From 7fad99d799e45b374172b5decc75623119a2e374 Mon Sep 17 00:00:00 2001 From: andetaille Date: Thu, 12 Dec 2024 17:03:46 +0100 Subject: [PATCH] =?UTF-8?q?Premi=C3=A8re=20version=20probl=C3=A8me=20des?= =?UTF-8?q?=20toboggans?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- fiches/toboggan-fiche.tex | 23 ++++- index.pdf | Bin 307827 -> 379498 bytes index_avec_fiches.pdf | Bin 314156 -> 373428 bytes src/toboggan.tex | 207 +++++++++++++++++++++++++++++++++++++- 4 files changed, 223 insertions(+), 7 deletions(-) diff --git a/fiches/toboggan-fiche.tex b/fiches/toboggan-fiche.tex index df50ad4..306a96f 100644 --- a/fiches/toboggan-fiche.tex +++ b/fiches/toboggan-fiche.tex @@ -1,5 +1,24 @@ \section*{Eléments de réponse} -\q (Facile) Première réponse +\q a) (facile) $H\geq \lceil N/2\rceil$; \\ +b) (facile) tout le temps\\ +c) (facile) $H\geq N-1$\\ +d) (facile) $H\geq \lfloor N/2\rfloor$\\ +e) (?) -\q (Moyen) Deuxieme réponse +\q a.(?) \\ +b. (très difficile) Pour qu’Emmy puisse créer un spectacle homogène, il faut et il suffit que $N$ soit une puissance de 2 et que $H\geq N-1$. + +\q (ouvert) + +\q a) (facile) Conditions nécessaires sur un spectacle : $\sum x_i=1$, \( 2^H x_{i} \in \mathbb{N} \)\\ (ouvert) condition suffisante pour que $(x_1,...,x_n)$ soit un spectacle.\\ +b) ??\\ +c) (ouvert) + +\q a) (moyen -- à vérifier!) Par densité des dyadiques, Emmy peut réaliser $(x_1,...,x_n)$ en apparence pour tous $x_i\in[0,1]$ avec $\sum x_i=1$.\\ +b) (ouvert) + +\q (?) + + +\q (ouvert) \ No newline at end of file diff --git a/index.pdf b/index.pdf index ad333a4a9891d5ba76f789e57bc31e9798fe729d..68dc0ed09899296cad018dc7d1ba1c53d4b76565 100644 GIT binary patch delta 264303 zcmV)MK)Ap2!yJ4%i_yqKZSV_^_;eeY%0h3^~JHJC$Qj1R9Ngzd*j(Z}{e?C;Sh-FK?ebzZRn0 zNX<-P#OC(XMrI{5Ty2aI%xrF}&9BMr^_!dRzi!_EE-yC1Fex&PxVd54I#Y$a- zt5uZnjhouAHpz`0KZ69~V8?oYSG%^2NN!w5@Ek6@@cbH-0-m|dz2K7R0tA;OD+=V6@ZK{+?1XJ5I<2<=}{ox8h8tlM%IDGT&@9)Fv?G>TRfb-)=CB&<%o0sqD zKys<0nTxsUO2K5Qz;rp|rii{mMeN~jYRKMNFFjmlurVMvXQdS3)+@Ash{|Phzi_q6 zRWfw#Ru;(^c2)h6t9(~C@q!z}*Q1$;24uD(x!ytg0= zA&lMh1p;w+7Pb(7Jggdj8@{`)Z6P8D+wNR|U~1s^%hZ407BZo$6&(*zil&}*t!XSz=|mMj{F+^8%53p zsxH!nhLlJLa*6elD#3j)nZ!-rO9HRUF`fZ7QVJ)-)Q82+MMsK7T(barOP9T)JOsi4 zW1%93mhOg71<1~}RbAPU;9!8dp(CGl;M?C&YWJ|`=%_|fdgrwaxy4n2*2M2jRtz34 zV^qM|kteJj;bRbg^{6WpwqqazJmjcH{NkqHiPoy59<7S@LYkHqkg!k_r>1t81uGi$ zE_fzr6zRz#qmY6H6^*GGsS+U25f*402z0~Zb`@{m*TWdr6jK26F;&cBiUs%&o^cdA~VqaD^_G8O|c45m|K2VdD70YYhigGJX@J}?of?|gBHR;IM@ z--S_F0A95K)F?IXFq81xV&+CB#LWjhRsqV88D+@EdUPZV2tBO1Em)*fXB8hIFYu$O zFxyZe0E`U3PBrBrT|OlxUcrb&aPK}%14@9KiGWl+0%qGC(1CZp4#6I5G|WjNv(`3s zG!EjtBSM3Ji(DvhdeGLc)=8|zGWRjh_l-N8;vJyWo_x8$Z8HsF z^+jVb^Ule$^UlfVL4~^t!nm*Rrq~R54hR#WHucV72ca%sE(RnRp%cycT9-G96;j13EHJrh16#`NEa=G>Nr78`=xN}q*3P95!^8$l0?V@ethvTn zP3CCN@C^Y53-3xG=@ow#V8CL4$oC=Qal?0jxLu517se1}$+F^TjBnq*vGLQF1MB+z z*~K%17{+1rrE5{f9eK#^W}=YTK+(4KU}HD*n)oh@STH#BkbHk2EktL|a$%O{0Mj8i6BFHV|A5*wXESQau0EtaQgJYE>(=Weti2LESLST9ObCciYHRr_+{2RF#= z5&D@6%YAPvO2-A&U!g3(JRP1$C-7M}ki_MGM=(XML$96Hjr&saqhxwoYK&Q8=j54K;*?uT30H}$?HPzj@fNO^vth$>)K%^$T? 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index 77df04a..ce4d0fe 100644 --- a/src/toboggan.tex +++ b/src/toboggan.tex @@ -1,9 +1,206 @@ -\section{Titre} +\section{Drôles de toboggans} -Énoncé +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 $N$. +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~$K$ ; + \item des paires de toboggans voisins qui se croisent sans se rencontrer et échangent leurs sorties, qu'on nommera \emph{paire de tuyaux de type \( \mathcal{Z} \)} : c’est-à-dire que ce qui rentre en position $K$ sort en position $K+1$, et ce qui rentre en position $K+1$ sort en position $K$ ; + \item des paires de toboggans voisins qui fusionnent en un seul puis se séparent, qu'on nommera \emph{paire de tuyaux de type \( \mathcal{X} \)} : c’est-à-dire que ce qui rentre en position $K$ s'additionne avec ce qui rentre en position $ K+1 $, et le total ressort en proportions égales en $ K $ et $ K+1$. +\end{itemize} +% Pour le spectacle, on va faire entrer $2^H$ litres d’eau dans l’une des entrées et elle va descendre les toboggans en suivant le trajet, et en se séparant en deux quantités égales dans les tuyaux de sortie dans la situation (c). +À chaque mètre, Emmy place une rangée composée d'une combinaison de son choix de tels tuyaux. -\q Première question +\begin{figure}[h] + \centering + \begin{tikzpicture} + \draw [very thick] (0,10) -- (0,7); + \draw [very thick] (2,10) -- (4,7); + \draw [line width=6pt, white] (4,10) -- (2,7); + \draw [very thick] (4,10) -- (2,7); + \draw [very thick] (6,10) -- (6,7); + \draw [very thick] (8,10) -- (8,7); + % + \draw (0,8.5) node [right] {\( I \)}; + \draw (3,8.5) node [right] {\( \mathcal{Z} \)}; + \draw (6,8.5) node [right] {\( I \)}; + \draw (8,8.5) node [right] {\( I \)}; + % + \draw [very thick] (0,7) -- (2,4); + \draw [very thick] (2,7) -- (0,4); + \draw [very thick] (4,7) -- (4,4); + \draw [very thick] (6,7) -- (8,4); + \draw [very thick] (8,7) -- (6,4); + % + \draw (1,5.5) node [right] {\( \mathcal{X} \)}; + \draw (4,5.5) node [right] {\( I \)}; + \draw (7,5.5) node [right] {\( \mathcal{X} \)}; + % + \draw [very thick] (0,4) -- (0,1); + \draw [very thick] (2,4) -- (4,1); + \draw [very thick] (4,4) -- (2,1); + \draw [very thick] (6,4) -- (8,1); + \draw [line width=6pt, white] (8,4) -- (6,1); + \draw [very thick] (8,4) -- (6,1); + % + \draw (0,2.5) node [right] {\( I \)}; + \draw (3,2.5) node [right] {\( \mathcal{X} \)}; + \draw (7,2.5) node [right] {\( \mathcal{Z} \)}; + % + \draw (0,10) node (rect) [blue, fill=white, minimum height=2em] {\( 1/2\)}; + \draw (2,10) node (rect) [blue, fill=white, minimum height=2em] {\( 0\)}; + \draw (4,10) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + \draw (6,10) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + \draw (8,10) node (rect) [blue, fill=white, minimum height=2em] {\( 0\)}; + % + \draw (0,7) node (rect) [blue, fill=white, minimum height=2em] {\( 1/2\)}; + \draw (2,7) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + \draw (4,7) node (rect) [blue, fill=white, minimum height=2em] {\( 0\)}; + \draw (6,7) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + \draw (8,7) node (rect) [blue, fill=white, minimum height=2em] {\( 0\)}; + % + \draw (0,4) node (rect) [blue, fill=white, minimum height=2em] {\( 3/8\)}; + \draw (2,4) node (rect) [blue, fill=white, minimum height=2em] {\( 3/8\)}; + \draw (4,4) node (rect) [blue, fill=white, minimum height=2em] {\( 0\)}; + \draw (6,4) node (rect) [blue, fill=white, minimum height=2em] {\( 1/8\)}; + \draw (8,4) node (rect) [blue, fill=white, minimum height=2em] {\( 1/8\)}; + % + \draw (0,1) node (rect) [blue, fill=white, minimum height=2em] {\( 3/8\)}; + \draw (2,1) node (rect) [blue, fill=white, minimum height=2em] {\( 3/16\)}; + \draw (4,1) node (rect) [blue, fill=white, minimum height=2em] {\( 3/16\)}; + \draw (6,1) node (rect) [blue, fill=white, minimum height=2em] {\( 1/8\)}; + \draw (8,1) node (rect) [blue, fill=white, minimum height=2em] {\( 1/8\)}; + \end{tikzpicture} + \caption{Une tuyauterie de toboggans de hauteur \( H = 3 \) avec \( N = 5 \) entrées. Les quantités d'eau à chaque étage sont indiquées en bleu, et le type de tuyau est indiqué à droite de chaque tuyau ou paire de tuyaux.} + \label{fig:ToboggansSansY} +\end{figure} +Pour le spectacle, on fait entrer une quantité totale de $1$ litre d'eau dans les entrées, selon une répartition à fixer ultérieurement. Ensuite, l'eau descend dans les toboggans en suivant les règles ci-dessus. Enfin, l'eau débouche dans les tuyaux de sortie. +On trouvera un exemple sur la figure~\ref{fig:ToboggansSansY}. -\q Deuxième question +\medskip -\q Proposer et étudier d'autres pistes de recherche. \ No newline at end of file + + +\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 +\begin{enumerate} + \item elle peut choisir où rentre l’eau ; + \item elle fait rentrer la même quantité d’eau dans tous les tuyaux ; + \item l’eau rentre dans le tuyau en position 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 +\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. +\end{enumerate} + + + +\q Pour cette question uniquement, le décor penche légèrement et, en traversant une paire de tuyaux de type \( \mathcal{X} \), l’eau qui rentre en position $K$ ou $K+1$ ressort en proportion $P$ dans la position $K$ et $(1-P)$ dans la position $K+1$, avec \( 0 \leq P \leq 1 \). Reprendre la question 2. dans ce cadre. + + + +\q Emmy choisit que toute l'eau rentre dans le tuyau 1 uniquement. On appelle \emph{spectacle} la liste des quantités d’eau $(x_1, ..., x_N)$ qui sortent de chaque tuyau, où $x_i$ est la quantité d'eau qui sort du tuyau numéroté $i$. Par exemple, le spectacle correspondant à la figure 1 est: $\left(\frac{3}{8}; \frac{3}{16}; \frac{3}{16}; \frac{1}{8}; \frac{1}{8}\right)$. +\begin{enumerate} + \item En fonction de $N$ et $H$, combien de spectacles différents Emmy peut-elle réaliser ? + \item Emmy a appris qu'un collègue facétieux a décidé de mélanger les entrées juste avant le spectacle, après la construction de la tuyauterie (l'eau pourrait donc rentrer par n'importe quel tuyau, pas nécessairement le~$1$). + Elle souhaite construire une tuyauterie qui réalise le spectacle qu'elle a choisi, indépendamment du tuyau par lequel son collègue fera rentrer l'eau. + Combien de spectacles peut-elle réaliser dans ces conditions, toujours en fonction de $ N $ et $ H $ ? + \item Suite à une pénurie de matériel, Emmy ne dispose que d'une quantité limitée $M$ de paires de tuyaux de type \( \mathcal{X} \). + En contrepartie, elle peut choisir $H$ aussi grand qu'elle le souhaite, les tuyaux (ou paires de tuyaux) \( I \) et \( \mathcal{Z} \) étant quant à eux disponibles en abondance. + Combien de spectacles différents peut-elle réaliser dans ces conditions, en fonction de $M$ et $N$ ? + (Heureusement, son collègue a décidé de la laisser tranquille pour cette fois.) +\end{enumerate} + + + +\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. +\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 $H$. +\end{enumerate} + + + +\q Le collègue d'Emmy est de retour pour l'ennuyer, 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 \( H \) ? + + + +\q Proposer et étudier d'autres pistes de recherche. + + + +\begin{figure}[h] + \centering + ~ + \hfill + \begin{tikzpicture} + \draw [very thick] (0,6) -- (1,4.5); + \draw [very thick] (2,6) -- (0,3); + \draw [very thick] (4,6) -- (4,3); + % + \draw [very thick] (0,3) -- (0,0); + \draw [very thick] (2,3) -- (4,0); + \draw [line width=6pt, white] (4,3) -- (2,0); + \draw [very thick] (4,3) -- (2,0); + % + \draw (1,4.5) node [right] {\( \mathcal{Y} \)}; + \draw (4,4.5) node [right] {\( I \)}; + % + \draw (0,1.5) node [right] {\( I \)}; + \draw (3,1.5) node [right] {\( \mathcal{Z} \)}; + % + \draw (0,6) node (rect) [blue, fill=white, minimum height=2em] {\( 1/2\)}; + \draw (2,6) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + \draw (4,6) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + % + \draw (0,3) node (rect) [blue, fill=white, minimum height=2em] {\( 3/4\)}; + \draw (2,3) node (rect) [blue, fill=white, minimum height=2em] {\( 0\)}; + \draw (4,3) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + % + \draw (0,0) node (rect) [blue, fill=white, minimum height=2em] {\( 3/4\)}; + \draw (2,0) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + \draw (4,0) node (rect) [blue, fill=white, minimum height=2em] {\( 0\)}; + \end{tikzpicture} + \hfill + \begin{tikzpicture} + \draw [very thick] (0,6) -- (2,3); + \draw [very thick] (2,6) -- (1,4.5); + \draw [very thick] (4,6) -- (4,3); + % + \draw [very thick] (0,3) -- (0,0); + \draw [very thick] (2,3) -- (4,0); + \draw [very thick] (4,3) -- (2,0); + % + \draw (1,4.5) node [right] {\reflectbox{\( \mathcal{Y} \)}}; + \draw (4,4.5) node [right] {\( I \)}; + % + \draw (0,1.5) node [right] {\( I \)}; + \draw (3,1.5) node [right] {\( \mathcal{X} \)}; + % + \draw (0,6) node (rect) [blue, fill=white, minimum height=2em] {\( 1/2\)}; + \draw (2,6) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + \draw (4,6) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + % + \draw (0,3) node (rect) [blue, fill=white, minimum height=2em] {\( 0\)}; + \draw (2,3) node (rect) [blue, fill=white, minimum height=2em] {\( 3/4\)}; + \draw (4,3) node (rect) [blue, fill=white, minimum height=2em] {\( 1/4\)}; + % + \draw (0,0) node (rect) [blue, fill=white, minimum height=2em] {\( 0\)}; + \draw (2,0) node (rect) [blue, fill=white, minimum height=2em] {\( 1/2\)}; + \draw (4,0) node (rect) [blue, fill=white, minimum height=2em] {\( 1/2\)}; + \end{tikzpicture} + \hfill + ~ + \caption{Portions de tuyauteries avec des tuyaux \( \mathcal{Y} \) et \reflectbox{\(\mathcal{Y}\)}} + \label{fig:ToboggansAvecY} +\end{figure} \ No newline at end of file

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