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\newcommand{\titre}{Le car de ramassage}
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    \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.
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\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 & ⑩ rue du Puits & Ⓐ Rond-point de la Gare & Ⓚ Place aux Herbes \\ \hline
① rue Neuve & ⑪ rue des Jardins & Ⓑ Carrefour de l'Hôpital & Ⓛ Carrefour du Lavoir \\ \hline
② chemin des Vignes & ⑫ rue des Lilas & Ⓒ Placette des Ormes & Ⓜ Place du Marché \\ \hline
③ chemin du Lavoir & ⑬ quai aux Fleurs & Ⓓ Place de l'Église & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ rue du Stade & ⑭ boulevard du Parc & Ⓔ Rond-point du Pont & \textbf{a} l'école \\ \hline
⑤ rue Pasteur & ⑮ chemin Creux & Ⓕ Carrefour des Tilleuls & \textbf{b} le gymnase \\ \hline
⑥ allée des Tilleuls & ⑯ rue de la Mairie & Ⓖ Carrefour Saint-Roch & \textbf{c} la bibliothèque \\ \hline
⑦ rue Haute & ⑰ rue du Moulin & Ⓗ Carrefour du Moulin & \textbf{d} la salle des fêtes \\ \hline
⑧ passage du Marché & ⑱ impasse Verte & Ⓘ Carrefour de la Mairie &  \\ \hline
⑨ allée des Peupliers & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Rond-point des Vaches &  \\ \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}
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\filldraw[fill=white, draw=black, line width=0.5pt] (8.604,10.532) -- (8.576,10.675) -- (8.495,10.796) -- (8.374,10.877) -- (8.231,10.905) -- (8.088,10.877) -- (7.967,10.796) -- (7.886,10.675) -- (7.858,10.532) -- (7.886,10.389) -- (7.967,10.268) -- (8.088,10.187) -- (8.231,10.159) -- (8.374,10.187) -- (8.495,10.268) -- (8.576,10.389) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.005,10.402) -- (10.976,10.545) -- (10.896,10.666) -- (10.775,10.747) -- (10.632,10.775) -- (10.489,10.747) -- (10.368,10.666) -- (10.287,10.545) -- (10.258,10.402) -- (10.287,10.259) -- (10.368,10.138) -- (10.489,10.057) -- (10.632,10.029) -- (10.775,10.057) -- (10.896,10.138) -- (10.976,10.259) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.536,9.98) -- (2.508,10.123) -- (2.427,10.244) -- (2.306,10.325) -- (2.163,10.353) -- (2.02,10.325) -- (1.899,10.244) -- (1.818,10.123) -- (1.79,9.98) -- (1.818,9.837) -- (1.899,9.716) -- (2.02,9.635) -- (2.163,9.607) -- (2.306,9.635) -- (2.427,9.716) -- (2.508,9.837) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.528,7.224) -- (8.499,7.367) -- (8.418,7.488) -- (8.297,7.569) -- (8.154,7.597) -- (8.012,7.569) -- (7.891,7.488) -- (7.81,7.367) -- (7.781,7.224) -- (7.81,7.081) -- (7.891,6.96) -- (8.012,6.879) -- (8.154,6.851) -- (8.297,6.879) -- (8.418,6.96) -- (8.499,7.081) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.756,7.089) -- (1.728,7.231) -- (1.647,7.352) -- (1.526,7.433) -- (1.383,7.462) -- (1.24,7.433) -- (1.119,7.352) -- (1.038,7.231) -- (1.009,7.089) -- (1.038,6.946) -- (1.119,6.825) -- (1.24,6.744) -- (1.383,6.715) -- (1.526,6.744) -- (1.647,6.825) -- (1.728,6.946) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.351,6.967) -- (5.323,7.11) -- (5.242,7.231) -- (5.121,7.312) -- (4.978,7.34) -- (4.835,7.312) -- (4.714,7.231) -- (4.633,7.11) -- (4.605,6.967) -- (4.633,6.824) -- (4.714,6.703) -- (4.835,6.622) -- (4.978,6.594) -- (5.121,6.622) -- (5.242,6.703) -- (5.323,6.824) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.756,6.839) -- (11.728,6.982) -- (11.647,7.103) -- (11.526,7.184) -- (11.383,7.212) -- (11.24,7.184) -- (11.119,7.103) -- (11.038,6.982) -- (11.009,6.839) -- (11.038,6.696) -- (11.119,6.575) -- (11.24,6.494) -- (11.383,6.466) -- (11.526,6.494) -- (11.647,6.575) -- (11.728,6.696) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.105,3.807) -- (8.077,3.95) -- (7.996,4.071) -- (7.875,4.152) -- (7.732,4.18) -- (7.589,4.152) -- (7.468,4.071) -- (7.387,3.95) -- (7.359,3.807) -- (7.387,3.664) -- (7.468,3.543) -- (7.589,3.462) -- (7.732,3.434) -- (7.875,3.462) -- (7.996,3.543) -- (8.077,3.664) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.062,3.744) -- (5.034,3.887) -- (4.953,4.008) -- (4.832,4.089) -- (4.689,4.117) -- (4.546,4.089) -- (4.425,4.008) -- (4.344,3.887) -- (4.316,3.744) -- (4.344,3.601) -- (4.425,3.48) -- (4.546,3.399) -- (4.689,3.371) -- (4.832,3.399) -- (4.953,3.48) -- (5.034,3.601) -- cycle;
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\node at (9.447,10.752) {\scalebox{1.617}{\small 3}};
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\node at (8.714,8.866) {\scalebox{1.617}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.236,8.284) -- (11.355,8.303) -- (11.463,8.358) -- (11.548,8.443) -- (11.602,8.55) -- (11.621,8.669) -- (11.602,8.788) -- (11.548,8.895) -- (11.463,8.98) -- (11.355,9.035) -- (11.236,9.053) -- (11.236,9.053) -- (11.118,9.035) -- (11.01,8.98) -- (10.925,8.895) -- (10.871,8.788) -- (10.852,8.669) -- (10.871,8.55) -- (10.925,8.443) -- (11.01,8.358) -- (11.118,8.303) -- (11.236,8.284) -- cycle;
\node at (11.236,8.669) {\scalebox{1.617}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.356,10.486) -- (3.475,10.505) -- (3.582,10.56) -- (3.667,10.645) -- (3.722,10.752) -- (3.741,10.871) -- (3.722,10.989) -- (3.667,11.097) -- (3.582,11.182) -- (3.475,11.236) -- (3.356,11.255) -- (3.356,11.255) -- (3.237,11.236) -- (3.13,11.182) -- (3.045,11.097) -- (2.991,10.989) -- (2.972,10.871) -- (2.991,10.752) -- (3.045,10.645) -- (3.13,10.56) -- (3.237,10.505) -- (3.356,10.486) -- cycle;
\node at (3.356,10.871) {\scalebox{1.617}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.025,8.082) -- (2.144,8.101) -- (2.251,8.155) -- (2.336,8.24) -- (2.391,8.347) -- (2.409,8.466) -- (2.391,8.585) -- (2.336,8.692) -- (2.251,8.777) -- (2.144,8.832) -- (2.025,8.851) -- (2.025,8.851) -- (1.906,8.832) -- (1.799,8.777) -- (1.714,8.692) -- (1.659,8.585) -- (1.64,8.466) -- (1.659,8.347) -- (1.714,8.24) -- (1.799,8.155) -- (1.906,8.101) -- (2.025,8.082) -- cycle;
\node at (2.025,8.466) {\scalebox{1.617}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.83,7.162) -- (9.949,7.181) -- (10.056,7.235) -- (10.141,7.321) -- (10.196,7.428) -- (10.215,7.547) -- (10.196,7.665) -- (10.141,7.773) -- (10.056,7.858) -- (9.949,7.912) -- (9.83,7.931) -- (9.83,7.931) -- (9.711,7.912) -- (9.604,7.858) -- (9.519,7.773) -- (9.464,7.665) -- (9.445,7.547) -- (9.464,7.428) -- (9.519,7.321) -- (9.604,7.235) -- (9.711,7.181) -- (9.83,7.162) -- cycle;
\node at (9.83,7.547) {\scalebox{1.617}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.184,5.101) -- (8.303,5.12) -- (8.41,5.175) -- (8.495,5.26) -- (8.55,5.367) -- (8.568,5.486) -- (8.55,5.605) -- (8.495,5.712) -- (8.41,5.797) -- (8.303,5.851) -- (8.184,5.87) -- (8.184,5.87) -- (8.065,5.851) -- (7.958,5.797) -- (7.873,5.712) -- (7.818,5.605) -- (7.799,5.486) -- (7.818,5.367) -- (7.873,5.26) -- (7.958,5.175) -- (8.065,5.12) -- (8.184,5.101) -- cycle;
\node at (8.184,5.486) {\scalebox{1.617}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.379,7.176) -- (3.498,7.195) -- (3.605,7.249) -- (3.69,7.334) -- (3.745,7.442) -- (3.764,7.56) -- (3.745,7.679) -- (3.69,7.786) -- (3.605,7.872) -- (3.498,7.926) -- (3.379,7.945) -- (3.017,7.945) -- (2.899,7.926) -- (2.791,7.872) -- (2.706,7.786) -- (2.652,7.679) -- (2.633,7.56) -- (2.652,7.442) -- (2.706,7.334) -- (2.791,7.249) -- (2.899,7.195) -- (3.017,7.176) -- cycle;
\node at (3.198,7.56) {\scalebox{1.617}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.909,5.027) -- (2.028,5.046) -- (2.135,5.1) -- (2.22,5.185) -- (2.275,5.292) -- (2.294,5.411) -- (2.275,5.53) -- (2.22,5.637) -- (2.135,5.722) -- (2.028,5.777) -- (1.909,5.796) -- (1.547,5.796) -- (1.429,5.777) -- (1.321,5.722) -- (1.236,5.637) -- (1.182,5.53) -- (1.163,5.411) -- (1.182,5.292) -- (1.236,5.185) -- (1.321,5.1) -- (1.429,5.046) -- (1.547,5.027) -- cycle;
\node at (1.728,5.411) {\scalebox{1.617}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.706,7.225) -- (6.824,7.244) -- (6.932,7.299) -- (7.017,7.384) -- (7.071,7.491) -- (7.09,7.61) -- (7.071,7.729) -- (7.017,7.836) -- (6.932,7.921) -- (6.824,7.976) -- (6.706,7.994) -- (6.344,7.994) -- (6.225,7.976) -- (6.118,7.921) -- (6.033,7.836) -- (5.978,7.729) -- (5.959,7.61) -- (5.978,7.491) -- (6.033,7.384) -- (6.118,7.299) -- (6.225,7.244) -- (6.344,7.225) -- cycle;
\node at (6.525,7.61) {\scalebox{1.617}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.264,4.949) -- (5.383,4.968) -- (5.491,5.022) -- (5.576,5.107) -- (5.63,5.214) -- (5.649,5.333) -- (5.63,5.452) -- (5.576,5.559) -- (5.491,5.644) -- (5.383,5.699) -- (5.264,5.718) -- (4.903,5.718) -- (4.784,5.699) -- (4.677,5.644) -- (4.592,5.559) -- (4.537,5.452) -- (4.518,5.333) -- (4.537,5.214) -- (4.592,5.107) -- (4.677,5.022) -- (4.784,4.968) -- (4.903,4.949) -- cycle;
\node at (5.084,5.333) {\scalebox{1.617}{\small 13}};
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\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 & A, C, E, F, J, L & 6 \\ \hline
clairs & B, D, G, H, I, K, M & 7 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{7} 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 13 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 6 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{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 la \textbf{rue Pasteur ⑤} vers le nord jusqu'à Placette des Ormes \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin du Lavoir ③}, vers l'ouest, jusqu'à Carrefour de l'Hôpital \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Neuve ①}, vers l'ouest, jusqu'à Rond-point de la Gare \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Tilleuls ⑥}, vers l'ouest, jusqu'à Place de l'Église \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Haute ⑦}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Jardins ⑪}, vers le sud, jusqu'à Place aux Herbes \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Moulin ⑰}, vers l'est, jusqu'à Rond-point des Vaches \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{quai aux Fleurs ⑬}, vers le nord, jusqu'à Carrefour Saint-Roch \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Lilas ⑫}, vers l'est, jusqu'à Rond-point du Pont \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Peupliers ⑨}, vers le sud, jusqu'à Carrefour de la Mairie \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin Creux ⑮}, vers l'est, jusqu'à Carrefour du Lavoir \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{impasse Verte ⑱}, vers le sud, jusqu'à Place du Marché \textbf{(M)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 13 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}
