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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 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 & ⑩ rue du Pont & Ⓐ Place du Marché & Ⓚ Place Sainte-Croix \\ \hline
① rue des Jardins & ⑪ rue Pasteur & Ⓑ Rond-point de la Gare & Ⓛ Fontaine Notre-Dame \\ \hline
② quai aux Fleurs & ⑫ rue du Moulin & Ⓒ Place du Vieux-Puits & Ⓜ Place de la Fontaine \\ \hline
③ sentier des Prés & ⑬ rue du Puits & Ⓓ Rond-point du Pont & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ rue de la Mairie & ⑭ passage du Marché & Ⓔ Carrefour du Moulin & \textbf{a} l'école \\ \hline
⑤ rue Verte & ⑮ rue du Four & Ⓕ Carrefour Saint-Roch & \textbf{b} la salle des fêtes \\ \hline
⑥ rue Haute & ⑯ chemin des Vignes & Ⓖ Rond-point des Vaches & \textbf{c} la mairie \\ \hline
⑦ rue des Écoles & ⑰ rue Basse & Ⓗ Placette des Ormes & \textbf{d} l'église \\ \hline
⑧ allée des Tilleuls & ⑱ avenue de la Gare & Ⓘ Carrefour des Tilleuls &  \\ \hline
⑨ rue Neuve & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Carrefour du Lavoir &  \\ \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] (2.016,11.552) -- (1.987,11.695) -- (1.906,11.816) -- (1.785,11.897) -- (1.642,11.925) -- (1.5,11.897) -- (1.379,11.816) -- (1.298,11.695) -- (1.269,11.552) -- (1.298,11.409) -- (1.379,11.288) -- (1.5,11.207) -- (1.642,11.179) -- (1.785,11.207) -- (1.906,11.288) -- (1.987,11.409) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.537,11.269) -- (4.509,11.412) -- (4.428,11.533) -- (4.307,11.614) -- (4.164,11.642) -- (4.021,11.614) -- (3.9,11.533) -- (3.819,11.412) -- (3.791,11.269) -- (3.819,11.126) -- (3.9,11.005) -- (4.021,10.924) -- (4.164,10.896) -- (4.307,10.924) -- (4.428,11.005) -- (4.509,11.126) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.664,11.064) -- (11.636,11.207) -- (11.555,11.328) -- (11.434,11.409) -- (11.291,11.437) -- (11.148,11.409) -- (11.027,11.328) -- (10.946,11.207) -- (10.918,11.064) -- (10.946,10.921) -- (11.027,10.8) -- (11.148,10.719) -- (11.291,10.691) -- (11.434,10.719) -- (11.555,10.8) -- (11.636,10.921) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.06,11.035) -- (8.031,11.178) -- (7.95,11.299) -- (7.829,11.38) -- (7.686,11.409) -- (7.544,11.38) -- (7.423,11.299) -- (7.342,11.178) -- (7.313,11.035) -- (7.342,10.893) -- (7.423,10.772) -- (7.544,10.691) -- (7.686,10.662) -- (7.829,10.691) -- (7.95,10.772) -- (8.031,10.893) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.702,8.152) -- (8.674,8.294) -- (8.593,8.415) -- (8.472,8.496) -- (8.329,8.525) -- (8.186,8.496) -- (8.065,8.415) -- (7.984,8.294) -- (7.956,8.152) -- (7.984,8.009) -- (8.065,7.888) -- (8.186,7.807) -- (8.329,7.778) -- (8.472,7.807) -- (8.593,7.888) -- (8.674,8.009) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.677,7.932) -- (4.649,8.075) -- (4.568,8.196) -- (4.447,8.277) -- (4.304,8.305) -- (4.161,8.277) -- (4.04,8.196) -- (3.959,8.075) -- (3.931,7.932) -- (3.959,7.789) -- (4.04,7.668) -- (4.161,7.587) -- (4.304,7.559) -- (4.447,7.587) -- (4.568,7.668) -- (4.649,7.789) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.855,7.684) -- (1.826,7.827) -- (1.745,7.948) -- (1.624,8.029) -- (1.481,8.058) -- (1.339,8.029) -- (1.217,7.948) -- (1.137,7.827) -- (1.108,7.684) -- (1.137,7.542) -- (1.217,7.421) -- (1.339,7.34) -- (1.481,7.311) -- (1.624,7.34) -- (1.745,7.421) -- (1.826,7.542) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.461,7.362) -- (11.433,7.505) -- (11.352,7.626) -- (11.231,7.707) -- (11.088,7.736) -- (10.945,7.707) -- (10.824,7.626) -- (10.743,7.505) -- (10.715,7.362) -- (10.743,7.22) -- (10.824,7.099) -- (10.945,7.018) -- (11.088,6.989) -- (11.231,7.018) -- (11.352,7.099) -- (11.433,7.22) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.677,4.925) -- (4.649,5.068) -- (4.568,5.189) -- (4.447,5.27) -- (4.304,5.298) -- (4.161,5.27) -- (4.04,5.189) -- (3.959,5.068) -- (3.931,4.925) -- (3.959,4.782) -- (4.04,4.661) -- (4.161,4.58) -- (4.304,4.552) -- (4.447,4.58) -- (4.568,4.661) -- (4.649,4.782) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.782,4.123) -- (7.754,4.266) -- (7.673,4.387) -- (7.552,4.468) -- (7.409,4.496) -- (7.266,4.468) -- (7.145,4.387) -- (7.064,4.266) -- (7.036,4.123) -- (7.064,3.98) -- (7.145,3.859) -- (7.266,3.778) -- (7.409,3.75) -- (7.552,3.778) -- (7.673,3.859) -- (7.754,3.98) -- cycle;
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\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.806,1.638) -- (7.777,1.781) -- (7.696,1.902) -- (7.575,1.983) -- (7.432,2.011) -- (7.29,1.983) -- (7.169,1.902) -- (7.088,1.781) -- (7.059,1.638) -- (7.088,1.495) -- (7.169,1.374) -- (7.29,1.293) -- (7.432,1.265) -- (7.575,1.293) -- (7.696,1.374) -- (7.777,1.495) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.803,1.552) -- (4.775,1.695) -- (4.694,1.816) -- (4.573,1.897) -- (4.43,1.925) -- (4.287,1.897) -- (4.166,1.816) -- (4.085,1.695) -- (4.057,1.552) -- (4.085,1.409) -- (4.166,1.288) -- (4.287,1.207) -- (4.43,1.179) -- (4.573,1.207) -- (4.694,1.288) -- (4.775,1.409) -- cycle;
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\node at (5.941,11.391) {\scalebox{1.606}{\small 3}};
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\node at (4.756,9.622) {\scalebox{1.606}{\small 4}};
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\node at (11.421,9.201) {\scalebox{1.606}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.484,11.198) -- (9.603,11.217) -- (9.71,11.272) -- (9.796,11.357) -- (9.85,11.464) -- (9.869,11.583) -- (9.85,11.702) -- (9.796,11.809) -- (9.71,11.894) -- (9.603,11.949) -- (9.484,11.968) -- (9.484,11.968) -- (9.366,11.949) -- (9.258,11.894) -- (9.173,11.809) -- (9.119,11.702) -- (9.1,11.583) -- (9.119,11.464) -- (9.173,11.357) -- (9.258,11.272) -- (9.366,11.217) -- (9.484,11.198) -- cycle;
\node at (9.484,11.583) {\scalebox{1.606}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.264,9.266) -- (8.383,9.285) -- (8.49,9.34) -- (8.575,9.425) -- (8.63,9.532) -- (8.649,9.651) -- (8.63,9.769) -- (8.575,9.877) -- (8.49,9.962) -- (8.383,10.016) -- (8.264,10.035) -- (8.264,10.035) -- (8.145,10.016) -- (8.038,9.962) -- (7.953,9.877) -- (7.898,9.769) -- (7.88,9.651) -- (7.898,9.532) -- (7.953,9.425) -- (8.038,9.34) -- (8.145,9.285) -- (8.264,9.266) -- cycle;
\node at (8.264,9.651) {\scalebox{1.606}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.847,7.856) -- (9.966,7.875) -- (10.073,7.929) -- (10.158,8.014) -- (10.213,8.122) -- (10.231,8.241) -- (10.213,8.359) -- (10.158,8.467) -- (10.073,8.552) -- (9.966,8.606) -- (9.847,8.625) -- (9.847,8.625) -- (9.728,8.606) -- (9.621,8.552) -- (9.536,8.467) -- (9.481,8.359) -- (9.462,8.241) -- (9.481,8.122) -- (9.536,8.014) -- (9.621,7.929) -- (9.728,7.875) -- (9.847,7.856) -- cycle;
\node at (9.847,8.241) {\scalebox{1.606}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.078,5.705) -- (8.196,5.724) -- (8.304,5.779) -- (8.389,5.864) -- (8.443,5.971) -- (8.462,6.09) -- (8.443,6.208) -- (8.389,6.316) -- (8.304,6.401) -- (8.196,6.455) -- (8.078,6.474) -- (8.078,6.474) -- (7.959,6.455) -- (7.852,6.401) -- (7.767,6.316) -- (7.712,6.208) -- (7.693,6.09) -- (7.712,5.971) -- (7.767,5.864) -- (7.852,5.779) -- (7.959,5.724) -- (8.078,5.705) -- cycle;
\node at (8.078,6.09) {\scalebox{1.606}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.468,8.207) -- (6.586,8.226) -- (6.694,8.281) -- (6.779,8.366) -- (6.833,8.473) -- (6.852,8.592) -- (6.833,8.71) -- (6.779,8.818) -- (6.694,8.903) -- (6.586,8.957) -- (6.468,8.976) -- (6.106,8.976) -- (5.987,8.957) -- (5.88,8.903) -- (5.795,8.818) -- (5.74,8.71) -- (5.721,8.592) -- (5.74,8.473) -- (5.795,8.366) -- (5.88,8.281) -- (5.987,8.226) -- (6.106,8.207) -- cycle;
\node at (6.287,8.592) {\scalebox{1.606}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.745,6.044) -- (4.864,6.063) -- (4.971,6.117) -- (5.057,6.202) -- (5.111,6.31) -- (5.13,6.428) -- (5.111,6.547) -- (5.057,6.654) -- (4.971,6.74) -- (4.864,6.794) -- (4.745,6.813) -- (4.384,6.813) -- (4.265,6.794) -- (4.158,6.74) -- (4.072,6.654) -- (4.018,6.547) -- (3.999,6.428) -- (4.018,6.31) -- (4.072,6.202) -- (4.158,6.117) -- (4.265,6.063) -- (4.384,6.044) -- cycle;
\node at (4.564,6.428) {\scalebox{1.606}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.097,7.157) -- (3.216,7.176) -- (3.323,7.23) -- (3.408,7.316) -- (3.463,7.423) -- (3.482,7.542) -- (3.463,7.66) -- (3.408,7.768) -- (3.323,7.853) -- (3.216,7.907) -- (3.097,7.926) -- (2.735,7.926) -- (2.616,7.907) -- (2.509,7.853) -- (2.424,7.768) -- (2.369,7.66) -- (2.351,7.542) -- (2.369,7.423) -- (2.424,7.316) -- (2.509,7.23) -- (2.616,7.176) -- (2.735,7.157) -- cycle;
\node at (2.916,7.542) {\scalebox{1.606}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.065,4.951) -- (2.184,4.97) -- (2.291,5.024) -- (2.376,5.109) -- (2.431,5.216) -- (2.45,5.335) -- (2.431,5.454) -- (2.376,5.561) -- (2.291,5.646) -- (2.184,5.701) -- (2.065,5.72) -- (1.703,5.72) -- (1.585,5.701) -- (1.477,5.646) -- (1.392,5.561) -- (1.338,5.454) -- (1.319,5.335) -- (1.338,5.216) -- (1.392,5.109) -- (1.477,5.024) -- (1.585,4.97) -- (1.703,4.951) -- cycle;
\node at (1.884,5.335) {\scalebox{1.606}{\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 & B, C, E, G, I, L & 6 \\ \hline
clairs & A, D, F, H, J, 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{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 le \textbf{chemin des Vignes ⑯} vers le sud jusqu'à Fontaine Notre-Dame \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{avenue de la Gare ⑱}, vers l'ouest, jusqu'à Place de la Fontaine \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Four ⑮}, vers le nord, jusqu'à Carrefour des Tilleuls \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Pasteur ⑪}, vers le nord, jusqu'à Carrefour Saint-Roch \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Pont ⑩}, vers l'est, jusqu'à Carrefour du Moulin \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Tilleuls ⑧}, vers l'est, jusqu'à Placette des Ormes \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Verte ⑤}, vers le nord, jusqu'à Place du Vieux-Puits \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Haute ⑥}, vers l'ouest, jusqu'à Rond-point du Pont \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{sentier des Prés ③}, vers l'ouest, jusqu'à Rond-point de la Gare \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Jardins ①}, vers l'ouest, jusqu'à Place du Marché \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{quai aux Fleurs ②}, vers le sud, jusqu'à Rond-point des Vaches \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue du Puits ⑬}, vers le sud, jusqu'à Place Sainte-Croix \textbf{(K)}.\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}
