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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 M, 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 chemin du Lavoir & Ⓑ Placette des Ormes & Ⓛ Rond-point du Pont \\ \hline
① sentier des Prés & ⑪ allée des Tilleuls & Ⓒ Carrefour Saint-Roch & Ⓜ Place aux Herbes \\ \hline
② allée des Peupliers & ⑫ rue des Écoles & Ⓓ Fontaine Notre-Dame & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ rue du Moulin & ⑬ quai aux Fleurs & Ⓔ Rond-point de la Poste & \textbf{a} l'école \\ \hline
④ chemin des Vignes & ⑭ boulevard du Parc & Ⓕ Rond-point de la Gare & \textbf{b} la pharmacie \\ \hline
⑤ passage du Marché & ⑮ venelle du Chat & Ⓖ Rond-point des Vaches & \textbf{c} la gendarmerie \\ \hline
⑥ rue des Jardins & ⑯ chemin de Ronde & Ⓗ Carrefour de la Mairie & \textbf{d} le cimetière \\ \hline
⑦ chemin Creux & ⑰ rue Neuve & Ⓘ Carrefour des Tilleuls &  \\ \hline
⑧ rue Verte & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Place des Tanneurs &  \\ \hline
⑨ rue du Pont & Ⓐ Coin du Lavoir & Ⓚ 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}
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\filldraw[fill=white, draw=black, line width=0.5pt] (7.962,11.014) -- (7.934,11.156) -- (7.853,11.278) -- (7.732,11.358) -- (7.589,11.387) -- (7.446,11.358) -- (7.325,11.278) -- (7.244,11.156) -- (7.216,11.014) -- (7.244,10.871) -- (7.325,10.75) -- (7.446,10.669) -- (7.589,10.64) -- (7.732,10.669) -- (7.853,10.75) -- (7.934,10.871) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.301,8.749) -- (11.272,8.891) -- (11.191,9.012) -- (11.07,9.093) -- (10.927,9.122) -- (10.785,9.093) -- (10.664,9.012) -- (10.583,8.891) -- (10.554,8.749) -- (10.583,8.606) -- (10.664,8.485) -- (10.785,8.404) -- (10.927,8.375) -- (11.07,8.404) -- (11.191,8.485) -- (11.272,8.606) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.132,8.439) -- (2.103,8.582) -- (2.022,8.703) -- (1.901,8.784) -- (1.758,8.813) -- (1.615,8.784) -- (1.494,8.703) -- (1.413,8.582) -- (1.385,8.439) -- (1.413,8.297) -- (1.494,8.175) -- (1.615,8.095) -- (1.758,8.066) -- (1.901,8.095) -- (2.022,8.175) -- (2.103,8.297) -- cycle;
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\filldraw[fill=white, draw=black, line width=0.5pt] (5.665,8.259) -- (5.636,8.402) -- (5.555,8.523) -- (5.434,8.604) -- (5.291,8.632) -- (5.149,8.604) -- (5.028,8.523) -- (4.947,8.402) -- (4.918,8.259) -- (4.947,8.116) -- (5.028,7.995) -- (5.149,7.914) -- (5.291,7.886) -- (5.434,7.914) -- (5.555,7.995) -- (5.636,8.116) -- cycle;
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\filldraw[fill=white, draw=black, line width=0.5pt] (9.346,8.402) -- (9.465,8.421) -- (9.572,8.476) -- (9.657,8.561) -- (9.712,8.668) -- (9.731,8.787) -- (9.712,8.906) -- (9.657,9.013) -- (9.572,9.098) -- (9.465,9.153) -- (9.346,9.172) -- (9.346,9.172) -- (9.227,9.153) -- (9.12,9.098) -- (9.035,9.013) -- (8.981,8.906) -- (8.962,8.787) -- (8.981,8.668) -- (9.035,8.561) -- (9.12,8.476) -- (9.227,8.421) -- (9.346,8.402) -- cycle;
\node at (9.346,8.787) {\scalebox{1.596}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.552,8.394) -- (6.671,8.413) -- (6.778,8.467) -- (6.863,8.552) -- (6.918,8.659) -- (6.937,8.778) -- (6.918,8.897) -- (6.863,9.004) -- (6.778,9.089) -- (6.671,9.144) -- (6.552,9.163) -- (6.552,9.163) -- (6.433,9.144) -- (6.326,9.089) -- (6.241,9.004) -- (6.186,8.897) -- (6.167,8.778) -- (6.186,8.659) -- (6.241,8.552) -- (6.326,8.467) -- (6.433,8.413) -- (6.552,8.394) -- cycle;
\node at (6.552,8.778) {\scalebox{1.596}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.176,6.068) -- (5.295,6.087) -- (5.403,6.142) -- (5.488,6.227) -- (5.542,6.334) -- (5.561,6.453) -- (5.542,6.572) -- (5.488,6.679) -- (5.403,6.764) -- (5.295,6.818) -- (5.176,6.837) -- (5.176,6.837) -- (5.058,6.818) -- (4.95,6.764) -- (4.865,6.679) -- (4.811,6.572) -- (4.792,6.453) -- (4.811,6.334) -- (4.865,6.227) -- (4.95,6.142) -- (5.058,6.087) -- (5.176,6.068) -- cycle;
\node at (5.176,6.453) {\scalebox{1.596}{\small 9}};
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\node at (3.844,5.507) {\scalebox{1.596}{\small 10}};
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\node at (2.15,3.626) {\scalebox{1.596}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.518,5.031) -- (9.637,5.049) -- (9.744,5.104) -- (9.829,5.189) -- (9.884,5.296) -- (9.902,5.415) -- (9.884,5.534) -- (9.829,5.641) -- (9.744,5.726) -- (9.637,5.781) -- (9.518,5.8) -- (9.156,5.8) -- (9.037,5.781) -- (8.93,5.726) -- (8.845,5.641) -- (8.79,5.534) -- (8.771,5.415) -- (8.79,5.296) -- (8.845,5.189) -- (8.93,5.104) -- (9.037,5.049) -- (9.156,5.031) -- cycle;
\node at (9.337,5.415) {\scalebox{1.596}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.359,2.986) -- (8.478,3.004) -- (8.585,3.059) -- (8.67,3.144) -- (8.725,3.251) -- (8.744,3.37) -- (8.725,3.489) -- (8.67,3.596) -- (8.585,3.681) -- (8.478,3.736) -- (8.359,3.755) -- (7.997,3.755) -- (7.879,3.736) -- (7.771,3.681) -- (7.686,3.596) -- (7.632,3.489) -- (7.613,3.37) -- (7.632,3.251) -- (7.686,3.144) -- (7.771,3.059) -- (7.879,3.004) -- (7.997,2.986) -- cycle;
\node at (8.178,3.37) {\scalebox{1.596}{\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, D, E, I, J, K, M & 7 \\ \hline
clairs & B, C, F, G, H, L & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{7} carrefours sombres et \textbf{6} clairs. Il y en a donc \textbf{1 de plus} du côté des sombres.}\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 7 carrefours sombres et 6 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{M} :}\par\addvspace{0.4em}
\newpage
\noindent \textbf{Le trajet du car}\par\addvspace{0.35em}
\noindent Le car sort de l'école et prend le \textbf{quai aux Fleurs ⑬} vers le nord jusqu'à Carrefour de la Mairie \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{boulevard du Parc ⑭}, vers l'ouest, jusqu'à Carrefour des Tilleuls \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{venelle du Chat ⑮}, vers le sud, jusqu'à Rond-point du Pont \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin de Ronde ⑯}, vers l'ouest, jusqu'à Place du Marché \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Tilleuls ⑪}, vers le nord, jusqu'à Rond-point des Vaches \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Jardins ⑥}, vers le nord, jusqu'à Fontaine Notre-Dame \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{passage du Marché ⑤}, vers l'est, jusqu'à Rond-point de la Gare \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Verte ⑧}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Moulin ③}, vers le nord, jusqu'à Placette des Ormes \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Peupliers ②}, vers l'est, jusqu'à Coin du Lavoir \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ①}, vers le sud, jusqu'à Carrefour Saint-Roch \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin des Vignes ④}, vers le sud, jusqu'à Place des Tanneurs \textbf{(J)}.\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}
