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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 G, 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 venelle du Chat & Ⓑ Place de la Fontaine & Ⓛ Place du Marché \\ \hline
① rue Neuve & ⑪ rue du Moulin & Ⓒ Rond-point des Vaches & Ⓜ Rond-point du Stade \\ \hline
② rue Basse & ⑫ rue des Écoles & Ⓓ Carrefour du Moulin & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ passage du Marché & ⑬ allée des Peupliers & Ⓔ Rond-point du Pont & \textbf{a} l'école \\ \hline
④ rue de l'Église & ⑭ rue du Pont & Ⓕ Coin du Lavoir & \textbf{b} la bibliothèque \\ \hline
⑤ rue des Jardins & ⑮ boulevard du Parc & Ⓖ Carrefour Saint-Roch & \textbf{c} le gymnase \\ \hline
⑥ rue du Stade & ⑯ allée des Tilleuls & Ⓗ Carrefour de l'Hôpital & \textbf{d} le lavoir \\ \hline
⑦ chemin Creux & ⑰ impasse du Vieux-Four & Ⓘ Rond-point de la Gare &  \\ \hline
⑧ impasse des Roses & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Fontaine Notre-Dame &  \\ \hline
⑨ rue Pasteur & Ⓐ Place des Tanneurs & Ⓚ 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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\draw[line width=0.5pt] (1.336,4.949) -- (1.503,4.925) -- (1.67,4.905) -- (1.838,4.889) -- (2.006,4.877) -- (2.174,4.869) -- (2.343,4.865) -- (2.512,4.865) -- (2.68,4.869) -- (2.849,4.877) -- (3.017,4.888) -- (3.184,4.904) -- (3.352,4.924) -- (3.519,4.947) -- (3.685,4.975);
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\filldraw[fill=white, draw=black, line width=0.5pt] (1.567,11.523) -- (1.539,11.666) -- (1.458,11.787) -- (1.337,11.868) -- (1.194,11.897) -- (1.051,11.868) -- (0.93,11.787) -- (0.849,11.666) -- (0.821,11.523) -- (0.849,11.381) -- (0.93,11.26) -- (1.051,11.179) -- (1.194,11.15) -- (1.337,11.179) -- (1.458,11.26) -- (1.539,11.381) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.867,10.981) -- (7.839,11.124) -- (7.758,11.245) -- (7.637,11.326) -- (7.494,11.354) -- (7.351,11.326) -- (7.23,11.245) -- (7.149,11.124) -- (7.121,10.981) -- (7.149,10.838) -- (7.23,10.717) -- (7.351,10.636) -- (7.494,10.608) -- (7.637,10.636) -- (7.758,10.717) -- (7.839,10.838) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.038,10.857) -- (4.009,10.999) -- (3.928,11.121) -- (3.807,11.201) -- (3.664,11.23) -- (3.521,11.201) -- (3.4,11.121) -- (3.32,10.999) -- (3.291,10.857) -- (3.32,10.714) -- (3.4,10.593) -- (3.521,10.512) -- (3.664,10.483) -- (3.807,10.512) -- (3.928,10.593) -- (4.009,10.714) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.671,8.549) -- (4.643,8.692) -- (4.562,8.813) -- (4.441,8.894) -- (4.298,8.923) -- (4.155,8.894) -- (4.034,8.813) -- (3.953,8.692) -- (3.924,8.549) -- (3.953,8.406) -- (4.034,8.285) -- (4.155,8.204) -- (4.298,8.176) -- (4.441,8.204) -- (4.562,8.285) -- (4.643,8.406) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.463,8.39) -- (7.435,8.533) -- (7.354,8.654) -- (7.233,8.735) -- (7.09,8.763) -- (6.947,8.735) -- (6.826,8.654) -- (6.745,8.533) -- (6.717,8.39) -- (6.745,8.247) -- (6.826,8.126) -- (6.947,8.045) -- (7.09,8.017) -- (7.233,8.045) -- (7.354,8.126) -- (7.435,8.247) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.019,8.007) -- (10.99,8.15) -- (10.909,8.271) -- (10.788,8.352) -- (10.645,8.38) -- (10.503,8.352) -- (10.382,8.271) -- (10.301,8.15) -- (10.272,8.007) -- (10.301,7.864) -- (10.382,7.743) -- (10.503,7.662) -- (10.645,7.634) -- (10.788,7.662) -- (10.909,7.743) -- (10.99,7.864) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.76,7.939) -- (1.732,8.081) -- (1.651,8.203) -- (1.53,8.283) -- (1.387,8.312) -- (1.244,8.283) -- (1.123,8.203) -- (1.042,8.081) -- (1.014,7.939) -- (1.042,7.796) -- (1.123,7.675) -- (1.244,7.594) -- (1.387,7.565) -- (1.53,7.594) -- (1.651,7.675) -- (1.732,7.796) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.966,5.071) -- (7.937,5.214) -- (7.856,5.335) -- (7.735,5.416) -- (7.592,5.445) -- (7.45,5.416) -- (7.329,5.335) -- (7.248,5.214) -- (7.219,5.071) -- (7.248,4.928) -- (7.329,4.807) -- (7.45,4.726) -- (7.592,4.698) -- (7.735,4.726) -- (7.856,4.807) -- (7.937,4.928) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.058,4.975) -- (4.03,5.118) -- (3.949,5.239) -- (3.828,5.32) -- (3.685,5.348) -- (3.542,5.32) -- (3.421,5.239) -- (3.34,5.118) -- (3.312,4.975) -- (3.34,4.832) -- (3.421,4.711) -- (3.542,4.63) -- (3.685,4.602) -- (3.828,4.63) -- (3.949,4.711) -- (4.03,4.832) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.709,4.949) -- (1.681,5.092) -- (1.6,5.213) -- (1.479,5.294) -- (1.336,5.322) -- (1.193,5.294) -- (1.072,5.213) -- (0.991,5.092) -- (0.963,4.949) -- (0.991,4.806) -- (1.072,4.685) -- (1.193,4.604) -- (1.336,4.576) -- (1.479,4.604) -- (1.6,4.685) -- (1.681,4.806) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.189,2.068) -- (4.16,2.211) -- (4.079,2.332) -- (3.958,2.413) -- (3.815,2.441) -- (3.673,2.413) -- (3.552,2.332) -- (3.471,2.211) -- (3.442,2.068) -- (3.471,1.925) -- (3.552,1.804) -- (3.673,1.723) -- (3.815,1.695) -- (3.958,1.723) -- (4.079,1.804) -- (4.16,1.925) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.486,1.623) -- (10.458,1.766) -- (10.377,1.887) -- (10.256,1.968) -- (10.113,1.996) -- (9.97,1.968) -- (9.849,1.887) -- (9.768,1.766) -- (9.74,1.623) -- (9.768,1.48) -- (9.849,1.359) -- (9.97,1.278) -- (10.113,1.25) -- (10.256,1.278) -- (10.377,1.359) -- (10.458,1.48) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.848,1.523) -- (7.819,1.666) -- (7.739,1.787) -- (7.617,1.868) -- (7.475,1.897) -- (7.332,1.868) -- (7.211,1.787) -- (7.13,1.666) -- (7.101,1.523) -- (7.13,1.381) -- (7.211,1.26) -- (7.332,1.179) -- (7.475,1.15) -- (7.617,1.179) -- (7.739,1.26) -- (7.819,1.381) -- cycle;
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\filldraw[fill=white, draw=black, line width=0.5pt] (1.822,9.375) -- (1.941,9.394) -- (2.048,9.449) -- (2.133,9.534) -- (2.188,9.641) -- (2.207,9.76) -- (2.188,9.879) -- (2.133,9.986) -- (2.048,10.071) -- (1.941,10.125) -- (1.822,10.144) -- (1.822,10.144) -- (1.703,10.125) -- (1.596,10.071) -- (1.511,9.986) -- (1.457,9.879) -- (1.438,9.76) -- (1.457,9.641) -- (1.511,9.534) -- (1.596,9.449) -- (1.703,9.394) -- (1.822,9.375) -- cycle;
\node at (1.822,9.76) {\scalebox{1.53}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.805,9.377) -- (6.924,9.396) -- (7.031,9.45) -- (7.116,9.535) -- (7.171,9.643) -- (7.19,9.761) -- (7.171,9.88) -- (7.116,9.987) -- (7.031,10.073) -- (6.924,10.127) -- (6.805,10.146) -- (6.805,10.146) -- (6.686,10.127) -- (6.579,10.073) -- (6.494,9.987) -- (6.44,9.88) -- (6.421,9.761) -- (6.44,9.643) -- (6.494,9.535) -- (6.579,9.45) -- (6.686,9.396) -- (6.805,9.377) -- cycle;
\node at (6.805,9.761) {\scalebox{1.53}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.562,11.077) -- (5.68,11.095) -- (5.788,11.15) -- (5.873,11.235) -- (5.927,11.342) -- (5.946,11.461) -- (5.927,11.58) -- (5.873,11.687) -- (5.788,11.772) -- (5.68,11.827) -- (5.562,11.846) -- (5.562,11.846) -- (5.443,11.827) -- (5.336,11.772) -- (5.251,11.687) -- (5.196,11.58) -- (5.177,11.461) -- (5.196,11.342) -- (5.251,11.235) -- (5.336,11.15) -- (5.443,11.095) -- (5.562,11.077) -- cycle;
\node at (5.562,11.461) {\scalebox{1.53}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.215,9.552) -- (4.334,9.57) -- (4.441,9.625) -- (4.526,9.71) -- (4.581,9.817) -- (4.6,9.936) -- (4.581,10.055) -- (4.526,10.162) -- (4.441,10.247) -- (4.334,10.302) -- (4.215,10.321) -- (4.215,10.321) -- (4.097,10.302) -- (3.989,10.247) -- (3.904,10.162) -- (3.85,10.055) -- (3.831,9.936) -- (3.85,9.817) -- (3.904,9.71) -- (3.989,9.625) -- (4.097,9.57) -- (4.215,9.552) -- cycle;
\node at (4.215,9.936) {\scalebox{1.53}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.722,8.584) -- (5.841,8.603) -- (5.948,8.658) -- (6.033,8.743) -- (6.088,8.85) -- (6.107,8.969) -- (6.088,9.088) -- (6.033,9.195) -- (5.948,9.28) -- (5.841,9.334) -- (5.722,9.353) -- (5.722,9.353) -- (5.603,9.334) -- (5.496,9.28) -- (5.411,9.195) -- (5.357,9.088) -- (5.338,8.969) -- (5.357,8.85) -- (5.411,8.743) -- (5.496,8.658) -- (5.603,8.603) -- (5.722,8.584) -- cycle;
\node at (5.722,8.969) {\scalebox{1.53}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.223,6.338) -- (4.342,6.357) -- (4.449,6.411) -- (4.534,6.496) -- (4.588,6.604) -- (4.607,6.722) -- (4.588,6.841) -- (4.534,6.948) -- (4.449,7.034) -- (4.342,7.088) -- (4.223,7.107) -- (4.223,7.107) -- (4.104,7.088) -- (3.997,7.034) -- (3.912,6.948) -- (3.857,6.841) -- (3.838,6.722) -- (3.857,6.604) -- (3.912,6.496) -- (3.997,6.411) -- (4.104,6.357) -- (4.223,6.338) -- cycle;
\node at (4.223,6.722) {\scalebox{1.53}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.925,8.343) -- (9.043,8.362) -- (9.151,8.416) -- (9.236,8.501) -- (9.29,8.609) -- (9.309,8.728) -- (9.29,8.846) -- (9.236,8.954) -- (9.151,9.039) -- (9.043,9.093) -- (8.925,9.112) -- (8.925,9.112) -- (8.806,9.093) -- (8.699,9.039) -- (8.614,8.954) -- (8.559,8.846) -- (8.54,8.728) -- (8.559,8.609) -- (8.614,8.501) -- (8.699,8.416) -- (8.806,8.362) -- (8.925,8.343) -- cycle;
\node at (8.925,8.728) {\scalebox{1.53}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.584,6.383) -- (7.703,6.402) -- (7.81,6.456) -- (7.896,6.541) -- (7.95,6.649) -- (7.969,6.767) -- (7.95,6.886) -- (7.896,6.993) -- (7.81,7.079) -- (7.703,7.133) -- (7.584,7.152) -- (7.584,7.152) -- (7.466,7.133) -- (7.358,7.079) -- (7.273,6.993) -- (7.219,6.886) -- (7.2,6.767) -- (7.219,6.649) -- (7.273,6.541) -- (7.358,6.456) -- (7.466,6.402) -- (7.584,6.383) -- cycle;
\node at (7.584,6.767) {\scalebox{1.53}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.919,8.356) -- (3.038,8.375) -- (3.145,8.429) -- (3.23,8.514) -- (3.285,8.622) -- (3.304,8.74) -- (3.285,8.859) -- (3.23,8.966) -- (3.145,9.051) -- (3.038,9.106) -- (2.919,9.125) -- (2.557,9.125) -- (2.438,9.106) -- (2.331,9.051) -- (2.246,8.966) -- (2.192,8.859) -- (2.173,8.74) -- (2.192,8.622) -- (2.246,8.514) -- (2.331,8.429) -- (2.438,8.375) -- (2.557,8.356) -- cycle;
\node at (2.738,8.74) {\scalebox{1.53}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.804,6.055) -- (1.923,6.074) -- (2.03,6.128) -- (2.115,6.213) -- (2.169,6.32) -- (2.188,6.439) -- (2.169,6.558) -- (2.115,6.665) -- (2.03,6.75) -- (1.923,6.805) -- (1.804,6.824) -- (1.442,6.824) -- (1.323,6.805) -- (1.216,6.75) -- (1.131,6.665) -- (1.076,6.558) -- (1.057,6.439) -- (1.076,6.32) -- (1.131,6.213) -- (1.216,6.128) -- (1.323,6.074) -- (1.442,6.055) -- cycle;
\node at (1.623,6.439) {\scalebox{1.53}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.245,2.895) -- (8.364,2.914) -- (8.471,2.969) -- (8.556,3.054) -- (8.611,3.161) -- (8.63,3.28) -- (8.611,3.399) -- (8.556,3.506) -- (8.471,3.591) -- (8.364,3.645) -- (8.245,3.664) -- (7.883,3.664) -- (7.764,3.645) -- (7.657,3.591) -- (7.572,3.506) -- (7.518,3.399) -- (7.499,3.28) -- (7.518,3.161) -- (7.572,3.054) -- (7.657,2.969) -- (7.764,2.914) -- (7.883,2.895) -- cycle;
\node at (8.064,3.28) {\scalebox{1.53}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.814,4.862) -- (5.933,4.881) -- (6.04,4.935) -- (6.125,5.02) -- (6.18,5.128) -- (6.199,5.246) -- (6.18,5.365) -- (6.125,5.472) -- (6.04,5.557) -- (5.933,5.612) -- (5.814,5.631) -- (5.452,5.631) -- (5.333,5.612) -- (5.226,5.557) -- (5.141,5.472) -- (5.086,5.365) -- (5.068,5.246) -- (5.086,5.128) -- (5.141,5.02) -- (5.226,4.935) -- (5.333,4.881) -- (5.452,4.862) -- cycle;
\node at (5.633,5.246) {\scalebox{1.53}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.435,3.16) -- (4.554,3.178) -- (4.661,3.233) -- (4.746,3.318) -- (4.801,3.425) -- (4.82,3.544) -- (4.801,3.663) -- (4.746,3.77) -- (4.661,3.855) -- (4.554,3.91) -- (4.435,3.929) -- (4.073,3.929) -- (3.954,3.91) -- (3.847,3.855) -- (3.762,3.77) -- (3.708,3.663) -- (3.689,3.544) -- (3.708,3.425) -- (3.762,3.318) -- (3.847,3.233) -- (3.954,3.178) -- (4.073,3.16) -- cycle;
\node at (4.254,3.544) {\scalebox{1.53}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.529,4.865) -- (2.647,4.884) -- (2.755,4.938) -- (2.84,5.024) -- (2.894,5.131) -- (2.913,5.25) -- (2.894,5.368) -- (2.84,5.476) -- (2.755,5.561) -- (2.647,5.615) -- (2.529,5.634) -- (2.167,5.634) -- (2.048,5.615) -- (1.941,5.561) -- (1.856,5.476) -- (1.801,5.368) -- (1.782,5.25) -- (1.801,5.131) -- (1.856,5.024) -- (1.941,4.938) -- (2.048,4.884) -- (2.167,4.865) -- cycle;
\node at (2.348,5.25) {\scalebox{1.53}{\small 15}};
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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 & C, E, G, I, M & 5 \\ \hline
clairs & A, B, D, F, H, J, K, L & 8 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{5} carrefours sombres et \textbf{8} clairs. Il y en a donc \textbf{3 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 5 carrefours sombres et 8 clairs — c'est impossible.}\par\addvspace{0.4em}
\noindent \textbf{4.}\par\addvspace{0.35em}
\noindent{\color{red!80!black}\bfseries \textbf{Non plus.} Une tournée n'a qu'un départ et qu'une arrivée : elle ne peut absorber qu'\textbf{un} carrefour clair en trop. Il y en a 3.}\par\addvspace{0.4em}
\end{document}
