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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 J, 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 rue de l'Église & Ⓑ Coin du Lavoir & Ⓚ Carrefour Saint-Roch \\ \hline
① chemin des Vignes & ⑩ rue du Four & Ⓒ Rond-point de la Poste & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
② rue des Écoles & ⑪ rue du Stade & Ⓓ Rond-point du Stade & \textbf{a} l'école \\ \hline
③ rue de la Mairie & ⑫ chemin du Lavoir & Ⓔ Carrefour des Tilleuls & \textbf{b} le gymnase \\ \hline
④ rue Basse & ⑬ rue Neuve & Ⓕ Place du Marché & \textbf{c} l'église \\ \hline
⑤ chemin Creux & ⑭ sentier des Prés & Ⓖ Rond-point des Vaches & \textbf{d} le marché couvert \\ \hline
⑥ rue des Jardins & ⑮ rue du Pont & Ⓗ Carrefour du Lavoir &  \\ \hline
⑦ rue Verte & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Place de la Fontaine &  \\ \hline
⑧ rue des Lilas & Ⓐ Place aux Herbes & Ⓙ Place Sainte-Croix &  \\ \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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\noindent{\small\itshape Deux carrefours reliés par une rue n'ont jamais la même couleur.}\par\addvspace{0.4em}
\noindent \textbf{1.} Le coloriage est \textbf{forcé} : une fois la couleur d'un carrefour choisie, celle de tous les autres suit, de proche en proche.\par\addvspace{0.35em}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{|c|c|c|}
\hline
\cellcolor{blue!7}\bfseries Couleur & \cellcolor{blue!7}\bfseries Carrefours & \cellcolor{blue!7}\bfseries Nombre \\ \hline
sombres & B, D, E, H, K & 5 \\ \hline
clairs & A, C, F, G, I, J & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{6} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\par\addvspace{0.4em}
\noindent \textbf{2.}\par\addvspace{0.35em}
\noindent{\color{green!45!black}\textbf{Le raisonnement clé.} Chaque rue relie un carrefour sombre à un carrefour clair. Donc à chaque fois que le car roule, il \textbf{change de couleur} : ses arrêts sont sombre, clair, sombre, clair\dots{} \textbf{en alternance}.}\par\addvspace{0.4em}
\noindent{\color{red!80!black}\bfseries Une tournée qui dessert les 11 carrefours alterne donc les deux couleurs d'un bout à l'autre : elle en utilise \textbf{autant de chaque, à un près}. Et si elle en utilise un de plus d'une couleur, c'est forcément celle de son \textbf{départ} et de son \textbf{arrivée}.}\par\addvspace{0.4em}
\noindent \textbf{3.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non.} Un circuit revient à son point de départ : il compte autant d'arrêts sombres que de clairs. Or ce quartier a 5 carrefours sombres et 6 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Oui}, à condition de ne pas revenir. Un trajet possible au départ de \textbf{J} :}\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 Neuve ⑬} vers le nord jusqu'à Carrefour du Lavoir \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin du Lavoir ⑫}, vers l'est, jusqu'à Place du Marché \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de la Mairie ③}, vers le nord, jusqu'à Coin du Lavoir \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ④}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Jardins ⑥}, vers l'ouest, jusqu'à Rond-point du Stade \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin des Vignes ①}, vers l'ouest, jusqu'à Place aux Herbes \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Écoles ②}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ⑧}, vers l'est, jusqu'à Rond-point des Vaches \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Stade ⑪}, vers le sud, jusqu'à Carrefour Saint-Roch \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ⑭}, vers l'ouest, jusqu'à Place de la Fontaine \textbf{(I)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 11 carrefours ont été desservis une fois chacun. Le car termine loin de l'école : c'est permis ici, mais pas à la question 3.}\par\addvspace{0.4em}
\end{document}
