From ec38ac88aae16e06ecc403ca4207d76ce53688f6 Mon Sep 17 00:00:00 2001 From: Guillaume Garnier Date: Tue, 7 Jan 2025 20:47:27 +0100 Subject: [PATCH] Version finale --- index.aux | 40 ++++++++++----------- index.fdb_latexmk | 38 ++++++++++---------- index.fls | 72 +++++++++++++++++++------------------- index.log | 56 +++++++++++++++-------------- index.out | 2 +- index.pdf | Bin 369875 -> 370490 bytes index.synctex.gz | Bin 147590 -> 151800 bytes index.tex | 5 ++- index.toc | 6 ++-- src/appat.tex | 15 ++++---- src/bacteries.tex | 29 ++++++++------- src/descente-bus.tex | 30 +++++++++------- src/gentillesse.tex | 23 ++++++------ src/inegalites-graphe.tex | 12 +++---- src/points-colories.tex | 34 +++++++++--------- src/tarte.tex | 30 ++++++++-------- src/toboggan.tex | 18 +++++----- 17 files changed, 210 insertions(+), 200 deletions(-) diff --git a/index.aux b/index.aux index 941178c..28d81c9 100644 --- a/index.aux +++ b/index.aux @@ -24,38 +24,38 @@ \babel@aux{french}{} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{Préambule}}{1}{section*.1}\protected@file@percent } \@writefile{toc}{\contentsline {section}{\tocsection {}{}{Notations}}{1}{section*.3}\protected@file@percent } -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Un exemple de propagation, qui montre que le nombre 3 est réalisable. 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z8zMP`z(v&}@pVUVG_vPx)Bf>XB`b@|bD4;kj@#BV&x=vQSYmh$Ml!uzODvY*&M=!H z4vHuTZ55>vQAYYzmoBlT|1rovgCHD@2ZKrYdLTSo`te-lGntb`dTt*WgRH}>yCGV0 zBy^P~5N#+rqpB(gE=yV=h}`Wofj}>v+^hGhcw!AbtuJ(qLleHA2x_6+zThLZ-7T}x zr%5({S@G|7@n2fary28pVJF`vce-Z*{j4%0oz@`(=-YPg*|v=ApSS!?huoisR^B8I zTX{eKJQMdmE!Mk?pQjSfe4l5sx$KNwpFe8~?}@V1vOQ(_{B-L7e2eKgl9k=_eP@q@ zJQ8%Cu#voz@K5cBfA;Bz|NOgu{No?66#Mq;-~Rp|e)H?^-~aZf-~Ih>e){!)|4;wr zzy3-Dru^%_|K0!o&7c0m-~RDWKm1KDKm51f|I;7-_81qq(P?&02T*&hX4Qo diff --git a/index.tex b/index.tex index d74e51d..deef87a 100644 --- a/index.tex +++ b/index.tex @@ -108,7 +108,10 @@ Ces problèmes sont distribués sous licence \texttt{CC-BY-SA 4.0}. En cas de qu \begin{tabular}{ll} $\{a_1,a_2,\dots, a_n\}$ & ensemble contenant les éléments $a_1, a_2, \dots, a_n$ \\ $\N = \{0,1,2,\ldots\}$ & ensemble des nombres entiers naturels \\ -$\ln$ & logarithme népérien +$\mathbb{R}$ & ensemble des nombres réels \\ +$\mathbb{R}_+$ & ensemble des nombres réels strictement positifs ou nuls \\ +$\ln$ & logarithme népérien\\ +$|\cdot|$ & valeur absolue \end{tabular} \restoregeometry diff --git a/index.toc b/index.toc index 87b8fac..4354e90 100644 --- a/index.toc +++ b/index.toc @@ -3,9 +3,9 @@ \contentsline {section}{\tocsection {}{}{Notations}}{1}{section*.3}% \contentsline {section}{\tocsection {}{1}{Une bonne humeur contagieuse}}{2}{section.1}% \contentsline {section}{\tocsection {}{2}{Drôles de toboggans}}{3}{section.2}% -\contentsline {section}{\tocsection {}{3}{Plats à tarte gradués}}{6}{section.3}% -\contentsline {section}{\tocsection {}{4}{Transformation de papillons}}{7}{section.4}% +\contentsline {section}{\tocsection {}{3}{Plats à tarte gradués}}{5}{section.3}% +\contentsline {section}{\tocsection {}{4}{Transformation de papillons}}{8}{section.4}% \contentsline {section}{\tocsection {}{5}{Gerrymandering}}{8}{section.5}% \contentsline {section}{\tocsection {}{6}{Le cauchemar de la ligne 20-25}}{9}{section.6}% -\contentsline {section}{\tocsection {}{7}{Le poids des graphes}}{11}{section.7}% +\contentsline {section}{\tocsection {}{7}{Taxes routières}}{11}{section.7}% \contentsline {section}{\tocsection {}{8}{Points colorés sur un cercle}}{13}{section.8}% diff --git a/src/appat.tex b/src/appat.tex index ff24ced..ee4693f 100644 --- a/src/appat.tex +++ b/src/appat.tex @@ -1,14 +1,13 @@ \section{Gerrymandering} -Elbridge cherche à déplacer les capitales des différents districts vers les emplacements demandés par son parti, en le moins d'années possible. +Afin de gagner les élections, Elbridge cherche à déplacer les capitales des différents districts vers les emplacements demandés par son parti, en le moins d'années possible. Soit $P$ une partie du plan, qui représente un pays, et $n \ge 2$. -On appelle \textbf{configuration} un choix de $n$ points $A_1$, $A_2$, $\ldots$, $A_n$, qui représentent les capitales des districts. -A chaque configuration est associée un \textbf{découpage} de $P$, où le district $D_i$ est constituée de l'ensemble des points strictement plus proches de $A_i$ que de tous les autres points. +On appelle \textbf{configuration} un choix de $n$ points distincts $A_1$, $A_2$, $\ldots$, $A_n$, qui représentent les capitales des districts où pour une configuration donnée, on découpe $P$ en $n$ parties $D_1, \ldots, D_n$ que l'on appelle des districts : le district $D_i$ est constitué de l'ensemble des points strictement plus proches de $A_i$ que de tous les autres points. -Chaque année, Elbridge peut déplacer, simultanément, chaque capitale $A_i$ vers un nouvel emplacement $A_i' \in D_i$. -On dit alors que $(A_1',\ldots,A_n')$ est \textbf{réalisable} à partir de $(A_1,\ldots,A_n)$ en 1 année. -Plus généralement, on définit, pour une configuration $C'$, le fait d'être réalisable à partir de $C$ comme le fait qu'il existe $a \in \N$ tels que $C'$ soit réalisable à partir de $C$ en $a$ années. +Chaque année, Elbridge peut déplacer, simultanément, chaque capitale $A_i$ vers un nouvel emplacement $A_i' \in D_i$. +On dit alors que $(A_1',\ldots,A_n')$ est \textbf{réalisable} à partir de $(A_1,\ldots,A_n)$ en 1 année. Ensuite le découpage de $P$ en $n$ districts est refait en fonction de cette nouvelle configuration. +Plus généralement, on définit, pour une configuration $C'$, le fait d'être réalisable à partir de $C$ comme le fait qu'il existe $a \in \N$ tel que $C'$ soit réalisable à partir de $C$ en $a$ années. Voir Fig. \ref{fig:gerry}. @@ -79,7 +78,7 @@ Voir Fig. \ref{fig:gerry}. \end{figure} -Dans un premier temps, on se place dans le cas où $P$ est un cercle centré en l'origine. +Dans un premier temps, on se place dans le cas où $P$ est un cercle centré en l'origine. Par conséquent toutes les capitales sont situées à une même distance du centre de $P$. \q A partir d'une configuration donnée, quelles configurations sont réalisables ? @@ -100,4 +99,4 @@ Dans la question 3, $M$ est un demi-plan. \q Généraliser au cas des dimensions supérieures. -\q Proposer et étudier d'autres directions de recherche. \ No newline at end of file +\q Proposer et étudier d'autres pistes de recherche. \ No newline at end of file diff --git a/src/bacteries.tex b/src/bacteries.tex index 25b429a..753de4d 100644 --- a/src/bacteries.tex +++ b/src/bacteries.tex @@ -1,30 +1,33 @@ \section{Transformation de papillons} -Pour embellir les différents tournois du \tfjm, le comité national d'organisation décide de faire un élevage de $N$ papillons ($N$ impair). À l’origine, le papillon numéro $i$ a une envergure égale à $x_i$ cm. Chaque jour, certains papillons subissent une transformation qui modifie leur envergure. La manière de choisir le papillon qui subit la transformation peut varier (par exemple, le plus petit, le plus grand, un choix aléatoire, ou le papillon avec l’envergure médiane). +Pour embellir les différents tournois du \tfjm, le comité national d'organisation décide de faire un élevage de $N$ papillons. À l’origine, le papillon numéro $i$ a une envergure égale à $x_i$ centimètres. Chaque jour, certains papillons subissent une transformation qui modifie leur envergure. -\q Dans cette question seulement, on suppose que tous les papillons ont une envergure de $1$ cm. Chaque jour, seul l'un des papillons d'envergure maximale voit celle-ci être divisée par deux. Combien de temps faudra-t-il pour que tous les papillons aient une envergure strictement inférieure à $0.5$ cm ? Et pour $0.1$ cm ? Que se passe-t-il si la transformation réduit l'envergure d'un des papillons de taille médiane ? +\q Dans cette question uniquement, on suppose que tous les papillons ont une envergure initiale de $1$ cm. +\begin{enumerate} + \item Chaque jour, un des papillons d'envergure maximale voit son envergure être divisée par deux. Combien de temps faudra-t-il pour que tous les papillons aient une envergure strictement inférieure à $0,5$ cm ? Et pour $0,1$ cm ? + \item Supposons que $N$ est impair. Que se passe-t-il si la transformation divise par deux l'envergure d'un des papillons d'envergure médiane ? +\end{enumerate} -\q Désormais, on suppose que les transformations alternent : +\q Supposons que $N$ est impair. Désormais, on suppose que les transformations alternent : \begin{itemize} - \item La première transformation s'applique à l'un des papillons ayant une envergure médiane, qui perd alors la moitié de sa taille. - \item La seconde transformation s'applique à l'un des papillons ayant une envergure médiane, qui gagne alors la moitié de sa taille. + \item La première transformation s'applique à l'un des papillons ayant une envergure médiane, qui perd alors la moitié de son envergure. + \item La seconde transformation s'applique à l'un des papillons ayant une envergure médiane, qui gagne alors la moitié de son envergure. \end{itemize} -Est-il possible que l'un des papillons atteigne une taille arbitrairement grande ? +Est-il vrai que, pour tout $M \in \R$, que l'un des papillon va finir par dépasser la taille M ? -\q Que devient la question précédente si la seconde transformation multiplie la taille du papillon par $2$ à la place ? +\q Reprendre la question précédente si la seconde transformation multiplie l'envergure du papillon par $2$ à la place. -\medskip +\bigskip Nous faisons désormais l'hypothèse que tous les papillons se transforment en même temps. \q Chaque jour, chaque papillon se transforme en deux papillons: Le premier hérite de 80\% de l'envergure du parent, et le second de 125\%. -Soit $x \in [0, \infty[$. Estimer la proportion de papillons ayant une taille supérieure à $x$ le $n$-ème jour ? Et la proportion de papillons ayant une taille inférieure à $x$ ? (\textit{On pourra commencer par regarder des valeurs particulières de $x$}) +Soit $x \in \R_+$. Estimer la proportion de papillons ayant une envergure supérieure à $x$ le $n$-ième jour. +\q Chaque jour, chaque papillon se transforme en deux papillons, ce qui augmente. Le premier hérite de 80\% de l'envergure de son parent, et le second de 125\% de l'envergure de son grand-parent. Puisqu'il n'y a pas de grand parent à la première transformation, on supposera que le grand-parent a la même envergure que le parent. +Soit $x \in \R_+$. Quelle est la proportion de papillons ayant une envergure strictement supérieure à $x$ le $n$-ème jour ? (\textit{On pourra commencer par regarder des valeurs particulières de $x$}) -\q Chaque jour, chaque papillon se transforme en deux papillons. Le premier hérite de 80\% de l'envergure de son parent, et le second de 125\% de l'envergure de son grand-parent. (Puisqu'il n'y a pas de grand parent à la première transformation, on supposera que le grand parent a la même envergure que le parent.) -Soit $x \in [0, \infty[$. Quelle est la proportion de papillons ayant une taille strictement supérieure à $x$ le $n$-ème jour ? Et la proportion de papillons ayant une taille strictement inférieure à $x$ ? (\textit{On pourra commencer par regarder des valeurs particulières de $x$}) - -\q Chaque jour, chaque papillon se transforme en deux papillons. La taille des nouveaux papillons est tirée aléatoirement selon une loi de probabilité continue. Peut-on retrouver cette loi de probabilité en observant assez longtemps l'évolution des papillons ? +\q Chaque jour, chaque papillon se transforme en deux papillons. Le pourcentage d'évolution de l'envergure des nouveaux papillons par rapport à leurs parents est tiré aléatoirement selon une loi de probabilité discrète fixée. Peut-on retrouver cette loi de probabilité en observant assez longtemps l'évolution des papillons ? \q Proposer et étudier d'autres pistes de recherche. diff --git a/src/descente-bus.tex b/src/descente-bus.tex index 5904c22..83c7a7d 100644 --- a/src/descente-bus.tex +++ b/src/descente-bus.tex @@ -5,24 +5,28 @@ On assiste notamment à des scènes spectaculaires où un bus est tellement bond Face à ce problème, la société des transports TRAP a mandaté Antoine pour effectuer une analyse d'efficacité et proposer des pistes d'amélioration. La situation étant plutôt complexe (c'est le moins qu'on puisse dire !), Antoine décide de travailler sur un modèle simplifié. Il considère la ligne 20-25 comme une ligne droite, avec le dépôt situé en \( 0 \), puis un arrêt en chaque entier \( n \geq 1 \) (le dépôt n'est donc pas considéré comme un arrêt). -Après quelques observations, il lui apparaît que les bus se déplacent à une vitesse moyenne maximale \( V_{0} \), mais que leur vitesse moyenne diminue à mesure qu'ils chargent des passagers. -Antoine conjecture que la vitesse moyenne d'un bus contenant \( k \) passagers (le chauffeur ne compte pas comme un passager) est donnée par +Après quelques observations, il lui apparaît que les bus se déplacent à une vitesse moyenne initiale \( V_{0} \), mais que leur vitesse moyenne diminue à mesure qu'ils chargent des passagers. +Antoine observe que la vitesse moyenne d'un bus contenant \( k \) passagers (le chauffeur ne compte pas comme un passager) est donnée par \begin{equation} \label{eq:VitesseBus} V_{k} = \frac{V_{0}}{1+\ln{(k+1)}}\text{.} \end{equation} (Antoine considère que les bus se déplacent constamment à leur vitesse moyenne, et le temps passé aux arrêts est compris dans cette vitesse moyenne. Autrement dit, il fait comme si les bus se déplaçaient constamment à vitesse \( V_{k} \), et lorsqu'ils atteignent un arrêt, ils embarquent tous les passagers qui s'y trouvent et changent de vitesse instantanément.) -Les bus ne peuvent pas se dépasser : lorsqu'un bus rattrape son prédécesseur, il le suit à la même vitesse que lui (y compris si cela implique de rouler à une vitesse inférieure à sa vitesse moyenne), et en arrivant à un arrêt, les passagers sont répartis équitablement entre les bus (s'il devait y avoir plus de bus que de passagers, en supposant qu'il y a \( N \) passagers, chacun des \( N \) bus les moins remplis recevrait un passager). -Par ailleurs, on suppose que les bus ont une capacité de transport infinie. +On suppose que les passagers ne descendent jamais, sauf lorsqu'ils arrivent à un terminus. + +\medskip + +Les bus ne peuvent pas se dépasser : lorsqu'un bus rattrape son prédécesseur, il le suit à la même vitesse que lui (y compris si cela implique de rouler à une vitesse inférieure à sa vitesse moyenne), et en arrivant à un arrêt, les passagers sont répartis équitablement entre les bus (s'il devait y avoir plus de bus que de passagers, en supposant qu'il y a \( N \) passagers, chacun des \( N \) bus les moins remplis recevrait un passager). Si deux bus ont le même nombre de passagers, on commencera par remplir le bus qui est arrivé en premier. +Par ailleurs, on suppose que les bus ont une capacité de transport infinie. Aussi, on suppose que les bus partent vides du dépôt. \q Dans un premier temps, Antoine s'intéresse à ce qui se produit aux heures de pointe. -Il considère qu'il y a \( N \) passagers à chaque arrêt, et que ceux-ci ne se remplissent pas par la suite (ce qui correspond à dire qu'il y a tant de passagers aux arrêts que le remplissage sur la période étudiée est négligeable). +Il considère qu'il y a \( N \) passagers à chaque arrêt, et que ceux-ci ne se remplissent pas par la suite. % %\begin{enumerate} % \item\label{item:DeuxBusRattrapent} - Deux bus quittent le dépôt, le premier au temps \( t = 0 \), le second au temps \( t = 1 \). + Deux bus quittent le dépôt, le premier au temps \( t = 0 \), le second démarrant quand le premier arrive au premier arrêt. Finissent-ils par se rattraper ? % \item Antoine se demande si cette réponse dépend de sa modélisation de la vitesse. % Reprendre le point~\ref{item:DeuxBusRattrapent} si \( V_{k} \) peut être donnée par une expression quelconque, et non nécessairement celle en~\eqref{eq:VitesseBus}. @@ -45,7 +49,7 @@ Il considère donc les variantes suivantes. \begin{enumerate} \item Au temps \( t = T \), le bus de tête est immobilisé pendant un intervalle de temps de \( \frac{1}{10} \) en raison d'un embouteillage, avant de reprendre à vitesse normale. - \item Au temps \( t = T \), une quantité \( q \) de passagers arrive à l'arrêt situé en \( n = 10 \) en plus du remplissage normal (par exemple, suite à l'arrivée d'un TGV, dont les passagers souhaitent rallier la cité universitaire en bus\dots). + \item Au temps \( t = T \), une quantité \( q \) de passagers arrive à l'arrêt situé en \( n = 10 \) en plus du remplissage normal. \end{enumerate} Toujours dans le cas de deux bus séparés d'une unité de temps, évaluer l'impact des deux perturbations ci-dessus sur la suite de leur parcours. @@ -54,7 +58,7 @@ Toujours dans le cas de deux bus séparés d'une unité de temps, évaluer l'imp \q Antoine souhaite à présent concevoir une stratégie pour retenir les bus aux heures de pointe, afin d'éviter qu'ils ne se rattrapent. On suppose donc à présent que, lorsqu'ils arrivent aux arrêts (et uniquement à ce moment), les bus peuvent s'arrêter et attendre un temps arbitraire avant de repartir à leur vitesse normale. -Une \emph{stratégie} est une façon pour les bus de décider, en connaissance de la position de tous les bus et du nombre de passagers qu'ils transportent, du temps à attendre lorsqu'ils arrivent à un arrêt. +Une \emph{stratégie} est une façon pour le conducteur du bus de décider, en connaissance de la position de tous les bus, du nombre de passagers qu'ils transportent et du nombre de passagers à chaque arrêt, du temps à attendre lorsqu'ils arrivent à un arrêt. %Dans le cadre de la question 1a), existe-t-il une stratégie pour retenir les bus de façon à ce que le second ne rattrape jamais le premier ? %Le cas échéant, une telle stratégie peut-elle être choisie de façon à améliorer le temps de parcours voyageur (c'est-à-dire que certains passagers iraient de l'arrêt où ils sont montés à un arrêt ultérieur plus vite que si cette stratégie n'était pas mise en \oe uvre) ? % @@ -70,24 +74,24 @@ On suppose que deux bus circulent sur la ligne, partant tous deux du dépôt en \item Existe-il une stratégie pour retenir les bus aux arrêts de façon à éviter qu'ils ne se rattrapent ? \item Que se passe-t-il dans le cas de \( m \) bus, chaque bus après le premier partant lorsque le précédent a atteint le premier arrêt ? \end{enumerate} -Discuter de l'optimalité d'une éventuelle stratégie par rapport au temps de parcours voyageur (c’est-à-dire qu'on cherche à maximiser le temps écoulé entre l'arrivée d'un passager à un arrêt, et le moment où le bus le dépose au terminus). +Discuter de l'optimalité d'une éventuelle stratégie par rapport au temps de parcours voyageur (c’est-à-dire qu'on cherche à minimiser le temps écoulé entre l'arrivée d'un passager à un arrêt, et le moment où le bus le dépose au terminus). \q Antoine cherche à explorer une dernière idée pour améliorer le temps de parcours voyageur. -On se place à nouveau dans le cadre de la question 2, avec deux bus circulant, et où les arrêts sont initialement vides et se remplissent progressivement. +On se place à nouveau dans le cadre de la question 2, avec deux bus circulant, et où les arrêts sont initialement vides et se remplissent progressivement. On suppose qu'il y a une infinité dénombrable d'arrêts indexés par les entiers naturels. Antoine propose que le premier bus desserve uniquement les arrêts impairs, et le second uniquement les arrêts pairs. \begin{enumerate} - \item Cette stratégie présente-t-elle un gain en termes de temps de parcours voyageur ? + \item Cette stratégie présente-t-elle un gain en terme de temps de parcours voyageur ? Le quantifier aussi précisément que possible. - \item Pour pousser son idée encore plus loin son idée, Antoine suppose à présent qu'un bus démarre à chaque temps \( t \) entier, et que les bus parcourent les arrêts de \( k \) en \( k \). + \item Pour pousser son idée encore plus loin, Antoine suppose à présent qu'un bus démarre à chaque temps \( t \) entier, et que les bus parcourent les arrêts de \( k \) en \( k \). Plus précisément, le premier bus dessert les arrêts multiples de \( k \), le suivant les arrêts multiples de \( k \) plus \( 1 \), et ainsi de suite. Après \( k \) bus, le schéma se répète : le \( (k+1) \)-ème bus dessert les arrêts multiples de \( k \), le \( (k+2) \)-ème bus dessert les arrêts multiples de \( k \) plus \( 1 \), et ainsi de suite. Quantifier le gain éventuel en termes de temps de parcours voyageur, et examiner ce qui se produit lorsque \( k \to +\infty \). \item Antoine décide de pousser son idée encore plus loin, et d'ajouter des arrêts, pour répartir les voyageurs sur davantage d'arrêts. Il suppose donc qu'il y a à présent un arrêt en \( \frac{n}{k} \) pour chaque entier \( n \geq 1 \), et que les voyageurs arrivent aux arrêts à un taux de \( \frac{\rho}{k} \) passagers par unité de temps. On considère toujours qu'un bus quitte le dépôt à chaque unité de temps. - Quantifier l'impact de ce changement sur le temps de parcours voyageur, à la fois dans le cas où chaque bus dessert tous les arrêts ou dans le cas où les bus desservent les arrêts de \( k \) en \( k \). + Quantifier l'impact de ce changement sur le temps de parcours voyageur, à la fois dans le cas où chaque bus dessert tous les arrêts et dans le cas où les bus desservent les arrêts de \( k \) en \( k \). On s'intéressera particulièrement à ce qui se produit à la limite lorsque \( k \to +\infty \). \end{enumerate} diff --git a/src/gentillesse.tex b/src/gentillesse.tex index 7926bf7..d37bf0a 100644 --- a/src/gentillesse.tex +++ b/src/gentillesse.tex @@ -2,16 +2,16 @@ Au pays des Merveilles, les lutins sont de mauvaise humeur. Un seul lutin parviendra-t-il à tous leur redonner le sourire ? -Le Pays des Merveilles, noté $M$, est constitué de $n \ge 3$ lutins, qui peuvent chacun être de bonne ou mauvaise humeur. Chaque paire de lutins est soit amie, soit inconnue. +Le pays des Merveilles, noté $M$, est constitué de $n \ge 3$ lutins, qui peuvent chacun être de bonne ou mauvaise humeur. Chaque paire de lutins est soit amie, soit inconnue. Chaque jour, chaque lutin de bonne humeur sourit simultanément à un certain nombre de ses amis (ce nombre sera précisé dans la suite). Un lutin de mauvaise humeur qui reçoit au moins un sourire devient de bonne humeur, et pourra se mettre à sourire à partir du jour suivant. -En partant d'un lutin $\ell$ donné, on dit qu'un nombre $j \in N$ est \emph{réalisable} s'il est possible que tous les lutins deviennent de bonne humeur à partir du $j$-ième jour si, initialement, le lutin $\ell$ est le seul à être de bonne humeur. +En partant d'un lutin $\ell$ donné, on dit qu'un nombre $j \in \mathbb{N}$ est \emph{réalisable} s'il est possible que tous les lutins deviennent de bonne humeur à partir du $j$-ième jour si, initialement, le lutin $\ell$ est le seul à être de bonne humeur. Par ailleurs, on dit que le nombre $\infty$ est réalisable s'il est possible qu'une situation où tous les lutins sont de bonne humeur n'arrive jamais. -Par exemple, si on suppose que chaque lutin sourit à exactement 1 de ses amis (n'importe lequel), que le pays des Merveilles $M$ est constitué de $3$ lutins, où un lutin est ami aux deux autres, et qu'on part du lutin ami aux deux autres, alors tous les nombres entiers supérieurs ou égaux à 2 sont réalisables. Voir Figure \ref{fig:lutins}. +Par exemple, si on suppose que chaque lutin sourit à exactement 1 de ses amis (n'importe lequel), que le pays des Merveilles $M$ est constitué de $3$ lutins, où un lutin est ami avec les deux autres, et qu'on part du lutin ami aux deux autres, alors tous les nombres entiers supérieurs ou égaux à 2 sont réalisables. Voir Figure \ref{fig:lutins}. -\begin{figure} +\begin{figure}[h] \centering \begin{tikzpicture} % Les nœuds avec des sourires @@ -78,7 +78,7 @@ Par exemple, si on suppose que chaque lutin sourit à exactement 1 de ses amis ( \caption{Un exemple de propagation, qui montre que le nombre 3 est réalisable. -Les disques correspondent aux lutins, ils sont remplis pour les lutins de bonne humeur. Les flèches indiquent à quel lutin chacun sourit.} +Les disques correspondent aux lutins, ils sont remplis pour les lutins de bonne humeur. Deux lutins sont reliés s'ils sont amis. Les flèches indiquent à quel lutin chacun sourit.} \label{fig:lutins} \end{figure} @@ -87,11 +87,11 @@ On note $A(M,\ell)$ l'ensemble des nombres réalisables. Dans l'exemple précéd \medskip -\q Dans un premier temps, un lutin sourit à exactement deux de ses amis, en priorité aux lutins de mauvaise humeur (c'est-à-dire qu'il ne peut pas sourire à un lutin de bonne humeur s'il ne sourit pas à tous ses amis de mauvaise humeur). Déterminer $A(M,\ell)$ (qu'on notera $A_1(M,\ell)$ pour éviter les ambiguités) si le pays des Merveilles est un réseau avec $a\in\N$ lignes et $b\in\N$ colonnes, avec $a,b \ge 2$, comme dans la Figure~\ref{fig:reseau_lutin}. On pourra distinguer les cas suivants : +\q Dans un premier temps, un lutin sourit à exactement deux de ses amis, en priorité aux lutins de mauvaise humeur (c'est-à-dire qu'il ne peut pas sourire à un lutin de bonne humeur s'il ne sourit pas à tous ses amis de mauvaise humeur). Déterminer $A(M,\ell)$ (qu'on notera $A_1(M,\ell)$ pour éviter les ambiguités) si le pays des Merveilles est un reseau d'amitié avec $a\in\N$ lignes et $b\in\N$ colonnes, avec $a,b \ge 2$, comme dans la Figure~\ref{fig:reseau_lutin}. Dans ce réseau, les sommets représentent les lutins et les arêtes représentent les liens d'amitié. On pourra distinguer les cas suivants : \begin{enumerate} -\item Si le lutin initial $\ell$ a 2 amis (c'est-à-dire, dans un coin); -\item Si le lutin initial $\ell$ a 3 amis (c'est-à-dire, sur un côté); -\item Si le lutin initial $\ell$ a 4 amis (c'est-à-dire, ni sur un coin ni sur un côté). +\item Si le lutin initial $\ell$ a 2 amis (c'est-à-dire dans un coin); +\item Si le lutin initial $\ell$ a 3 amis (c'est-à-dire sur un côté); +\item Si le lutin initial $\ell$ a 4 amis (c'est-à-dire ni sur un coin ni sur un côté). \end{enumerate} @@ -143,11 +143,10 @@ On note~$\tau$ la variable aléatoire correspondant au numéro du premier jour o \q Calculer, en fonction de $n$ et $p$, l’espérance de $\tau$ : \begin{enumerate} \item Si chaque lutin est ami avec tous les autres; -\item Si les lutin sont numérotés de $1$ à $n$, et que chaque lutin $\ell_k$ a deux amis : les lutins $\ell_{k-1}$ et $\ell_{k+1}$ (les lutins $\ell_1$ et $\ell_n$ sont également amis). -\item Si $n$ est pair et la population des lutins est séparée en deux clans de taille $n/2$, tels qu'un lutin d'un clan est ami avec tous les lutins de l'autre clan, et seulement avec ceux-là. Pour cette question, déterminer également, à $p$ fixé, la limite de l’espérance de $\tau$ quand $n \to \infty$ ; +\item Si les lutins sont numérotés de $1$ à $n$, et que chaque lutin $\ell_k$ a deux amis : les lutins $\ell_{k-1}$ et $\ell_{k+1}$ (les lutins $\ell_1$ et $\ell_n$ sont également amis). +\item Si $n$ est pair et la population des lutins est séparée en deux clans de taille $n/2$, tels que chaque lutin d'un clan est ami avec tous les lutins de l'autre clan, et seulement avec ceux-là. Pour cette question, déterminer également, à $p$ fixé, la limite de l’espérance de $\tau$ quand $n \to \infty$ ; \item Si $M$ est le pays décrit dans la question 1 (ici, le résultat peut dépendre de $a$, $b$ et du lutin de départ). \end{enumerate} - \q On fixe $p \in ]0,1]$, $n \ge 3$ et $k \ge n-1$. On considère les pays des Merveilles $M$ avec $n$ lutins, $k$ paires d'amis, et tels que, pour tous lutins $\ell$ et $\ell'$, il existe une chaîne d'amitié qui les relie. Parmi ces pays, que peut valoir, au maximum, l’espérance de $\tau$: diff --git a/src/inegalites-graphe.tex b/src/inegalites-graphe.tex index 00b24d3..fbba5b8 100644 --- a/src/inegalites-graphe.tex +++ b/src/inegalites-graphe.tex @@ -2,8 +2,8 @@ Dans un pays lointain, le roi Louis XLIX-III veut taxer les routes pour maximiser ses revenus. L'Association et Syndicat des Mécontents Routiers (ASMR) essaie de réduire autant que possible les frais pour le peuple. -Dans ce pays, il y a $n$ villes ($n\in \mathbb{N}^*$). Certaines villes sont reliées par une route, formant ainsi le \emph{système routier}. Le système des taxes est le suivant : toutes les villes se voient attribuer un numéro de 1 à $n$ (chaque numéro est utilisé exactement une fois). La \emph{taxe} à payer pour une route reliant une ville de numéro $i$ à une ville de numéro $j$ est la maximum entre $i$ et $j$. -Le \emph{coût total} du système routier est la somme de toutes les taxes à payer pour chacune des routes. Ce coût total dépend de la manière de numéroter les villes. Figure \ref{Fig1} montre deux exemples. +Dans ce pays, il y a $n$ villes ($n\in \mathbb{N}^*$). Certaines villes sont reliées par une route, formant ainsi le \emph{système routier}. Le système des taxes est le suivant : toutes les villes se voient attribuer un numéro de 1 à $n$ (chaque numéro est utilisé exactement une fois). La \emph{taxe} à payer pour une route reliant une ville de numéro $i$ à une ville de numéro $j$ est le maximum entre $i$ et $j$. +Le \emph{coût total} du système routier est la somme de toutes les taxes à payer pour chacune des routes. Ce coût total dépend de la manière de numéroter les villes. La Figure \ref{Fig1} montre deux exemples. \begin{figure}[!ht] \centering @@ -137,13 +137,13 @@ Le \emph{coût total} du système routier est la somme de toutes les taxes à pa Après moultes grèves, le roi et l'ASMR s'accordent sur la manière suivante d'attribuer les numéros aux villes : à tour de rôle ils vont attribuer un numéro entre 1 et $n$ à une ville. Ils n'ont pas le droit de réattribuer un nombre qui a déjà été utilisé, et ils ne peuvent pas attribuer un nombre à une ville qui ait déjà un numéro. Le roi commence. Le but pour le roi est d'obtenir le coût total le plus grand possible, tandis que l'ASMR cherche à obtenir le coût total le plus petit possible. -\q En reprenant les systèmes routiers des questions précédentes, décrire les stratégies du roi et de l'ASMR. Quel est le coût total du système routier quand les deux attribuent les numéros d'une manière optimale ? +\q En reprenant les systèmes routiers des questions précédentes, décrire les stratégies du roi et de l'ASMR. Quel est le coût total du système routier quand les deux attribuent les numéros d'une manière optimale ? Quel est le plus grand coût que le roi peut s'assurer d'obtenir quelle que soit la manière de jouer de l'ASMR ? De même, quel est le plus petit coût que l'ASMR peut s'assurer d'obtenir quelle que soit la manière dont le roi attribue les numéros ?" \medskip -Le roi Louis XLIX-III abuse de son pouvoir pour changer la taxe d'une route. Au lieu d'utiliser le maximum des deux numéros aux villes extrémales, il utilise une fonction $f$. Le coût total du système routier reste la somme des taxes de toutes les routes. +Le roi Louis XLIX-III abuse de son pouvoir pour changer la taxe sur les route. Au lieu d'utiliser le maximum des deux numéros aux villes extrémales, il utilise une fonction $f$. Le coût total du système routier reste la somme des taxes de toutes les routes. -\q Reprendre les questions précédentes où le roi utilise pour la fonction $f$ le produit. +\q Reprendre les questions précédentes où le roi utilise pour la fonction $f$ le produit des numéros. -\q Reprendre les questions précédentes où le roi utilise pour la fonction $f$ le plus petit commun multiple. +\q Reprendre les questions précédentes où le roi utilise pour la fonction $f$ le plus petit commun multiple des numéros. \q Proposer et étudier d'autres pistes de recherche. \ No newline at end of file diff --git a/src/points-colories.tex b/src/points-colories.tex index bf635ea..73d0f0a 100644 --- a/src/points-colories.tex +++ b/src/points-colories.tex @@ -1,12 +1,12 @@ \section{Points colorés sur un cercle} -Lucie a inventé un jeu. Les règles sont les suivantes. +Lucie a inventé un jeu pour deux joueurs. Les règles sont les suivantes. -Le jeu se déroule sur un cercle. Au départ, les points du cercle sont non colorés. L'un des deux adversaires est désigné pour jouer en premier. Chacun leur tour, Lucie et son adversaire jouent (c'est à dire, choisissent) un point, qui sera alors coloré en leur couleur respective : Orange pour Lucie, Bleu pour son adversaire. Lorsqu'ils jouent, il leur est interdit de choisir un point qui a déjà été coloré par l'un d'eux. Lucie convient à l'avance du \textit{nombre de coups} que la partie durera. Tous deux jouent le même nombre de coups, de sorte que le nombre de coups est un entier pair, noté $2n$. Par exemple, si le nombre de coups vaut $2n = 6$ coups, ils joueront $n=3$ coups chacun. La partie s'arrête donc lorsque les $2n$ coups sont joués. +Le jeu se déroule sur un cercle. Au départ, les points du cercle sont non colorés. L'un des deux adversaires est désigné pour jouer en premier. Chacun leur tour, Lucie et son adversaire choisissent un point, qui sera alors coloré en leur couleur respective : Orange pour Lucie, Bleu pour son adversaire. Lorsqu'ils jouent, il leur est interdit de choisir un point qui a déjà été coloré par l'un d'eux. Lucie convient à l'avance du \textbf{nombre de coups} que la partie durera. Tous deux jouent le même nombre de coups, de sorte que le nombre de coups est un entier pair, noté $2n$. Par exemple, si le nombre de coups vaut $2n = 6$ coups, ils joueront $n=3$ coups chacun. La partie s'arrête donc lorsque les $2n$ coups sont joués. \medskip -À la fin de la partie, le cercle est découpé en arcs de cercle dont les extrémités sont soit orange, soit bleu. Dans une telle configuration, un \textit{arc primitif} est un arc dont les deux extrémités sont colorées (en orange ou en bleu) et dont tous les autres points ne sont pas colorés (par exemple, le cercle tout entier, vu comme un arc de cercle, n'est jamais primitif). Les arcs primitifs dont les deux extrémités sont orange sont alors colorés en orange, et ceux dont les extrémités sont toutes deux bleues sont colorés en bleu. Le gagnant est alors celui étant parvenu à former l'arc de cercle non nécessairement primitif le plus long entièrement coloré de sa propre couleur. S'il y a égalité de tels arcs, ou s'il n'en existe aucun, la partie est déclarée nulle. +À la fin de la partie, le cercle est découpé en arcs de cercle dont les extrémités sont soit orange, soit bleu. Dans une telle configuration, un \textbf{arc primitif} est un arc dont les deux extrémités sont colorées (en orange ou en bleue) et dont aucun autre n'est coloré (par exemple, le cercle tout entier, vu comme un arc de cercle, n'est jamais primitif). Les arcs primitifs dont les deux extrémités sont orange sont alors colorés en orange, et ceux dont les extrémités sont toutes deux bleues sont colorés en bleu. Le gagnant est alors celui étant parvenu à former l'arc de cercle non nécessairement primitif le plus long entièrement coloré de sa propre couleur. S'il y a égalité de tels arcs, ou s'il n'en existe aucun, la partie est déclarée nulle. \begin{figure}[h] \centering @@ -39,7 +39,7 @@ Le jeu se déroule sur un cercle. Au départ, les points du cercle sont non colo \draw[very thick, bleuAnimath] (B2) arc[start angle=319, end angle=375, radius=1.5cm]; \foreach \angle in {30, 300, 150} { - \filldraw[fill=bleuAnimath] (\angle:1.5cm) circle (2pt); + \filldraw[fill=orangeAnimath] (\angle:1.5cm) circle (2pt); } \foreach \angle in {219, 319, 15} { @@ -76,23 +76,23 @@ Le jeu se déroule sur un cercle. Au départ, les points du cercle sont non colo } \end{scope} \end{tikzpicture} - \caption{Deux fins de partie pour $2n = 6$ et une pour $2n = 10$. À gauche, il n'y a pas d'arcs colorés (pas d'arc primitif) ; la partie est donc nulle. Au centre, Lucie (en orange) gagne : elle a réussi à construire un arc de taille maximale. À droite, l'adversaire (en bleu) gagne car il a formé un arc (non primitif) bleu de taille maximale.} + \caption{Deux fins de partie pour $2n = 6$ et une pour $2n = 10$. À gauche, il n'y a pas d'arcs colorés; la partie est donc nulle. Au centre, Lucie (en orange) gagne : elle a réussi à construire un arc de taille maximale. À droite, l'adversaire (en bleu) gagne car il a formé un arc (non primitif) bleu de taille maximale.} \label{fig:Exemple} \end{figure} -Dans tout le problème, on appelle \textit{stratégie} une manière déterministe de décrire quoi jouer en fonction des coups qui ont été joués précédemment. Autrement dit, une stratégie est un algorithme qui indique quel coup jouer en fonction de la situation courante, de sorte que, dans deux situations identiques, il indiquera toujours le même coup à jouer. +Dans tout le problème, on appelle \textbf{stratégie} une manière déterministe de décrire quoi jouer en fonction des coups qui ont été joués précédemment. Autrement dit, une stratégie est un algorithme qui indique quel coup jouer en fonction de la situation courante, de sorte que, dans deux situations identiques, il indiquera toujours le même coup à jouer. Comme Lucie n'aime pas perdre, elle commence par se choisir pour adversaire l'Idiot du Village. Ce dernier portant bien son nom, il joue ses coups aléatoirement, sans réfléchir. Chaque coup joué suit alors une loi uniforme sur le cercle. Lucie cherche alors des stratégies qui maximisent sa probabilité de gagner contre cet adversaire. Lucie et son adversaire conviennent de commencer par fixer $2n = 4$. -\q Montrer que si Lucie laisse son adversaire jouer en premier, elle dispose d'une stratégie lui permettant de gagner à tous les coups. +\q Est-ce que si Lucie laisse son adversaire jouer en premier, elle dispose d'une stratégie lui permettant de gagner avec certitude ? -\q Après qu'elle ait gagné une partie, son adversaire la laisse jouer en premier. +\q Après qu'elle gagne une partie, son adversaire la laisse jouer en premier. \begin{enumerate} - \item Lucie dispose-t-elle dispose d'une stratégie lui permettant de gagner quoi qu'il advienne ? + \item Lucie dispose-t-elle d'une stratégie lui permettant de gagner quoi qu'il advienne ? \item Étudier l'ensemble des $ p \in [0,1] $ tels qu'il existe une stratégie permettant à Lucie de gagner avec une probabilité exactement $p$. @@ -104,28 +104,28 @@ Lucie et son adversaire conviennent de commencer par fixer $2n = 4$. \begin{enumerate} - \item Reprendre la question précédente en étudiant dans ce nouveau contexte l'ensemble des $ p \in [0,1] $ tels qu'il existe une stratégie permettant à Lucie de gagner avec une probabilité exactement $p$. On pourra commencer par le cas $2n = 6$. + \item Reprendre la question précédente pour $2n>4$. On pourra commencer par le cas $2n=6$. \item Même question pour des probabilités de ne pas perdre. \end{enumerate} -Lucie propose un pacte à son adversaire. Ils conviennent d'un entier $p \geqslant n $, et les règles sont changées de sorte que l'adversaire de Lucie place $p$ points plutôt que $n$. Comme la règle du tour-par-tour est alors difficile à appliquer, ils conviennent que l'adversaire de Lucie placera tous ses points en premier. Ce dernier joue toujours aléatoirement sur le cercle, mais avant Lucie, de sorte que cette dernière a alors toute la liberté de choisir où placer ses points. Lucie a donc plus d'information que son adversaire, mais en contrepartie, ce dernier peut placer plus de points qu'elle. +Lucie propose de changer les règles. Ils conviennent d'un entier $k$, et les règles sont changées de sorte que l'adversaire de Lucie place $k$ points plutôt que $n$. L'adversaire de Lucie placera tous ses points en premier. Ce dernier joue toujours aléatoirement sur le cercle, mais avant Lucie, de sorte que cette dernière a alors toute la liberté de choisir où placer ses points. Lucie a donc plus d'information que son adversaire, mais en contrepartie, ce dernier peut placer plus de points qu'elle. -\q Déterminer des conditions nécessaires et suffisantes sur $n,p$ pour que Lucie dispose d'une stratégie lui permettant de gagner avec probabilité 1. +\q En fonction de $n$ et $k$, Lucie dispose-t-elle d'une stratégie lui permettant de gagner avec probabilité 1 ? \medskip -Pour essayer, Lucie et son adversaire reprennent exactement la même configuration que la précédente, mais en échangeant les rôles. Lucie place $p$ points, son adversaire en place $n$. Ce dernier joue toujours aléatoirement, et Lucie place tous ses points en premier. De plus, cette fois-ci, on n'impose plus $ p \geqslant n $. +Pour essayer, Lucie et son adversaire reprennent exactement la même configuration que la précédente, mais en échangeant les rôles. Lucie place $k$ points, son adversaire en place $n$. Ce dernier joue toujours aléatoirement, et Lucie place tous ses points en premier. -\q Étudier l'existence d'une configuration de ses $p$ points maximisant sa probabilité de gagner. +\q Étudier l'ensemble des $ p \in [0,1] $ tels qu'il existe une stratégie permettant à Lucie de gagner avec une probabilité exactement $p$. \medskip -Fatiguée de jouer avec l'Idiot du Village, Lucie se trouve un adversaire à sa taille : Lucien. L'un des deux joueurs est désigné pour jouer en premier, et $ 2n \in \mathbb N^* $ est fixé. La règle du tour-par-tour est alors appliquée. Lucien commence à jouer. +Fatiguée de jouer avec l'Idiot du Village, Lucie se trouve un adversaire à sa taille : Lucien. L'un des deux joueurs est désigné pour jouer en premier, et $ 2n \in \mathbb N$ est fixé. La règle du tour-par-tour est alors appliquée. Lucien commence à jouer. -\q L'un des deux joueurs dispose-t-il d'une stratégie gagnante ? +\q L'un des deux joueurs dispose-t-il d'une stratégie lui permettant de gagner à coup sûr ? Si oui, en décrire une. -\q Lucie se demande : que dire des questions précédentes si elle avait convenu dès le départ que le gagnant était non pas celui ayant l'arc le plus long, mais celui étant parvenu à maximiser la somme des longueurs des arcs primitifs de sa couleur ? +\q Reprendre le problème si Lucie avait convenu dès le départ que le gagnant n'était non pas celui ayant l'arc le plus long, mais celui étant parvenu à maximiser la somme des longueurs des arcs primitifs de sa couleur. \q Proposer et étudier d'autres pistes de recherche. \ No newline at end of file diff --git a/src/tarte.tex b/src/tarte.tex index 0339c8e..08fcd71 100644 --- a/src/tarte.tex +++ b/src/tarte.tex @@ -1,17 +1,17 @@ \section{Plats à tarte gradués} Lors d'un tournoi de tartes, différentes familles veulent faire des découpages en parts égales. -Chaque famille va fabriquer son propre moule à tarte sur lequel elle va indiquer des graduations qui correspondent aux endroits où il faut découper entre le centre et la graduation pour former des parts égales. C'est à dire que pour un nombre $u \in \mathbb{N}^*$, on va faire apparaître $u$ fois un trait avec le numéro $u$ indiqué sur le bord du moule à tarte formant un $u$-gone régulier inscrit dans le cercle. +Chaque famille va fabriquer son propre moule à tarte sur lequel elle va indiquer des graduations qui correspondent aux endroits où il faut découper entre le centre et la graduation pour former des parts égales. C'est-à-dire que pour un nombre $u \in \mathbb{N}^*$, on va faire apparaître $u$ fois un trait avec le numéro $u$ indiqué sur le bord du moule à tarte formant un $u$-gone régulier inscrit dans le cercle. -Bien sûr, un même moule doit pouvoir avoir plusieurs graduations parce que le nombre $u$ de parts de tarte qu'on voudra faire n'est pas toujours le même suivant le nombre de convives. On note alors $N$ le nombre de numéros de graduations différentes qu'on veut faire apparaître et $S=\{u_1,\cdots,u_n\}$ l'ensemble des numéros qu'on veut faire apparaître. +Bien sûr, un même moule doit pouvoir avoir plusieurs graduations parce que le nombre $u$ de parts de tarte qu'on voudra faire n'est pas toujours le même suivant le nombre de convives. On note alors $N \geq 2$ le nombre de numéros de graduations différentes qu'on veut faire apparaître et $S=\{u_1,\cdots,u_N\}$ l'ensemble des numéros qu'on veut faire apparaître. On peut identifier le moule à tarte à un cercle (car on ne veut faire que des tartes rondes) et les graduations à des points de ce cercle. -Par exemple, le plat à tarte de graduations $S=\{1,2,3,4,6\}$ peut être gradué en +Par exemple, pour $S=\{1,2,3,4,6\}$, il est possible d'utiliser 10 graduations, avec au plus 2 au même endroit, comme illustré par la figure suivante. \definecolor{ffxfqq}{rgb}{1,0.5,0} \definecolor{qqqqff}{rgb}{0,0,1} -\begin{figure}[ht]\label{Tarte2346} +\begin{figure}[ht] \vspace{-2em} \begin{center} \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm] @@ -71,6 +71,7 @@ Par exemple, le plat à tarte de graduations $S=\{1,2,3,4,6\}$ peut être gradu \end{tikzpicture} \end{center} \vspace{-5em} +\label{Tarte2346} \caption{Cercle en bleu, le bord du plat à tarte et ses graduations\\En noir, les endroits où il faudra couper en fonction du nombre de parts\\En orange, les $u$-gones réguliers pour placer les graduations.} \end{figure} @@ -78,21 +79,20 @@ Il y a donc $10$ graduations sur le plat à tarte % de la Figure~\ref{Tarte2346} avec au plus $2$ au même endroit sur cet exemple. +\q On veut $N=2$ graduations, c'est-à-dire que $S=\{a,b\}$ avec $a,b \in \mathbb{N}^*$ distincts. Quel est le nombre minimal de graduations à mettre sur le plat à tarte afin d'être capable de couper la tarte à la fois en $a$ parts égales ou $b$ parts égales? -\q Soit $a$ et $b$ deux entiers naturels distincts. On prend $N=2$, $u_1 = a$ et $u_2 = b$ de sorte que $S = \{a,b\}$. Quel est le nombre minimal de graduations à mettre sur le moule à tarte afin d'être capable de couper la tarte à la fois en $a$ parts égales ou en $b$ parts égales? - -\q On veut maintenant $N=3$ graduations, c'est-à-dire que $S=\{a,b,c\}$ avec $a,b,c \in \mathbb{N}^*$ deux à deux distincts. Quel est le nombre minimal de graduation à mettre sur le plat à tarte afin d'être capable de couper la tarte à la fois en $a$ parts égales, $b$ parts égales ou $c$ parts égales? +\q On veut maintenant $N=3$ graduations, c'est-à-dire que $S=\{a,b,c\}$ avec $a,b,c \in \mathbb{N}^*$ deux à deux distincts. Quel est le nombre minimal de graduations à mettre sur le plat à tarte afin d'être capable de couper la tarte à la fois en $a$ parts égales, $b$ parts égales ou $c$ parts égales? %\q Par manque de place, on veut mettre au plus deux fois la même graduation en un point donné. Quelle est alors le nombre minimal de graduations à mettre sur le plat à tarte pour permettre les découpages $S =\{a,b,c\}$? \medskip -Désormais, il y a différentes familles qui s'affrontent dans un tournoi de graduations de moules à tarte. Chacun famille doit choisir $N$ valeurs distinctes à graduer $S =\{u_1,\dots,u_N\}$. +Désormais, il y a différentes familles qui s'affrontent dans un tournoi de graduations de moules à tarte. Chaque famille doit choisir $N$ valeurs distinctes à graduer $S =\{u_1,\dots,u_N\}$. \begin{itemize} \item La famille \textbf{Première} choisit de graduer les $N$ premiers nombres premiers, soit $S^{p}_N = \{p_1,p_2,\cdots,p_N\}$. \item La famille \textbf{Géométrique} choisit un entier $a \geqslant 2$ et de graduer les puissances de $a$, soit $S^g_N = \{1,a,a^2,\cdots,a^{N-1}\} $. \item La famille \textbf{Complète} choisit de graduer les $N$ premiers nombres entiers naturels, soit $S^c_N = \{1,2,\cdots,N\}$. -\item La famille \textbf{Divisée} choisit le plus petit entier $\alpha_N$ qui admet exactement $N$ diviseurs $\delta_{1,\alpha_N}, \dots, \delta_{N,\alpha_N}$ et choisit de graduer les $N$ diviseurs de $\alpha_N$, soit $S^d_N = \{\delta_{1,\alpha_N},\cdots,\delta_{N,\alpha_N}\}$. +\item La famille \textbf{Divisée} choisit le plus petit entier $\alpha_N$ qui admet exactement $N$ diviseurs $d_{1,\alpha_N}, \dots, d_{N,\alpha_N}$ et choisit de graduer les $N$ diviseurs de $\alpha_N$, soit $S^d_N = \{d_{1,\alpha_N},\cdots,d_{N,d_N}\}$. \end{itemize} Par exemple, si $N=6$, on a: @@ -103,7 +103,7 @@ Par exemple, si $N=6$, on a: \item $S^d_6 = \{1,2,3,4,6,12\}$ car $\alpha_6 = 12$. \end{itemize} -\q Les familles choisissent qu'il existe un point d'origine du cercle sur lequelle toutes les graduations de apparaissent. La famille Première (resp. Géométrique, Complète, Divisée) compte alors le nombre de graduations $G^p_N$ (resp.~$G^g_N$, $G^c_N$, $G^d_N$) qui apparaissent sur tout le tour du plat à tarte avec les nombres de $S^p_N$ (resp. $S^g_N$, $S^c_N$, $S^d_N$). Combien y aura-t-il alors de graduations: +\q Les familles choisissent un point initial sur le cercle sur lequel toutes les graduations apparaissent. La famille Première (resp. Géométrique, Complète, Divisée) compte alors le nombre de graduations $G^p_N$ (resp.~$G^g_N$, $G^c_N$, $G^d_N$) qui apparaissent sur tout le tour du plat à tarte avec les nombres de $S^p_N$ (resp. $S^g_N$, $S^c_N$, $S^d_N$). Combien y aura-t-il alors de graduations: a) $G^p_N$ pour la famille Première; @@ -111,15 +111,15 @@ b) $G^g_N$ pour la famille Géométrique; c) $G^c_N$ pour la famille Complète; -d) $G^d_N$ pour la famille Divisée. +d) $G^d_N$ pour la famille Divisée ? % en fonction des éléments de $S_N^p$ (resp. $S^g_N, S^c_N, S^d_N$). -Donner une valeur exacte ou un encadrement aussi précis que possible des valeurs $G^*_N$ pour ces quatre familles. +Donner une valeur exacte ou un encadrement aussi précis que possible de ces valeurs pour les quatre familles. \medskip -Les épaisseurs des traits de graduation posent des problèmes de lecture. Les familles veulent donc mettre le moins possible de graduations sur le plat à tartes. Elles se demandent si toutes les faire démarrer à la même origine est vraiment le plus efficace. +Les familles cherchent à mettre le moins possible de graduations sur le plat à tarte. Elles se demandent si toutes les faire démarrer à la même origine est vraiment le plus efficace. \q Peut-on trouver un exemple d'une famille $S_N^u = \{u_1,\cdots,u_N\}$ pour lequel il est possible de faire mieux quand les graduations ne démarrent pas toutes au même endroit @@ -127,7 +127,7 @@ a) si $N=3$? b) si $N=4$? -c) pour un $N$ assez grand? +c) pour une autre valeur de $N$ ? d) Dans ce cas, estimer aussi précisément que possible le nombre $\widehat{G}^u_N$ de graduations qu'on peut mettre en fonction de $N$ et de $S=\{u_1,\dots,u_N\}$. @@ -135,7 +135,7 @@ d) Dans ce cas, estimer aussi précisément que possible le nombre $\widehat{G}^ \q Lorsque ce minimum est atteint, pour chacune des quatre familles précédemment décrite, combien doit-on mettre de graduations au même endroit au minimum? -\q Existe-t-il un entier $N$ tel que pour $n > N$, l'ordre entre les nombres optimaux de graduations que peuvent mettre les quatre familles $\widehat{G}^p_n, \widehat{G}^g_n, \widehat{G}^c_n, \widehat{G}^d_n$ fini par être toujours le même. Si oui, quel est cet ordre? +\q Existe-t-il un entier $N$ tel que pour $n > N$, l'ordre entre les nombres optimaux de graduations que peuvent mettre les quatre familles $\widehat{G}^p_n, \widehat{G}^g_n, \widehat{G}^c_n, \widehat{G}^d_n$ finit par être toujours le même. Si oui, quel est cet ordre? \q Existe-t-il une suite $(u_n)_{n \in \mathbb{N}^*}$ telle que pour tout $N \in \mathbb{N}^*$, on a $\widehat{G}_N^u \leqslant \min(\widehat{G}^p_N,\widehat{G}^g_N,\widehat{G}^c_N,\widehat{G}^d_N)$? diff --git a/src/toboggan.tex b/src/toboggan.tex index ab202c2..f790879 100644 --- a/src/toboggan.tex +++ b/src/toboggan.tex @@ -1,6 +1,6 @@ \section{Drôles de toboggans} -Dans un centre de loisirs, la direction a confié à l’ingénieure Emmy la construction d'une haute tuyauterie de toboggans de hauteur $H\in\N$ mètres. En haut, ses tuyaux présentent $N$ entrées alignées et numérotées de $1$ à $N$, de même qu’en bas il y a $N$ sorties alignées et numérotées de $1$ à $N$. +Dans un centre de loisirs aquatique, la direction a confié à l’ingénieure Emmy la construction d'une haute tuyauterie de toboggans de hauteur $H\in\N$ mètres. En haut, ses tuyaux présentent $N \geq 2$ entrées alignées et numérotées de $1$ à $N$ de même qu’en bas il y a $N$ sorties alignées et numérotées de $1$ à $N$. Pour que ce soit plus amusant, Emmy dispose des tuyaux suivants: \begin{itemize} \item des tuyaux droits, qu'on nommera \emph{tuyaux de type \( I \)}, tels que ce qui rentre en position $K$ sort également en position~$K$ ; @@ -78,22 +78,20 @@ On trouvera un exemple sur la figure~\ref{fig:ToboggansSansY}. \medskip -\q Emmy souhaite que de l'eau sorte par toutes les sorties en même temps. Pour quelle(s) valeur(s) de $N$ et $H$ cela est-il possible si +\q Emmy voudrait qu’il sorte de l’eau de tous les toboggans. Pour quelles valeurs de $N$ et $H$ est-il possible de construire des toboggans qui font sortir de l’eau par toutes les sorties si \begin{enumerate} - \item toute l'eau rentre dans un seul tuyau, et Emmy peut choisir lequel ; + \item toute l'eau rentre dans un seul tuyau, et Emmy peut choisir dans lequel ; \item elle fait rentrer la même quantité d’eau dans tous les tuyaux ; - \item l’eau rentre dans le tuyau en position 1 ; + \item toute l’eau rentre dans le tuyau 1 ; \item $1/2$ litre d’eau rentre dans le tuyau 1 et $1/2$ litre d’eau rentre dans le tuyau $N$. %\item %on fait rentrer n’importe quelle quantité d’eau n’importe où (somme égale à $2^H$ litres d’eau) et la sortie est la même quoi qu’on fasse. %l'eau peut être répartie de façon arbitraire entre les entrées, et la sortie ne dépend pas du choix de la répartition. \end{enumerate} - - -\q Pour que le spectacle soit grandiose, Emmy voudrait qu’il sorte la même quantité d’eau de tous les toboggans. Pour quelles valeurs de $N$ et $H$ est-il possible de construire des toboggans qui font sortir la même quantité d’eau par toutes les sorties si +\q Pour que le spectacle soit grandiose, Emmy voudrait qu’il sorte la même quantité d’eau de tous les toboggans, et ce par toutes les sorties. Pour quelles valeurs de $N$ et $H$ est-il possible de construire des toboggans qui font sortir la même quantité d’eau par toutes les sorties si \begin{enumerate} \item l’eau rentre dans le tuyau 1 ; - \item la répartition de l'eau est arbitraire, et la construction du toboggan est indépendante de la répartition. Plus précisément, Emmy souhaite construire une tuyauterie de toboggans telle que, pour tout choix de répartition de l'eau en entrée, la tuyauterie fasse sortir la même quantité d'eau par toutes les sorties. + \item la répartition de l'eau est arbitraire, et la construction du toboggan est indépendante de la répartition. Autrement dit, Emmy souhaite construire une tuyauterie de toboggans telle que, pour tout choix de répartition de l'eau en entrée, la tuyauterie fasse sortir la même quantité d'eau par toutes les sorties. \end{enumerate} @@ -118,7 +116,7 @@ On trouvera un exemple sur la figure~\ref{fig:ToboggansSansY}. \q Emmy s'aperçoit que, si les spectateurs sont placés trop loin, ceux-ci ne peuvent pas distinguer deux spectacles qui diffèrent par une quantité d'eau suffisamment petite. Plus précisément, étant donné un \( \varepsilon > 0 \) fixé, on dit qu'Emmy réalise le spectacle $(x_1, ..., x_N)$ \emph{en apparence} lorsque la quantité d'eau $(y_1, ..., y_N)$ qui sort effectivement des tuyaux vérifie \( \lvert x_i - y_i \rvert \leq \varepsilon \) pour tout \( i \in \{1,\dotsc,N \} \). -On suppose toujours que Emmy fait rentrer toute l'eau dans le tuyau 1. +On suppose toujours qu'Emmy fait rentrer toute l'eau dans le tuyau 1. \begin{enumerate} \item En fonction de \( \varepsilon > 0 \) et \( N \), quels spectacles Emmy peut-elle réaliser en apparence, si elle peut choisir $H$ comme elle le souhaite ? \item Reprendre le point a) si on fixe également $H$. @@ -126,7 +124,7 @@ On suppose toujours que Emmy fait rentrer toute l'eau dans le tuyau 1. -\q Le collègue d'Emmy est de retour pour l'ennuyer, mais cette fois-ci, Emmy dispose de nouveaux tuyaux pour l'aider. +\q Le collègue d'Emmy est de retour pour l'ennuyer. Là encore, il prévoit de mélanger les entrées juste avant le spectacle, mais cette fois-ci, Emmy dispose de nouveaux tuyaux pour l'aider. Plus précisément, elle dispose de tuyaux en $Y$ à gauche, dits de type \( \mathcal{Y} \) (respectivement à droite, dits de type \reflectbox{\(\mathcal{Y}\)}\footnote{Si vous utilisez \LaTeX, ce symbole peut s'obtenir en composant \texttt{\textbackslash reflectbox\{\$\textbackslash mathcal\{Y\}\$\}}, la commande \texttt{reflectbox} donnant l'image miroir de son argument.}), qui rassemblent l'eau rentrant en position \( K \) et \( K+1 \) pour la faire ressortir entièrement en \( K \) (respectivement en \( K+1 \)). On trouvera une illustration à la figure~\ref{fig:ToboggansAvecY}. Combien de spectacles Emmy peut-elle réaliser (de façon exacte) dans ces conditions, en fonction de \( N \) et \( H \) ?