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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 B, 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 Pasteur & Ⓐ Rond-point de la Gare & Ⓚ Carrefour des Tilleuls \\ \hline
① rue Verte & ⑪ avenue de la Gare & Ⓑ Carrefour du Moulin & Ⓛ Coin du Lavoir \\ \hline
② rue des Jardins & ⑫ sentier des Prés & Ⓒ Place de la Fontaine & Ⓜ Place de l'Église \\ \hline
③ chemin des Vignes & ⑬ rue des Écoles & Ⓓ Place Sainte-Croix & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
④ rue Haute & ⑭ allée des Peupliers & Ⓔ Fontaine Notre-Dame & \textbf{a} l'école \\ \hline
⑤ venelle du Chat & ⑮ chemin de Ronde & Ⓕ Rond-point de la Poste & \textbf{b} le lavoir \\ \hline
⑥ rue du Stade & ⑯ rue du Four & Ⓖ Carrefour de l'Hôpital & \textbf{c} l'église \\ \hline
⑦ rue du Pont & ⑰ rue de la Mairie & Ⓗ Rond-point du Pont & \textbf{d} la gendarmerie \\ \hline
⑧ boulevard du Parc & ⑱ allée des Tilleuls & Ⓘ Rond-point du Stade &  \\ \hline
⑨ rue de la Poste & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Place du Vieux-Puits &  \\ \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] (8.213,7.975) -- (8.418,7.931) -- (8.625,7.893) -- (8.832,7.859) -- (9.041,7.831) -- (9.25,7.807) -- (9.459,7.788) -- (9.669,7.774) -- (9.879,7.765) -- (10.089,7.761) -- (10.299,7.762) -- (10.509,7.768) -- (10.719,7.779) -- (10.929,7.795) -- (11.138,7.815);
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\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.163,11.547) -- (5.134,11.689) -- (5.053,11.81) -- (4.932,11.891) -- (4.789,11.92) -- (4.647,11.891) -- (4.526,11.81) -- (4.445,11.689) -- (4.416,11.547) -- (4.445,11.404) -- (4.526,11.283) -- (4.647,11.202) -- (4.789,11.173) -- (4.932,11.202) -- (5.053,11.283) -- (5.134,11.404) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.133,11.276) -- (8.105,11.419) -- (8.024,11.54) -- (7.903,11.621) -- (7.76,11.649) -- (7.617,11.621) -- (7.496,11.54) -- (7.415,11.419) -- (7.387,11.276) -- (7.415,11.133) -- (7.496,11.012) -- (7.617,10.931) -- (7.76,10.903) -- (7.903,10.931) -- (8.024,11.012) -- (8.105,11.133) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (1.646,11.247) -- (1.617,11.39) -- (1.536,11.511) -- (1.415,11.592) -- (1.272,11.621) -- (1.13,11.592) -- (1.008,11.511) -- (0.928,11.39) -- (0.899,11.247) -- (0.928,11.105) -- (1.008,10.983) -- (1.13,10.903) -- (1.272,10.874) -- (1.415,10.903) -- (1.536,10.983) -- (1.617,11.105) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.008,11.238) -- (10.98,11.381) -- (10.899,11.502) -- (10.778,11.583) -- (10.635,11.611) -- (10.492,11.583) -- (10.371,11.502) -- (10.29,11.381) -- (10.262,11.238) -- (10.29,11.095) -- (10.371,10.974) -- (10.492,10.893) -- (10.635,10.865) -- (10.778,10.893) -- (10.899,10.974) -- (10.98,11.095) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (2.024,8.482) -- (1.995,8.625) -- (1.914,8.746) -- (1.793,8.827) -- (1.651,8.856) -- (1.508,8.827) -- (1.387,8.746) -- (1.306,8.625) -- (1.277,8.482) -- (1.306,8.34) -- (1.387,8.219) -- (1.508,8.138) -- (1.651,8.109) -- (1.793,8.138) -- (1.914,8.219) -- (1.995,8.34) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.184,8.189) -- (5.156,8.332) -- (5.075,8.453) -- (4.954,8.534) -- (4.811,8.562) -- (4.668,8.534) -- (4.547,8.453) -- (4.466,8.332) -- (4.438,8.189) -- (4.466,8.046) -- (4.547,7.925) -- (4.668,7.844) -- (4.811,7.816) -- (4.954,7.844) -- (5.075,7.925) -- (5.156,8.046) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.586,7.975) -- (8.558,8.118) -- (8.477,8.239) -- (8.356,8.32) -- (8.213,8.348) -- (8.07,8.32) -- (7.949,8.239) -- (7.868,8.118) -- (7.84,7.975) -- (7.868,7.832) -- (7.949,7.711) -- (8.07,7.63) -- (8.213,7.602) -- (8.356,7.63) -- (8.477,7.711) -- (8.558,7.832) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.511,7.815) -- (11.483,7.958) -- (11.402,8.079) -- (11.281,8.16) -- (11.138,8.189) -- (10.995,8.16) -- (10.874,8.079) -- (10.793,7.958) -- (10.765,7.815) -- (10.793,7.673) -- (10.874,7.551) -- (10.995,7.471) -- (11.138,7.442) -- (11.281,7.471) -- (11.402,7.551) -- (11.483,7.673) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (5.654,5.105) -- (5.626,5.248) -- (5.545,5.369) -- (5.424,5.45) -- (5.281,5.479) -- (5.138,5.45) -- (5.017,5.369) -- (4.936,5.248) -- (4.908,5.105) -- (4.936,4.963) -- (5.017,4.841) -- (5.138,4.761) -- (5.281,4.732) -- (5.424,4.761) -- (5.545,4.841) -- (5.626,4.963) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.34,5.038) -- (11.312,5.18) -- (11.231,5.301) -- (11.11,5.382) -- (10.967,5.411) -- (10.824,5.382) -- (10.703,5.301) -- (10.622,5.18) -- (10.594,5.038) -- (10.622,4.895) -- (10.703,4.774) -- (10.824,4.693) -- (10.967,4.664) -- (11.11,4.693) -- (11.231,4.774) -- (11.312,4.895) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.555,4.981) -- (8.526,5.124) -- (8.445,5.245) -- (8.324,5.326) -- (8.182,5.355) -- (8.039,5.326) -- (7.918,5.245) -- (7.837,5.124) -- (7.808,4.981) -- (7.837,4.839) -- (7.918,4.717) -- (8.039,4.637) -- (8.182,4.608) -- (8.324,4.637) -- (8.445,4.717) -- (8.526,4.839) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.56,1.574) -- (8.531,1.716) -- (8.45,1.838) -- (8.329,1.918) -- (8.186,1.947) -- (8.044,1.918) -- (7.923,1.838) -- (7.842,1.716) -- (7.813,1.574) -- (7.842,1.431) -- (7.923,1.31) -- (8.044,1.229) -- (8.186,1.2) -- (8.329,1.229) -- (8.45,1.31) -- (8.531,1.431) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (5.489,1.547) -- (5.46,1.689) -- (5.379,1.81) -- (5.258,1.891) -- (5.116,1.92) -- (4.973,1.891) -- (4.852,1.81) -- (4.771,1.689) -- (4.742,1.547) -- (4.771,1.404) -- (4.852,1.283) -- (4.973,1.202) -- (5.116,1.173) -- (5.258,1.202) -- (5.379,1.283) -- (5.46,1.404) -- cycle;
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\node at (5.323,9.871) {\scalebox{1.588}{\small 2}};
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\node at (9.201,11.523) {\scalebox{1.588}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.504,9.312) -- (8.622,9.331) -- (8.73,9.385) -- (8.815,9.47) -- (8.869,9.578) -- (8.888,9.696) -- (8.869,9.815) -- (8.815,9.922) -- (8.73,10.007) -- (8.622,10.062) -- (8.504,10.081) -- (8.504,10.081) -- (8.385,10.062) -- (8.278,10.007) -- (8.193,9.922) -- (8.138,9.815) -- (8.119,9.696) -- (8.138,9.578) -- (8.193,9.47) -- (8.278,9.385) -- (8.385,9.331) -- (8.504,9.312) -- cycle;
\node at (8.504,9.696) {\scalebox{1.588}{\small 4}};
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\node at (3.011,11.635) {\scalebox{1.588}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (1.956,9.548) -- (2.075,9.567) -- (2.182,9.622) -- (2.268,9.707) -- (2.322,9.814) -- (2.341,9.933) -- (2.322,10.051) -- (2.268,10.159) -- (2.182,10.244) -- (2.075,10.298) -- (1.956,10.317) -- (1.956,10.317) -- (1.838,10.298) -- (1.73,10.244) -- (1.645,10.159) -- (1.591,10.051) -- (1.572,9.933) -- (1.591,9.814) -- (1.645,9.707) -- (1.73,9.622) -- (1.838,9.567) -- (1.956,9.548) -- cycle;
\node at (1.956,9.933) {\scalebox{1.588}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.126,9.177) -- (11.245,9.196) -- (11.352,9.251) -- (11.437,9.336) -- (11.491,9.443) -- (11.51,9.562) -- (11.491,9.681) -- (11.437,9.788) -- (11.352,9.873) -- (11.245,9.927) -- (11.126,9.946) -- (11.126,9.946) -- (11.007,9.927) -- (10.9,9.873) -- (10.815,9.788) -- (10.76,9.681) -- (10.741,9.562) -- (10.76,9.443) -- (10.815,9.336) -- (10.9,9.251) -- (11.007,9.196) -- (11.126,9.177) -- cycle;
\node at (11.126,9.562) {\scalebox{1.588}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (3.278,8.464) -- (3.397,8.483) -- (3.504,8.538) -- (3.589,8.623) -- (3.644,8.73) -- (3.663,8.849) -- (3.644,8.968) -- (3.589,9.075) -- (3.504,9.16) -- (3.397,9.215) -- (3.278,9.233) -- (3.278,9.233) -- (3.159,9.215) -- (3.052,9.16) -- (2.967,9.075) -- (2.913,8.968) -- (2.894,8.849) -- (2.913,8.73) -- (2.967,8.623) -- (3.052,8.538) -- (3.159,8.483) -- (3.278,8.464) -- cycle;
\node at (3.278,8.849) {\scalebox{1.588}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.527,7.941) -- (6.646,7.96) -- (6.753,8.014) -- (6.838,8.099) -- (6.893,8.206) -- (6.912,8.325) -- (6.893,8.444) -- (6.838,8.551) -- (6.753,8.636) -- (6.646,8.691) -- (6.527,8.71) -- (6.527,8.71) -- (6.408,8.691) -- (6.301,8.636) -- (6.216,8.551) -- (6.161,8.444) -- (6.143,8.325) -- (6.161,8.206) -- (6.216,8.099) -- (6.301,8.014) -- (6.408,7.96) -- (6.527,7.941) -- cycle;
\node at (6.527,8.325) {\scalebox{1.588}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.734,6.34) -- (5.853,6.359) -- (5.96,6.413) -- (6.045,6.499) -- (6.1,6.606) -- (6.119,6.725) -- (6.1,6.843) -- (6.045,6.951) -- (5.96,7.036) -- (5.853,7.09) -- (5.734,7.109) -- (5.372,7.109) -- (5.254,7.09) -- (5.146,7.036) -- (5.061,6.951) -- (5.007,6.843) -- (4.988,6.725) -- (5.007,6.606) -- (5.061,6.499) -- (5.146,6.413) -- (5.254,6.359) -- (5.372,6.34) -- cycle;
\node at (5.553,6.725) {\scalebox{1.588}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.829,7.006) -- (9.948,7.025) -- (10.055,7.079) -- (10.14,7.164) -- (10.195,7.272) -- (10.213,7.39) -- (10.195,7.509) -- (10.14,7.616) -- (10.055,7.702) -- (9.948,7.756) -- (9.829,7.775) -- (9.467,7.775) -- (9.348,7.756) -- (9.241,7.702) -- (9.156,7.616) -- (9.101,7.509) -- (9.082,7.39) -- (9.101,7.272) -- (9.156,7.164) -- (9.241,7.079) -- (9.348,7.025) -- (9.467,7.006) -- cycle;
\node at (9.648,7.39) {\scalebox{1.588}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.886,6.088) -- (9.005,6.107) -- (9.112,6.162) -- (9.197,6.247) -- (9.252,6.354) -- (9.271,6.473) -- (9.252,6.592) -- (9.197,6.699) -- (9.112,6.784) -- (9.005,6.838) -- (8.886,6.857) -- (8.524,6.857) -- (8.405,6.838) -- (8.298,6.784) -- (8.213,6.699) -- (8.159,6.592) -- (8.14,6.473) -- (8.159,6.354) -- (8.213,6.247) -- (8.298,6.162) -- (8.405,6.107) -- (8.524,6.088) -- cycle;
\node at (8.705,6.473) {\scalebox{1.588}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.502,6.025) -- (11.621,6.044) -- (11.728,6.099) -- (11.814,6.184) -- (11.868,6.291) -- (11.887,6.41) -- (11.868,6.529) -- (11.814,6.636) -- (11.728,6.721) -- (11.621,6.776) -- (11.502,6.794) -- (11.141,6.794) -- (11.022,6.776) -- (10.915,6.721) -- (10.829,6.636) -- (10.775,6.529) -- (10.756,6.41) -- (10.775,6.291) -- (10.829,6.184) -- (10.915,6.099) -- (11.022,6.044) -- (11.141,6.025) -- cycle;
\node at (11.322,6.41) {\scalebox{1.588}{\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, D, E, G, I, J, L & 7 \\ \hline
clairs & B, C, F, H, K, 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{Non plus.} Avec un carrefour sombre de plus, la tournée doit \textbf{partir et arriver} sur un carrefour sombre. Or l'école est en \textbf{B}, qui est clair.}\par\addvspace{0.4em}
\noindent{\color{green!45!black}Une tournée existe pourtant dans ce quartier : il suffirait d'installer l'école sur un carrefour sombre. Ce n'est pas le plan qui est en cause, c'est l'endroit d'où l'on part.}\par\addvspace{0.4em}
\end{document}
