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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 C, 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 chemin du Lavoir & Ⓑ Place du Marché & Ⓛ Place Sainte-Croix \\ \hline
① rue du Stade & ⑪ rue Basse & Ⓒ Place du Vieux-Puits & Ⓜ Place de l'Église \\ \hline
② venelle du Chat & ⑫ allée des Tilleuls & Ⓓ Place des Tanneurs & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ rue du Moulin & ⑬ passage du Marché & Ⓔ Rond-point de la Gare & \textbf{a} l'école \\ \hline
④ rue des Écoles & ⑭ rue du Four & Ⓕ Carrefour Saint-Roch & \textbf{b} la bibliothèque \\ \hline
⑤ rue Pasteur & ⑮ rue des Jardins & Ⓖ Carrefour des Tilleuls & \textbf{c} la boulangerie \\ \hline
⑥ rue Verte & ⑯ chemin des Vignes & Ⓗ Carrefour du Lavoir & \textbf{d} la pharmacie \\ \hline
⑦ rue de la Mairie & ⑰ impasse des Cerisiers & Ⓘ Rond-point du Stade &  \\ \hline
⑧ rue de l'Église & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Carrefour du Moulin &  \\ \hline
⑨ rue Haute & Ⓐ Place de la Fontaine & Ⓚ Carrefour de la Mairie &  \\ \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] (5.037,4.058) -- (5.063,3.881) -- (5.085,3.702) -- (5.103,3.523) -- (5.117,3.344) -- (5.126,3.164) -- (5.132,2.985) -- (5.133,2.805) -- (5.129,2.625) -- (5.122,2.445) -- (5.11,2.266) -- (5.095,2.087) -- (5.074,1.908) -- (5.05,1.73) -- (5.022,1.553);
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\filldraw[fill=white, draw=black, line width=0.5pt] (2.066,10.361) -- (2.038,10.504) -- (1.957,10.625) -- (1.836,10.706) -- (1.693,10.734) -- (1.55,10.706) -- (1.429,10.625) -- (1.348,10.504) -- (1.32,10.361) -- (1.348,10.218) -- (1.429,10.097) -- (1.55,10.016) -- (1.693,9.988) -- (1.836,10.016) -- (1.957,10.097) -- (2.038,10.218) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.606,10.164) -- (5.578,10.307) -- (5.497,10.428) -- (5.376,10.508) -- (5.233,10.537) -- (5.09,10.508) -- (4.969,10.428) -- (4.888,10.307) -- (4.86,10.164) -- (4.888,10.021) -- (4.969,9.9) -- (5.09,9.819) -- (5.233,9.79) -- (5.376,9.819) -- (5.497,9.9) -- (5.578,10.021) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.703,7.383) -- (1.675,7.526) -- (1.594,7.647) -- (1.473,7.728) -- (1.33,7.756) -- (1.187,7.728) -- (1.066,7.647) -- (0.985,7.526) -- (0.957,7.383) -- (0.985,7.24) -- (1.066,7.119) -- (1.187,7.038) -- (1.33,7.01) -- (1.473,7.038) -- (1.594,7.119) -- (1.675,7.24) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.639,7.322) -- (11.61,7.465) -- (11.529,7.586) -- (11.408,7.667) -- (11.266,7.696) -- (11.123,7.667) -- (11.002,7.586) -- (10.921,7.465) -- (10.892,7.322) -- (10.921,7.18) -- (11.002,7.059) -- (11.123,6.978) -- (11.266,6.949) -- (11.408,6.978) -- (11.529,7.059) -- (11.61,7.18) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.225,7.146) -- (5.196,7.289) -- (5.115,7.41) -- (4.994,7.491) -- (4.851,7.52) -- (4.709,7.491) -- (4.588,7.41) -- (4.507,7.289) -- (4.478,7.146) -- (4.507,7.004) -- (4.588,6.883) -- (4.709,6.802) -- (4.851,6.773) -- (4.994,6.802) -- (5.115,6.883) -- (5.196,7.004) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.142,7.003) -- (8.114,7.146) -- (8.033,7.267) -- (7.912,7.348) -- (7.769,7.376) -- (7.626,7.348) -- (7.505,7.267) -- (7.424,7.146) -- (7.396,7.003) -- (7.424,6.86) -- (7.505,6.739) -- (7.626,6.658) -- (7.769,6.63) -- (7.912,6.658) -- (8.033,6.739) -- (8.114,6.86) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (10.791,4.112) -- (10.762,4.255) -- (10.681,4.376) -- (10.56,4.457) -- (10.418,4.485) -- (10.275,4.457) -- (10.154,4.376) -- (10.073,4.255) -- (10.044,4.112) -- (10.073,3.969) -- (10.154,3.848) -- (10.275,3.767) -- (10.418,3.738) -- (10.56,3.767) -- (10.681,3.848) -- (10.762,3.969) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.41,4.058) -- (5.382,4.201) -- (5.301,4.322) -- (5.18,4.403) -- (5.037,4.431) -- (4.894,4.403) -- (4.773,4.322) -- (4.692,4.201) -- (4.664,4.058) -- (4.692,3.915) -- (4.773,3.794) -- (4.894,3.713) -- (5.037,3.685) -- (5.18,3.713) -- (5.301,3.794) -- (5.382,3.915) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.163,3.843) -- (8.134,3.985) -- (8.053,4.107) -- (7.932,4.187) -- (7.789,4.216) -- (7.647,4.187) -- (7.526,4.107) -- (7.445,3.985) -- (7.416,3.843) -- (7.445,3.7) -- (7.526,3.579) -- (7.647,3.498) -- (7.789,3.469) -- (7.932,3.498) -- (8.053,3.579) -- (8.134,3.7) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.395,1.553) -- (5.366,1.696) -- (5.286,1.817) -- (5.164,1.898) -- (5.022,1.926) -- (4.879,1.898) -- (4.758,1.817) -- (4.677,1.696) -- (4.648,1.553) -- (4.677,1.41) -- (4.758,1.289) -- (4.879,1.208) -- (5.022,1.18) -- (5.164,1.208) -- (5.286,1.289) -- (5.366,1.41) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.817,1.538) -- (7.789,1.68) -- (7.708,1.801) -- (7.587,1.882) -- (7.444,1.911) -- (7.301,1.882) -- (7.18,1.801) -- (7.099,1.68) -- (7.071,1.538) -- (7.099,1.395) -- (7.18,1.274) -- (7.301,1.193) -- (7.444,1.164) -- (7.587,1.193) -- (7.708,1.274) -- (7.789,1.395) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.047,1.449) -- (2.018,1.592) -- (1.937,1.713) -- (1.816,1.794) -- (1.673,1.822) -- (1.531,1.794) -- (1.409,1.713) -- (1.329,1.592) -- (1.3,1.449) -- (1.329,1.306) -- (1.409,1.185) -- (1.531,1.104) -- (1.673,1.076) -- (1.816,1.104) -- (1.937,1.185) -- (2.018,1.306) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.703,1.338) -- (11.675,1.48) -- (11.594,1.601) -- (11.473,1.682) -- (11.33,1.711) -- (11.187,1.682) -- (11.066,1.601) -- (10.985,1.48) -- (10.957,1.338) -- (10.985,1.195) -- (11.066,1.074) -- (11.187,0.993) -- (11.33,0.964) -- (11.473,0.993) -- (11.594,1.074) -- (11.675,1.195) -- cycle;
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\node at (5.299,8.623) {\scalebox{1.611}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.126,7.409) -- (3.245,7.428) -- (3.352,7.483) -- (3.437,7.568) -- (3.492,7.675) -- (3.511,7.794) -- (3.492,7.912) -- (3.437,8.02) -- (3.352,8.105) -- (3.245,8.159) -- (3.126,8.178) -- (3.126,8.178) -- (3.007,8.159) -- (2.9,8.105) -- (2.815,8.02) -- (2.761,7.912) -- (2.742,7.794) -- (2.761,7.675) -- (2.815,7.568) -- (2.9,7.483) -- (3.007,7.428) -- (3.126,7.409) -- cycle;
\node at (3.126,7.794) {\scalebox{1.611}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.081,5.269) -- (11.2,5.288) -- (11.307,5.343) -- (11.392,5.428) -- (11.447,5.535) -- (11.465,5.654) -- (11.447,5.773) -- (11.392,5.88) -- (11.307,5.965) -- (11.2,6.02) -- (11.081,6.038) -- (11.081,6.038) -- (10.962,6.02) -- (10.855,5.965) -- (10.77,5.88) -- (10.715,5.773) -- (10.696,5.654) -- (10.715,5.535) -- (10.77,5.428) -- (10.855,5.343) -- (10.962,5.288) -- (11.081,5.269) -- cycle;
\node at (11.081,5.654) {\scalebox{1.611}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.335,7.194) -- (6.454,7.213) -- (6.561,7.268) -- (6.646,7.353) -- (6.701,7.46) -- (6.72,7.579) -- (6.701,7.698) -- (6.646,7.805) -- (6.561,7.89) -- (6.454,7.945) -- (6.335,7.964) -- (6.335,7.964) -- (6.216,7.945) -- (6.109,7.89) -- (6.024,7.805) -- (5.969,7.698) -- (5.951,7.579) -- (5.969,7.46) -- (6.024,7.353) -- (6.109,7.268) -- (6.216,7.213) -- (6.335,7.194) -- cycle;
\node at (6.335,7.579) {\scalebox{1.611}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.201,5.233) -- (5.319,5.252) -- (5.427,5.307) -- (5.512,5.392) -- (5.566,5.499) -- (5.585,5.618) -- (5.566,5.737) -- (5.512,5.844) -- (5.427,5.929) -- (5.319,5.983) -- (5.201,6.002) -- (5.201,6.002) -- (5.082,5.983) -- (4.975,5.929) -- (4.89,5.844) -- (4.835,5.737) -- (4.816,5.618) -- (4.835,5.499) -- (4.89,5.392) -- (4.975,5.307) -- (5.082,5.252) -- (5.201,5.233) -- cycle;
\node at (5.201,5.618) {\scalebox{1.611}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.469,7.305) -- (9.588,7.324) -- (9.695,7.379) -- (9.78,7.464) -- (9.835,7.571) -- (9.854,7.69) -- (9.835,7.809) -- (9.78,7.916) -- (9.695,8.001) -- (9.588,8.055) -- (9.469,8.074) -- (9.469,8.074) -- (9.35,8.055) -- (9.243,8.001) -- (9.158,7.916) -- (9.103,7.809) -- (9.085,7.69) -- (9.103,7.571) -- (9.158,7.464) -- (9.243,7.379) -- (9.35,7.324) -- (9.469,7.305) -- cycle;
\node at (9.469,7.69) {\scalebox{1.611}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.033,5.04) -- (8.152,5.059) -- (8.259,5.113) -- (8.344,5.198) -- (8.399,5.306) -- (8.418,5.424) -- (8.399,5.543) -- (8.344,5.65) -- (8.259,5.735) -- (8.152,5.79) -- (8.033,5.809) -- (8.033,5.809) -- (7.915,5.79) -- (7.807,5.735) -- (7.722,5.65) -- (7.668,5.543) -- (7.649,5.424) -- (7.668,5.306) -- (7.722,5.198) -- (7.807,5.113) -- (7.915,5.059) -- (8.033,5.04) -- cycle;
\node at (8.033,5.424) {\scalebox{1.611}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.534,2.498) -- (11.653,2.517) -- (11.76,2.571) -- (11.846,2.656) -- (11.9,2.764) -- (11.919,2.882) -- (11.9,3.001) -- (11.846,3.108) -- (11.76,3.194) -- (11.653,3.248) -- (11.534,3.267) -- (11.173,3.267) -- (11.054,3.248) -- (10.947,3.194) -- (10.861,3.108) -- (10.807,3.001) -- (10.788,2.882) -- (10.807,2.764) -- (10.861,2.656) -- (10.947,2.571) -- (11.054,2.517) -- (11.173,2.498) -- cycle;
\node at (11.353,2.882) {\scalebox{1.611}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.615,3.836) -- (6.734,3.855) -- (6.841,3.909) -- (6.926,3.994) -- (6.981,4.101) -- (7.0,4.22) -- (6.981,4.339) -- (6.926,4.446) -- (6.841,4.531) -- (6.734,4.586) -- (6.615,4.605) -- (6.253,4.605) -- (6.135,4.586) -- (6.027,4.531) -- (5.942,4.446) -- (5.888,4.339) -- (5.869,4.22) -- (5.888,4.101) -- (5.942,3.994) -- (6.027,3.909) -- (6.135,3.855) -- (6.253,3.836) -- cycle;
\node at (6.434,4.22) {\scalebox{1.611}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.698,2.418) -- (5.817,2.437) -- (5.924,2.491) -- (6.009,2.577) -- (6.064,2.684) -- (6.083,2.803) -- (6.064,2.921) -- (6.009,3.029) -- (5.924,3.114) -- (5.817,3.168) -- (5.698,3.187) -- (5.336,3.187) -- (5.217,3.168) -- (5.11,3.114) -- (5.025,3.029) -- (4.971,2.921) -- (4.952,2.803) -- (4.971,2.684) -- (5.025,2.577) -- (5.11,2.491) -- (5.217,2.437) -- (5.336,2.418) -- cycle;
\node at (5.517,2.803) {\scalebox{1.611}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.256,3.867) -- (9.375,3.886) -- (9.482,3.94) -- (9.567,4.025) -- (9.622,4.132) -- (9.641,4.251) -- (9.622,4.37) -- (9.567,4.477) -- (9.482,4.562) -- (9.375,4.617) -- (9.256,4.636) -- (8.894,4.636) -- (8.776,4.617) -- (8.668,4.562) -- (8.583,4.477) -- (8.529,4.37) -- (8.51,4.251) -- (8.529,4.132) -- (8.583,4.025) -- (8.668,3.94) -- (8.776,3.886) -- (8.894,3.867) -- cycle;
\node at (9.075,4.251) {\scalebox{1.611}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.513,2.348) -- (7.631,2.367) -- (7.739,2.422) -- (7.824,2.507) -- (7.878,2.614) -- (7.897,2.733) -- (7.878,2.852) -- (7.824,2.959) -- (7.739,3.044) -- (7.631,3.099) -- (7.513,3.117) -- (7.151,3.117) -- (7.032,3.099) -- (6.925,3.044) -- (6.84,2.959) -- (6.785,2.852) -- (6.766,2.733) -- (6.785,2.614) -- (6.84,2.507) -- (6.925,2.422) -- (7.032,2.367) -- (7.151,2.348) -- cycle;
\node at (7.332,2.733) {\scalebox{1.611}{\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 & B, C, F, G, H, K, L & 7 \\ \hline
clairs & A, D, E, I, J, M & 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{C} :}\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{venelle du Chat ②} vers le nord jusqu'à Place de la Fontaine \textbf{(A)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Stade ①}, vers l'est, jusqu'à Place du Marché \textbf{(B)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Moulin ③}, vers le sud, jusqu'à Rond-point de la Gare \textbf{(E)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Verte ⑥}, vers l'est, jusqu'à Carrefour Saint-Roch \textbf{(F)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans la \textbf{rue de l'Église ⑧}, vers l'est, jusqu'à Place des Tanneurs \textbf{(D)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue Pasteur ⑤}, vers le sud, jusqu'à Carrefour des Tilleuls \textbf{(G)}.\par\addvspace{0.35em}
\noindent Il continue tout droit dans le \textbf{chemin du Lavoir ⑩}, vers le sud, jusqu'à Place de l'Église \textbf{(M)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans le \textbf{chemin des Vignes ⑯}, vers l'ouest, jusqu'à Carrefour de la Mairie \textbf{(K)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans la \textbf{rue du Four ⑭}, vers le nord, jusqu'à Rond-point du Stade \textbf{(I)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans la \textbf{rue Basse ⑪}, vers l'ouest, jusqu'à Carrefour du Lavoir \textbf{(H)}.\par\addvspace{0.35em}
\noindent Il tourne à gauche dans l'\textbf{allée des Tilleuls ⑫}, vers le sud, jusqu'à Carrefour du Moulin \textbf{(J)}.\par\addvspace{0.35em}
\noindent Il tourne à droite dans l'\textbf{impasse des Cerisiers ⑰}, vers l'ouest, jusqu'à Place Sainte-Croix \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}
