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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 K, 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 du Pont & Ⓐ Rond-point de la Poste & Ⓙ Placette des Ormes \\ \hline
① sentier des Prés & ⑩ chemin des Vignes & Ⓑ Place aux Herbes & Ⓚ Coin du Lavoir \\ \hline
② rue du Puits & ⑪ chemin Creux & Ⓒ Place des Tanneurs & Ⓛ Fontaine Notre-Dame \\ \hline
③ rue des Écoles & ⑫ rue de la Poste & Ⓓ Carrefour du Moulin & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ passage du Marché & ⑬ rue de l'Église & Ⓔ Carrefour Saint-Roch & \textbf{a} l'école \\ \hline
⑤ allée des Tilleuls & ⑭ rue Verte & Ⓕ Rond-point du Stade & \textbf{b} le lavoir \\ \hline
⑥ rue du Stade & ⑮ rue de la Mairie & Ⓖ Place du Marché & \textbf{c} la boulangerie \\ \hline
⑦ rue Haute & ⑯ rue Pasteur & Ⓗ Carrefour des Tilleuls & \textbf{d} la mairie \\ \hline
⑧ rue des Jardins & \cellcolor{black!32}\bfseries Les carrefours & Ⓘ Rond-point du Pont &  \\ \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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\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, E, H, J, L & 6 \\ \hline
clairs & A, D, F, G, I, K & 6 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{6} 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 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.} Si l'on retire le carrefour \textbf{H}, le quartier tombe en 2 morceaux : toute route de l'un à l'autre passe par lui. Un circuit qui reviendrait à l'école devrait donc s'y arrêter au moins \textbf{deux fois} — c'est interdit.}\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{K} :}\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 ⑯} vers l'ouest jusqu'à Fontaine Notre-Dame \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Poste ⑫}, vers le nord, jusqu'à Place du Marché \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin Creux ⑪}, vers l'est, jusqu'à Carrefour des Tilleuls \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Haute ⑦}, vers le nord, jusqu'à Carrefour du Moulin \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{passage du Marché ④}, vers le nord, jusqu'à Place aux Herbes \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Écoles ③}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{sentier des Prés ①}, vers l'est, jusqu'à Place des Tanneurs \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Tilleuls ⑤}, vers le sud, jusqu'à Rond-point du Stade \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Jardins ⑧}, vers l'ouest, jusqu'à Carrefour Saint-Roch \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Pont ⑨}, vers le sud, jusqu'à Rond-point du Pont \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de la Mairie ⑮}, vers l'est, jusqu'à Placette des Ormes \textbf{(J)}.\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries Les 12 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}
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