\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 234}

\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=234}\\[2pt]
    {\small\ttfamily\bfseries\color{red} HAMI0234}
\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=0.6035, line join=round, line cap=round]
\draw[line width=0.5pt] (6.26,6.999) -- (6.251,6.783) -- (6.246,6.567) -- (6.247,6.35) -- (6.253,6.134) -- (6.264,5.917) -- (6.28,5.701) -- (6.301,5.486) -- (6.327,5.271) -- (6.358,5.056) -- (6.395,4.843) -- (6.436,4.63) -- (6.482,4.419) -- (6.533,4.208) -- (6.589,3.999);
\draw[line width=0.5pt] (1.785,6.963) -- (1.956,6.946) -- (2.127,6.924) -- (2.297,6.898) -- (2.467,6.869) -- (2.636,6.835) -- (2.804,6.798) -- (2.971,6.756) -- (3.138,6.711) -- (3.303,6.661) -- (3.466,6.608) -- (3.629,6.551) -- (3.79,6.49) -- (3.95,6.426) -- (4.108,6.358);
\draw[line width=0.5pt] (1.785,6.963) -- (1.727,6.776) -- (1.674,6.587) -- (1.625,6.398) -- (1.58,6.207) -- (1.54,6.015) -- (1.504,5.822) -- (1.473,5.628) -- (1.447,5.434) -- (1.425,5.239) -- (1.407,5.043) -- (1.395,4.848) -- (1.386,4.652) -- (1.383,4.456) -- (1.384,4.26);
\draw[line width=0.5pt] (4.108,6.358) -- (4.253,6.427) -- (4.4,6.492) -- (4.549,6.554) -- (4.699,6.613) -- (4.851,6.668) -- (5.003,6.719) -- (5.157,6.767) -- (5.312,6.811) -- (5.468,6.852) -- (5.625,6.889) -- (5.783,6.922) -- (5.942,6.952) -- (6.101,6.977) -- (6.26,6.999);
\draw[line width=0.5pt] (4.108,6.358) -- (4.078,6.188) -- (4.053,6.018) -- (4.033,5.847) -- (4.016,5.675) -- (4.003,5.504) -- (3.994,5.332) -- (3.989,5.159) -- (3.988,4.987) -- (3.992,4.815) -- (3.999,4.643) -- (4.01,4.471) -- (4.026,4.299) -- (4.045,4.128) -- (4.069,3.958);
\draw[line width=0.5pt] (11.384,4.548) -- (11.391,4.333) -- (11.392,4.118) -- (11.388,3.903) -- (11.379,3.688) -- (11.365,3.473) -- (11.346,3.259) -- (11.322,3.045) -- (11.293,2.831) -- (11.259,2.619) -- (11.221,2.407) -- (11.177,2.196) -- (11.128,1.986) -- (11.074,1.778) -- (11.015,1.571);
\draw[line width=0.5pt] (8.598,4.44) -- (8.796,4.418) -- (8.995,4.399) -- (9.195,4.386) -- (9.395,4.377) -- (9.595,4.373) -- (9.795,4.374) -- (9.995,4.379) -- (10.195,4.39) -- (10.395,4.404) -- (10.594,4.424) -- (10.793,4.448) -- (10.991,4.477) -- (11.188,4.51) -- (11.384,4.548);
\draw[line width=0.5pt] (1.384,4.26) -- (1.578,4.268) -- (1.771,4.271) -- (1.965,4.27) -- (2.159,4.264) -- (2.353,4.254) -- (2.546,4.239) -- (2.739,4.22) -- (2.931,4.196) -- (3.123,4.167) -- (3.314,4.134) -- (3.504,4.097) -- (3.693,4.055) -- (3.882,4.009) -- (4.069,3.958);
\draw[line width=0.5pt] (1.384,4.26) -- (1.365,4.117) -- (1.349,3.974) -- (1.337,3.83) -- (1.328,3.686) -- (1.322,3.542) -- (1.32,3.398) -- (1.321,3.254) -- (1.325,3.11) -- (1.333,2.965) -- (1.345,2.822) -- (1.359,2.678) -- (1.377,2.535) -- (1.399,2.393) -- (1.424,2.251);
\draw[line width=0.5pt] (6.589,3.999) -- (6.727,4.053) -- (6.866,4.103) -- (7.006,4.149) -- (7.147,4.193) -- (7.289,4.233) -- (7.432,4.269) -- (7.575,4.303) -- (7.72,4.333) -- (7.865,4.359) -- (8.011,4.382) -- (8.157,4.402) -- (8.304,4.418) -- (8.45,4.431) -- (8.598,4.44);
\draw[line width=0.5pt] (6.589,3.999) -- (6.544,3.828) -- (6.503,3.656) -- (6.467,3.482) -- (6.434,3.308) -- (6.405,3.133) -- (6.381,2.957) -- (6.36,2.781) -- (6.344,2.604) -- (6.332,2.427) -- (6.324,2.25) -- (6.32,2.073) -- (6.32,1.896) -- (6.325,1.718) -- (6.334,1.542);
\draw[line width=0.5pt] (4.069,3.958) -- (4.247,3.988) -- (4.426,4.015) -- (4.605,4.037) -- (4.785,4.055) -- (4.966,4.068) -- (5.146,4.078) -- (5.327,4.083) -- (5.508,4.084) -- (5.689,4.08) -- (5.87,4.072) -- (6.051,4.061) -- (6.231,4.044) -- (6.41,4.024) -- (6.589,3.999);
\draw[line width=0.5pt] (4.069,3.958) -- (4.012,3.793) -- (3.96,3.626) -- (3.912,3.458) -- (3.867,3.289) -- (3.827,3.119) -- (3.79,2.948) -- (3.758,2.777) -- (3.73,2.604) -- (3.705,2.431) -- (3.685,2.258) -- (3.669,2.084) -- (3.657,1.909) -- (3.649,1.735) -- (3.646,1.561);
\draw[line width=0.5pt] (1.424,2.251) -- (1.589,2.226) -- (1.753,2.197) -- (1.917,2.165) -- (2.08,2.129) -- (2.242,2.089) -- (2.403,2.045) -- (2.563,1.997) -- (2.722,1.946) -- (2.88,1.891) -- (3.036,1.832) -- (3.191,1.769) -- (3.344,1.703) -- (3.496,1.634) -- (3.646,1.561);
\draw[line width=0.5pt] (3.646,1.561) -- (3.836,1.53) -- (4.027,1.504) -- (4.218,1.482) -- (4.411,1.465) -- (4.603,1.452) -- (4.796,1.444) -- (4.989,1.44) -- (5.182,1.441) -- (5.375,1.446) -- (5.567,1.456) -- (5.76,1.471) -- (5.952,1.49) -- (6.143,1.513) -- (6.334,1.542);
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.42,6.999) -- (6.399,7.079) -- (6.34,7.138) -- (6.26,7.159) -- (6.18,7.138) -- (6.122,7.079) -- (6.1,6.999) -- (6.122,6.919) -- (6.18,6.861) -- (6.26,6.839) -- (6.34,6.861) -- (6.399,6.919) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.945,6.963) -- (1.924,7.043) -- (1.865,7.102) -- (1.785,7.123) -- (1.705,7.102) -- (1.647,7.043) -- (1.625,6.963) -- (1.647,6.883) -- (1.705,6.824) -- (1.785,6.803) -- (1.865,6.824) -- (1.924,6.883) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.268,6.358) -- (4.246,6.438) -- (4.188,6.496) -- (4.108,6.518) -- (4.028,6.496) -- (3.969,6.438) -- (3.948,6.358) -- (3.969,6.278) -- (4.028,6.219) -- (4.108,6.198) -- (4.188,6.219) -- (4.246,6.278) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.544,4.548) -- (11.523,4.628) -- (11.464,4.687) -- (11.384,4.708) -- (11.304,4.687) -- (11.246,4.628) -- (11.224,4.548) -- (11.246,4.468) -- (11.304,4.41) -- (11.384,4.388) -- (11.464,4.41) -- (11.523,4.468) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (8.758,4.44) -- (8.736,4.52) -- (8.678,4.579) -- (8.598,4.6) -- (8.518,4.579) -- (8.459,4.52) -- (8.438,4.44) -- (8.459,4.36) -- (8.518,4.302) -- (8.598,4.28) -- (8.678,4.302) -- (8.736,4.36) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.544,4.26) -- (1.523,4.34) -- (1.464,4.399) -- (1.384,4.42) -- (1.304,4.399) -- (1.246,4.34) -- (1.224,4.26) -- (1.246,4.18) -- (1.304,4.122) -- (1.384,4.1) -- (1.464,4.122) -- (1.523,4.18) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.749,3.999) -- (6.728,4.079) -- (6.669,4.138) -- (6.589,4.159) -- (6.509,4.138) -- (6.451,4.079) -- (6.429,3.999) -- (6.451,3.919) -- (6.509,3.861) -- (6.589,3.839) -- (6.669,3.861) -- (6.728,3.919) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.229,3.958) -- (4.207,4.038) -- (4.149,4.097) -- (4.069,4.118) -- (3.989,4.097) -- (3.93,4.038) -- (3.909,3.958) -- (3.93,3.878) -- (3.989,3.819) -- (4.069,3.798) -- (4.149,3.819) -- (4.207,3.878) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (1.584,2.251) -- (1.562,2.331) -- (1.504,2.389) -- (1.424,2.411) -- (1.344,2.389) -- (1.285,2.331) -- (1.264,2.251) -- (1.285,2.171) -- (1.344,2.112) -- (1.424,2.091) -- (1.504,2.112) -- (1.562,2.171) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (11.175,1.571) -- (11.154,1.651) -- (11.095,1.71) -- (11.015,1.731) -- (10.935,1.71) -- (10.877,1.651) -- (10.855,1.571) -- (10.877,1.491) -- (10.935,1.433) -- (11.015,1.411) -- (11.095,1.433) -- (11.154,1.491) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (3.806,1.561) -- (3.784,1.641) -- (3.726,1.699) -- (3.646,1.721) -- (3.566,1.699) -- (3.507,1.641) -- (3.486,1.561) -- (3.507,1.481) -- (3.566,1.422) -- (3.646,1.401) -- (3.726,1.422) -- (3.784,1.481) -- cycle;
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.494,1.542) -- (6.472,1.622) -- (6.414,1.68) -- (6.334,1.702) -- (6.254,1.68) -- (6.195,1.622) -- (6.174,1.542) -- (6.195,1.462) -- (6.254,1.403) -- (6.334,1.382) -- (6.414,1.403) -- (6.472,1.462) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.693,5.135) -- (6.814,5.154) -- (6.924,5.21) -- (7.011,5.297) -- (7.067,5.407) -- (7.087,5.529) -- (7.067,5.65) -- (7.011,5.76) -- (6.924,5.847) -- (6.814,5.903) -- (6.693,5.923) -- (6.693,5.923) -- (6.571,5.903) -- (6.461,5.847) -- (6.374,5.76) -- (6.318,5.65) -- (6.299,5.529) -- (6.318,5.407) -- (6.374,5.297) -- (6.461,5.21) -- (6.571,5.154) -- (6.693,5.135) -- cycle;
\node at (6.693,5.529) {\scalebox{1.657}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.872,5.981) -- (2.994,6.0) -- (3.103,6.056) -- (3.191,6.143) -- (3.247,6.253) -- (3.266,6.375) -- (3.247,6.497) -- (3.191,6.606) -- (3.103,6.694) -- (2.994,6.75) -- (2.872,6.769) -- (2.872,6.769) -- (2.75,6.75) -- (2.64,6.694) -- (2.553,6.606) -- (2.497,6.497) -- (2.478,6.375) -- (2.497,6.253) -- (2.553,6.143) -- (2.64,6.056) -- (2.75,6.0) -- (2.872,5.981) -- cycle;
\node at (2.872,6.375) {\scalebox{1.657}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.863,5.176) -- (1.985,5.196) -- (2.094,5.252) -- (2.181,5.339) -- (2.237,5.449) -- (2.257,5.57) -- (2.237,5.692) -- (2.181,5.802) -- (2.094,5.889) -- (1.985,5.945) -- (1.863,5.964) -- (1.863,5.964) -- (1.741,5.945) -- (1.631,5.889) -- (1.544,5.802) -- (1.488,5.692) -- (1.469,5.57) -- (1.488,5.449) -- (1.544,5.339) -- (1.631,5.252) -- (1.741,5.196) -- (1.863,5.176) -- cycle;
\node at (1.863,5.57) {\scalebox{1.657}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.045,6.751) -- (5.167,6.77) -- (5.276,6.826) -- (5.364,6.913) -- (5.42,7.023) -- (5.439,7.145) -- (5.42,7.266) -- (5.364,7.376) -- (5.276,7.463) -- (5.167,7.519) -- (5.045,7.539) -- (5.045,7.539) -- (4.923,7.519) -- (4.813,7.463) -- (4.726,7.376) -- (4.67,7.266) -- (4.651,7.145) -- (4.67,7.023) -- (4.726,6.913) -- (4.813,6.826) -- (4.923,6.77) -- (5.045,6.751) -- cycle;
\node at (5.045,7.145) {\scalebox{1.657}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.595,4.772) -- (3.717,4.791) -- (3.827,4.847) -- (3.914,4.934) -- (3.97,5.044) -- (3.989,5.166) -- (3.97,5.288) -- (3.914,5.397) -- (3.827,5.484) -- (3.717,5.54) -- (3.595,5.56) -- (3.595,5.56) -- (3.473,5.54) -- (3.364,5.484) -- (3.276,5.397) -- (3.221,5.288) -- (3.201,5.166) -- (3.221,5.044) -- (3.276,4.934) -- (3.364,4.847) -- (3.473,4.791) -- (3.595,4.772) -- cycle;
\node at (3.595,5.166) {\scalebox{1.657}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.932,2.699) -- (11.053,2.718) -- (11.163,2.774) -- (11.25,2.861) -- (11.306,2.971) -- (11.325,3.093) -- (11.306,3.215) -- (11.25,3.325) -- (11.163,3.412) -- (11.053,3.468) -- (10.932,3.487) -- (10.932,3.487) -- (10.81,3.468) -- (10.7,3.412) -- (10.613,3.325) -- (10.557,3.215) -- (10.538,3.093) -- (10.557,2.971) -- (10.613,2.861) -- (10.7,2.774) -- (10.81,2.718) -- (10.932,2.699) -- cycle;
\node at (10.932,3.093) {\scalebox{1.657}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.011,3.592) -- (10.132,3.611) -- (10.242,3.667) -- (10.329,3.754) -- (10.385,3.864) -- (10.404,3.986) -- (10.385,4.108) -- (10.329,4.217) -- (10.242,4.304) -- (10.132,4.36) -- (10.011,4.38) -- (10.011,4.38) -- (9.889,4.36) -- (9.779,4.304) -- (9.692,4.217) -- (9.636,4.108) -- (9.617,3.986) -- (9.636,3.864) -- (9.692,3.754) -- (9.779,3.667) -- (9.889,3.611) -- (10.011,3.592) -- cycle;
\node at (10.011,3.986) {\scalebox{1.657}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.783,4.217) -- (2.905,4.237) -- (3.014,4.293) -- (3.102,4.38) -- (3.157,4.489) -- (3.177,4.611) -- (3.157,4.733) -- (3.102,4.843) -- (3.014,4.93) -- (2.905,4.986) -- (2.783,5.005) -- (2.783,5.005) -- (2.661,4.986) -- (2.551,4.93) -- (2.464,4.843) -- (2.408,4.733) -- (2.389,4.611) -- (2.408,4.489) -- (2.464,4.38) -- (2.551,4.293) -- (2.661,4.237) -- (2.783,4.217) -- cycle;
\node at (2.783,4.611) {\scalebox{1.657}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.715,2.868) -- (1.837,2.887) -- (1.946,2.943) -- (2.033,3.03) -- (2.089,3.14) -- (2.109,3.261) -- (2.089,3.383) -- (2.033,3.493) -- (1.946,3.58) -- (1.837,3.636) -- (1.715,3.655) -- (1.715,3.655) -- (1.593,3.636) -- (1.483,3.58) -- (1.396,3.493) -- (1.34,3.383) -- (1.321,3.261) -- (1.34,3.14) -- (1.396,3.03) -- (1.483,2.943) -- (1.593,2.887) -- (1.715,2.868) -- cycle;
\node at (1.715,3.261) {\scalebox{1.657}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.925,4.696) -- (8.047,4.716) -- (8.157,4.771) -- (8.244,4.859) -- (8.3,4.968) -- (8.319,5.09) -- (8.3,5.212) -- (8.244,5.322) -- (8.157,5.409) -- (8.047,5.465) -- (7.925,5.484) -- (7.555,5.484) -- (7.433,5.465) -- (7.323,5.409) -- (7.236,5.322) -- (7.18,5.212) -- (7.161,5.09) -- (7.18,4.968) -- (7.236,4.859) -- (7.323,4.771) -- (7.433,4.716) -- (7.555,4.696) -- cycle;
\node at (7.74,5.09) {\scalebox{1.657}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.937,2.346) -- (7.059,2.366) -- (7.169,2.422) -- (7.256,2.509) -- (7.312,2.618) -- (7.331,2.74) -- (7.312,2.862) -- (7.256,2.972) -- (7.169,3.059) -- (7.059,3.115) -- (6.937,3.134) -- (6.567,3.134) -- (6.445,3.115) -- (6.335,3.059) -- (6.248,2.972) -- (6.192,2.862) -- (6.173,2.74) -- (6.192,2.618) -- (6.248,2.509) -- (6.335,2.422) -- (6.445,2.366) -- (6.567,2.346) -- cycle;
\node at (6.752,2.74) {\scalebox{1.657}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.506,4.083) -- (5.628,4.102) -- (5.738,4.158) -- (5.825,4.245) -- (5.881,4.355) -- (5.9,4.477) -- (5.881,4.598) -- (5.825,4.708) -- (5.738,4.795) -- (5.628,4.851) -- (5.506,4.87) -- (5.135,4.87) -- (5.014,4.851) -- (4.904,4.795) -- (4.817,4.708) -- (4.761,4.598) -- (4.742,4.477) -- (4.761,4.355) -- (4.817,4.245) -- (4.904,4.158) -- (5.014,4.102) -- (5.135,4.083) -- cycle;
\node at (5.321,4.477) {\scalebox{1.657}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.331,2.314) -- (4.453,2.334) -- (4.563,2.39) -- (4.65,2.477) -- (4.706,2.587) -- (4.725,2.708) -- (4.706,2.83) -- (4.65,2.94) -- (4.563,3.027) -- (4.453,3.083) -- (4.331,3.102) -- (3.961,3.102) -- (3.839,3.083) -- (3.729,3.027) -- (3.642,2.94) -- (3.586,2.83) -- (3.567,2.708) -- (3.586,2.587) -- (3.642,2.477) -- (3.729,2.39) -- (3.839,2.334) -- (3.961,2.314) -- cycle;
\node at (4.146,2.708) {\scalebox{1.657}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.865,1.979) -- (2.987,1.999) -- (3.097,2.055) -- (3.184,2.142) -- (3.24,2.252) -- (3.259,2.373) -- (3.24,2.495) -- (3.184,2.605) -- (3.097,2.692) -- (2.987,2.748) -- (2.865,2.767) -- (2.494,2.767) -- (2.373,2.748) -- (2.263,2.692) -- (2.176,2.605) -- (2.12,2.495) -- (2.101,2.373) -- (2.12,2.252) -- (2.176,2.142) -- (2.263,2.055) -- (2.373,1.999) -- (2.494,1.979) -- cycle;
\node at (2.68,2.373) {\scalebox{1.657}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.171,0.652) -- (5.293,0.672) -- (5.403,0.728) -- (5.49,0.815) -- (5.546,0.925) -- (5.565,1.046) -- (5.546,1.168) -- (5.49,1.278) -- (5.403,1.365) -- (5.293,1.421) -- (5.171,1.44) -- (4.801,1.44) -- (4.679,1.421) -- (4.569,1.365) -- (4.482,1.278) -- (4.426,1.168) -- (4.407,1.046) -- (4.426,0.925) -- (4.482,0.815) -- (4.569,0.728) -- (4.679,0.672) -- (4.801,0.652) -- cycle;
\node at (4.986,1.046) {\scalebox{1.657}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (3.808,6.058) -- (4.408,6.058) -- (4.408,6.658) -- (3.808,6.658) -- cycle;
\node at (4.108,6.358) {\scalebox{1.657}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (12.161,3.079) -- (12.761,3.079) -- (12.761,3.679) -- (12.161,3.679) -- cycle;
\node at (12.461,3.379) {\scalebox{1.657}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.252,5.169) -- (4.852,5.169) -- (4.852,5.769) -- (4.252,5.769) -- cycle;
\node at (4.552,5.469) {\scalebox{1.657}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.816,6.111) -- (7.416,6.111) -- (7.416,6.711) -- (6.816,6.711) -- cycle;
\node at (7.116,6.411) {\scalebox{1.657}{\small d}};
\node at (6.405,7.602) {\scalebox{1.657}{\small A}};
\node at (1.296,7.343) {\scalebox{1.657}{\small B}};
\node at (3.78,6.884) {\scalebox{1.657}{\small C}};
\node at (12.002,4.602) {\scalebox{1.657}{\small D}};
\node at (9.212,4.522) {\scalebox{1.657}{\small E}};
\node at (0.765,4.293) {\scalebox{1.657}{\small F}};
\node at (7.209,3.977) {\scalebox{1.657}{\small G}};
\node at (3.449,3.925) {\scalebox{1.657}{\small H}};
\node at (0.855,2.004) {\scalebox{1.657}{\small I}};
\node at (11.58,1.316) {\scalebox{1.657}{\small J}};
\node at (3.268,1.069) {\scalebox{1.657}{\small K}};
\node at (6.519,0.95) {\scalebox{1.657}{\small L}};
\draw[line width=0.5pt] (12.684,6.049) -- (12.684,7.149);
\draw[line width=0.5pt] (12.453,6.764) -- (12.684,7.149) -- (12.915,6.764);
\draw[line width=0.5pt] (12.382,6.599) -- (12.987,6.599);
\node at (12.684,7.534) {\scalebox{1.657}{\small N}};
\end{tikzpicture}
\end{center}
\noindent \textbf{Le car part de l'école, au carrefour C, 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 & ⑨ boulevard du Parc & Ⓑ Place de l'Église & Ⓚ Carrefour Saint-Roch \\ \hline
① chemin du Lavoir & ⑩ rue du Four & Ⓒ Carrefour du Lavoir & Ⓛ Place des Tanneurs \\ \hline
② rue du Pont & ⑪ rue des Écoles & Ⓓ Fontaine Notre-Dame & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ rue Neuve & ⑫ avenue de la Gare & Ⓔ Place Sainte-Croix & \textbf{a} l'école \\ \hline
④ chemin Creux & ⑬ rue du Puits & Ⓕ Rond-point du Pont & \textbf{b} le gymnase \\ \hline
⑤ rue du Stade & ⑭ sentier des Prés & Ⓖ Carrefour des Tilleuls & \textbf{c} le lavoir \\ \hline
⑥ impasse des Cerisiers & ⑮ rue de l'Église & Ⓗ Carrefour de l'Hôpital & \textbf{d} le marché couvert \\ \hline
⑦ allée des Peupliers & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Place aux Herbes &  \\ \hline
⑧ rue Verte & Ⓐ Place de la Fontaine & Ⓙ Place du Marché &  \\ \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=0.6183, line join=round, line cap=round]
\draw[line width=0.5pt] (6.26,6.999) -- (6.251,6.783) -- (6.246,6.567) -- (6.247,6.35) -- (6.253,6.134) -- (6.264,5.917) -- (6.28,5.701) -- (6.301,5.486) -- (6.327,5.271) -- (6.358,5.056) -- (6.395,4.843) -- (6.436,4.63) -- (6.482,4.419) -- (6.533,4.208) -- (6.589,3.999);
\draw[line width=0.5pt] (1.785,6.963) -- (1.956,6.946) -- (2.127,6.924) -- (2.297,6.898) -- (2.467,6.869) -- (2.636,6.835) -- (2.804,6.798) -- (2.971,6.756) -- (3.138,6.711) -- (3.303,6.661) -- (3.466,6.608) -- (3.629,6.551) -- (3.79,6.49) -- (3.95,6.426) -- (4.108,6.358);
\draw[line width=0.5pt] (1.785,6.963) -- (1.727,6.776) -- (1.674,6.587) -- (1.625,6.398) -- (1.58,6.207) -- (1.54,6.015) -- (1.504,5.822) -- (1.473,5.628) -- (1.447,5.434) -- (1.425,5.239) -- (1.407,5.043) -- (1.395,4.848) -- (1.386,4.652) -- (1.383,4.456) -- (1.384,4.26);
\draw[line width=0.5pt] (4.108,6.358) -- (4.253,6.427) -- (4.4,6.492) -- (4.549,6.554) -- (4.699,6.613) -- (4.851,6.668) -- (5.003,6.719) -- (5.157,6.767) -- (5.312,6.811) -- (5.468,6.852) -- (5.625,6.889) -- (5.783,6.922) -- (5.942,6.952) -- (6.101,6.977) -- (6.26,6.999);
\draw[line width=0.5pt] (4.108,6.358) -- (4.078,6.188) -- (4.053,6.018) -- (4.033,5.847) -- (4.016,5.675) -- (4.003,5.504) -- (3.994,5.332) -- (3.989,5.159) -- (3.988,4.987) -- (3.992,4.815) -- (3.999,4.643) -- (4.01,4.471) -- (4.026,4.299) -- (4.045,4.128) -- (4.069,3.958);
\draw[line width=0.5pt] (11.384,4.548) -- (11.391,4.333) -- (11.392,4.118) -- (11.388,3.903) -- (11.379,3.688) -- (11.365,3.473) -- (11.346,3.259) -- (11.322,3.045) -- (11.293,2.831) -- (11.259,2.619) -- (11.221,2.407) -- (11.177,2.196) -- (11.128,1.986) -- (11.074,1.778) -- (11.015,1.571);
\draw[line width=0.5pt] (8.598,4.44) -- (8.796,4.418) -- (8.995,4.399) -- (9.195,4.386) -- (9.395,4.377) -- (9.595,4.373) -- (9.795,4.374) -- (9.995,4.379) -- (10.195,4.39) -- (10.395,4.404) -- (10.594,4.424) -- (10.793,4.448) -- (10.991,4.477) -- (11.188,4.51) -- (11.384,4.548);
\draw[line width=0.5pt] (1.384,4.26) -- (1.578,4.268) -- (1.771,4.271) -- (1.965,4.27) -- (2.159,4.264) -- (2.353,4.254) -- (2.546,4.239) -- (2.739,4.22) -- (2.931,4.196) -- (3.123,4.167) -- (3.314,4.134) -- (3.504,4.097) -- (3.693,4.055) -- (3.882,4.009) -- (4.069,3.958);
\draw[line width=0.5pt] (1.384,4.26) -- (1.365,4.117) -- (1.349,3.974) -- (1.337,3.83) -- (1.328,3.686) -- (1.322,3.542) -- (1.32,3.398) -- (1.321,3.254) -- (1.325,3.11) -- (1.333,2.965) -- (1.345,2.822) -- (1.359,2.678) -- (1.377,2.535) -- (1.399,2.393) -- (1.424,2.251);
\draw[line width=0.5pt] (6.589,3.999) -- (6.727,4.053) -- (6.866,4.103) -- (7.006,4.149) -- (7.147,4.193) -- (7.289,4.233) -- (7.432,4.269) -- (7.575,4.303) -- (7.72,4.333) -- (7.865,4.359) -- (8.011,4.382) -- (8.157,4.402) -- (8.304,4.418) -- (8.45,4.431) -- (8.598,4.44);
\draw[line width=0.5pt] (6.589,3.999) -- (6.544,3.828) -- (6.503,3.656) -- (6.467,3.482) -- (6.434,3.308) -- (6.405,3.133) -- (6.381,2.957) -- (6.36,2.781) -- (6.344,2.604) -- (6.332,2.427) -- (6.324,2.25) -- (6.32,2.073) -- (6.32,1.896) -- (6.325,1.718) -- (6.334,1.542);
\draw[line width=0.5pt] (4.069,3.958) -- (4.247,3.988) -- (4.426,4.015) -- (4.605,4.037) -- (4.785,4.055) -- (4.966,4.068) -- (5.146,4.078) -- (5.327,4.083) -- (5.508,4.084) -- (5.689,4.08) -- (5.87,4.072) -- (6.051,4.061) -- (6.231,4.044) -- (6.41,4.024) -- (6.589,3.999);
\draw[line width=0.5pt] (4.069,3.958) -- (4.012,3.793) -- (3.96,3.626) -- (3.912,3.458) -- (3.867,3.289) -- (3.827,3.119) -- (3.79,2.948) -- (3.758,2.777) -- (3.73,2.604) -- (3.705,2.431) -- (3.685,2.258) -- (3.669,2.084) -- (3.657,1.909) -- (3.649,1.735) -- (3.646,1.561);
\draw[line width=0.5pt] (1.424,2.251) -- (1.589,2.226) -- (1.753,2.197) -- (1.917,2.165) -- (2.08,2.129) -- (2.242,2.089) -- (2.403,2.045) -- (2.563,1.997) -- (2.722,1.946) -- (2.88,1.891) -- (3.036,1.832) -- (3.191,1.769) -- (3.344,1.703) -- (3.496,1.634) -- (3.646,1.561);
\draw[line width=0.5pt] (3.646,1.561) -- (3.836,1.53) -- (4.027,1.504) -- (4.218,1.482) -- (4.411,1.465) -- (4.603,1.452) -- (4.796,1.444) -- (4.989,1.44) -- (5.182,1.441) -- (5.375,1.446) -- (5.567,1.456) -- (5.76,1.471) -- (5.952,1.49) -- (6.143,1.513) -- (6.334,1.542);
\filldraw[fill=white, draw=black, line width=0.5pt] (6.633,6.999) -- (6.605,7.142) -- (6.524,7.263) -- (6.403,7.344) -- (6.26,7.372) -- (6.117,7.344) -- (5.996,7.263) -- (5.915,7.142) -- (5.887,6.999) -- (5.915,6.856) -- (5.996,6.735) -- (6.117,6.654) -- (6.26,6.626) -- (6.403,6.654) -- (6.524,6.735) -- (6.605,6.856) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.158,6.963) -- (2.13,7.106) -- (2.049,7.227) -- (1.928,7.308) -- (1.785,7.336) -- (1.642,7.308) -- (1.521,7.227) -- (1.44,7.106) -- (1.412,6.963) -- (1.44,6.82) -- (1.521,6.699) -- (1.642,6.618) -- (1.785,6.59) -- (1.928,6.618) -- (2.049,6.699) -- (2.13,6.82) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.481,6.358) -- (4.452,6.5) -- (4.371,6.621) -- (4.25,6.702) -- (4.108,6.731) -- (3.965,6.702) -- (3.844,6.621) -- (3.763,6.5) -- (3.734,6.358) -- (3.763,6.215) -- (3.844,6.094) -- (3.965,6.013) -- (4.108,5.984) -- (4.25,6.013) -- (4.371,6.094) -- (4.452,6.215) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.757,4.548) -- (11.729,4.691) -- (11.648,4.812) -- (11.527,4.893) -- (11.384,4.922) -- (11.241,4.893) -- (11.12,4.812) -- (11.039,4.691) -- (11.011,4.548) -- (11.039,4.405) -- (11.12,4.284) -- (11.241,4.203) -- (11.384,4.175) -- (11.527,4.203) -- (11.648,4.284) -- (11.729,4.405) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.971,4.44) -- (8.942,4.583) -- (8.862,4.704) -- (8.74,4.785) -- (8.598,4.814) -- (8.455,4.785) -- (8.334,4.704) -- (8.253,4.583) -- (8.224,4.44) -- (8.253,4.298) -- (8.334,4.177) -- (8.455,4.096) -- (8.598,4.067) -- (8.74,4.096) -- (8.862,4.177) -- (8.942,4.298) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.757,4.26) -- (1.729,4.403) -- (1.648,4.524) -- (1.527,4.605) -- (1.384,4.633) -- (1.241,4.605) -- (1.12,4.524) -- (1.039,4.403) -- (1.011,4.26) -- (1.039,4.117) -- (1.12,3.996) -- (1.241,3.915) -- (1.384,3.887) -- (1.527,3.915) -- (1.648,3.996) -- (1.729,4.117) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (6.963,3.999) -- (6.934,4.142) -- (6.853,4.263) -- (6.732,4.344) -- (6.589,4.373) -- (6.447,4.344) -- (6.326,4.263) -- (6.245,4.142) -- (6.216,3.999) -- (6.245,3.857) -- (6.326,3.735) -- (6.447,3.655) -- (6.589,3.626) -- (6.732,3.655) -- (6.853,3.735) -- (6.934,3.857) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.442,3.958) -- (4.413,4.101) -- (4.332,4.222) -- (4.211,4.303) -- (4.069,4.331) -- (3.926,4.303) -- (3.805,4.222) -- (3.724,4.101) -- (3.695,3.958) -- (3.724,3.815) -- (3.805,3.694) -- (3.926,3.613) -- (4.069,3.585) -- (4.211,3.613) -- (4.332,3.694) -- (4.413,3.815) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.797,2.251) -- (1.768,2.393) -- (1.687,2.514) -- (1.566,2.595) -- (1.424,2.624) -- (1.281,2.595) -- (1.16,2.514) -- (1.079,2.393) -- (1.05,2.251) -- (1.079,2.108) -- (1.16,1.987) -- (1.281,1.906) -- (1.424,1.877) -- (1.566,1.906) -- (1.687,1.987) -- (1.768,2.108) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.388,1.571) -- (11.36,1.714) -- (11.279,1.835) -- (11.158,1.916) -- (11.015,1.944) -- (10.872,1.916) -- (10.751,1.835) -- (10.67,1.714) -- (10.642,1.571) -- (10.67,1.428) -- (10.751,1.307) -- (10.872,1.226) -- (11.015,1.198) -- (11.158,1.226) -- (11.279,1.307) -- (11.36,1.428) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.019,1.561) -- (3.99,1.703) -- (3.909,1.824) -- (3.788,1.905) -- (3.646,1.934) -- (3.503,1.905) -- (3.382,1.824) -- (3.301,1.703) -- (3.272,1.561) -- (3.301,1.418) -- (3.382,1.297) -- (3.503,1.216) -- (3.646,1.187) -- (3.788,1.216) -- (3.909,1.297) -- (3.99,1.418) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.707,1.542) -- (6.678,1.684) -- (6.598,1.805) -- (6.476,1.886) -- (6.334,1.915) -- (6.191,1.886) -- (6.07,1.805) -- (5.989,1.684) -- (5.96,1.542) -- (5.989,1.399) -- (6.07,1.278) -- (6.191,1.197) -- (6.334,1.168) -- (6.476,1.197) -- (6.598,1.278) -- (6.678,1.399) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (6.683,5.143) -- (6.802,5.162) -- (6.909,5.217) -- (6.994,5.302) -- (7.049,5.409) -- (7.068,5.528) -- (7.049,5.646) -- (6.994,5.754) -- (6.909,5.839) -- (6.802,5.893) -- (6.683,5.912) -- (6.683,5.912) -- (6.564,5.893) -- (6.457,5.839) -- (6.372,5.754) -- (6.318,5.646) -- (6.299,5.528) -- (6.318,5.409) -- (6.372,5.302) -- (6.457,5.217) -- (6.564,5.162) -- (6.683,5.143) -- cycle;
\node at (6.683,5.528) {\scalebox{1.617}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.874,5.999) -- (2.993,6.018) -- (3.1,6.073) -- (3.185,6.158) -- (3.24,6.265) -- (3.259,6.384) -- (3.24,6.503) -- (3.185,6.61) -- (3.1,6.695) -- (2.993,6.75) -- (2.874,6.769) -- (2.874,6.769) -- (2.756,6.75) -- (2.648,6.695) -- (2.563,6.61) -- (2.509,6.503) -- (2.49,6.384) -- (2.509,6.265) -- (2.563,6.158) -- (2.648,6.073) -- (2.756,6.018) -- (2.874,5.999) -- cycle;
\node at (2.874,6.384) {\scalebox{1.617}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.854,5.187) -- (1.972,5.206) -- (2.08,5.261) -- (2.165,5.346) -- (2.219,5.453) -- (2.238,5.572) -- (2.219,5.69) -- (2.165,5.798) -- (2.08,5.883) -- (1.972,5.937) -- (1.854,5.956) -- (1.854,5.956) -- (1.735,5.937) -- (1.628,5.883) -- (1.542,5.798) -- (1.488,5.69) -- (1.469,5.572) -- (1.488,5.453) -- (1.542,5.346) -- (1.628,5.261) -- (1.735,5.206) -- (1.854,5.187) -- cycle;
\node at (1.854,5.572) {\scalebox{1.617}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.048,6.751) -- (5.166,6.77) -- (5.274,6.825) -- (5.359,6.91) -- (5.413,7.017) -- (5.432,7.136) -- (5.413,7.255) -- (5.359,7.362) -- (5.274,7.447) -- (5.166,7.501) -- (5.048,7.52) -- (5.048,7.52) -- (4.929,7.501) -- (4.822,7.447) -- (4.736,7.362) -- (4.682,7.255) -- (4.663,7.136) -- (4.682,7.017) -- (4.736,6.91) -- (4.822,6.825) -- (4.929,6.77) -- (5.048,6.751) -- cycle;
\node at (5.048,7.136) {\scalebox{1.617}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.605,4.781) -- (3.723,4.8) -- (3.831,4.855) -- (3.916,4.94) -- (3.97,5.047) -- (3.989,5.166) -- (3.97,5.284) -- (3.916,5.392) -- (3.831,5.477) -- (3.723,5.531) -- (3.605,5.55) -- (3.605,5.55) -- (3.486,5.531) -- (3.379,5.477) -- (3.293,5.392) -- (3.239,5.284) -- (3.22,5.166) -- (3.239,5.047) -- (3.293,4.94) -- (3.379,4.855) -- (3.486,4.8) -- (3.605,4.781) -- cycle;
\node at (3.605,5.166) {\scalebox{1.617}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.941,2.707) -- (11.06,2.726) -- (11.167,2.781) -- (11.252,2.866) -- (11.307,2.973) -- (11.325,3.092) -- (11.307,3.211) -- (11.252,3.318) -- (11.167,3.403) -- (11.06,3.458) -- (10.941,3.476) -- (10.941,3.476) -- (10.822,3.458) -- (10.715,3.403) -- (10.63,3.318) -- (10.575,3.211) -- (10.556,3.092) -- (10.575,2.973) -- (10.63,2.866) -- (10.715,2.781) -- (10.822,2.726) -- (10.941,2.707) -- cycle;
\node at (10.941,3.092) {\scalebox{1.617}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (10.01,3.611) -- (10.129,3.629) -- (10.236,3.684) -- (10.321,3.769) -- (10.376,3.876) -- (10.395,3.995) -- (10.376,4.114) -- (10.321,4.221) -- (10.236,4.306) -- (10.129,4.361) -- (10.01,4.38) -- (10.01,4.38) -- (9.891,4.361) -- (9.784,4.306) -- (9.699,4.221) -- (9.644,4.114) -- (9.626,3.995) -- (9.644,3.876) -- (9.699,3.769) -- (9.784,3.684) -- (9.891,3.629) -- (10.01,3.611) -- cycle;
\node at (10.01,3.995) {\scalebox{1.617}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.782,4.217) -- (2.901,4.236) -- (3.008,4.291) -- (3.093,4.376) -- (3.147,4.483) -- (3.166,4.602) -- (3.147,4.721) -- (3.093,4.828) -- (3.008,4.913) -- (2.901,4.968) -- (2.782,4.986) -- (2.782,4.986) -- (2.663,4.968) -- (2.556,4.913) -- (2.471,4.828) -- (2.416,4.721) -- (2.397,4.602) -- (2.416,4.483) -- (2.471,4.376) -- (2.556,4.291) -- (2.663,4.236) -- (2.782,4.217) -- cycle;
\node at (2.782,4.602) {\scalebox{1.617}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.705,2.877) -- (1.824,2.896) -- (1.931,2.95) -- (2.016,3.035) -- (2.071,3.142) -- (2.09,3.261) -- (2.071,3.38) -- (2.016,3.487) -- (1.931,3.572) -- (1.824,3.627) -- (1.705,3.646) -- (1.705,3.646) -- (1.587,3.627) -- (1.479,3.572) -- (1.394,3.487) -- (1.34,3.38) -- (1.321,3.261) -- (1.34,3.142) -- (1.394,3.035) -- (1.479,2.95) -- (1.587,2.896) -- (1.705,2.877) -- cycle;
\node at (1.705,3.261) {\scalebox{1.617}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.924,4.688) -- (8.043,4.707) -- (8.15,4.762) -- (8.235,4.847) -- (8.29,4.954) -- (8.309,5.073) -- (8.29,5.192) -- (8.235,5.299) -- (8.15,5.384) -- (8.043,5.438) -- (7.924,5.457) -- (7.562,5.457) -- (7.443,5.438) -- (7.336,5.384) -- (7.251,5.299) -- (7.196,5.192) -- (7.178,5.073) -- (7.196,4.954) -- (7.251,4.847) -- (7.336,4.762) -- (7.443,4.707) -- (7.562,4.688) -- cycle;
\node at (7.743,5.073) {\scalebox{1.617}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.924,2.357) -- (7.042,2.375) -- (7.15,2.43) -- (7.235,2.515) -- (7.289,2.622) -- (7.308,2.741) -- (7.289,2.86) -- (7.235,2.967) -- (7.15,3.052) -- (7.042,3.107) -- (6.924,3.126) -- (6.562,3.126) -- (6.443,3.107) -- (6.336,3.052) -- (6.251,2.967) -- (6.196,2.86) -- (6.177,2.741) -- (6.196,2.622) -- (6.251,2.515) -- (6.336,2.43) -- (6.443,2.375) -- (6.562,2.357) -- cycle;
\node at (6.743,2.741) {\scalebox{1.617}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.502,4.083) -- (5.621,4.101) -- (5.728,4.156) -- (5.813,4.241) -- (5.868,4.348) -- (5.886,4.467) -- (5.868,4.586) -- (5.813,4.693) -- (5.728,4.778) -- (5.621,4.833) -- (5.502,4.852) -- (5.14,4.852) -- (5.021,4.833) -- (4.914,4.778) -- (4.829,4.693) -- (4.774,4.586) -- (4.755,4.467) -- (4.774,4.348) -- (4.829,4.241) -- (4.914,4.156) -- (5.021,4.101) -- (5.14,4.083) -- cycle;
\node at (5.321,4.467) {\scalebox{1.617}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.318,2.325) -- (4.437,2.344) -- (4.544,2.399) -- (4.629,2.484) -- (4.683,2.591) -- (4.702,2.71) -- (4.683,2.829) -- (4.629,2.936) -- (4.544,3.021) -- (4.437,3.076) -- (4.318,3.094) -- (3.956,3.094) -- (3.837,3.076) -- (3.73,3.021) -- (3.645,2.936) -- (3.59,2.829) -- (3.571,2.71) -- (3.59,2.591) -- (3.645,2.484) -- (3.73,2.399) -- (3.837,2.344) -- (3.956,2.325) -- cycle;
\node at (4.137,2.71) {\scalebox{1.617}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.858,1.98) -- (2.977,1.999) -- (3.084,2.053) -- (3.169,2.138) -- (3.224,2.246) -- (3.243,2.364) -- (3.224,2.483) -- (3.169,2.59) -- (3.084,2.676) -- (2.977,2.73) -- (2.858,2.749) -- (2.496,2.749) -- (2.377,2.73) -- (2.27,2.676) -- (2.185,2.59) -- (2.13,2.483) -- (2.112,2.364) -- (2.13,2.246) -- (2.185,2.138) -- (2.27,2.053) -- (2.377,1.999) -- (2.496,1.98) -- cycle;
\node at (2.677,2.364) {\scalebox{1.617}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.167,0.671) -- (5.286,0.69) -- (5.393,0.745) -- (5.478,0.83) -- (5.533,0.937) -- (5.552,1.056) -- (5.533,1.174) -- (5.478,1.282) -- (5.393,1.367) -- (5.286,1.421) -- (5.167,1.44) -- (4.805,1.44) -- (4.686,1.421) -- (4.579,1.367) -- (4.494,1.282) -- (4.439,1.174) -- (4.421,1.056) -- (4.439,0.937) -- (4.494,0.83) -- (4.579,0.745) -- (4.686,0.69) -- (4.805,0.671) -- cycle;
\node at (4.986,1.056) {\scalebox{1.617}{\small 15}};
\filldraw[fill=yellow!35, draw=black, line width=0.5pt] (3.808,6.058) -- (4.408,6.058) -- (4.408,6.658) -- (3.808,6.658) -- cycle;
\node at (4.108,6.358) {\scalebox{1.617}{\small a}};
\filldraw[fill=black!14, draw=black, line width=0.5pt] (12.161,3.079) -- (12.761,3.079) -- (12.761,3.679) -- (12.161,3.679) -- cycle;
\node at (12.461,3.379) {\scalebox{1.617}{\small b}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (4.252,5.169) -- (4.852,5.169) -- (4.852,5.769) -- (4.252,5.769) -- cycle;
\node at (4.552,5.469) {\scalebox{1.617}{\small c}};
\filldraw[fill=red!16, draw=black, line width=0.5pt] (6.816,6.111) -- (7.416,6.111) -- (7.416,6.711) -- (6.816,6.711) -- cycle;
\node at (7.116,6.411) {\scalebox{1.617}{\small d}};
\node at (6.405,7.602) {\scalebox{1.617}{\small A}};
\node at (1.296,7.343) {\scalebox{1.617}{\small B}};
\node at (3.78,6.884) {\scalebox{1.617}{\small C}};
\node at (12.002,4.602) {\scalebox{1.617}{\small D}};
\node at (9.212,4.522) {\scalebox{1.617}{\small E}};
\node at (0.765,4.293) {\scalebox{1.617}{\small F}};
\node at (7.209,3.977) {\scalebox{1.617}{\small G}};
\node at (3.449,3.925) {\scalebox{1.617}{\small H}};
\node at (0.855,2.004) {\scalebox{1.617}{\small I}};
\node at (11.58,1.316) {\scalebox{1.617}{\small J}};
\node at (3.268,1.069) {\scalebox{1.617}{\small K}};
\node at (6.519,0.95) {\scalebox{1.617}{\small L}};
\draw[line width=0.5pt] (12.684,6.049) -- (12.684,7.149);
\draw[line width=0.5pt] (12.453,6.764) -- (12.684,7.149) -- (12.915,6.764);
\draw[line width=0.5pt] (12.382,6.599) -- (12.987,6.599);
\node at (12.684,7.534) {\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 & C, D, F, G, K & 5 \\ \hline
clairs & A, B, E, H, I, J, L & 7 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{7} clairs. Il y en a donc \textbf{2 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 12 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 7 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non plus.} Une tournée n'a qu'un départ et qu'une arrivée : elle ne peut absorber qu'\textbf{un} carrefour clair en trop. Il y en a 2.}\par\addvspace{0.4em}
\end{document}
