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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 & ⑩ rue des Jardins & Ⓐ Place de la Fontaine & Ⓚ Place du Marché \\ \hline
① rue des Lilas & ⑪ sentier des Prés & Ⓑ Rond-point de la Gare & Ⓛ Carrefour Saint-Roch \\ \hline
② chemin Creux & ⑫ avenue de la Gare & Ⓒ Place Sainte-Croix & Ⓜ Place aux Herbes \\ \hline
③ rue Verte & ⑬ rue de la Mairie & Ⓓ Place des Tanneurs & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ rue des Écoles & ⑭ rue du Four & Ⓔ Carrefour de la Mairie & \textbf{a} l'école \\ \hline
⑤ quai aux Fleurs & ⑮ boulevard du Parc & Ⓕ Place du Vieux-Puits & \textbf{b} la pharmacie \\ \hline
⑥ allée des Peupliers & ⑯ venelle du Chat & Ⓖ Carrefour de l'Hôpital & \textbf{c} la salle des fêtes \\ \hline
⑦ rue Haute & ⑰ chemin de Ronde & Ⓗ Carrefour du Lavoir & \textbf{d} la bibliothèque \\ \hline
⑧ rue Pasteur & ⑱ rue Neuve & Ⓘ Carrefour du Moulin &  \\ \hline
⑨ allée des Tilleuls & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Rond-point de la Poste &  \\ \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] (8.393,10.28) -- (8.364,10.422) -- (8.283,10.544) -- (8.162,10.624) -- (8.019,10.653) -- (7.876,10.624) -- (7.755,10.544) -- (7.675,10.422) -- (7.646,10.28) -- (7.675,10.137) -- (7.755,10.016) -- (7.876,9.935) -- (8.019,9.906) -- (8.162,9.935) -- (8.283,10.016) -- (8.364,10.137) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.556,10.273) -- (11.528,10.416) -- (11.447,10.537) -- (11.326,10.618) -- (11.183,10.646) -- (11.04,10.618) -- (10.919,10.537) -- (10.838,10.416) -- (10.81,10.273) -- (10.838,10.13) -- (10.919,10.009) -- (11.04,9.928) -- (11.183,9.9) -- (11.326,9.928) -- (11.447,10.009) -- (11.528,10.13) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.685,7.889) -- (10.657,8.031) -- (10.576,8.152) -- (10.455,8.233) -- (10.312,8.262) -- (10.169,8.233) -- (10.048,8.152) -- (9.967,8.031) -- (9.939,7.889) -- (9.967,7.746) -- (10.048,7.625) -- (10.169,7.544) -- (10.312,7.515) -- (10.455,7.544) -- (10.576,7.625) -- (10.657,7.746) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.631,7.83) -- (4.602,7.973) -- (4.521,8.094) -- (4.4,8.175) -- (4.258,8.203) -- (4.115,8.175) -- (3.994,8.094) -- (3.913,7.973) -- (3.884,7.83) -- (3.913,7.687) -- (3.994,7.566) -- (4.115,7.485) -- (4.258,7.457) -- (4.4,7.485) -- (4.521,7.566) -- (4.602,7.687) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.782,7.709) -- (1.754,7.852) -- (1.673,7.973) -- (1.552,8.054) -- (1.409,8.083) -- (1.266,8.054) -- (1.145,7.973) -- (1.064,7.852) -- (1.036,7.709) -- (1.064,7.566) -- (1.145,7.445) -- (1.266,7.364) -- (1.409,7.336) -- (1.552,7.364) -- (1.673,7.445) -- (1.754,7.566) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.213,7.495) -- (8.185,7.637) -- (8.104,7.758) -- (7.983,7.839) -- (7.84,7.868) -- (7.697,7.839) -- (7.576,7.758) -- (7.495,7.637) -- (7.467,7.495) -- (7.495,7.352) -- (7.576,7.231) -- (7.697,7.15) -- (7.84,7.121) -- (7.983,7.15) -- (8.104,7.231) -- (8.185,7.352) -- cycle;
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\filldraw[fill=white, draw=black, line width=0.5pt] (1.769,4.503) -- (1.741,4.646) -- (1.66,4.767) -- (1.539,4.848) -- (1.396,4.877) -- (1.253,4.848) -- (1.132,4.767) -- (1.051,4.646) -- (1.023,4.503) -- (1.051,4.36) -- (1.132,4.239) -- (1.253,4.158) -- (1.396,4.13) -- (1.539,4.158) -- (1.66,4.239) -- (1.741,4.36) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.907,4.376) -- (7.879,4.519) -- (7.798,4.64) -- (7.677,4.721) -- (7.534,4.749) -- (7.391,4.721) -- (7.27,4.64) -- (7.189,4.519) -- (7.161,4.376) -- (7.189,4.233) -- (7.27,4.112) -- (7.391,4.031) -- (7.534,4.003) -- (7.677,4.031) -- (7.798,4.112) -- (7.879,4.233) -- cycle;
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\node at (9.602,10.53) {\scalebox{1.593}{\small 3}};
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\node at (8.428,8.855) {\scalebox{1.593}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.01,8.6) -- (11.129,8.619) -- (11.236,8.674) -- (11.321,8.759) -- (11.376,8.866) -- (11.395,8.985) -- (11.376,9.104) -- (11.321,9.211) -- (11.236,9.296) -- (11.129,9.351) -- (11.01,9.369) -- (11.01,9.369) -- (10.891,9.351) -- (10.784,9.296) -- (10.699,9.211) -- (10.644,9.104) -- (10.626,8.985) -- (10.644,8.866) -- (10.699,8.759) -- (10.784,8.674) -- (10.891,8.619) -- (11.01,8.6) -- cycle;
\node at (11.01,8.985) {\scalebox{1.593}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.099,7.808) -- (6.217,7.827) -- (6.325,7.882) -- (6.41,7.967) -- (6.464,8.074) -- (6.483,8.193) -- (6.464,8.312) -- (6.41,8.419) -- (6.325,8.504) -- (6.217,8.559) -- (6.099,8.578) -- (6.099,8.578) -- (5.98,8.559) -- (5.873,8.504) -- (5.788,8.419) -- (5.733,8.312) -- (5.714,8.193) -- (5.733,8.074) -- (5.788,7.967) -- (5.873,7.882) -- (5.98,7.827) -- (6.099,7.808) -- cycle;
\node at (6.099,8.193) {\scalebox{1.593}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.524,6.069) -- (4.643,6.087) -- (4.75,6.142) -- (4.835,6.227) -- (4.89,6.334) -- (4.908,6.453) -- (4.89,6.572) -- (4.835,6.679) -- (4.75,6.764) -- (4.643,6.819) -- (4.524,6.838) -- (4.524,6.838) -- (4.405,6.819) -- (4.298,6.764) -- (4.213,6.679) -- (4.158,6.572) -- (4.139,6.453) -- (4.158,6.334) -- (4.213,6.227) -- (4.298,6.142) -- (4.405,6.087) -- (4.524,6.069) -- cycle;
\node at (4.524,6.453) {\scalebox{1.593}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.812,7.887) -- (2.931,7.906) -- (3.038,7.96) -- (3.123,8.045) -- (3.178,8.153) -- (3.196,8.271) -- (3.178,8.39) -- (3.123,8.497) -- (3.038,8.583) -- (2.931,8.637) -- (2.812,8.656) -- (2.812,8.656) -- (2.693,8.637) -- (2.586,8.583) -- (2.501,8.497) -- (2.446,8.39) -- (2.427,8.271) -- (2.446,8.153) -- (2.501,8.045) -- (2.586,7.96) -- (2.693,7.906) -- (2.812,7.887) -- cycle;
\node at (2.812,8.271) {\scalebox{1.593}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.655,5.721) -- (1.774,5.74) -- (1.881,5.794) -- (1.966,5.879) -- (2.021,5.986) -- (2.039,6.105) -- (2.021,6.224) -- (1.966,6.331) -- (1.881,6.416) -- (1.774,6.471) -- (1.655,6.49) -- (1.655,6.49) -- (1.536,6.471) -- (1.429,6.416) -- (1.344,6.331) -- (1.289,6.224) -- (1.27,6.105) -- (1.289,5.986) -- (1.344,5.879) -- (1.429,5.794) -- (1.536,5.74) -- (1.655,5.721) -- cycle;
\node at (1.655,6.105) {\scalebox{1.593}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.355,6.694) -- (9.474,6.713) -- (9.581,6.768) -- (9.666,6.853) -- (9.72,6.96) -- (9.739,7.079) -- (9.72,7.198) -- (9.666,7.305) -- (9.581,7.39) -- (9.474,7.444) -- (9.355,7.463) -- (8.993,7.463) -- (8.874,7.444) -- (8.767,7.39) -- (8.682,7.305) -- (8.627,7.198) -- (8.608,7.079) -- (8.627,6.96) -- (8.682,6.853) -- (8.767,6.768) -- (8.874,6.713) -- (8.993,6.694) -- cycle;
\node at (9.174,7.079) {\scalebox{1.593}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.122,5.526) -- (8.241,5.545) -- (8.348,5.599) -- (8.433,5.684) -- (8.488,5.792) -- (8.507,5.91) -- (8.488,6.029) -- (8.433,6.136) -- (8.348,6.222) -- (8.241,6.276) -- (8.122,6.295) -- (7.76,6.295) -- (7.641,6.276) -- (7.534,6.222) -- (7.449,6.136) -- (7.395,6.029) -- (7.376,5.91) -- (7.395,5.792) -- (7.449,5.684) -- (7.534,5.599) -- (7.641,5.545) -- (7.76,5.526) -- cycle;
\node at (7.941,5.91) {\scalebox{1.593}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.181,4.854) -- (6.3,4.873) -- (6.407,4.927) -- (6.493,5.013) -- (6.547,5.12) -- (6.566,5.239) -- (6.547,5.357) -- (6.493,5.465) -- (6.407,5.55) -- (6.3,5.604) -- (6.181,5.623) -- (5.82,5.623) -- (5.701,5.604) -- (5.593,5.55) -- (5.508,5.465) -- (5.454,5.357) -- (5.435,5.239) -- (5.454,5.12) -- (5.508,5.013) -- (5.593,4.927) -- (5.701,4.873) -- (5.82,4.854) -- cycle;
\node at (6.0,5.239) {\scalebox{1.593}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.737,3.133) -- (4.856,3.152) -- (4.963,3.207) -- (5.048,3.292) -- (5.103,3.399) -- (5.122,3.518) -- (5.103,3.637) -- (5.048,3.744) -- (4.963,3.829) -- (4.856,3.884) -- (4.737,3.902) -- (4.375,3.902) -- (4.256,3.884) -- (4.149,3.829) -- (4.064,3.744) -- (4.009,3.637) -- (3.991,3.518) -- (4.009,3.399) -- (4.064,3.292) -- (4.149,3.207) -- (4.256,3.152) -- (4.375,3.133) -- cycle;
\node at (4.556,3.518) {\scalebox{1.593}{\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, C, F, G, H, K, M & 7 \\ \hline
clairs & B, D, E, I, J, 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{Non plus.} Avec un carrefour sombre de plus, la tournée doit \textbf{partir et arriver} sur un carrefour sombre. Or l'école est en \textbf{J}, qui est clair.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Une tournée existe pourtant dans ce quartier : il suffirait d'installer l'école sur un carrefour sombre. Ce n'est pas le plan qui est en cause, c'est l'endroit d'où l'on part.}\par\addvspace{0.4em}
\end{document}
