{\small d}};
\node at (0.993,12.081) {\scalebox{1.326}{\small A}};
\node at (5.28,11.763) {\scalebox{1.326}{\small B}};
\node at (0.765,9.094) {\scalebox{1.326}{\small C}};
\node at (8.603,8.637) {\scalebox{1.326}{\small D}};
\node at (5.77,8.534) {\scalebox{1.326}{\small E}};
\node at (9.267,5.427) {\scalebox{1.326}{\small F}};
\node at (1.498,5.009) {\scalebox{1.326}{\small G}};
\node at (5.265,4.381) {\scalebox{1.326}{\small H}};
\node at (1.956,1.684) {\scalebox{1.326}{\small I}};
\node at (5.439,0.95) {\scalebox{1.326}{\small J}};
\draw[line width=0.5pt] (9.969,10.611) -- (9.969,11.711);
\draw[line width=0.5pt] (9.738,11.326) -- (9.969,11.711) -- (10.2,11.326);
\draw[line width=0.5pt] (9.667,11.161) -- (10.272,11.161);
\node at (9.969,12.096) {\scalebox{1.326}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour F, 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 boulevard du Parc & Ⓑ Place des Tanneurs & Ⓙ Place Sainte-Croix \\ \hline
① allée des Peupliers & ⑨ rue du Stade & Ⓒ Rond-point du Pont & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
② rue du Pont & ⑩ rue Neuve & Ⓓ Placette des Ormes & \textbf{a} l'école \\ \hline
③ avenue de la Gare & ⑪ rue Pasteur & Ⓔ Carrefour du Lavoir & \textbf{b} le marché couvert \\ \hline
④ rue des Écoles & ⑫ allée des Tilleuls & Ⓕ Coin du Lavoir & \textbf{c} la mairie \\ \hline
⑤ rue du Puits & ⑬ passage du Marché & Ⓖ Rond-point de la Gare & \textbf{d} le gymnase \\ \hline
⑥ rue de la Poste & \cellcolor{black!32}\bfseries Les carrefours & Ⓗ Carrefour des Tilleuls &  \\ \hline
⑦ venelle du Chat & Ⓐ Place du Marché & Ⓘ Fontaine Notre-Dame &  \\ \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=0.7726, line join=round, line cap=round]
\draw[line width=0.5pt] (1.331,11.561) -- (1.599,11.49) -- (1.869,11.425) -- (2.141,11.367) -- (2.413,11.315) -- (2.687,11.269) -- (2.963,11.23) -- (3.238,11.198) -- (3.515,11.171) -- (3.792,11.152) -- (4.07,11.139) -- (4.347,11.132) -- (4.625,11.132) -- (4.902,11.138) -- (5.18,11.151);
\draw[line width=0.5pt] (1.331,11.561) -- (1.359,11.363) -- (1.382,11.163) -- (1.4,10.963) -- (1.413,10.763) -- (1.422,10.562) -- (1.426,10.361) -- (1.425,10.16) -- (1.42,9.959) -- (1.41,9.759) -- (1.395,9.559) -- (1.376,9.359) -- (1.351,9.159) -- (1.323,8.961) -- (1.289,8.763);
\draw[line width=0.5pt] (5.18,11.151) -- (5.161,10.928) -- (5.148,10.705) -- (5.14,10.481) -- (5.138,10.256) -- (5.141,10.032) -- (5.149,9.808) -- (5.162,9.584) -- (5.181,9.361) -- (5.205,9.138) -- (5.234,8.916) -- (5.268,8.694) -- (5.307,8.474) -- (5.352,8.254) -- (5.402,8.036);
\draw[line width=0.5pt] (1.289,8.763) -- (1.588,8.756) -- (1.888,8.743) -- (2.186,8.722) -- (2.485,8.694) -- (2.783,8.659) -- (3.079,8.618) -- (3.375,8.569) -- (3.67,8.513) -- (3.963,8.451) -- (4.255,8.381) -- (4.545,8.305) -- (4.832,8.222) -- (5.118,8.132) -- (5.402,8.036);
\draw[line width=0.5pt] (1.289,8.763) -- (1.304,8.511) -- (1.325,8.26) -- (1.352,8.009) -- (1.385,7.758) -- (1.423,7.509) -- (1.468,7.261) -- (1.518,7.013) -- (1.575,6.767) -- (1.637,6.522) -- (1.704,6.279) -- (1.778,6.038) -- (1.857,5.798) -- (1.941,5.56) -- (2.032,5.325);
\draw[line width=0.5pt] (8.039,8.381) -- (8.114,8.19) -- (8.184,7.998) -- (8.251,7.804) -- (8.312,7.608) -- (8.369,7.411) -- (8.421,7.212) -- (8.469,7.013) -- (8.512,6.812) -- (8.55,6.61) -- (8.583,6.408) -- (8.612,6.205) -- (8.636,6.001) -- (8.655,5.797) -- (8.669,5.592);
\draw[line width=0.5pt] (5.402,8.036) -- (5.592,8.031) -- (5.783,8.031) -- (5.974,8.036) -- (6.164,8.045) -- (6.355,8.059) -- (6.545,8.077) -- (6.734,8.1) -- (6.923,8.127) -- (7.112,8.158) -- (7.299,8.194) -- (7.486,8.234) -- (7.671,8.279) -- (7.856,8.328) -- (8.039,8.381);
\draw[line width=0.5pt] (5.402,8.036) -- (5.411,7.815) -- (5.416,7.593) -- (5.415,7.372) -- (5.409,7.151) -- (5.398,6.929) -- (5.382,6.709) -- (5.36,6.488) -- (5.334,6.268) -- (5.302,6.049) -- (5.265,5.831) -- (5.223,5.613) -- (5.175,5.397) -- (5.123,5.182) -- (5.066,4.968);
\draw[line width=0.5pt] (2.032,5.325) -- (2.243,5.266) -- (2.455,5.213) -- (2.669,5.164) -- (2.884,5.121) -- (3.1,5.083) -- (3.317,5.049) -- (3.534,5.021) -- (3.752,4.998) -- (3.971,4.98) -- (4.19,4.968) -- (4.409,4.96) -- (4.628,4.957) -- (4.847,4.96) -- (5.066,4.968);
\draw[line width=0.5pt] (2.032,5.325) -- (2.079,5.108) -- (2.122,4.891) -- (2.16,4.672) -- (2.192,4.453) -- (2.219,4.233) -- (2.241,4.012) -- (2.258,3.791) -- (2.27,3.569) -- (2.276,3.347) -- (2.277,3.126) -- (2.273,2.904) -- (2.264,2.682) -- (2.25,2.461) -- (2.23,2.24);
\draw[line width=0.5pt] (5.066,4.968) -- (5.328,4.973) -- (5.59,4.984) -- (5.852,5.001) -- (6.113,5.025) -- (6.374,5.054) -- (6.634,5.09) -- (6.893,5.132) -- (7.151,5.179) -- (7.408,5.233) -- (7.664,5.293) -- (7.918,5.359) -- (8.17,5.431) -- (8.421,5.509) -- (8.669,5.592);
\draw[line width=0.5pt] (5.066,4.968) -- (5.122,4.73) -- (5.173,4.49) -- (5.218,4.249) -- (5.257,4.007) -- (5.291,3.764) -- (5.319,3.52) -- (5.341,3.276) -- (5.358,3.031) -- (5.368,2.786) -- (5.373,2.541) -- (5.373,2.296) -- (5.366,2.05) -- (5.354,1.806) -- (5.336,1.561);
\draw[line width=0.5pt] (2.23,2.24) -- (2.442,2.158) -- (2.657,2.081) -- (2.873,2.009) -- (3.092,1.942) -- (3.311,1.881) -- (3.532,1.824) -- (3.755,1.773) -- (3.978,1.727) -- (4.203,1.686) -- (4.428,1.65) -- (4.654,1.62) -- (4.881,1.595) -- (5.108,1.575) -- (5.336,1.561);
\filldraw[fill=white, draw=black, line width=0.5pt] (1.704,11.561) -- (1.676,11.704) -- (1.595,11.825) -- (1.474,11.906) -- (1.331,11.935) -- (1.188,11.906) -- (1.067,11.825) -- (0.986,11.704) -- (0.958,11.561) -- (0.986,11.419) -- (1.067,11.297) -- (1.188,11.217) -- (1.331,11.188) -- (1.474,11.217) -- (1.595,11.297) -- (1.676,11.419) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.553,11.151) -- (5.524,11.294) -- (5.444,11.415) -- (5.322,11.496) -- (5.18,11.525) -- (5.037,11.496) -- (4.916,11.415) -- (4.835,11.294) -- (4.806,11.151) -- (4.835,11.009) -- (4.916,10.888) -- (5.037,10.807) -- (5.18,10.778) -- (5.322,10.807) -- (5.444,10.888) -- (5.524,11.009) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.662,8.763) -- (1.634,8.906) -- (1.553,9.027) -- (1.432,9.108) -- (1.289,9.136) -- (1.146,9.108) -- (1.025,9.027) -- (0.944,8.906) -- (0.916,8.763) -- (0.944,8.62) -- (1.025,8.499) -- (1.146,8.418) -- (1.289,8.39) -- (1.432,8.418) -- (1.553,8.499) -- (1.634,8.62) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.412,8.381) -- (8.383,8.524) -- (8.302,8.645) -- (8.181,8.726) -- (8.039,8.754) -- (7.896,8.726) -- (7.775,8.645) -- (7.694,8.524) -- (7.665,8.381) -- (7.694,8.238) -- (7.775,8.117) -- (7.896,8.036) -- (8.039,8.008) -- (8.181,8.036) -- (8.302,8.117) -- (8.383,8.238) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.775,8.036) -- (5.747,8.179) -- (5.666,8.3) -- (5.545,8.381) -- (5.402,8.409) -- (5.259,8.381) -- (5.138,8.3) -- (5.057,8.179) -- (5.028,8.036) -- (5.057,7.893) -- (5.138,7.772) -- (5.259,7.691) -- (5.402,7.663) -- (5.545,7.691) -- (5.666,7.772) -- (5.747,7.893) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (9.042,5.592) -- (9.014,5.735) -- (8.933,5.856) -- (8.812,5.937) -- (8.669,5.966) -- (8.526,5.937) -- (8.405,5.856) -- (8.324,5.735) -- (8.296,5.592) -- (8.324,5.45) -- (8.405,5.328) -- (8.526,5.248) -- (8.669,5.219) -- (8.812,5.248) -- (8.933,5.328) -- (9.014,5.45) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.405,5.325) -- (2.376,5.468) -- (2.296,5.589) -- (2.175,5.67) -- (2.032,5.698) -- (1.889,5.67) -- (1.768,5.589) -- (1.687,5.468) -- (1.658,5.325) -- (1.687,5.182) -- (1.768,5.061) -- (1.889,4.98) -- (2.032,4.952) -- (2.175,4.98) -- (2.296,5.061) -- (2.376,5.182) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.439,4.968) -- (5.411,5.111) -- (5.33,5.232) -- (5.209,5.313) -- (5.066,5.341) -- (4.923,5.313) -- (4.802,5.232) -- (4.721,5.111) -- (4.693,4.968) -- (4.721,4.825) -- (4.802,4.704) -- (4.923,4.623) -- (5.066,4.595) -- (5.209,4.623) -- (5.33,4.704) -- (5.411,4.825) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.603,2.24) -- (2.575,2.383) -- (2.494,2.504) -- (2.373,2.585) -- (2.23,2.613) -- (2.087,2.585) -- (1.966,2.504) -- (1.885,2.383) -- (1.857,2.24) -- (1.885,2.097) -- (1.966,1.976) -- (2.087,1.895) -- (2.23,1.867) -- (2.373,1.895) -- (2.494,1.976) -- (2.575,2.097) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.709,1.561) -- (5.68,1.704) -- (5.599,1.825) -- (5.478,1.906) -- (5.336,1.935) -- (5.193,1.906) -- (5.072,1.825) -- (4.991,1.704) -- (4.962,1.561) -- (4.991,1.419) -- (5.072,1.297) -- (5.193,1.217) -- (5.336,1.188) -- (5.478,1.217) -- (5.599,1.297) -- (5.68,1.419) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (3.279,11.195) -- (3.398,11.214) -- (3.505,11.269) -- (3.59,11.354) -- (3.645,11.461) -- (3.664,11.58) -- (3.645,11.699) -- (3.59,11.806) -- (3.505,11.891) -- (3.398,11.946) -- (3.279,11.965) -- (3.279,11.965) -- (3.16,11.946) -- (3.053,11.891) -- (2.968,11.806) -- (2.913,11.699) -- (2.895,11.58) -- (2.913,11.461) -- (2.968,11.354) -- (3.053,11.269) -- (3.16,11.214) -- (3.279,11.195) -- cycle;
\node at (3.279,11.58) {\scalebox{1.294}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.81,9.77) -- (1.929,9.789) -- (2.036,9.843) -- (2.121,9.929) -- (2.176,10.036) -- (2.194,10.155) -- (2.176,10.273) -- (2.121,10.381) -- (2.036,10.466) -- (1.929,10.52) -- (1.81,10.539) -- (1.81,10.539) -- (1.691,10.52) -- (1.584,10.466) -- (1.499,10.381) -- (1.444,10.273) -- (1.425,10.155) -- (1.444,10.036) -- (1.499,9.929) -- (1.584,9.843) -- (1.691,9.789) -- (1.81,9.77) -- cycle;
\node at (1.81,10.155) {\scalebox{1.294}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.546,9.227) -- (5.665,9.246) -- (5.772,9.301) -- (5.857,9.386) -- (5.911,9.493) -- (5.93,9.612) -- (5.911,9.731) -- (5.857,9.838) -- (5.772,9.923) -- (5.665,9.977) -- (5.546,9.996) -- (5.546,9.996) -- (5.427,9.977) -- (5.32,9.923) -- (5.235,9.838) -- (5.18,9.731) -- (5.161,9.612) -- (5.18,9.493) -- (5.235,9.386) -- (5.32,9.301) -- (5.427,9.246) -- (5.546,9.227) -- cycle;
\node at (5.546,9.612) {\scalebox{1.294}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.442,8.563) -- (3.561,8.582) -- (3.668,8.636) -- (3.753,8.722) -- (3.808,8.829) -- (3.827,8.948) -- (3.808,9.066) -- (3.753,9.174) -- (3.668,9.259) -- (3.561,9.313) -- (3.442,9.332) -- (3.442,9.332) -- (3.323,9.313) -- (3.216,9.259) -- (3.131,9.174) -- (3.077,9.066) -- (3.058,8.948) -- (3.077,8.829) -- (3.131,8.722) -- (3.216,8.636) -- (3.323,8.582) -- (3.442,8.563) -- cycle;
\node at (3.442,8.948) {\scalebox{1.294}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.894,6.71) -- (2.013,6.729) -- (2.12,6.783) -- (2.205,6.868) -- (2.26,6.976) -- (2.279,7.094) -- (2.26,7.213) -- (2.205,7.32) -- (2.12,7.405) -- (2.013,7.46) -- (1.894,7.479) -- (1.894,7.479) -- (1.776,7.46) -- (1.668,7.405) -- (1.583,7.32) -- (1.529,7.213) -- (1.51,7.094) -- (1.529,6.976) -- (1.583,6.868) -- (1.668,6.783) -- (1.776,6.729) -- (1.894,6.71) -- cycle;
\node at (1.894,7.094) {\scalebox{1.294}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.844,6.713) -- (8.963,6.732) -- (9.07,6.786) -- (9.155,6.871) -- (9.21,6.979) -- (9.228,7.097) -- (9.21,7.216) -- (9.155,7.324) -- (9.07,7.409) -- (8.963,7.463) -- (8.844,7.482) -- (8.844,7.482) -- (8.725,7.463) -- (8.618,7.409) -- (8.533,7.324) -- (8.478,7.216) -- (8.459,7.097) -- (8.478,6.979) -- (8.533,6.871) -- (8.618,6.786) -- (8.725,6.732) -- (8.844,6.713) -- cycle;
\node at (8.844,7.097) {\scalebox{1.294}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.684,8.096) -- (6.803,8.115) -- (6.91,8.17) -- (6.996,8.255) -- (7.05,8.362) -- (7.069,8.481) -- (7.05,8.6) -- (6.996,8.707) -- (6.91,8.792) -- (6.803,8.847) -- (6.684,8.865) -- (6.684,8.865) -- (6.566,8.847) -- (6.458,8.792) -- (6.373,8.707) -- (6.319,8.6) -- (6.3,8.481) -- (6.319,8.362) -- (6.373,8.255) -- (6.458,8.17) -- (6.566,8.115) -- (6.684,8.096) -- cycle;
\node at (6.684,8.481) {\scalebox{1.294}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.743,6.062) -- (5.861,6.08) -- (5.969,6.135) -- (6.054,6.22) -- (6.108,6.327) -- (6.127,6.446) -- (6.108,6.565) -- (6.054,6.672) -- (5.969,6.757) -- (5.861,6.812) -- (5.743,6.831) -- (5.743,6.831) -- (5.624,6.812) -- (5.517,6.757) -- (5.431,6.672) -- (5.377,6.565) -- (5.358,6.446) -- (5.377,6.327) -- (5.431,6.22) -- (5.517,6.135) -- (5.624,6.08) -- (5.743,6.062) -- cycle;
\node at (5.743,6.446) {\scalebox{1.294}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.579,5.019) -- (3.698,5.037) -- (3.805,5.092) -- (3.89,5.177) -- (3.945,5.284) -- (3.963,5.403) -- (3.945,5.522) -- (3.89,5.629) -- (3.805,5.714) -- (3.698,5.769) -- (3.579,5.788) -- (3.579,5.788) -- (3.46,5.769) -- (3.353,5.714) -- (3.268,5.629) -- (3.213,5.522) -- (3.194,5.403) -- (3.213,5.284) -- (3.268,5.177) -- (3.353,5.092) -- (3.46,5.037) -- (3.579,5.019) -- cycle;
\node at (3.579,5.403) {\scalebox{1.294}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.823,3.431) -- (2.942,3.45) -- (3.049,3.504) -- (3.134,3.589) -- (3.188,3.697) -- (3.207,3.815) -- (3.188,3.934) -- (3.134,4.041) -- (3.049,4.126) -- (2.942,4.181) -- (2.823,4.2) -- (2.461,4.2) -- (2.342,4.181) -- (2.235,4.126) -- (2.15,4.041) -- (2.095,3.934) -- (2.076,3.815) -- (2.095,3.697) -- (2.15,3.589) -- (2.235,3.504) -- (2.342,3.45) -- (2.461,3.431) -- cycle;
\node at (2.642,3.815) {\scalebox{1.294}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.009,5.126) -- (7.127,5.145) -- (7.235,5.199) -- (7.32,5.284) -- (7.374,5.392) -- (7.393,5.51) -- (7.374,5.629) -- (7.32,5.736) -- (7.235,5.822) -- (7.127,5.876) -- (7.009,5.895) -- (6.647,5.895) -- (6.528,5.876) -- (6.421,5.822) -- (6.336,5.736) -- (6.281,5.629) -- (6.262,5.51) -- (6.281,5.392) -- (6.336,5.284) -- (6.421,5.199) -- (6.528,5.145) -- (6.647,5.126) -- cycle;
\node at (6.828,5.51) {\scalebox{1.294}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.906,2.922) -- (6.024,2.94) -- (6.132,2.995) -- (6.217,3.08) -- (6.271,3.187) -- (6.29,3.306) -- (6.271,3.425) -- (6.217,3.532) -- (6.132,3.617) -- (6.024,3.672) -- (5.906,3.691) -- (5.544,3.691) -- (5.425,3.672) -- (5.318,3.617) -- (5.233,3.532) -- (5.178,3.425) -- (5.159,3.306) -- (5.178,3.187) -- (5.233,3.08) -- (5.318,2.995) -- (5.425,2.94) -- (5.544,2.922) -- cycle;
\node at (5.725,3.306) {\scalebox{1.294}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.018,1.764) -- (4.137,1.783) -- (4.244,1.837) -- (4.329,1.922) -- (4.383,2.03) -- (4.402,2.148) -- (4.383,2.267) -- (4.329,2.374) -- (4.244,2.459) -- (4.137,2.514) -- (4.018,2.533) -- (3.656,2.533) -- (3.537,2.514) -- (3.43,2.459) -- (3.345,2.374) -- (3.29,2.267) -- (3.271,2.148) -- (3.29,2.03) -- (3.345,1.922) -- (3.43,1.837) -- (3.537,1.783) -- (3.656,1.764) -- cycle;
\node at (3.837,2.148) {\scalebox{1.294}{\small 13}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (8.369,5.292) -- (8.969,5.292) -- (8.969,5.892) -- (8.369,5.892) -- cycle;
\node at (8.669,5.592) {\scalebox{1.294}{\small a}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.504,2.592) -- (2.104,2.592) -- (2.104,3.192) -- (1.504,3.192) -- cycle;
\node at (1.804,2.892) {\scalebox{1.294}{\small b}};
\filldraw[fill=green!18, draw=black, line width=0.5pt] (7.446,4.397) -- (8.046,4.397) -- (8.046,4.997) -- (7.446,4.997) -- cycle;
\node at (7.746,4.697) {\scalebox{1.294}{\small c}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (2.069,4.592) -- (2.669,4.592) -- (2.669,5.192) -- (2.069,5.192) -- cycle;
\node at (2.369,4.892) {\scalebox{1.294}{\small d}};
\node at (0.993,12.081) {\scalebox{1.294}{\small A}};
\node at (5.28,11.763) {\scalebox{1.294}{\small B}};
\node at (0.765,9.094) {\scalebox{1.294}{\small C}};
\node at (8.603,8.637) {\scalebox{1.294}{\small D}};
\node at (5.77,8.534) {\scalebox{1.294}{\small E}};
\node at (9.267,5.427) {\scalebox{1.294}{\small F}};
\node at (1.498,5.009) {\scalebox{1.294}{\small G}};
\node at (5.265,4.381) {\scalebox{1.294}{\small H}};
\node at (1.956,1.684) {\scalebox{1.294}{\small I}};
\node at (5.439,0.95) {\scalebox{1.294}{\small J}};
\draw[line width=0.5pt] (9.969,10.611) -- (9.969,11.711);
\draw[line width=0.5pt] (9.738,11.326) -- (9.969,11.711) -- (10.2,11.326);
\draw[line width=0.5pt] (9.667,11.161) -- (10.272,11.161);
\node at (9.969,12.096) {\scalebox{1.294}{\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, C, D, H, I & 5 \\ \hline
clairs & A, E, F, G, J & 5 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{5} clairs. Il y en a \textbf{autant} de chaque côté.}\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 10 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{Oui.} Un circuit existe :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend la \textbf{rue de la Poste ⑥} vers le nord jusqu'à Placette des Ormes \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{venelle du Chat ⑦}, vers l'ouest, jusqu'à Carrefour du Lavoir \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{avenue de la Gare ③}, vers le nord, jusqu'à Place des Tanneurs \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{allée des Peupliers ①}, vers l'ouest, jusqu'à Place du Marché \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Pont ②}, vers le sud, jusqu'à Rond-point du Pont \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Puits ⑤}, vers le sud, jusqu'à Rond-point de la Gare \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Neuve ⑩}, vers le sud, jusqu'à Fontaine Notre-Dame \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{passage du Marché ⑬}, vers l'est, jusqu'à Place Sainte-Croix \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{allée des Tilleuls ⑫}, vers le nord, jusqu'à Carrefour des Tilleuls \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Pasteur ⑪}, vers l'est, jusqu'à Coin du Lavoir \textbf{(F)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Le car est passé une fois par chaque carrefour et se retrouve à l'école.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, et c'est déjà fait : le trajet ci-dessus dessert tous les carrefours. Il suffit au car de s'arrêter avant le dernier trajet de retour.}\par\addvspace{0.4em}
\end{document}
