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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 Verte & Ⓑ Place des Tanneurs & Ⓛ Place aux Herbes \\ \hline
① boulevard du Parc & ⑪ rue des Écoles & Ⓒ Carrefour des Tilleuls & Ⓜ Placette des Ormes \\ \hline
② rue de la Mairie & ⑫ rue des Lilas & Ⓓ Place de la Fontaine & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ chemin Creux & ⑬ avenue de la Gare & Ⓔ Carrefour du Lavoir & \textbf{a} l'école \\ \hline
④ allée des Tilleuls & ⑭ quai aux Fleurs & Ⓕ Carrefour de la Mairie & \textbf{b} la boulangerie \\ \hline
⑤ rue des Jardins & ⑮ rue du Pont & Ⓖ Place de l'Église & \textbf{c} le gymnase \\ \hline
⑥ rue de l'Église & ⑯ chemin des Vignes & Ⓗ Carrefour du Moulin & \textbf{d} le lavoir \\ \hline
⑦ rue Neuve & ⑰ impasse des Roses & Ⓘ Rond-point des Vaches &  \\ \hline
⑧ passage du Marché & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Place du Vieux-Puits &  \\ \hline
⑨ rue du Four & Ⓐ Coin du Lavoir & Ⓚ 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=white, draw=black, line width=0.5pt] (8.593,11.309) -- (8.564,11.452) -- (8.483,11.573) -- (8.362,11.654) -- (8.219,11.682) -- (8.077,11.654) -- (7.955,11.573) -- (7.875,11.452) -- (7.846,11.309) -- (7.875,11.166) -- (7.955,11.045) -- (8.077,10.964) -- (8.219,10.936) -- (8.362,10.964) -- (8.483,11.045) -- (8.564,11.166) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.072,11.213) -- (5.043,11.356) -- (4.962,11.477) -- (4.841,11.558) -- (4.698,11.586) -- (4.556,11.558) -- (4.435,11.477) -- (4.354,11.356) -- (4.325,11.213) -- (4.354,11.07) -- (4.435,10.949) -- (4.556,10.868) -- (4.698,10.84) -- (4.841,10.868) -- (4.962,10.949) -- (5.043,11.07) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.716,8.265) -- (1.688,8.408) -- (1.607,8.529) -- (1.486,8.61) -- (1.343,8.639) -- (1.2,8.61) -- (1.079,8.529) -- (0.998,8.408) -- (0.97,8.265) -- (0.998,8.122) -- (1.079,8.001) -- (1.2,7.921) -- (1.343,7.892) -- (1.486,7.921) -- (1.607,8.001) -- (1.688,8.122) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.043,8.091) -- (8.015,8.234) -- (7.934,8.355) -- (7.813,8.436) -- (7.67,8.464) -- (7.527,8.436) -- (7.406,8.355) -- (7.325,8.234) -- (7.297,8.091) -- (7.325,7.948) -- (7.406,7.827) -- (7.527,7.746) -- (7.67,7.718) -- (7.813,7.746) -- (7.934,7.827) -- (8.015,7.948) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.368,7.865) -- (5.339,8.008) -- (5.258,8.129) -- (5.137,8.21) -- (4.994,8.238) -- (4.851,8.21) -- (4.73,8.129) -- (4.65,8.008) -- (4.621,7.865) -- (4.65,7.722) -- (4.73,7.601) -- (4.851,7.52) -- (4.994,7.492) -- (5.137,7.52) -- (5.258,7.601) -- (5.339,7.722) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.465,7.568) -- (11.436,7.711) -- (11.356,7.832) -- (11.234,7.913) -- (11.092,7.941) -- (10.949,7.913) -- (10.828,7.832) -- (10.747,7.711) -- (10.718,7.568) -- (10.747,7.425) -- (10.828,7.304) -- (10.949,7.223) -- (11.092,7.195) -- (11.234,7.223) -- (11.356,7.304) -- (11.436,7.425) -- cycle;
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\filldraw[fill=white, draw=black, line width=0.5pt] (8.245,4.439) -- (8.217,4.582) -- (8.136,4.703) -- (8.015,4.784) -- (7.872,4.812) -- (7.729,4.784) -- (7.608,4.703) -- (7.527,4.582) -- (7.499,4.439) -- (7.527,4.296) -- (7.608,4.175) -- (7.729,4.094) -- (7.872,4.065) -- (8.015,4.094) -- (8.136,4.175) -- (8.217,4.296) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.578,4.43) -- (11.549,4.573) -- (11.469,4.694) -- (11.347,4.775) -- (11.205,4.803) -- (11.062,4.775) -- (10.941,4.694) -- (10.86,4.573) -- (10.831,4.43) -- (10.86,4.287) -- (10.941,4.166) -- (11.062,4.085) -- (11.205,4.057) -- (11.347,4.085) -- (11.469,4.166) -- (11.549,4.287) -- cycle;
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\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.578,1.579) -- (1.549,1.722) -- (1.469,1.843) -- (1.347,1.924) -- (1.205,1.952) -- (1.062,1.924) -- (0.941,1.843) -- (0.86,1.722) -- (0.831,1.579) -- (0.86,1.436) -- (0.941,1.315) -- (1.062,1.234) -- (1.205,1.206) -- (1.347,1.234) -- (1.469,1.315) -- (1.549,1.436) -- cycle;
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\filldraw[fill=white, draw=black, line width=0.5pt] (6.444,11.406) -- (6.563,11.425) -- (6.67,11.48) -- (6.756,11.565) -- (6.81,11.672) -- (6.829,11.791) -- (6.81,11.91) -- (6.756,12.017) -- (6.67,12.102) -- (6.563,12.157) -- (6.444,12.175) -- (6.444,12.175) -- (6.326,12.157) -- (6.218,12.102) -- (6.133,12.017) -- (6.079,11.91) -- (6.06,11.791) -- (6.079,11.672) -- (6.133,11.565) -- (6.218,11.48) -- (6.326,11.425) -- (6.444,11.406) -- cycle;
\node at (6.444,11.791) {\scalebox{1.596}{\small 4}};
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\node at (3.227,8.598) {\scalebox{1.596}{\small 6}};
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\node at (9.417,8.068) {\scalebox{1.596}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.306,5.91) -- (8.424,5.929) -- (8.532,5.983) -- (8.617,6.068) -- (8.671,6.176) -- (8.69,6.294) -- (8.671,6.413) -- (8.617,6.52) -- (8.532,6.606) -- (8.424,6.66) -- (8.306,6.679) -- (8.306,6.679) -- (8.187,6.66) -- (8.08,6.606) -- (7.994,6.52) -- (7.94,6.413) -- (7.921,6.294) -- (7.94,6.176) -- (7.994,6.068) -- (8.08,5.983) -- (8.187,5.929) -- (8.306,5.91) -- cycle;
\node at (8.306,6.294) {\scalebox{1.596}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.309,7.866) -- (6.428,7.885) -- (6.535,7.94) -- (6.62,8.025) -- (6.675,8.132) -- (6.694,8.251) -- (6.675,8.37) -- (6.62,8.477) -- (6.535,8.562) -- (6.428,8.617) -- (6.309,8.635) -- (6.309,8.635) -- (6.19,8.617) -- (6.083,8.562) -- (5.998,8.477) -- (5.943,8.37) -- (5.925,8.251) -- (5.943,8.132) -- (5.998,8.025) -- (6.083,7.94) -- (6.19,7.885) -- (6.309,7.866) -- cycle;
\node at (6.309,8.251) {\scalebox{1.596}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.598,5.858) -- (5.717,5.877) -- (5.824,5.932) -- (5.909,6.017) -- (5.964,6.124) -- (5.983,6.243) -- (5.964,6.362) -- (5.909,6.469) -- (5.824,6.554) -- (5.717,6.609) -- (5.598,6.627) -- (5.236,6.627) -- (5.118,6.609) -- (5.01,6.554) -- (4.925,6.469) -- (4.871,6.362) -- (4.852,6.243) -- (4.871,6.124) -- (4.925,6.017) -- (5.01,5.932) -- (5.118,5.877) -- (5.236,5.858) -- cycle;
\node at (5.417,6.243) {\scalebox{1.596}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.584,5.624) -- (11.703,5.642) -- (11.81,5.697) -- (11.895,5.782) -- (11.95,5.889) -- (11.968,6.008) -- (11.95,6.127) -- (11.895,6.234) -- (11.81,6.319) -- (11.703,6.374) -- (11.584,6.393) -- (11.222,6.393) -- (11.103,6.374) -- (10.996,6.319) -- (10.911,6.234) -- (10.856,6.127) -- (10.838,6.008) -- (10.856,5.889) -- (10.911,5.782) -- (10.996,5.697) -- (11.103,5.642) -- (11.222,5.624) -- cycle;
\node at (11.403,6.008) {\scalebox{1.596}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.562,4.685) -- (6.681,4.704) -- (6.788,4.758) -- (6.873,4.843) -- (6.928,4.95) -- (6.947,5.069) -- (6.928,5.188) -- (6.873,5.295) -- (6.788,5.38) -- (6.681,5.435) -- (6.562,5.454) -- (6.2,5.454) -- (6.081,5.435) -- (5.974,5.38) -- (5.889,5.295) -- (5.835,5.188) -- (5.816,5.069) -- (5.835,4.95) -- (5.889,4.843) -- (5.974,4.758) -- (6.081,4.704) -- (6.2,4.685) -- cycle;
\node at (6.381,5.069) {\scalebox{1.596}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.408,2.947) -- (5.527,2.966) -- (5.634,3.021) -- (5.719,3.106) -- (5.773,3.213) -- (5.792,3.332) -- (5.773,3.451) -- (5.719,3.558) -- (5.634,3.643) -- (5.527,3.697) -- (5.408,3.716) -- (5.046,3.716) -- (4.927,3.697) -- (4.82,3.643) -- (4.735,3.558) -- (4.68,3.451) -- (4.661,3.332) -- (4.68,3.213) -- (4.735,3.106) -- (4.82,3.021) -- (4.927,2.966) -- (5.046,2.947) -- cycle;
\node at (5.227,3.332) {\scalebox{1.596}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.721,4.572) -- (9.839,4.591) -- (9.947,4.645) -- (10.032,4.73) -- (10.086,4.838) -- (10.105,4.956) -- (10.086,5.075) -- (10.032,5.182) -- (9.947,5.268) -- (9.839,5.322) -- (9.721,5.341) -- (9.359,5.341) -- (9.24,5.322) -- (9.133,5.268) -- (9.048,5.182) -- (8.993,5.075) -- (8.974,4.956) -- (8.993,4.838) -- (9.048,4.73) -- (9.133,4.645) -- (9.24,4.591) -- (9.359,4.572) -- cycle;
\node at (9.54,4.956) {\scalebox{1.596}{\small 14}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.523,2.609) -- (7.641,2.628) -- (7.749,2.682) -- (7.834,2.768) -- (7.888,2.875) -- (7.907,2.994) -- (7.888,3.112) -- (7.834,3.22) -- (7.749,3.305) -- (7.641,3.359) -- (7.523,3.378) -- (7.161,3.378) -- (7.042,3.359) -- (6.935,3.305) -- (6.85,3.22) -- (6.795,3.112) -- (6.776,2.994) -- (6.795,2.875) -- (6.85,2.768) -- (6.935,2.682) -- (7.042,2.628) -- (7.161,2.609) -- cycle;
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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, D, E, H, J, L, M & 7 \\ \hline
clairs & A, B, F, G, I, K & 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 la \textbf{rue des Écoles ⑪} vers le nord jusqu'à Place de l'Église \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Neuve ⑦}, vers l'ouest, jusqu'à Carrefour du Lavoir \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin Creux ③}, vers le nord, jusqu'à Place des Tanneurs \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{allée des Tilleuls ④}, vers l'ouest, jusqu'à Carrefour des Tilleuls \textbf{(C)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{boulevard du Parc ①}, vers l'ouest, jusqu'à Coin du Lavoir \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de la Mairie ②}, vers le sud, jusqu'à Place de la Fontaine \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue de l'Église ⑥}, vers l'est, jusqu'à Carrefour de la Mairie \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Verte ⑩}, vers le sud, jusqu'à Carrefour du Moulin \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue des Lilas ⑫}, vers l'est, jusqu'à Rond-point des Vaches \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Pont ⑮}, vers le sud, jusqu'à Placette des Ormes \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin des Vignes ⑯}, vers l'ouest, jusqu'à Rond-point du Pont \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans l'\textbf{impasse des Roses ⑰}, vers l'ouest, jusqu'à Place aux Herbes \textbf{(L)}.\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}
