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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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\noindent \textbf{Le car part de l'école, au carrefour I, 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 Stade & Ⓐ Place de la Fontaine & Ⓚ Place de l'Église \\ \hline
① avenue de la Gare & ⑪ rue Neuve & Ⓑ Fontaine Notre-Dame & Ⓛ Placette des Ormes \\ \hline
② rue Pasteur & ⑫ sentier des Prés & Ⓒ Rond-point de la Poste & Ⓜ Coin du Lavoir \\ \hline
③ chemin Creux & ⑬ chemin du Lavoir & Ⓓ Carrefour Saint-Roch & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ rue des Jardins & ⑭ rue Basse & Ⓔ Place Sainte-Croix & \textbf{a} l'école \\ \hline
⑤ rue de l'Église & ⑮ venelle du Chat & Ⓕ Rond-point du Pont & \textbf{b} la mairie \\ \hline
⑥ rue du Puits & ⑯ allée des Peupliers & Ⓖ Rond-point des Vaches & \textbf{c} le lavoir \\ \hline
⑦ rue de la Mairie & ⑰ quai aux Fleurs & Ⓗ Carrefour des Tilleuls & \textbf{d} la gendarmerie \\ \hline
⑧ rue du Moulin & ⑱ rue Verte & Ⓘ Place du Vieux-Puits &  \\ \hline
⑨ rue des Lilas & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Rond-point de la Gare &  \\ \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] (7.737,11.224) -- (7.708,11.367) -- (7.627,11.488) -- (7.506,11.569) -- (7.364,11.597) -- (7.221,11.569) -- (7.1,11.488) -- (7.019,11.367) -- (6.99,11.224) -- (7.019,11.081) -- (7.1,10.96) -- (7.221,10.879) -- (7.364,10.851) -- (7.506,10.879) -- (7.627,10.96) -- (7.708,11.081) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.712,8.739) -- (10.684,8.882) -- (10.603,9.003) -- (10.482,9.084) -- (10.339,9.112) -- (10.196,9.084) -- (10.075,9.003) -- (9.994,8.882) -- (9.966,8.739) -- (9.994,8.596) -- (10.075,8.475) -- (10.196,8.394) -- (10.339,8.366) -- (10.482,8.394) -- (10.603,8.475) -- (10.684,8.596) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.752,8.454) -- (1.724,8.597) -- (1.643,8.718) -- (1.522,8.799) -- (1.379,8.827) -- (1.236,8.799) -- (1.115,8.718) -- (1.034,8.597) -- (1.006,8.454) -- (1.034,8.311) -- (1.115,8.19) -- (1.236,8.109) -- (1.379,8.081) -- (1.522,8.109) -- (1.643,8.19) -- (1.724,8.311) -- cycle;
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\filldraw[fill=white, draw=black, line width=0.5pt] (5.018,8.24) -- (4.989,8.382) -- (4.908,8.504) -- (4.787,8.584) -- (4.644,8.613) -- (4.502,8.584) -- (4.381,8.504) -- (4.3,8.382) -- (4.271,8.24) -- (4.3,8.097) -- (4.381,7.976) -- (4.502,7.895) -- (4.644,7.866) -- (4.787,7.895) -- (4.908,7.976) -- (4.989,8.097) -- cycle;
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\filldraw[fill=black!38, draw=black, line width=0.5pt] (10.655,5.495) -- (10.626,5.637) -- (10.545,5.758) -- (10.424,5.839) -- (10.282,5.868) -- (10.139,5.839) -- (10.018,5.758) -- (9.937,5.637) -- (9.908,5.495) -- (9.937,5.352) -- (10.018,5.231) -- (10.139,5.15) -- (10.282,5.121) -- (10.424,5.15) -- (10.545,5.231) -- (10.626,5.352) -- cycle;
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\node at (8.865,11.863) {\scalebox{1.506}{\small 4}};
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\node at (7.879,9.88) {\scalebox{1.506}{\small 5}};
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\node at (10.828,7.108) {\scalebox{1.506}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.028,8.211) -- (3.147,8.23) -- (3.254,8.285) -- (3.339,8.37) -- (3.394,8.477) -- (3.413,8.596) -- (3.394,8.715) -- (3.339,8.822) -- (3.254,8.907) -- (3.147,8.962) -- (3.028,8.98) -- (3.028,8.98) -- (2.909,8.962) -- (2.802,8.907) -- (2.717,8.822) -- (2.662,8.715) -- (2.643,8.596) -- (2.662,8.477) -- (2.717,8.37) -- (2.802,8.285) -- (2.909,8.23) -- (3.028,8.211) -- cycle;
\node at (3.028,8.596) {\scalebox{1.506}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.889,6.207) -- (2.008,6.225) -- (2.115,6.28) -- (2.2,6.365) -- (2.255,6.472) -- (2.273,6.591) -- (2.255,6.71) -- (2.2,6.817) -- (2.115,6.902) -- (2.008,6.957) -- (1.889,6.976) -- (1.889,6.976) -- (1.77,6.957) -- (1.663,6.902) -- (1.578,6.817) -- (1.523,6.71) -- (1.504,6.591) -- (1.523,6.472) -- (1.578,6.365) -- (1.663,6.28) -- (1.77,6.225) -- (1.889,6.207) -- cycle;
\node at (1.889,6.591) {\scalebox{1.506}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.162,7.722) -- (9.28,7.741) -- (9.388,7.795) -- (9.473,7.88) -- (9.527,7.987) -- (9.546,8.106) -- (9.527,8.225) -- (9.473,8.332) -- (9.388,8.417) -- (9.28,8.472) -- (9.162,8.491) -- (9.162,8.491) -- (9.043,8.472) -- (8.936,8.417) -- (8.85,8.332) -- (8.796,8.225) -- (8.777,8.106) -- (8.796,7.987) -- (8.85,7.88) -- (8.936,7.795) -- (9.043,7.741) -- (9.162,7.722) -- cycle;
\node at (9.162,8.106) {\scalebox{1.506}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.544,6.641) -- (8.663,6.659) -- (8.77,6.714) -- (8.855,6.799) -- (8.91,6.906) -- (8.928,7.025) -- (8.91,7.144) -- (8.855,7.251) -- (8.77,7.336) -- (8.663,7.391) -- (8.544,7.41) -- (8.182,7.41) -- (8.063,7.391) -- (7.956,7.336) -- (7.871,7.251) -- (7.816,7.144) -- (7.798,7.025) -- (7.816,6.906) -- (7.871,6.799) -- (7.956,6.714) -- (8.063,6.659) -- (8.182,6.641) -- cycle;
\node at (8.363,7.025) {\scalebox{1.506}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.469,7.439) -- (6.588,7.458) -- (6.695,7.513) -- (6.78,7.598) -- (6.835,7.705) -- (6.854,7.824) -- (6.835,7.942) -- (6.78,8.05) -- (6.695,8.135) -- (6.588,8.189) -- (6.469,8.208) -- (6.107,8.208) -- (5.989,8.189) -- (5.881,8.135) -- (5.796,8.05) -- (5.742,7.942) -- (5.723,7.824) -- (5.742,7.705) -- (5.796,7.598) -- (5.881,7.513) -- (5.989,7.458) -- (6.107,7.439) -- cycle;
\node at (6.288,7.824) {\scalebox{1.506}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.001,6.043) -- (5.12,6.062) -- (5.227,6.117) -- (5.313,6.202) -- (5.367,6.309) -- (5.386,6.428) -- (5.367,6.547) -- (5.313,6.654) -- (5.227,6.739) -- (5.12,6.794) -- (5.001,6.812) -- (4.64,6.812) -- (4.521,6.794) -- (4.413,6.739) -- (4.328,6.654) -- (4.274,6.547) -- (4.255,6.428) -- (4.274,6.309) -- (4.328,6.202) -- (4.413,6.117) -- (4.521,6.062) -- (4.64,6.043) -- cycle;
\node at (4.82,6.428) {\scalebox{1.506}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.592,4.518) -- (9.711,4.537) -- (9.818,4.591) -- (9.903,4.676) -- (9.958,4.783) -- (9.977,4.902) -- (9.958,5.021) -- (9.903,5.128) -- (9.818,5.213) -- (9.711,5.268) -- (9.592,5.287) -- (9.23,5.287) -- (9.111,5.268) -- (9.004,5.213) -- (8.919,5.128) -- (8.865,5.021) -- (8.846,4.902) -- (8.865,4.783) -- (8.919,4.676) -- (9.004,4.591) -- (9.111,4.537) -- (9.23,4.518) -- cycle;
\node at (9.411,4.902) {\scalebox{1.506}{\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, B, E, F, I, J, M & 7 \\ \hline
clairs & C, D, G, H, K, 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{I} :}\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 du Puits ⑥} vers le nord jusqu'à Carrefour Saint-Roch \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin Creux ③}, vers le nord, jusqu'à Fontaine Notre-Dame \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Jardins ④}, vers l'ouest, jusqu'à Rond-point de la Poste \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{avenue de la Gare ①}, vers l'ouest, jusqu'à Place de la Fontaine \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Pasteur ②}, vers le sud, jusqu'à Rond-point des Vaches \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de la Mairie ⑦}, vers l'ouest, jusqu'à Place Sainte-Croix \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Moulin ⑧}, vers le sud, jusqu'à Place de l'Église \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{quai aux Fleurs ⑰}, vers l'est, jusqu'à Rond-point de la Gare \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Peupliers ⑯}, vers le sud, jusqu'à Placette des Ormes \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Verte ⑱}, vers l'est, jusqu'à Coin du Lavoir \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ⑭}, vers le nord, jusqu'à Carrefour des Tilleuls \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Stade ⑩}, vers le nord, jusqu'à Rond-point du Pont \textbf{(F)}.\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}
