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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 H, 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 aux Herbes & Ⓚ Place du Marché \\ \hline
① allée des Tilleuls & ⑪ rue des Lilas & Ⓑ Coin du Lavoir & Ⓛ Carrefour du Lavoir \\ \hline
② rue Haute & ⑫ quai aux Fleurs & Ⓒ Carrefour du Moulin & Ⓜ Place de la Fontaine \\ \hline
③ rue Basse & ⑬ rue des Écoles & Ⓓ Carrefour de l'Hôpital & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ passage du Marché & ⑭ rue Verte & Ⓔ Rond-point de la Poste & \textbf{a} l'école \\ \hline
⑤ rue Pasteur & ⑮ boulevard du Parc & Ⓕ Carrefour des Tilleuls & \textbf{b} l'église \\ \hline
⑥ rue du Puits & ⑯ rue Neuve & Ⓖ Carrefour de la Mairie & \textbf{c} le lavoir \\ \hline
⑦ rue du Stade & ⑰ rue de la Poste & Ⓗ Place du Vieux-Puits & \textbf{d} le marché couvert \\ \hline
⑧ rue de la Mairie & ⑱ chemin Creux & Ⓘ Place de l'Église &  \\ \hline
⑨ rue de l'Église & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Rond-point du Pont &  \\ \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=black!38, draw=black, line width=0.5pt] (4.486,10.785) -- (4.458,10.928) -- (4.377,11.049) -- (4.256,11.13) -- (4.113,11.159) -- (3.97,11.13) -- (3.849,11.049) -- (3.768,10.928) -- (3.74,10.785) -- (3.768,10.643) -- (3.849,10.522) -- (3.97,10.441) -- (4.113,10.412) -- (4.256,10.441) -- (4.377,10.522) -- (4.458,10.643) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.693,10.579) -- (7.665,10.722) -- (7.584,10.843) -- (7.463,10.924) -- (7.32,10.952) -- (7.177,10.924) -- (7.056,10.843) -- (6.975,10.722) -- (6.947,10.579) -- (6.975,10.436) -- (7.056,10.315) -- (7.177,10.234) -- (7.32,10.206) -- (7.463,10.234) -- (7.584,10.315) -- (7.665,10.436) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.811,8.226) -- (4.783,8.369) -- (4.702,8.49) -- (4.581,8.571) -- (4.438,8.599) -- (4.295,8.571) -- (4.174,8.49) -- (4.093,8.369) -- (4.065,8.226) -- (4.093,8.083) -- (4.174,7.962) -- (4.295,7.881) -- (4.438,7.853) -- (4.581,7.881) -- (4.702,7.962) -- (4.783,8.083) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.47,8.054) -- (8.442,8.197) -- (8.361,8.318) -- (8.24,8.399) -- (8.097,8.428) -- (7.954,8.399) -- (7.833,8.318) -- (7.752,8.197) -- (7.724,8.054) -- (7.752,7.912) -- (7.833,7.791) -- (7.954,7.71) -- (8.097,7.681) -- (8.24,7.71) -- (8.361,7.791) -- (8.442,7.912) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.927,7.345) -- (10.899,7.488) -- (10.818,7.609) -- (10.697,7.69) -- (10.554,7.719) -- (10.411,7.69) -- (10.29,7.609) -- (10.209,7.488) -- (10.181,7.345) -- (10.209,7.203) -- (10.29,7.082) -- (10.411,7.001) -- (10.554,6.972) -- (10.697,7.001) -- (10.818,7.082) -- (10.899,7.203) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.729,5.205) -- (7.701,5.348) -- (7.62,5.469) -- (7.499,5.55) -- (7.356,5.578) -- (7.213,5.55) -- (7.092,5.469) -- (7.011,5.348) -- (6.983,5.205) -- (7.011,5.062) -- (7.092,4.941) -- (7.213,4.86) -- (7.356,4.832) -- (7.499,4.86) -- (7.62,4.941) -- (7.701,5.062) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.596,4.918) -- (11.567,5.061) -- (11.486,5.182) -- (11.365,5.263) -- (11.223,5.291) -- (11.08,5.263) -- (10.959,5.182) -- (10.878,5.061) -- (10.849,4.918) -- (10.878,4.775) -- (10.959,4.654) -- (11.08,4.573) -- (11.223,4.545) -- (11.365,4.573) -- (11.486,4.654) -- (11.567,4.775) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.38,4.886) -- (2.351,5.028) -- (2.27,5.15) -- (2.149,5.23) -- (2.006,5.259) -- (1.863,5.23) -- (1.742,5.15) -- (1.662,5.028) -- (1.633,4.886) -- (1.662,4.743) -- (1.742,4.622) -- (1.863,4.541) -- (2.006,4.512) -- (2.149,4.541) -- (2.27,4.622) -- (2.351,4.743) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.719,4.288) -- (4.691,4.431) -- (4.61,4.552) -- (4.489,4.633) -- (4.346,4.662) -- (4.203,4.633) -- (4.082,4.552) -- (4.001,4.431) -- (3.973,4.288) -- (4.001,4.146) -- (4.082,4.025) -- (4.203,3.944) -- (4.346,3.915) -- (4.489,3.944) -- (4.61,4.025) -- (4.691,4.146) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.459,1.785) -- (8.431,1.928) -- (8.35,2.049) -- (8.229,2.13) -- (8.086,2.158) -- (7.943,2.13) -- (7.822,2.049) -- (7.741,1.928) -- (7.713,1.785) -- (7.741,1.642) -- (7.822,1.521) -- (7.943,1.44) -- (8.086,1.412) -- (8.229,1.44) -- (8.35,1.521) -- (8.431,1.642) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.77,1.712) -- (4.742,1.855) -- (4.661,1.976) -- (4.54,2.057) -- (4.397,2.085) -- (4.254,2.057) -- (4.133,1.976) -- (4.052,1.855) -- (4.024,1.712) -- (4.052,1.569) -- (4.133,1.448) -- (4.254,1.367) -- (4.397,1.339) -- (4.54,1.367) -- (4.661,1.448) -- (4.742,1.569) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.596,1.368) -- (1.567,1.511) -- (1.486,1.632) -- (1.365,1.713) -- (1.223,1.742) -- (1.08,1.713) -- (0.959,1.632) -- (0.878,1.511) -- (0.849,1.368) -- (0.878,1.226) -- (0.959,1.105) -- (1.08,1.024) -- (1.223,0.995) -- (1.365,1.024) -- (1.486,1.105) -- (1.567,1.226) -- cycle;
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\node at (5.75,11.198) {\scalebox{1.598}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.552,9.156) -- (4.67,9.175) -- (4.778,9.23) -- (4.863,9.315) -- (4.917,9.422) -- (4.936,9.541) -- (4.917,9.66) -- (4.863,9.767) -- (4.778,9.852) -- (4.67,9.907) -- (4.552,9.925) -- (4.552,9.925) -- (4.433,9.907) -- (4.326,9.852) -- (4.24,9.767) -- (4.186,9.66) -- (4.167,9.541) -- (4.186,9.422) -- (4.24,9.315) -- (4.326,9.23) -- (4.433,9.175) -- (4.552,9.156) -- cycle;
\node at (4.552,9.541) {\scalebox{1.598}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.011,10.872) -- (9.13,10.891) -- (9.237,10.945) -- (9.322,11.03) -- (9.377,11.137) -- (9.395,11.256) -- (9.377,11.375) -- (9.322,11.482) -- (9.237,11.567) -- (9.13,11.622) -- (9.011,11.641) -- (9.011,11.641) -- (8.892,11.622) -- (8.785,11.567) -- (8.7,11.482) -- (8.645,11.375) -- (8.626,11.256) -- (8.645,11.137) -- (8.7,11.03) -- (8.785,10.945) -- (8.892,10.891) -- (9.011,10.872) -- cycle;
\node at (9.011,11.256) {\scalebox{1.598}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.972,9.013) -- (8.091,9.032) -- (8.198,9.087) -- (8.283,9.172) -- (8.338,9.279) -- (8.357,9.398) -- (8.338,9.517) -- (8.283,9.624) -- (8.198,9.709) -- (8.091,9.764) -- (7.972,9.782) -- (7.972,9.782) -- (7.853,9.764) -- (7.746,9.709) -- (7.661,9.624) -- (7.606,9.517) -- (7.587,9.398) -- (7.606,9.279) -- (7.661,9.172) -- (7.746,9.087) -- (7.853,9.032) -- (7.972,9.013) -- cycle;
\node at (7.972,9.398) {\scalebox{1.598}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.293,8.291) -- (6.412,8.31) -- (6.519,8.364) -- (6.604,8.449) -- (6.658,8.557) -- (6.677,8.675) -- (6.658,8.794) -- (6.604,8.901) -- (6.519,8.986) -- (6.412,9.041) -- (6.293,9.06) -- (6.293,9.06) -- (6.174,9.041) -- (6.067,8.986) -- (5.982,8.901) -- (5.927,8.794) -- (5.908,8.675) -- (5.927,8.557) -- (5.982,8.449) -- (6.067,8.364) -- (6.174,8.31) -- (6.293,8.291) -- cycle;
\node at (6.293,8.675) {\scalebox{1.598}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.614,5.868) -- (4.733,5.886) -- (4.84,5.941) -- (4.925,6.026) -- (4.98,6.133) -- (4.999,6.252) -- (4.98,6.371) -- (4.925,6.478) -- (4.84,6.563) -- (4.733,6.618) -- (4.614,6.637) -- (4.614,6.637) -- (4.495,6.618) -- (4.388,6.563) -- (4.303,6.478) -- (4.248,6.371) -- (4.23,6.252) -- (4.248,6.133) -- (4.303,6.026) -- (4.388,5.941) -- (4.495,5.886) -- (4.614,5.868) -- cycle;
\node at (4.614,6.252) {\scalebox{1.598}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.461,7.786) -- (9.58,7.805) -- (9.687,7.86) -- (9.773,7.945) -- (9.827,8.052) -- (9.846,8.171) -- (9.827,8.29) -- (9.773,8.397) -- (9.687,8.482) -- (9.58,8.536) -- (9.461,8.555) -- (9.461,8.555) -- (9.343,8.536) -- (9.235,8.482) -- (9.15,8.397) -- (9.096,8.29) -- (9.077,8.171) -- (9.096,8.052) -- (9.15,7.945) -- (9.235,7.86) -- (9.343,7.805) -- (9.461,7.786) -- cycle;
\node at (9.461,8.171) {\scalebox{1.598}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.981,6.179) -- (8.1,6.198) -- (8.207,6.252) -- (8.292,6.337) -- (8.347,6.445) -- (8.366,6.563) -- (8.347,6.682) -- (8.292,6.789) -- (8.207,6.874) -- (8.1,6.929) -- (7.981,6.948) -- (7.981,6.948) -- (7.862,6.929) -- (7.755,6.874) -- (7.67,6.789) -- (7.615,6.682) -- (7.597,6.563) -- (7.615,6.445) -- (7.67,6.337) -- (7.755,6.252) -- (7.862,6.198) -- (7.981,6.179) -- cycle;
\node at (7.981,6.563) {\scalebox{1.598}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.54,5.877) -- (11.659,5.896) -- (11.766,5.95) -- (11.851,6.035) -- (11.906,6.143) -- (11.925,6.261) -- (11.906,6.38) -- (11.851,6.487) -- (11.766,6.573) -- (11.659,6.627) -- (11.54,6.646) -- (11.178,6.646) -- (11.059,6.627) -- (10.952,6.573) -- (10.867,6.487) -- (10.812,6.38) -- (10.794,6.261) -- (10.812,6.143) -- (10.867,6.035) -- (10.952,5.95) -- (11.059,5.896) -- (11.178,5.877) -- cycle;
\node at (11.359,6.261) {\scalebox{1.598}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.487,4.901) -- (9.606,4.92) -- (9.713,4.974) -- (9.798,5.059) -- (9.852,5.167) -- (9.871,5.285) -- (9.852,5.404) -- (9.798,5.511) -- (9.713,5.596) -- (9.606,5.651) -- (9.487,5.67) -- (9.125,5.67) -- (9.006,5.651) -- (8.899,5.596) -- (8.814,5.511) -- (8.759,5.404) -- (8.74,5.285) -- (8.759,5.167) -- (8.814,5.059) -- (8.899,4.974) -- (9.006,4.92) -- (9.125,4.901) -- cycle;
\node at (9.306,5.285) {\scalebox{1.598}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.419,3.221) -- (8.538,3.24) -- (8.645,3.294) -- (8.73,3.379) -- (8.785,3.487) -- (8.803,3.605) -- (8.785,3.724) -- (8.73,3.831) -- (8.645,3.916) -- (8.538,3.971) -- (8.419,3.99) -- (8.057,3.99) -- (7.938,3.971) -- (7.831,3.916) -- (7.746,3.831) -- (7.691,3.724) -- (7.673,3.605) -- (7.691,3.487) -- (7.746,3.379) -- (7.831,3.294) -- (7.938,3.24) -- (8.057,3.221) -- cycle;
\node at (8.238,3.605) {\scalebox{1.598}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.428,4.479) -- (3.547,4.497) -- (3.654,4.552) -- (3.739,4.637) -- (3.793,4.744) -- (3.812,4.863) -- (3.793,4.982) -- (3.739,5.089) -- (3.654,5.174) -- (3.547,5.229) -- (3.428,5.248) -- (3.066,5.248) -- (2.947,5.229) -- (2.84,5.174) -- (2.755,5.089) -- (2.7,4.982) -- (2.681,4.863) -- (2.7,4.744) -- (2.755,4.637) -- (2.84,4.552) -- (2.947,4.497) -- (3.066,4.479) -- cycle;
\node at (3.247,4.863) {\scalebox{1.598}{\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, H, J, K, M & 7 \\ \hline
clairs & C, D, F, G, I, 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{H} :}\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 des Jardins ⑩} vers le nord jusqu'à Carrefour des Tilleuls \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{allée des Tilleuls ①}, vers le nord, jusqu'à Place aux Herbes \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{passage du Marché ④}, vers l'ouest, jusqu'à Carrefour du Moulin \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue Haute ②}, vers l'ouest, jusqu'à Coin du Lavoir \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ③}, vers le sud, jusqu'à Carrefour de l'Hôpital \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue du Puits ⑥}, vers l'est, jusqu'à Rond-point de la Poste \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue de l'Église ⑨}, vers le sud, jusqu'à Carrefour de la Mairie \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{boulevard du Parc ⑮}, vers l'ouest, jusqu'à Rond-point du Pont \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue des Écoles ⑬}, vers l'ouest, jusqu'à Place de l'Église \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Verte ⑭}, vers le sud, jusqu'à Place de la Fontaine \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans le \textbf{chemin Creux ⑱}, vers l'est, jusqu'à Carrefour du Lavoir \textbf{(L)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de la Poste ⑰}, vers l'est, jusqu'à Place du Marché \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}
