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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 M, 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 de l'Église & Ⓑ Place de la Fontaine & Ⓛ Place des Tanneurs \\ \hline
① rue de la Poste & ⑪ impasse des Cerisiers & Ⓒ Place du Marché & Ⓜ Coin du Lavoir \\ \hline
② rue des Jardins & ⑫ avenue de la Gare & Ⓓ Rond-point du Pont & \cellcolor{black!32}\bfseries Les bâtiments \\ \hline
③ chemin Creux & ⑬ rue du Moulin & Ⓔ Carrefour des Tilleuls & \textbf{a} l'école \\ \hline
④ rue de la Mairie & ⑭ allée des Tilleuls & Ⓕ Carrefour du Lavoir & \textbf{b} le marché couvert \\ \hline
⑤ sentier des Prés & ⑮ rue Neuve & Ⓖ Rond-point de la Poste & \textbf{c} la pharmacie \\ \hline
⑥ rue du Pont & ⑯ passage du Marché & Ⓗ Carrefour du Moulin & \textbf{d} le gymnase \\ \hline
⑦ rue du Puits & ⑰ chemin de Ronde & Ⓘ Place aux Herbes &  \\ \hline
⑧ rue Basse & \cellcolor{black!32}\bfseries Les carrefours & Ⓙ Place du Vieux-Puits &  \\ \hline
⑨ chemin des Vignes & Ⓐ Placette des Ormes & Ⓚ 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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\draw[line width=0.5pt] (1.671,4.424) -- (1.609,4.213) -- (1.552,4.002) -- (1.5,3.788) -- (1.453,3.574) -- (1.411,3.358) -- (1.374,3.142) -- (1.342,2.925) -- (1.316,2.707) -- (1.294,2.488) -- (1.278,2.269) -- (1.267,2.05) -- (1.261,1.831) -- (1.26,1.611) -- (1.264,1.392);
\draw[line width=0.5pt] (4.532,1.804) -- (4.732,1.837) -- (4.932,1.866) -- (5.133,1.89) -- (5.335,1.909) -- (5.537,1.923) -- (5.739,1.933) -- (5.941,1.938) -- (6.144,1.938) -- (6.346,1.934) -- (6.548,1.924) -- (6.75,1.91) -- (6.952,1.891) -- (7.153,1.868) -- (7.353,1.839);
\draw[line width=0.5pt] (1.264,1.392) -- (1.5,1.385) -- (1.737,1.384) -- (1.973,1.389) -- (2.209,1.399) -- (2.445,1.415) -- (2.68,1.436) -- (2.915,1.463) -- (3.15,1.495) -- (3.383,1.533) -- (3.615,1.576) -- (3.847,1.625) -- (4.077,1.679) -- (4.305,1.739) -- (4.532,1.804);
\filldraw[fill=black!38, draw=black, line width=0.5pt] (10.943,10.952) -- (10.915,11.095) -- (10.834,11.216) -- (10.713,11.297) -- (10.57,11.325) -- (10.427,11.297) -- (10.306,11.216) -- (10.225,11.095) -- (10.197,10.952) -- (10.225,10.809) -- (10.306,10.688) -- (10.427,10.607) -- (10.57,10.579) -- (10.713,10.607) -- (10.834,10.688) -- (10.915,10.809) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (8.208,10.906) -- (8.179,11.049) -- (8.098,11.17) -- (7.977,11.251) -- (7.835,11.28) -- (7.692,11.251) -- (7.571,11.17) -- (7.49,11.049) -- (7.461,10.906) -- (7.49,10.764) -- (7.571,10.643) -- (7.692,10.562) -- (7.835,10.533) -- (7.977,10.562) -- (8.098,10.643) -- (8.179,10.764) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.553,8.183) -- (4.525,8.326) -- (4.444,8.447) -- (4.323,8.528) -- (4.18,8.556) -- (4.037,8.528) -- (3.916,8.447) -- (3.835,8.326) -- (3.807,8.183) -- (3.835,8.04) -- (3.916,7.919) -- (4.037,7.838) -- (4.18,7.81) -- (4.323,7.838) -- (4.444,7.919) -- (4.525,8.04) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (10.8,8.127) -- (10.772,8.27) -- (10.691,8.391) -- (10.57,8.472) -- (10.427,8.5) -- (10.284,8.472) -- (10.163,8.391) -- (10.082,8.27) -- (10.054,8.127) -- (10.082,7.984) -- (10.163,7.863) -- (10.284,7.782) -- (10.427,7.754) -- (10.57,7.782) -- (10.691,7.863) -- (10.772,7.984) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (8.044,7.541) -- (8.015,7.684) -- (7.934,7.805) -- (7.813,7.886) -- (7.67,7.914) -- (7.528,7.886) -- (7.407,7.805) -- (7.326,7.684) -- (7.297,7.541) -- (7.326,7.398) -- (7.407,7.277) -- (7.528,7.196) -- (7.67,7.168) -- (7.813,7.196) -- (7.934,7.277) -- (8.015,7.398) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (4.483,5.23) -- (4.454,5.373) -- (4.373,5.494) -- (4.252,5.575) -- (4.109,5.603) -- (3.967,5.575) -- (3.846,5.494) -- (3.765,5.373) -- (3.736,5.23) -- (3.765,5.087) -- (3.846,4.966) -- (3.967,4.885) -- (4.109,4.857) -- (4.252,4.885) -- (4.373,4.966) -- (4.454,5.087) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (11.637,4.859) -- (11.609,5.002) -- (11.528,5.123) -- (11.407,5.204) -- (11.264,5.232) -- (11.121,5.204) -- (11.0,5.123) -- (10.919,5.002) -- (10.891,4.859) -- (10.919,4.716) -- (11.0,4.595) -- (11.121,4.514) -- (11.264,4.486) -- (11.407,4.514) -- (11.528,4.595) -- (11.609,4.716) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (7.572,4.827) -- (7.544,4.97) -- (7.463,5.091) -- (7.342,5.172) -- (7.199,5.201) -- (7.056,5.172) -- (6.935,5.091) -- (6.854,4.97) -- (6.826,4.827) -- (6.854,4.685) -- (6.935,4.563) -- (7.056,4.483) -- (7.199,4.454) -- (7.342,4.483) -- (7.463,4.563) -- (7.544,4.685) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (2.044,4.424) -- (2.016,4.567) -- (1.935,4.688) -- (1.814,4.769) -- (1.671,4.797) -- (1.528,4.769) -- (1.407,4.688) -- (1.326,4.567) -- (1.298,4.424) -- (1.326,4.281) -- (1.407,4.16) -- (1.528,4.079) -- (1.671,4.051) -- (1.814,4.079) -- (1.935,4.16) -- (2.016,4.281) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (7.726,1.839) -- (7.698,1.982) -- (7.617,2.103) -- (7.496,2.184) -- (7.353,2.213) -- (7.21,2.184) -- (7.089,2.103) -- (7.008,1.982) -- (6.98,1.839) -- (7.008,1.697) -- (7.089,1.575) -- (7.21,1.495) -- (7.353,1.466) -- (7.496,1.495) -- (7.617,1.575) -- (7.698,1.697) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (4.906,1.804) -- (4.877,1.947) -- (4.796,2.068) -- (4.675,2.149) -- (4.532,2.177) -- (4.39,2.149) -- (4.269,2.068) -- (4.188,1.947) -- (4.159,1.804) -- (4.188,1.661) -- (4.269,1.54) -- (4.39,1.459) -- (4.532,1.431) -- (4.675,1.459) -- (4.796,1.54) -- (4.877,1.661) -- cycle;
\filldraw[fill=white, draw=black, line width=0.5pt] (11.198,1.395) -- (11.17,1.538) -- (11.089,1.659) -- (10.968,1.74) -- (10.825,1.768) -- (10.682,1.74) -- (10.561,1.659) -- (10.48,1.538) -- (10.452,1.395) -- (10.48,1.252) -- (10.561,1.131) -- (10.682,1.05) -- (10.825,1.021) -- (10.968,1.05) -- (11.089,1.131) -- (11.17,1.252) -- cycle;
\filldraw[fill=black!38, draw=black, line width=0.5pt] (1.637,1.392) -- (1.609,1.535) -- (1.528,1.656) -- (1.407,1.737) -- (1.264,1.765) -- (1.121,1.737) -- (1.0,1.656) -- (0.919,1.535) -- (0.891,1.392) -- (0.919,1.249) -- (1.0,1.128) -- (1.121,1.047) -- (1.264,1.019) -- (1.407,1.047) -- (1.528,1.128) -- (1.609,1.249) -- cycle;
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\node at (10.766,9.526) {\scalebox{1.603}{\small 1}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.194,11.042) -- (9.313,11.061) -- (9.42,11.115) -- (9.505,11.201) -- (9.56,11.308) -- (9.579,11.427) -- (9.56,11.545) -- (9.505,11.653) -- (9.42,11.738) -- (9.313,11.792) -- (9.194,11.811) -- (9.194,11.811) -- (9.075,11.792) -- (8.968,11.738) -- (8.883,11.653) -- (8.828,11.545) -- (8.809,11.427) -- (8.828,11.308) -- (8.883,11.201) -- (8.968,11.115) -- (9.075,11.061) -- (9.194,11.042) -- cycle;
\node at (9.194,11.427) {\scalebox{1.603}{\small 2}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.998,8.827) -- (8.117,8.846) -- (8.224,8.901) -- (8.309,8.986) -- (8.363,9.093) -- (8.382,9.212) -- (8.363,9.331) -- (8.309,9.438) -- (8.224,9.523) -- (8.117,9.578) -- (7.998,9.596) -- (7.998,9.596) -- (7.879,9.578) -- (7.772,9.523) -- (7.687,9.438) -- (7.632,9.331) -- (7.613,9.212) -- (7.632,9.093) -- (7.687,8.986) -- (7.772,8.901) -- (7.879,8.846) -- (7.998,8.827) -- cycle;
\node at (7.998,9.212) {\scalebox{1.603}{\small 3}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.021,8.0) -- (6.14,8.019) -- (6.247,8.073) -- (6.332,8.158) -- (6.387,8.266) -- (6.406,8.384) -- (6.387,8.503) -- (6.332,8.61) -- (6.247,8.695) -- (6.14,8.75) -- (6.021,8.769) -- (6.021,8.769) -- (5.902,8.75) -- (5.795,8.695) -- (5.71,8.61) -- (5.655,8.503) -- (5.637,8.384) -- (5.655,8.266) -- (5.71,8.158) -- (5.795,8.073) -- (5.902,8.019) -- (6.021,8.0) -- cycle;
\node at (6.021,8.384) {\scalebox{1.603}{\small 4}};
\filldraw[fill=white, draw=black, line width=0.5pt] (4.407,6.316) -- (4.526,6.335) -- (4.633,6.389) -- (4.718,6.474) -- (4.773,6.581) -- (4.792,6.7) -- (4.773,6.819) -- (4.718,6.926) -- (4.633,7.011) -- (4.526,7.066) -- (4.407,7.085) -- (4.407,7.085) -- (4.288,7.066) -- (4.181,7.011) -- (4.096,6.926) -- (4.042,6.819) -- (4.023,6.7) -- (4.042,6.581) -- (4.096,6.474) -- (4.181,6.389) -- (4.288,6.335) -- (4.407,6.316) -- cycle;
\node at (4.407,6.7) {\scalebox{1.603}{\small 5}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.353,6.238) -- (11.472,6.257) -- (11.579,6.312) -- (11.664,6.397) -- (11.719,6.504) -- (11.738,6.623) -- (11.719,6.742) -- (11.664,6.849) -- (11.579,6.934) -- (11.472,6.989) -- (11.353,7.007) -- (11.353,7.007) -- (11.234,6.989) -- (11.127,6.934) -- (11.042,6.849) -- (10.987,6.742) -- (10.968,6.623) -- (10.987,6.504) -- (11.042,6.397) -- (11.127,6.312) -- (11.234,6.257) -- (11.353,6.238) -- cycle;
\node at (11.353,6.623) {\scalebox{1.603}{\small 6}};
\filldraw[fill=white, draw=black, line width=0.5pt] (8.993,7.712) -- (9.112,7.731) -- (9.219,7.785) -- (9.304,7.87) -- (9.359,7.978) -- (9.378,8.096) -- (9.359,8.215) -- (9.304,8.322) -- (9.219,8.408) -- (9.112,8.462) -- (8.993,8.481) -- (8.993,8.481) -- (8.874,8.462) -- (8.767,8.408) -- (8.682,8.322) -- (8.627,8.215) -- (8.609,8.096) -- (8.627,7.978) -- (8.682,7.87) -- (8.767,7.785) -- (8.874,7.731) -- (8.993,7.712) -- cycle;
\node at (8.993,8.096) {\scalebox{1.603}{\small 7}};
\filldraw[fill=white, draw=black, line width=0.5pt] (7.925,5.714) -- (8.044,5.733) -- (8.151,5.788) -- (8.237,5.873) -- (8.291,5.98) -- (8.31,6.099) -- (8.291,6.218) -- (8.237,6.325) -- (8.151,6.41) -- (8.044,6.465) -- (7.925,6.484) -- (7.925,6.484) -- (7.807,6.465) -- (7.699,6.41) -- (7.614,6.325) -- (7.56,6.218) -- (7.541,6.099) -- (7.56,5.98) -- (7.614,5.873) -- (7.699,5.788) -- (7.807,5.733) -- (7.925,5.714) -- cycle;
\node at (7.925,6.099) {\scalebox{1.603}{\small 8}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.687,4.898) -- (5.806,4.917) -- (5.913,4.971) -- (5.998,5.056) -- (6.053,5.164) -- (6.072,5.283) -- (6.053,5.401) -- (5.998,5.509) -- (5.913,5.594) -- (5.806,5.648) -- (5.687,5.667) -- (5.687,5.667) -- (5.568,5.648) -- (5.461,5.594) -- (5.376,5.509) -- (5.322,5.401) -- (5.303,5.283) -- (5.322,5.164) -- (5.376,5.056) -- (5.461,4.971) -- (5.568,4.917) -- (5.687,4.898) -- cycle;
\node at (5.687,5.283) {\scalebox{1.603}{\small 9}};
\filldraw[fill=white, draw=black, line width=0.5pt] (5.025,3.197) -- (5.144,3.216) -- (5.251,3.27) -- (5.336,3.355) -- (5.391,3.463) -- (5.409,3.581) -- (5.391,3.7) -- (5.336,3.807) -- (5.251,3.893) -- (5.144,3.947) -- (5.025,3.966) -- (4.663,3.966) -- (4.544,3.947) -- (4.437,3.893) -- (4.352,3.807) -- (4.297,3.7) -- (4.278,3.581) -- (4.297,3.463) -- (4.352,3.355) -- (4.437,3.27) -- (4.544,3.216) -- (4.663,3.197) -- cycle;
\node at (4.844,3.581) {\scalebox{1.603}{\small 10}};
\filldraw[fill=white, draw=black, line width=0.5pt] (11.464,2.712) -- (11.583,2.731) -- (11.69,2.786) -- (11.775,2.871) -- (11.83,2.978) -- (11.849,3.097) -- (11.83,3.215) -- (11.775,3.323) -- (11.69,3.408) -- (11.583,3.462) -- (11.464,3.481) -- (11.102,3.481) -- (10.983,3.462) -- (10.876,3.408) -- (10.791,3.323) -- (10.737,3.215) -- (10.718,3.097) -- (10.737,2.978) -- (10.791,2.871) -- (10.876,2.786) -- (10.983,2.731) -- (11.102,2.712) -- cycle;
\node at (11.283,3.097) {\scalebox{1.603}{\small 11}};
\filldraw[fill=white, draw=black, line width=0.5pt] (9.408,5.011) -- (9.527,5.03) -- (9.634,5.084) -- (9.719,5.169) -- (9.774,5.277) -- (9.793,5.395) -- (9.774,5.514) -- (9.719,5.621) -- (9.634,5.706) -- (9.527,5.761) -- (9.408,5.78) -- (9.046,5.78) -- (8.927,5.761) -- (8.82,5.706) -- (8.735,5.621) -- (8.681,5.514) -- (8.662,5.395) -- (8.681,5.277) -- (8.735,5.169) -- (8.82,5.084) -- (8.927,5.03) -- (9.046,5.011) -- cycle;
\node at (9.227,5.395) {\scalebox{1.603}{\small 12}};
\filldraw[fill=white, draw=black, line width=0.5pt] (6.95,2.923) -- (7.068,2.941) -- (7.176,2.996) -- (7.261,3.081) -- (7.315,3.188) -- (7.334,3.307) -- (7.315,3.426) -- (7.261,3.533) -- (7.176,3.618) -- (7.068,3.673) -- (6.95,3.692) -- (6.588,3.692) -- (6.469,3.673) -- (6.362,3.618) -- (6.277,3.533) -- (6.222,3.426) -- (6.203,3.307) -- (6.222,3.188) -- (6.277,3.081) -- (6.362,2.996) -- (6.469,2.941) -- (6.588,2.923) -- cycle;
\node at (6.769,3.307) {\scalebox{1.603}{\small 13}};
\filldraw[fill=white, draw=black, line width=0.5pt] (2.62,5.165) -- (2.739,5.184) -- (2.846,5.238) -- (2.931,5.323) -- (2.986,5.43) -- (3.005,5.549) -- (2.986,5.668) -- (2.931,5.775) -- (2.846,5.86) -- (2.739,5.915) -- (2.62,5.934) -- (2.258,5.934) -- (2.14,5.915) -- (2.032,5.86) -- (1.947,5.775) -- (1.893,5.668) -- (1.874,5.549) -- (1.893,5.43) -- (1.947,5.323) -- (2.032,5.238) -- (2.14,5.184) -- (2.258,5.165) -- cycle;
\node at (2.439,5.549) {\scalebox{1.603}{\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, E, F, G, J, M & 6 \\ \hline
clairs & B, C, D, H, I, K, L & 7 \\ \hline
\end{tabular}
\end{center}
\noindent{\color{red!80!black}\bfseries Il y a \textbf{6} carrefours sombres et \textbf{7} clairs. Il y en a donc \textbf{1 de plus} du côté des clairs.}\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 6 carrefours sombres et 7 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 clair de plus, la tournée doit \textbf{partir et arriver} sur un carrefour clair. Or l'école est en \textbf{M}, qui est sombre.}\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 clair. 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}
