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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 avenue de la Gare & Ⓑ Place du Vieux-Puits & Ⓛ Placette des Ormes \\ \hline
① impasse des Roses & ⑪ rue Haute & Ⓒ Place de la Fontaine & Ⓜ Coin du Lavoir \\ \hline
② rue du Pont & ⑫ rue des Lilas & Ⓓ Carrefour de la Mairie & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ allée des Tilleuls & ⑬ rue Pasteur & Ⓔ Rond-point de la Poste & \textbf{a} l'école \\ \hline
④ rue des Écoles & ⑭ rue Basse & Ⓕ Rond-point du Stade & \textbf{b} le marché couvert \\ \hline
⑤ quai aux Fleurs & ⑮ chemin des Vignes & Ⓖ Rond-point des Vaches & \textbf{c} le stade \\ \hline
⑥ allée des Peupliers & ⑯ rue du Four & Ⓗ Carrefour du Moulin & \textbf{d} la bibliothèque \\ \hline
⑦ rue de l'Église & ⑰ rue des Jardins & Ⓘ Fontaine Notre-Dame &  \\ \hline
⑧ rue de la Mairie & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Carrefour de l'Hôpital &  \\ \hline
⑨ chemin de Ronde & Ⓐ Place du Marché & Ⓚ Carrefour Saint-Roch &  \\ \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] (8.495,10.717) -- (8.467,10.86) -- (8.386,10.981) -- (8.265,11.062) -- (8.122,11.091) -- (7.979,11.062) -- (7.858,10.981) -- (7.777,10.86) -- (7.749,10.717) -- (7.777,10.575) -- (7.858,10.454) -- (7.979,10.373) -- (8.122,10.344) -- (8.265,10.373) -- (8.386,10.454) -- (8.467,10.575) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.382,8.036) -- (2.354,8.179) -- (2.273,8.3) -- (2.152,8.381) -- (2.009,8.41) -- (1.866,8.381) -- (1.745,8.3) -- (1.664,8.179) -- (1.636,8.036) -- (1.664,7.893) -- (1.745,7.772) -- (1.866,7.692) -- (2.009,7.663) -- (2.152,7.692) -- (2.273,7.772) -- (2.354,7.893) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.023,7.815) -- (7.995,7.958) -- (7.914,8.079) -- (7.793,8.16) -- (7.65,8.188) -- (7.507,8.16) -- (7.386,8.079) -- (7.305,7.958) -- (7.277,7.815) -- (7.305,7.672) -- (7.386,7.551) -- (7.507,7.47) -- (7.65,7.442) -- (7.793,7.47) -- (7.914,7.551) -- (7.995,7.672) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.93,7.52) -- (4.901,7.663) -- (4.82,7.784) -- (4.699,7.864) -- (4.556,7.893) -- (4.414,7.864) -- (4.293,7.784) -- (4.212,7.663) -- (4.183,7.52) -- (4.212,7.377) -- (4.293,7.256) -- (4.414,7.175) -- (4.556,7.146) -- (4.699,7.175) -- (4.82,7.256) -- (4.901,7.377) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.757,5.076) -- (1.728,5.218) -- (1.647,5.339) -- (1.526,5.42) -- (1.383,5.449) -- (1.241,5.42) -- (1.12,5.339) -- (1.039,5.218) -- (1.01,5.076) -- (1.039,4.933) -- (1.12,4.812) -- (1.241,4.731) -- (1.383,4.702) -- (1.526,4.731) -- (1.647,4.812) -- (1.728,4.933) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.464,4.948) -- (5.436,5.09) -- (5.355,5.212) -- (5.234,5.292) -- (5.091,5.321) -- (4.948,5.292) -- (4.827,5.212) -- (4.746,5.09) -- (4.718,4.948) -- (4.746,4.805) -- (4.827,4.684) -- (4.948,4.603) -- (5.091,4.574) -- (5.234,4.603) -- (5.355,4.684) -- (5.436,4.805) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.321,4.39) -- (8.293,4.533) -- (8.212,4.654) -- (8.091,4.735) -- (7.948,4.763) -- (7.805,4.735) -- (7.684,4.654) -- (7.603,4.533) -- (7.574,4.39) -- (7.603,4.247) -- (7.684,4.126) -- (7.805,4.045) -- (7.948,4.017) -- (8.091,4.045) -- (8.212,4.126) -- (8.293,4.247) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.132,4.269) -- (11.104,4.412) -- (11.023,4.533) -- (10.902,4.614) -- (10.759,4.643) -- (10.616,4.614) -- (10.495,4.533) -- (10.414,4.412) -- (10.386,4.269) -- (10.414,4.127) -- (10.495,4.006) -- (10.616,3.925) -- (10.759,3.896) -- (10.902,3.925) -- (11.023,4.006) -- (11.104,4.127) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.345,2.006) -- (8.317,2.149) -- (8.236,2.27) -- (8.115,2.351) -- (7.972,2.379) -- (7.829,2.351) -- (7.708,2.27) -- (7.627,2.149) -- (7.599,2.006) -- (7.627,1.863) -- (7.708,1.742) -- (7.829,1.661) -- (7.972,1.633) -- (8.115,1.661) -- (8.236,1.742) -- (8.317,1.863) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.741,1.867) -- (5.712,2.01) -- (5.631,2.131) -- (5.51,2.212) -- (5.367,2.24) -- (5.225,2.212) -- (5.103,2.131) -- (5.023,2.01) -- (4.994,1.867) -- (5.023,1.724) -- (5.103,1.603) -- (5.225,1.522) -- (5.367,1.494) -- (5.51,1.522) -- (5.631,1.603) -- (5.712,1.724) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.496,1.543) -- (11.468,1.686) -- (11.387,1.807) -- (11.266,1.888) -- (11.123,1.916) -- (10.98,1.888) -- (10.859,1.807) -- (10.778,1.686) -- (10.75,1.543) -- (10.778,1.4) -- (10.859,1.279) -- (10.98,1.198) -- (11.123,1.17) -- (11.266,1.198) -- (11.387,1.279) -- (11.468,1.4) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.317,1.54) -- (2.288,1.683) -- (2.207,1.804) -- (2.086,1.885) -- (1.944,1.913) -- (1.801,1.885) -- (1.68,1.804) -- (1.599,1.683) -- (1.57,1.54) -- (1.599,1.397) -- (1.68,1.276) -- (1.801,1.195) -- (1.944,1.167) -- (2.086,1.195) -- (2.207,1.276) -- (2.288,1.397) -- cycle;
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\node at (8.041,6.124) {\scalebox{1.617}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.054,7.793) -- (6.173,7.812) -- (6.28,7.867) -- (6.366,7.952) -- (6.42,8.059) -- (6.439,8.178) -- (6.42,8.297) -- (6.366,8.404) -- (6.28,8.489) -- (6.173,8.543) -- (6.054,8.562) -- (6.054,8.562) -- (5.936,8.543) -- (5.828,8.489) -- (5.743,8.404) -- (5.689,8.297) -- (5.67,8.178) -- (5.689,8.059) -- (5.743,7.952) -- (5.828,7.867) -- (5.936,7.812) -- (6.054,7.793) -- cycle;
\node at (6.054,8.178) {\scalebox{1.617}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.094,5.905) -- (5.213,5.924) -- (5.32,5.979) -- (5.405,6.064) -- (5.46,6.171) -- (5.479,6.29) -- (5.46,6.409) -- (5.405,6.516) -- (5.32,6.601) -- (5.213,6.656) -- (5.094,6.674) -- (5.094,6.674) -- (4.975,6.656) -- (4.868,6.601) -- (4.783,6.516) -- (4.728,6.409) -- (4.71,6.29) -- (4.728,6.171) -- (4.783,6.064) -- (4.868,5.979) -- (4.975,5.924) -- (5.094,5.905) -- cycle;
\node at (5.094,6.29) {\scalebox{1.617}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.256,5.164) -- (3.375,5.183) -- (3.482,5.238) -- (3.567,5.323) -- (3.622,5.43) -- (3.64,5.549) -- (3.622,5.668) -- (3.567,5.775) -- (3.482,5.86) -- (3.375,5.915) -- (3.256,5.933) -- (3.256,5.933) -- (3.137,5.915) -- (3.03,5.86) -- (2.945,5.775) -- (2.89,5.668) -- (2.871,5.549) -- (2.89,5.43) -- (2.945,5.323) -- (3.03,5.238) -- (3.137,5.183) -- (3.256,5.164) -- cycle;
\node at (3.256,5.549) {\scalebox{1.617}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.897,2.96) -- (2.016,2.979) -- (2.124,3.034) -- (2.209,3.119) -- (2.263,3.226) -- (2.282,3.345) -- (2.263,3.464) -- (2.209,3.571) -- (2.124,3.656) -- (2.016,3.711) -- (1.897,3.729) -- (1.897,3.729) -- (1.779,3.711) -- (1.671,3.656) -- (1.586,3.571) -- (1.532,3.464) -- (1.513,3.345) -- (1.532,3.226) -- (1.586,3.119) -- (1.671,3.034) -- (1.779,2.979) -- (1.897,2.96) -- cycle;
\node at (1.897,3.345) {\scalebox{1.617}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.797,4.78) -- (6.916,4.799) -- (7.023,4.853) -- (7.108,4.938) -- (7.163,5.045) -- (7.182,5.164) -- (7.163,5.283) -- (7.108,5.39) -- (7.023,5.475) -- (6.916,5.53) -- (6.797,5.549) -- (6.435,5.549) -- (6.316,5.53) -- (6.209,5.475) -- (6.124,5.39) -- (6.069,5.283) -- (6.051,5.164) -- (6.069,5.045) -- (6.124,4.938) -- (6.209,4.853) -- (6.316,4.799) -- (6.435,4.78) -- cycle;
\node at (6.616,5.164) {\scalebox{1.617}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.666,3.046) -- (5.785,3.065) -- (5.892,3.119) -- (5.977,3.204) -- (6.032,3.311) -- (6.051,3.43) -- (6.032,3.549) -- (5.977,3.656) -- (5.892,3.741) -- (5.785,3.796) -- (5.666,3.815) -- (5.304,3.815) -- (5.185,3.796) -- (5.078,3.741) -- (4.993,3.656) -- (4.939,3.549) -- (4.92,3.43) -- (4.939,3.311) -- (4.993,3.204) -- (5.078,3.119) -- (5.185,3.065) -- (5.304,3.046) -- cycle;
\node at (5.485,3.43) {\scalebox{1.617}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.556,4.445) -- (9.675,4.464) -- (9.782,4.519) -- (9.867,4.604) -- (9.922,4.711) -- (9.94,4.83) -- (9.922,4.949) -- (9.867,5.056) -- (9.782,5.141) -- (9.675,5.196) -- (9.556,5.215) -- (9.194,5.215) -- (9.075,5.196) -- (8.968,5.141) -- (8.883,5.056) -- (8.828,4.949) -- (8.809,4.83) -- (8.828,4.711) -- (8.883,4.604) -- (8.968,4.519) -- (9.075,4.464) -- (9.194,4.445) -- cycle;
\node at (9.375,4.83) {\scalebox{1.617}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.658,2.809) -- (7.777,2.827) -- (7.884,2.882) -- (7.969,2.967) -- (8.024,3.074) -- (8.043,3.193) -- (8.024,3.312) -- (7.969,3.419) -- (7.884,3.504) -- (7.777,3.559) -- (7.658,3.578) -- (7.296,3.578) -- (7.177,3.559) -- (7.07,3.504) -- (6.985,3.419) -- (6.93,3.312) -- (6.912,3.193) -- (6.93,3.074) -- (6.985,2.967) -- (7.07,2.882) -- (7.177,2.827) -- (7.296,2.809) -- cycle;
\node at (7.477,3.193) {\scalebox{1.617}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.616,2.588) -- (11.734,2.606) -- (11.842,2.661) -- (11.927,2.746) -- (11.981,2.853) -- (12.0,2.972) -- (11.981,3.091) -- (11.927,3.198) -- (11.842,3.283) -- (11.734,3.338) -- (11.616,3.357) -- (11.254,3.357) -- (11.135,3.338) -- (11.028,3.283) -- (10.943,3.198) -- (10.888,3.091) -- (10.869,2.972) -- (10.888,2.853) -- (10.943,2.746) -- (11.028,2.661) -- (11.135,2.606) -- (11.254,2.588) -- cycle;
\node at (11.435,2.972) {\scalebox{1.617}{\small 14}};
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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, D, G, I, J, M & 7 \\ \hline
clairs & B, E, F, 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{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 ⑮} vers l'est jusqu'à Placette des Ormes \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ⑭}, vers le nord, jusqu'à Fontaine Notre-Dame \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ⑫}, vers l'ouest, jusqu'à Carrefour du Moulin \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{avenue de la Gare ⑩}, vers l'ouest, jusqu'à Rond-point des Vaches \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Haute ⑪}, vers le sud, jusqu'à Carrefour Saint-Roch \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue des Jardins ⑰}, vers l'ouest, jusqu'à Coin du Lavoir \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin de Ronde ⑨}, vers le nord, jusqu'à Rond-point du Stade \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Écoles ④}, vers le nord, jusqu'à Place de la Fontaine \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{allée des Tilleuls ③}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Peupliers ⑥}, vers l'est, jusqu'à Carrefour de la Mairie \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Pont ②}, vers le nord, jusqu'à Place du Vieux-Puits \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{impasse des Roses ①}, vers l'est, jusqu'à Place du Marché \textbf{(A)}.\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}
