From 0f2967a218f440ce738d7931f5f412f1407cd1d9 Mon Sep 17 00:00:00 2001 From: Timothee Rocquet Date: Wed, 3 Apr 2024 11:55:52 +0200 Subject: [PATCH] =?UTF-8?q?r=C3=A9solution=20du=20probl=C3=A8me=20de=20cha?= =?UTF-8?q?ngement=20de=20nom,=20ajout=20de=20la=20fiche=20probl=C3=A8me?= =?UTF-8?q?=20pour=20les=20cookies?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- fiches/cookies-fiche.tex | 88 ++++++++++++++++++++++ index.tex | 2 +- index_avec_fiches.tex | 17 +++-- src/anciennes_images/Pavage.png | Bin 62660 -> 0 bytes src/anciennes_images/Symetrie.png | Bin 9470 -> 0 bytes src/anciennes_images/Triangle 1.png | Bin 17346 -> 0 bytes src/anciennes_images/Triangle 2.png | Bin 30893 -> 0 bytes src/cookies.tex | 109 ++++++++++++++++++++++++++++ 8 files changed, 208 insertions(+), 8 deletions(-) create mode 100644 fiches/cookies-fiche.tex delete mode 100644 src/anciennes_images/Pavage.png delete mode 100644 src/anciennes_images/Symetrie.png delete mode 100644 src/anciennes_images/Triangle 1.png delete mode 100644 src/anciennes_images/Triangle 2.png create mode 100644 src/cookies.tex diff --git a/fiches/cookies-fiche.tex b/fiches/cookies-fiche.tex new file mode 100644 index 0000000..4b4b430 --- /dev/null +++ b/fiches/cookies-fiche.tex @@ -0,0 +1,88 @@ +\section*{Eléments de réponse} + +\q (facile) Le cas a) est simple, il suffit de mettre une quantité de pâte $R$ au centre du disque (segment de longueur 0). On donne une réponse positives dans chacun des autres cas au moyen de dessins : + + + +On peut même en déduire les résultas plus généraux suivants. Un \textbf{coin} est un bout du bord qui consiste en deux segments ayant un sommet en commun appelé la \textbf{pointe}. L'\textbf{angle intérieure} d'un coin est l'angle formé par les deux segments et qui se trouve à l'intérieur du cookie. + +\begin{lemme}[Lemme sur les coins]\label{lemme : coins} +Si un des bouts du cookie est formé d'un coin dont l'angle intérieure est plus petit strictement que 180, alors il faut qu'un des segments que Fabrice choisi passe par la pointe de ce coin. De plus, il faut qu'il dépose une quantité de pâte nulle à la pointe. +\end{lemme} + +\begin{proof} +Si on veut avoir une chance de remplir la forme, il faut au moins qu'il existe un cercle de centre $O$ et de rayon $r$ qui passe par la pointe $c$. Comme le coin à un angle intérieur strictement inférieur à 180 il sera compris strictement dans ce cercle. Sauf si $r=0$ et $O=c$, ce qui conclut. +\end{proof} + +\begin{rem} +On peut relâcher la définition de coin. En pratique on voudrait démontrer que si le bord n'est pas dérivable en un point, alors un segment devrait passer par ce coin. Ceci est faux comme on pourrait considérer un angle obtu. Il faut donc trouver une autre définition. +\end{rem} + +\begin{thm}[Théorème fondamentale du cookie]\label{thm : fonda} +Soit $f \colon \mathbb{R} \to \mathbb{R}^+$ une fonction de classe~$\mathcal{C}^1$. Fabrice peut remplir par de la pâte à cookie l'espace entre le graphe de $f$ et l'axe de réel en posant de la pâte uniquement sur l'axe des réels si et seulement si la fonction +\[g_f(x)=x+f(x)f'(x)\] +est croissante. +\end{thm} + +\begin{rem} +On peut a priori seulement demander que $f$ soit $\mathcal{D}^1$. Si $f$ est $\mathcal{C}^2$ (ou $\mathcal{D}^2$) on peut dériver une fois de plus et dire qu'il faut alors que +\[1+(f'(x))^2+f(x)f'(x) \geq 0.\] +\end{rem} + +\begin{rem} +On pourrait se demander si le cas des fraphes des fonctions \textbf{continues} $f \colon \R \to \R^+$ exhauste tous les cas que l'on peut remplir en mettant de la pâte sur l'axe des réels. Il se trouve que c'est le cas pour la raison suivante. Supposons que Fabrice dépose une quantité de pâte $p$ en un point donné $x \in \R$, ce point de pâte va s'étaler en un disque qui est le graphe de la fonction $f(y)=\sqrt{r^2-(y-x)^2}$. Si il dépose plus de pâte, la forme restante aura pour bord le graphe de la fonction qui vaut le maximum de fonctions de ce type là (une pour chaque point où il dépose de la pâte) et donc une fonction. +\end{rem} + +\begin{proof} + +\end{proof} + +Le théorème \ref{thm : fonda} nous renseigne sur une large classe de cookies de la manière suivante : on découpe le cookie en différente pièce ayant un axe de symétrie et de telle sorte que si on choisit cet axe de symétrie comme l'axe des réel, le bord de la pièce est une fonction qui vérifie la condition du théorème fondamental. Il serait intéressant de trouver des conditions sur la régularité du contour pour satisfaire automatiquement cette condition. C'est d'ailleurs l'esprit de la question \textbf{5.}. + +\q (Moyen) dans le cas a) c'est bien sûr 0 comme le segment que l'on a choisi dans l'exemple suivant est de longueur nulle. Dans les autres cas il faut faire un peu plus attention. Tout d'abord le résultat suivant est clair mais nécessaire. + +\begin{lemme} +Soit $S$ un segment qui est le bord d'une forme. Si Fabrice veut s'assure que le cookie passe par tous les points du segment $S$, il faut une quantité de pâte au moins aussi grande que celle du segment. +\end{lemme} + +\begin{proof} +Soit $x$ un point du segment $S$. Si un cercle de centre $O$ et de rayon $r$ passe par $x$, il faut que la droite $(Ox)$ soit perpendiculaire au segment $S$ (même preuve que le lemme \ref{lemme : coins}). Si +\[ I= \bigsqcup I_i \] +est une union de segment sur lesquels Fabrice dépose de la pâte et de telle sorte que $S$ soit le bord d'un cookie. Soit $\pi$ le projection orthogonale sur $S$ (enfin sur le droite qui prolonge le segment $S$). D'après ce que l'on vient de dire, on note remarque que +\[ S \subset \pi(I)=\bigcup \pi(I_i).\] +Mais la longueur de $\pi(I_i)$ est plus petite que le longueur de $I_i$, donc $\sum \vert I_i \vert \geq \sum \vert \pi(I_i) \vert \geq \vert S \vert$. +\end{proof} + +En pratique pour remplir une forme avec une quantité minimale de pâte il suffit de remplir un voisinnage du bord. + +\begin{lemme} +Soit $B$ le bord de la forme de cookie que Fabrice souhaite remplir. On suppose qu'il existe un voisinnage $V$ du bord dans la forme qui soit remplissable avec une quantité $p$ de pâte. Alors, il est possible de remplir l'ensemble du cookie avec une quantité $p$ de pâte. +\end{lemme} + +\begin{proof} +Soit $V_1 \subset V$ un voisinnage du bord dans la forme, on peut recouvrir le complémentaire de $V_1$ dans la forme par des boules de rayon $r$ et par compacité on peut en choisir un nombre fini. Cela ne nous coûte donc pas de pâte de remplir toute la forme une fois que l'on a remplit le voisinnage. +\end{proof} + +On a ensuite le lemme suivant, remarqué par Nicolas Fabiano. + +\begin{lemme}[Lemme de Nicolas] +Soit $S_1$ et $S_2$ deux segments qui sont symétriques l'un de l'autre par rapport à une droite $(d)$ (on suppose de plus qu'ils forment un angles $\alpha$) avec cette droite. Fabrice souhaite remplir un espaces entre ces deux droites (il y a en un à l'intérieure et un à l'extérieur voir dessin), il lui faudra alors les quantités suivantes de pâtes. +\begin{itemize} +\item[Cas 1 :] Si l'angle $\alpha$ est plus petit que $60$, alors il faut au moins $\vert S_1 \vert / \sin(\alpha)$, +\item[Cas 2 :] Sinon il faut au moins $2\vert S_1 \vert$ (c'est-à-dire que l'on ne peut pas faire mieux que de remplir un voisinnage des segments séparément). +\end{itemize} +\end{lemme} + +La preuve est une version plus élaborée de celle du lemme précédent. + +\q + +\q + +\q On peut appliquer le lemme \ref{lemme : coins} et créer un cookie qui a une infinité de coins (qui ne puissent pas être recouvert par un nombre fini de segments). Par exemple un triangle de Sierpinski. Cela fournit un contre-exemple pour tout $r \leq 0$. + +\q + +\q + +\q diff --git a/index.tex b/index.tex index be5a72d..f81deb8 100644 --- a/index.tex +++ b/index.tex @@ -136,7 +136,7 @@ $PPCM(u,v)$ & plus petit entier positif divisible à la fois par $u$ et $v$\\ \nextpb \input{src/piece_truquee.tex} \nextpb -\input{src/drôles_de_cookies.tex} +\input{src/cookies.tex} \nextpb \input{src/oracle.tex} \nextpb diff --git a/index_avec_fiches.tex b/index_avec_fiches.tex index e03e3a7..2c6d634 100644 --- a/index_avec_fiches.tex +++ b/index_avec_fiches.tex @@ -72,11 +72,6 @@ \newcommand{\q}{\stepcounter{question}\medskip \noindent\textbf{\thequestion.}\,} \newcommand{\qnospace}{\stepcounter{question}\medskip \noindent\textbf{\thequestion.}\, \vspace*{-6.5mm}} \renewcommand{\labelenumi}{\alph{enumi})} -\newtheorem{thm}{Théorème}[section] -\newtheorem{lemme}{Lemme}[section] -\newtheorem{rem}{Remarque}[section] -\newtheorem{ques}{Question}[section] -\newtheorem{coro}{Corollaire}[section] %----------------------------------------------------------------------------------------------- @@ -84,6 +79,14 @@ \definecolor{orangeAnimath}{RGB}{234,94,0} \definecolor{bleuAnimath}{RGB}{0,159,227} +%----------------------------------------------------------------------------------------------- + +\newtheorem{thm}{Théorème}[section] +\newtheorem{lemme}{Lemme}[section] +\newtheorem{rem}{Remarque}[section] +\newtheorem{ques}{Question}[section] +\newtheorem{coro}{Corollaire}[section] + \begin{document} \newgeometry{top=2cm, bottom=2cm, left=2cm, right=2cm} @@ -140,10 +143,10 @@ Le contenu de ces fiches doit rester confidentiel jusqu'au tournoi national ! 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$Y4kL&6#5Oz0!@%!|NB_Hc$!w2aGd$r?HHeYVTyeWO2KJF?^+oNE~ zboV%OR~0zJtkr=S$G2tz^9Y|UoAPgv1{b(JhGLC#mC0~Gd^8jOuq9O!m)DXA^t$s} zFbNy>xX^}pl_Xv&!YAfGw16R8-f}z3Vm8Xr&R_1$), les deux cercles qui constituent le bord de l'anneau étant inclus dans le cookie. +\end{enumerate} + +\begin{center} +\begin{tikzpicture} +\fill[orangeAnimath] (0,0) circle (1); +\draw (0,0) -- (1,0) node[midway,above]{$R$}; +\fill[bleuAnimath] (2,-0.5) -- ++(0,1) node[black,midway,left]{$a$} -- ++(2,0) node[black,midway,above]{$b$} -- ++(0,-1) -- cycle; +\fill[orangeAnimath] (5,-1) -- ++(1,2) node[black,midway,left]{$a$} -- ++(2,-2) node[black,midway,above right]{$b$} -- cycle node[black,midway,below]{$c$}; +\fill[bleuAnimath,even odd rule] (10,0) circle(0.8) circle(1.2); +\draw (10,0) -- ++(0:0.8) node[midway,below]{$R_1$}; +\draw (10,0) -- ++(30:1.2) node[midway,above]{$R_2$}; +\end{tikzpicture} +\end{center} + +\q La forme a) est-elle un cookie ? Même question pour chacune des formes b), c) et d). + +\medskip + +La \textbf{quantité de pâte} utilisée pour faire un cookie est la somme des longueurs des segments où Fabrice place de la pâte. + +\q Pour chacune des formes de la question précédente qui sont des cookies, avec quelles quantités de pâte Fabrice peut-il la réaliser ? + +\medskip + +La précision de la poche à douille de Fabrice étant limitée, la quantité de pâte qu'il dépose en $P$ ne peut pas être trop petite. Pour un $r \geq 0$ fixé, on dit que l'outil de Fabrice est de précision~$r$ lorsque $R(P) \geq r$ pour tout point $P$ placé par Fabrice. On appelle $r$-\textbf{cookie du plan}, ou plus simplement $r$-cookie, un cookie que Fabrice peut réaliser avec un outil de précision $r$. Les réponses aux questions suivantes vont donc dépendre de $r$. + +En particulier, les $0$-cookies sont exactement les cookies, et tout $r$-cookie est un cookie. + +\q Reprendre les questions précédentes dans le cas des $r$-cookies, en fonction de $r$. + +\q On suppose dans cette question que Fabrice réalise un $r$-cookie sans faire de segment de longueur $0$ et tel qu'il est impossible d'obtenir la même forme en utilisant strictement moins de pâte. Est-il possible qu'une répartition différente de la même quantité de pâte permette d'obtenir le même $r$-cookie, toujous sans segment de longueur $0$ ? + +%\q Si $R(P)=r$ ne dépend pas de P, quelles valeur peut prendre le rapport entre la quantité de pâte nécessaire et l'aire de la brioche après cuisson ? + +%\q Eric souhaite faire des brioches d'un seul tenant (il est toujours possible de faire un chemin de segments consécutifs dans la brioche qu'il obtient) [i.e. connexes par arcs + connexe ]. Parmi les points où Eric a placé de la pâte, est-il toujours possible de trouver un chemin de segments de pâte reliant deux points $P$ et $P'$. + +%\q En fait, Eric se rend compte qu'il regardait ses brioches gonfler par au-dessus mais qu'elle gonfle en fait dans l'espace et non pas dans le plan. La brioche occupe alors la réunion des boules de centre P et de rayon $R(P)$ (où les points $P$ où Eric place de la brioche sont toujours dans le plan). Quel est alors le volume que prend la brioche si la forme vue au-dessus est : +%\begin{enumerate} +% \item un disque de rayon $R$ ? +% \item un carré de côté $C$ ? +% \item Un rectangle de côté $a$ et $b$ ? +%\end{enumerate} + +\medskip + +Fabrice s'intéresse maintenant à la forme du bord de ses $r$-cookies. Pour cela on suppose qu'il dispose de deux fonctions continues $x : \R \to \R$ et~$y : \R \to \R$ qui vérifient les propriétés suivantes : + +\begin{itemize} + \item elles sont 1-périodiques, c'est-à-dire que $x(t+1) = x(t)$ et $y(t+1) = y(t)$ pour tout $t$ réel, + \item pour toutes valeurs de $t$ et $t'$, on a simultanément $x(t) = x(t')$ et $y(t) = y(t')$ si et seulemlent si la différence $t-t'$ est entière. +\end{itemize} + +Fabrice trace dans le plan l'ensemble $\Gamma$ des points de coordonnées $\left(x(t),y(t)\right)$, appelé \textbf{contour}. Il cherche maintenant à savoir si la partie du plan que le contour délimite (que l'on suppose bien définie) est un~$r$-cookie. + + +\q Existe-t-il un contour pour lequel $x$ et $y$ sont continues, mais pour lequel la partie délimitée n'est pas un~$r$-cookie ? Si oui, est-ce possible avec $x$ et $y$ dérivables ? Deux fois dérivables ? Trois fois dérivables ? + +\q Existe-t-il un contour pour lequel $x$ et $y$ sont continues, qui soit un cookie, mais qui ne soit un~$r$-cookie pour aucun $r>0$ ? Si oui, est-ce possible avec $x$ et $y$ dérivables ? Deux fois dérivables ? Trois fois dérivables ? + +%Existe-t-il des formes donc le contour est continu (je regarde l'intérieur d'un lacet simple continu) qui ne peuvent pas être obtenue avec un nombre fini de brioche ? + +%\q Existe-t-il des formes donc le contour est $C^1$ (je regarde l'intérieur d'un lacet simple continu) qui ne peuvent pas être obtenue avec un nombre fini de brioche ? Et si on est $C^1$ sauf en un nombre fini de points ? + +\q Donner d'autres conditions sur une forme pour que ce soit un $r$-cookie. + +%\q Maintenant on souhaite que Eric soit "proche" de la forme finale (l'idée serait d'approximer un coin par exemple). On suppose qu'il essaie d'obtenir la forme A mais qu'il obtienne la forme B, on définit alors un coefficient $\rho(A,B):=\rho(A \Delta B)/\rho(A)$ qui estime à quel point on est proche de la bonne forme. Si $\rho=0$ on dira que B recouvre presque partout A. + +\q Proposer et explorer d'autres pistes de recherche, par exemple en dimension $3$. \ No newline at end of file