diff --git a/fiches/points-colories-fiche.tex b/fiches/points-colories-fiche.tex
index 2ba0802..d835326 100644
--- a/fiches/points-colories-fiche.tex
+++ b/fiches/points-colories-fiche.tex
@@ -46,13 +46,15 @@
     
     \textbf{Facile :} On peut trouver des éléments de $S$ et démontrer que d'autres n'y sont pas. 
 
-    Par exemple la question précédente montre que $ 1 \notin S $. En prenant des stratégies arbitraires, on trouve des éléments de $S$ en calculant leurs probabilités associées. Il est impossible de lister dans ce document toutes les stratégies possibles, mais il est clair qu'on peut toujours en trouver, calculer les probabilités associées et en déduire des des éléments de $S$. 
+    Par exemple la réponse à la question précédente justifie que $ 1 \notin S $. En prenant des stratégies arbitraires, on trouve des éléments de $S$ en calculant leurs probabilités associées. Il est impossible de lister dans ce document toutes les stratégies possibles, mais il est clair qu'on peut toujours en trouver, calculer les probabilités associées et en déduire des éléments de $S$. 
 
     \textbf{Moyen :} On a $ \frac 3 4 \in S $.
 
     Démonstration :
 
-    On note $D$ la droite formée par le premier point placé par Alice et le centre du cercle. Après que l'adversaire ait joué son point $2$, Alice joue le symétrique de $2$ par rapport à $D$.
+    (une autre preuve est donnée un peu plus loin)
+
+    On note $D$ la droite formée par le premier point placé par Lucie et le centre du cercle. Après que l'adversaire ait joué son point $2$, Lucie joue le symétrique de $2$ par rapport à $D$.
     
     On définit $ \theta $ comme l'angle \textit{aigu} l'angle formé entre le premier point placé (par Lucie) et le second (par l'adversaire). Clairement, $\theta$ suit une loi uniforme sur $[0,\pi]$.
 
@@ -100,7 +102,77 @@
     
     Comme $ \theta $ est d'espérance $ \frac \pi 2 $, l'espérance vaut $ 1 - \frac{\pi/2}{2\pi} $, c'est à dire 3/4.
 
-    \textbf{Moyen/Difficile :} On a $]\frac 1 4, \frac 1 2[ \subset S$
+    \textbf{Moyen/Difficile :} On a $]0, \frac 1 4 ] \subset S$
+
+    Démonstration :
+
+    Soit $\alpha \in ]0,1[$. Si $\theta$ désigne toujours l'angle aigu formé entre le premier point placé (par Lucie) et le second (par l'adversaire), Lucie joue le point situé à l'angle $ 2\pi - \alpha \theta $ dans le même repère. 
+
+\begin{figure}[h]
+    \centering
+    \begin{tikzpicture}
+        \begin{scope}[xshift=10cm]
+            \draw (0,0) circle (1.5cm);
+            
+            \coordinate (A1) at (0:1.5cm);  
+            \coordinate (A2) at (120:1.5cm);
+            \coordinate (B1) at (160:1.5cm);
+            \coordinate (B2) at (320:1.5cm);
+
+            \node at (0:2.5cm)  {1er point}; 
+            \node at (160:1.8cm)  {$ \theta $}; 
+            \node at (290:2.25cm)  {$ 2\pi - \alpha \theta $}; 
+
+            % \draw[dotted] (0:5cm) -- (180:5cm);
+            
+            % \draw[very thick, orangeAnimath] (A1) arc[start angle=0, end angle=140, radius=1.5cm];
+            % \draw[very thick, bleuAnimath] (B1) arc[start angle=180, end angle=340, radius=1.5cm];
+            
+            \foreach \angle in {0, 290} {
+                \filldraw[fill=orangeAnimath] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm
+            }
+
+            \foreach \angle in {160 } {
+                \filldraw[fill=bleuAnimath] (\angle:1.5cm) circle (2pt); % Points sur le cercle de rayon 2 cm
+            }
+
+            % \draw[dotted] (0,0) -- (B1);
+            
+            % \draw[thin] (0.4,0) arc[start angle=0, end angle=140, radius=0.4cm];
+
+        \end{scope}
+    \end{tikzpicture}
+\end{figure}
+
+    En se repérant toujours dans le même repère, l'évènement \og gagner \fg revient à \og l'adversaire joue un point dont l'angle (dans le même repère) est dans $[\theta - \alpha \theta ,\min( \theta + \alpha \theta, 2\pi - \alpha \theta )]$\fg. 
+
+    Supposons $ \alpha \in ]0, \frac 1 2]$ (le cas $ \alpha = 0 $ n'est pas une stratégie valable car elle demanderait de jouer à l'angle $ 2\pi = 0 $ qui est par définition là où le tout premier point a été placé).
+
+    On a nécessairement $ \theta + \alpha \theta \leqslant \frac{3\pi}{2} \leqslant 2\pi - \alpha \theta $ (car $ \theta \in [0, \pi]$ ), l'intervalle précédent devient $[\theta - \alpha \theta , \theta + \alpha \theta ]$. 
+
+    L'adversaire joue dedans avec probabilité $ \frac{\theta + \alpha \theta - (\theta - \alpha \theta )}{2 \pi } = \alpha \frac \theta \pi  $, l'espérance vaut $ \frac \alpha 2 $, et cette probabilité de gagner avec cette stratégie décrit $]0,\frac 1 4]$ quand $ \alpha $ décrit $ ] 0, \frac 1 2 ] $.
+
+    \bigskip
+
+    On retrouve le $ 3/4 $ précédent si on prend $ \alpha \in ]\frac 1 2, 1] $ : on a alors nécessairement $ 2\pi - \alpha \theta \leqslant \frac{3\pi}{2} \leqslant \theta + \alpha \theta $, l'intervalle devient $[\theta - \alpha \theta , 2\pi - \alpha \theta]$, la probabilité de gagner vaut $\frac{1}{2\pi} (2\pi - \alpha \theta - (\theta - \alpha \theta )) = 1 - \frac{\theta}{2\pi}$ dont l'espérance vaut $ \frac 3 4 $.
+
+    En fait, le cas $ \alpha = 1 $ correspond précisément à la stratégie décrite un peu plus haut (jouer le symétrique du point adverse).
+
+    \bigskip
+
+    \textbf{Difficile :} Il y a pas mal de choses à dire et à faire, en considérant par exemple des variantes de la stratégie précédente, dépendant d'un paramètre qu'on fait varier, ce qui permet de trouver certains intervalles inclus dans l'ensemble cherché.
+
+    \textbf{Ouvert :} La détermination de l'ensemble complet. Notamment, trouver la borne supérieure de $S$ et établir des inclusions réciproques à celles établies ci-dessus a l'air assez difficile.
+
+    \item (Facile/Moyen/Difficile/Ouvert) La difficulté est la même que la question précédente mais toute la combinatoire change (Facile : trouver quelques petits résultats partiels, Moyen : Trouver des résultats partiels assez forts, Difficile : Aller très loin, Ouvert : Déterminer l'ensemble complet)
+     
+    Les raisonnements sont différents, la combinatoire change, mais on peut obtenir des résultats partiels comment avant, avec les mêmes ordres de difficulté. Il y a beaucoup de possibilités.
+
+    Comme précédemment, la détermination de l'ensemble complet, en particulier de sa borne supérieure, est difficile sinon ouverte.
+
+    Un exemple : 
+
+    \textbf{Moyen/Difficile :} Soit $S$ l'ensemble cherché ; on a $]\frac 1 4, \frac 1 2[ \subset S$
 
     Démonstration :
 
@@ -146,12 +218,6 @@ En se repérant toujours dans le même repère, l'évènement \og ne pas perdre
 
 La probabilité associée est $ \frac{\theta + \alpha \theta}{2\pi} $, l'espérance vaut $(1+\alpha)/4$, qui décrit $]\frac 1 4, \frac 1 2[$ quand $\alpha$ décrit $]0,1[$.
 
-    \textbf{Difficile :} Il y a pas mal de choses à dire et à faire, en considérant par exemple des variantes de la stratégie précédente, dépendant d'un paramètre qu'on fait varier, ce qui permet de trouver certains intervalles inclus dans l'ensemble cherché.
-
-    \textbf{Ouvert :} La détermination de l'ensemble complet. Notamment, trouver la borne supérieure de $S$ et établit des inclusions réciproques à celles établies ci-dessus a l'air assez difficile.
-
-    \item On peut faire des raisonnement assez similaires à ceux de la question précédentes pour obtenir des résultats partiels de la même forme. Les raisonnements sont différents, la combinatoire change, mais on peut obtenir des résultats aussi partiels que ceux de la question précédente, avec les mêmes ordres de difficulté. Je pense que les élèves vont imaginer tellement de stratégies différentes, et obtenir tellement de résultats (partiels) différents, qu'il n'est pas forcément nécessaire d'en montrer plus que ça et de développer des pages et des pages de calculs dans cette fiche solution. Simplement, voilà la tête des solutions qu'on peut donner à ces questions.
-
 \end{enumerate}
 
 \q (Difficile/Ouvert) On peut établir des résultats partiels, mais c'est plus difficile parce qu'on a beaucoup plus de mal à voir les choses vu que $n \geqslant 3$ tours sont joués. On peut trouver des intervalles inclus dans les ensembles recherchés en considérant certaines stratégies dépendant d'un paramètre qu'on fait varier, les inclusions réciproques étant plus difficiles, et la détermination complète de l'ensemble ouverte.
@@ -186,7 +252,7 @@ La probabilité associée est $ \frac{\theta + \alpha \theta}{2\pi} $, l'espéra
 
 Le fait qu'on considère les $\varepsilon$-approximations de ce polygône régulier fait qu'on considère un évènement de probabilité non-nulle. (L'évènement \og Il existe un polygone régulier $P$ à $n$ cotés inscrit dans le cercle tel qu'il y a, pour chaque sommet $S$ de $P$, un unique point joué par l'adversaire dans la boule $B(S,\varepsilon)$ \fg est de probabilité non-nulle. Pour le voir, il suffit de fixer un tel polygône $P_0$, et d'écrire l'évènement $E_{P_0}$ = \og Il y a, pour chaque sommet $S$ de $P_0$, un unique point joué par l'adversaire dans la boule $B(S,\varepsilon)$ \fg avec des unions, des intersections (et des permutations). L'évènement \og Il existe ... \fg est l'union des $E_{P_0}$, tous de probabilité non-nulle).
 
-Dans ce cas, il est clair que pour $\varepsilon$ assez petit, Alice va être obligée de : 
+Dans ce cas, il est clair que pour $\varepsilon$ assez petit, Lucie va être obligée de : 
 
 \begin{itemize}
 
@@ -216,13 +282,17 @@ L'évènement \og Elle est obligée de faire tout ça \fg est de probabilité st
 
 Je ne sais pas si c'est suffisant, je pense que non. Je n'ai pas trouvé d'autres conditions.
 
-\q Évidemment, elle a envie de jouer le (un) polygône à $p$ côtés inscrit dans le cercle. 
+\q Tout d'abord on peut commencer par s'intéresser à la borne supérieure de cet ensemble : Lucie dispose-t-elle d'une stratégie optimale ?
 
-La probabilité de gagner est un dénombrement au cours duquel il est très facile de s'exciter trop vite et de se tromper ! Cette question saura distinguer les équipes sachant raisonner droit de celles trébuchant au moindre obstacle.
+Évidemment, elle a envie de jouer le (un) polygône régulier à $p$ côtés inscrit dans le cercle. 
+
+La probabilité de gagner avec cette stratégie est un dénombrement au cours duquel il est très facile de s'exciter trop vite et de se tromper ! Si des équipes partent sur cette voie, on saura distinguer celles sachant raisonner droit de celles trébuchant au moindre obstacle.
 
 La réponse est $0$ si $ n < p $ et $ \displaystyle \frac{1}{p^n} \sum\limits_{k=1}^p (-1)^{k-1} (p-k)^n \binom{p}{k} $ sinon.
 
-Je pense qu'on doit pouvoir montrer que c'est optimal, en utilisant le fait que toute autre stratégie formera nécessairement un arc de longueur $ > \frac{2\pi}{p} $. Il faut montrer que ça diminue nécessairement les chances de gagner. Une piste possible est de noter $ \theta_1, ..., \theta_{p-1} $ les angles entre les différents placés, et de calculer explicitement la probabilité $ f(\theta_1, ..., \theta_{p-1}) $  de gagner (noter que $f$ est symétrique). Je pense que c'est faisable (sans être facile).
+Est-ce optimal ? Je pense qu'on doit pouvoir montrer que oui, en utilisant le fait que toute autre stratégie formera nécessairement un arc de longueur $ > \frac{2\pi}{p} $. Il faut montrer que ça diminue nécessairement les chances de gagner. Une piste possible est de noter $ \theta_1, ..., \theta_{p-1} $ les angles entre les différents points placés, et de calculer explicitement la probabilité $ f(\theta_1, ..., \theta_{p-1}) $  de gagner (noter que $f$ est symétrique). Je pense que c'est faisable (sans être facile).
+
+Ensuite, il faut encore chercher des configurations et leurs stratégies associées pour explorer l'ensemble cherché. Il y a beaucoup de choses que l'on peut faire !
 
 \q (Difficile/Ouvert) Appelons J1 le joueur qui joue en premier et J2 celui qui joue en deuxième, donc aussi en dernier.
 
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diff --git a/index.out b/index.out
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-\BOOKMARK [1][-]{section*.1}{Pr\351ambule}{}% 1
-\BOOKMARK [1][-]{section*.3}{Notations}{}% 2
-\BOOKMARK [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. Taxes routi\350res}{}% 9
-\BOOKMARK [1][-]{section.8}{8. Points color\351s sur un cercle}{}% 10
diff --git a/index.pdf b/index.pdf
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diff --git a/index.toc b/index.toc
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-\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}}{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}{Taxes routières}}{11}{section.7}%
-\contentsline {section}{\tocsection {}{8}{Points colorés sur un cercle}}{13}{section.8}%
diff --git a/src/points-colories.tex b/src/points-colories.tex
index 73d0f0a..37565cd 100644
--- a/src/points-colories.tex
+++ b/src/points-colories.tex
@@ -124,7 +124,7 @@ Pour essayer, Lucie et son adversaire reprennent exactement la même configurati
 
 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 lui permettant de gagner à coup sûr ? Si oui, en décrire une. 
+\q L'un des deux joueurs dispose-t-il d'une stratégie lui permettant de gagner à coup sûr ?
 
 \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.