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[1][-]{section.1}{1. Une bonne humeur contagieuse}{}% 3 +\BOOKMARK [1][-]{section.2}{2. Dr\364les de toboggans}{}% 4 +\BOOKMARK [1][-]{section.3}{3. Plats \340 tarte gradu\351s}{}% 5 +\BOOKMARK [1][-]{section.4}{4. Transformation de papillons}{}% 6 +\BOOKMARK [1][-]{section.5}{5. Gerrymandering}{}% 7 +\BOOKMARK [1][-]{section.6}{6. Le cauchemar de la ligne 20-25}{}% 8 +\BOOKMARK [1][-]{section.7}{7. Le poids des graphes}{}% 9 +\BOOKMARK [1][-]{section.8}{8. Points color\351s sur un cercle}{}% 10 diff --git a/index.pdf b/index.pdf new file mode 100644 index 0000000..972a5b7 Binary files /dev/null and b/index.pdf differ diff --git a/index.synctex.gz b/index.synctex.gz new file mode 100644 index 0000000..4f80229 Binary files /dev/null and b/index.synctex.gz differ diff --git a/index.toc b/index.toc new file mode 100644 index 0000000..87b8fac --- /dev/null +++ b/index.toc @@ -0,0 +1,11 @@ +\babel@toc {french}{} +\contentsline {section}{\tocsection {}{}{Préambule}}{1}{section*.1}% +\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 {}{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 {}{8}{Points colorés sur un cercle}}{13}{section.8}% diff --git a/src/bacteries.tex b/src/bacteries.tex index e193b4b..25b429a 100644 --- a/src/bacteries.tex +++ b/src/bacteries.tex @@ -17,14 +17,14 @@ Est-il possible que l'un des papillons atteigne une taille arbitrairement grande 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 120\%. -Soit $x \in [0, \infty[$. Quelle est 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 $x$}) +\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 $x$}) -\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 120\% de l'envergure de son grand-parent. +\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. 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. 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 Proposer et étudier d'autres pistes de recherche.