% !-*-coding:utf-8-*-

\chapter{Amériques-Antilles-Guyane}

\olympEntete{Grands pairs\ldots}{Des droites et des mots}{Additionnons des points}

\begin{Enonce}[Grands pairs\ldots]
  Dans ce problème, on ne considère que des nombres entiers naturels
  non nuls. Pour chacun de ces entiers, on numérote les chiffres de
  son écriture décimale de gauche à droite. Le premier chiffre de
  gauche ne peut être $0$.
  
  Par exemple, pour le nombre $3021$, le chiffre $3$ reçoit le numéro
  $1$, le  chiffre $0$ le numéro $2$, le chiffre $2$ le numéro $3$ et
  le chiffre $1$ le numéro $4$.

  On nomme « grand pair » tout nombre dont chaque chiffre en position
  paire, s’il y en a, est au moins aussi grand que ses chiffres
  directement voisins (s’il en a). On nomme « grand impair » tout
  nombre dont chaque chiffre en position impaire est au moins aussi
  grand que ses chiffres directement voisins (s’il en a).

  Par exemple :
  \begin{itemize}
  \item le nombre $3021$ est un grand impair, mais pas un grand pair ; 
  \item les nombres $3$, $2$, $7$ et $777$ sont à la fois des grands pairs et des grands
    impairs ; 
  \item le nombre $2019$ n’est ni un grand pair, ni un grand impair.
  \end{itemize}
  \begin{enumerate}
  \item Le nombre $\num{384957}$ est-il un grand pair ? Un grand
    impair ?
  \item Déterminer les nombres qui sont à la fois des grands pairs et
    des grands impairs.
  \item Parmi les nombres s’écrivant avec deux chiffres, y a-t-il
    davantage de grands pairs ou de grands impairs ?
  \item
    \begin{enumerate}
    \item Le nombre $3021$ peut-il s’écrire comme la somme de deux
      grands impairs ayant le même nombre de chiffres ?
    \item Le nombre $3021$ peut-il s’écrire comme la somme de deux
      grands pairs ayant le même nombre de chiffres ?
    \end{enumerate}
  \item Prouver que tout nombre entier peut s’écrire comme la somme de
    deux grands impairs (rien n’est ici imposé quant au nombre de
    chiffres de ces deux grands impairs).
  \item Démontrer que tout nombre grand impair strictement inférieur à
    $100$ peut s’écrire comme la somme de deux grands pairs (sans
    contrainte quant au nombre de chiffres de ces grands pairs).
  \item Déterminer le plus petit grand impair supérieur ou égal à $2$
    qui ne peut pas s’écrire comme la somme de deux grands pairs (sans
    contrainte quant au nombre de chiffres de ces grands pairs).

  \item Compléter le pseudocode ci-dessous (ou s’en inspirer) pour
    rédiger un algorithme (à retranscrire sur sa copie), qui, partant
    d’un tableau « T » représentant un nombre « N » (par exemple
    $\num{384 957}$) de « nb » chiffres (ici $6$), renvoie « 1 » si N est
    un grand pair, et « 0 » sinon.

    \begin{tcblisting}{top=0mm,bottom=0mm,left=0mm,right=0mm,colframe=colLogoFort,listing
      only, title={Exercice national, les grands pairs\ldots},
      fonttitle=\bfseries\small,colbacktitle=colLogoFort,
      listing options={language=python,style=afdmpython}}
nb = 6
T=[3,8,4,9,5,7]

resultat = 1
i = 1
... ... ...
while (resultat == 1) and (i <= nb) :
    ... ... ...
    i = i + 2

print(r)

    \end{tcblisting}
  \end{enumerate}
  \begin{Corrige}
    \begin{enumerate}
    \item Le nombre $\num{384957}$ est un nombre premier pair, mais
      n'est pas impair, car $8 > 4$.
    \item Ce sont les nombres qui sont constitués que du même
      chiffre. En effet, deux chiffres consécutifs $a$ et $b$ doivent
      vérifier $a \leq b$ et $a \geq b$, donc $a=b$.
    \item Les nombres de deux chiffres qui sont des grands pairs sont
      : $11$, $12$, $13$, $14$, $15$, $16$, $17$, $18$, $19$, $22$,
      $23$, $24$, $25$, $26$, $27$, $28$, $29$, $33$, $34$, $35$,
      $36$, $37$, $38$, $39$, $44$, $45$, $46$, $47$, $48$, $49$,
      $55$, $56$, $57$, $58$, $59$, $66$, $67$, $68$, $69$, $77$,
      $78$, $79$, $88$, $89$, $99$. Il y en a $45$.

      Les nombres de deux chiffres qui sont des grands pairs sont :
      $10$, $11$, $20$, $21$, $22$, $30$, $31$, $32$, $33$, $40$,
      $41$, $42$, $43$, $44$, $50$, $51$, $52$, $53$, $54$, $55$,
      $60$, $61$, $62$, $63$, $64$, $65$, $66$, $70$, $71$, $72$,
      $73$, $74$, $75$, $76$, $77$, $80$, $81$, $82$, $83$, $84$,
      $85$, $86$, $87$, $88$, $90$, $91$, $92$, $93$, $94$, $95$,
      $96$, $97$, $98$, $99$. Il y en a $54$.

      Il y a donc plus d'entiers grands impairs que d'entiers grands
      pairs.
    \item
      \begin{enumerate}
      \item $3021 = 2000 + 1021$, c'est donc bien la somme de deux
        entiers grands impairs.
      \item $3021 = 1509 + 1512$, c'est donc bien la somme de deux
        entiers grands pairs.
      \end{enumerate}
    \item $N = a_1a_{2}\cdots a_n = a_10a_30 \cdots b + a_20a_40\cdot c$
      avec $b = 0$ si $n$ est pair, $a_n$ sinon et $c = a_n$ si $n$
      est pair et $0$ sinon. $N$ est bien la somme de deux grands
      impairs. On peut écrire tous les nombres de $1$ chiffre, sous la
      forme de $1 + n-1$, sauf, $ n = 1$.
    \item Soit $n > 1$ un grand impair inférieur à $100$. Si $n$ est un
      nombre de $1$ chiffre, alors c'est la somme de $2$ grand
      pairs. Si $n \geq 10$, alors $n = 10a+b$ avec $a \geq b$.
      $n = (10(a-1)+a) + (10+b-a)$. Le premier nombre est bien un
      grand pair, et le deuxième n'a qu'un chiffre donc est aussi
      grand pair. 
    \item Il faut chercher plus grand que $100$. Mais $100 = 99 +1$,
      $101 = 99+2$, \ldots $108 = 99 +9$, donc ces nombres sont la
      somme de $2$ grands pairs.

      Pour $109$. Supposons que $109 =  10a+b + 10c+d$, avec $1 \leq a
      \leq b \leq 9$ et $ 1 \leq c \leq d \leq 9 $.

      $b+c = 9$ grâce à  un tableau à double entrée. Il n'y a donc pas
      de retenue.

      $10(a+b) = 100$, donc $a+b = 10$. Soit, $1 \leq a \leq 9-d \leq
      8$ et $ 1 \leq 10 - a \leq d \leq 8$. Ces deux inégalités
      s'excluent l'une et l'autre.

      Donc, $109$ est la valeur chérchée.
    \item On obtient le proogramme suivant.
      \begin{tcblisting}{top=0mm,bottom=0mm,left=0mm,right=0mm,colframe=colLogoFort,listing
      only, title={Exercice national, les grands pairs\ldots},
      fonttitle=\bfseries\small,colbacktitle=colLogoFort,
      listing options={language=python,style=afdmpython}}
nb = 6
N = 324957
T = []
for c in str(N):
    T.append(int(c))

r = 1
i = 1
while (r == 1) and (i <= nb) :
    if i == nb or i == nb -1 :
        r = 1
    else :
        if T[i-1] <= T[i] and T[i+1] <= T[i]:
            r = 1
        else :
            r = 0
    i = i + 2

print(r)

    \end{tcblisting}

    \end{enumerate}
  \end{Corrige}
\end{Enonce}

\begin{Enonce}[Des droites et des mots]
  Le plan est rapporté à un repère orthonormé d’origine $O$. Dans tout
  le problème, on s’intéresse aux demi-droites issues de $O$ et
  contenues dans le premier quadrant (ensemble des points du plan dont
  l’abscisse et l’ordonnée sont positives). À tout entier naturel $n$, on
  associe la droite d’équation $x =n$ et la droite d’équation $y=n$. L’ensemble
  de ces droites constitue un « quadrillage ».
  
  Toute demi-droite $(d)$ issue de $O$ et de pente strictement positive
  possède des points d’intersection notés $M_0$, $M_1$, $M_2$, \ldots,
  $M_n$, \ldots avec les droites du quadrillage, les points étant
  numérotés dans l’ordre croissant de leurs abscisses. À chaque point $M_i$
  on associe la lettre $H$, la lettre $V$ ou la lettre $C$ selon qu’il est le point
  d’intersection de avec une droite horizontale, une droite verticale
  ou les deux à la fois. On construit ainsi des « mots » de longueur
  infinie. Le mot correspondant à la figure \ref{fig_olymp_19Bd}
  débute par : $CHVHHVHVHH$.

  \begin{figure}[!h]
    \centering
    \begin{tikzpicture}
      \path [clip] (-1,-1) rectangle (4.5,7.5) ;
      \draw[lightgray] (0,0) grid (4,7) ;
      \foreach \x in {1,2,...,4}
      {
        \node[below] at (\x,0) {$\x$} ;
      }
      \foreach \y in {1,2,...,6}
      {
        \node[left] at (0,\y) {$\y$} ;
      }
      \draw[->,>=stealth] (0,0) -- (4,0) ;
      \draw[->,>=stealth] (0,0) -- (0,7) ;
      \node[below left] at (0,0) {$0$} ;
      \node[above left] at (0,0) {$M_0$} ;
      \draw[colLogo,line width=1pt,name path=d] (0,0) -- (4,6.2) ;
      \foreach \n/\l in {1/2,2/5,3/7}
      {
        \path[name path=\n] (\n,0) -- (\n,6) ;
        \fill [name intersections={of=d and \n,by=x}] (x) circle (1pt)
        node[right] {$M_\l$};
      }
      \foreach \n/\l in {1/1,2/3,3/4,4/6,5/8,6/9}
      {
        \path[name path=\n] (0,\n) -- (4,\n) ;
        \fill [name intersections={of=d and \n,by=x}] (x) circle (1pt)
        node[left] {$M_\l$};
      }

    \end{tikzpicture}
    \caption{Exemple}
    \label{fig_olymp_19Bd}
  \end{figure}
  \begin{enumerate}
  \item  Représenter, dans un repère orthonormé d’unité $\SI{1}{\centi\m}$, la
    demi-droite d’équation $y=1,5x$ et donner les huit premières
    lettres du mot qu’on peut lui associer.
  \item Pourquoi les mots associés aux demi-droites commencent-ils
    tous par la lettre $C$ ?
  \end{enumerate}
  Un mot est dit « périodique » si une séquence se répète indéfiniment
  à partir de la première lettre. On appelle « motif » plus petite
  séquence qui sé répète indéfiniment et « période » le nombre de
  lettres du motif.

  Par exemple, $CVVCVVCVVCVV\dots$ est un mot périodique de période
  $3$ dont le motif est $CVV$.
  \begin{enumerate}[resume]
  \item Déterminer et représenter la demi-droite qui donne naissance
    au mot périodique de période $3$ de motif $CVV$.
  \item 
    \begin{enumerate}
    \item Montrer que si on rencontre la séquence $VV$ dans le
      mot associé à une demi-droite, alors la pente de cette
      demi-doite est strictement inférieure à $1$.

      \item Que peut-on dire de la pente d’une demi-droite si on
      rencontre la séquence $HH$ dans le mot associé ?

      \item Tous les mots associés à une demi-droite commencent par
      $C$. Tout mot commençant par $C$ est-il associé à une
      demi-droite ?
    \end{enumerate}
  \item On suppose dans cette question que le point $M_6$ est associé
    à la septième lettre d’un mot de période $6$.  Quelles sont les
    droites pouvant conduire à ce résultat ? Quels mots leur sont
    associés ?

  \item Énoncer et prouver une condition nécessaire et suffisante
    portant sur la pente d’une demi-droite pour que le mot qui lui est
    associé soit périodique.

  \item On donne un entier naturel $p$ non nul. Est-il possible de
    trouver un mot périodique de période $p$ ?

  \item Soit $W$ le mot correspondant à la demi-droite de pente
    $\sqrt 2$. Pour tout entier naturel $n$, le point d’intersection
    de la demi-droite avec la droite d’équation $x=n$ a pour lettre
    associée $V$ : pourquoi ?  On appelle $F_W(n)$ le
    nombre de lettres $H$ précédant $V$ dans l’écriture de $W$. La
    suite de terme général $\frac{F_W(n)}{n}$ possède-t-elle
    une limite ?
  \end{enumerate}
  \begin{Corrige}
    \begin{enumerate}
    \item On obtient la figure \ref{fig_olymp_19BdC}.
      \begin{figure}[!h]
        \centering
        \begin{tikzpicture}
          \path [clip] (-1,-1) rectangle (4.5,7.5) ;
          \draw[lightgray] (0,0) grid (4,7) ;
          \foreach \x in {1,2,...,4}
          {
            \node[below] at (\x,0) {$\x$} ;
          } \foreach \y in {1,2,...,6}
          {
            \node[left] at (0,\y) {$\y$} ;
          }
          \draw[->,>=stealth] (0,0) -- (4,0) ;
          \draw[->,>=stealth] (0,0) -- (0,7) ;
          \node[below left] at (0,0) {$0$} ;
          \node[above left] at (0,0) {$M_0$} ;
          \draw[colLogo,line width=1pt,name path=d] (0,0) -- (4,6) ;
          \foreach \n/\l in {1/2,2/4,3/6}
          {
            \path[name path=\n] (\n,0)  -- (\n,6) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }
          \foreach \n/\l in {1/1,2/3,4/5,6/7}
          {
            \path[name path=\n] (0,\n) -- (4,\n) ;
            \fill [name intersections={of=d and \n,by=x}] (x)
            circle (1pt) node[left] {$M_\l$};
          }
         \end{tikzpicture}
         \caption{Motif $CHVHCHVH$}
         \label{fig_olymp_19BdC}
       \end{figure}
       Le mot obtenu est $CHVHCHVH$.
     \item Les mots associés commencent tous par $C$ car la droite
       passe par le point de coordonnées $ \coord(0,0)$.
     \item Le mot passe par les points de coordonnées $\coord(0,0)$ et
       $(3,1)$. C'est donc la droite d'équation $y = \frac{1}{3}x$,
       voir la figure \ref{fig_olymp_19BeC}.
       \begin{figure}[!h]
         \centering
         \begin{tikzpicture}
           \path [clip] (-1,-1) rectangle (4.5,2.5) ;
           \draw[lightgray] (0,0) grid (4,2) ;
           \foreach \x in {1,2,...,4}
           {
             \node[below] at (\x,0) {$\x$} ;
           }
           \foreach \y in {1,2}
           {
             \node[left] at (0,\y) {$\y$} ;
           }
           \draw[->,>=stealth] (0,0) -- (4,0) ;
           \draw[->,>=stealth] (0,0) -- (0,2) ;
           \node[below left] at (0,0) {$0$} ;
           \draw[colLogo,line width=1pt,name path=d] (0,0) -- (6,3) ;           
         \end{tikzpicture}
         \caption{Droite d'équation $y = \frac{1}{3}x$.}
         \label{fig_olymp_19BeC}
       \end{figure}
     \item
       \begin{enumerate}
       \item Si on rencontre le motif $VV$, alors la droite d'équation
         $y = ax$ coupe deux verticales consécutives $x =n$ et
         $x = n+1$, entre deux horizontales consécutives d'équation
         $y = p$ et $y = p+1$. La pente est donc forcément inférieure
         $ p+1 -p = 1$.
       \item Situation symétrique par rapport à la droite d'équation
         $y =x$. Donc la pente est supérie à $1$.
       \item Le mot qui commence par $CVVC$ détermine la droite
         d'équation $y \frac{1}{3}x$. Donc, il ne peut pas y avoir de
         droite qui commence par $CVVCH$.
       \end{enumerate}
     \item Commme le mot est de période $6$, le point $M_6$ a des
       coordonnées entières.
       \begin{itemize}
       \item Le point est en $\coord(6,0)$, impossible le motif est
         alors de période $1$ ;
       \item le point est en $\coord(6,1)$, ce la correspond au motif
         $CVVVVV$. 
       \item le point est en $\coord(6,2)$, impossible, motif
         périodique de période $3$.
       \item  le point est en $\coord(5,2)$, le motif est alors
         $CVVHVV$. 
       \item  le point est en $\coord(4,3)$, le motif est alors
         $CVHCHV$. 
       \item  le point est en $\coord(3,4)$, le motif est alors
         $CHVHVH$. 
       \item  le point est en $\coord(2,5)$, le motif est alors
         $CHHVHHV$. 
       \item  le point est en $\coord(1,6)$, le motif est alors
         $CHHHHHH$.          
       \end{itemize}

       \begin{figure}[!h]
        \centering
        \begin{tikzpicture}
          \draw[lightgray] (0,0) grid (7,7) ;
          \foreach \x in {1,2,...,7}
          {
            \node[below] at (\x,0) {$\x$} ;
          } \foreach \y in {1,2,...,7}
          {
            \node[left] at (0,\y) {$\y$} ;
          }
          \draw[->,>=stealth] (0,0) -- (7,0) ;
          \draw[->,>=stealth] (0,0) -- (0,7) ;
          \node[below left] at (0,0) {$0$} ;
          \node[above left] at (0,0) {$M_0$} ;
          \path [clip] (0,0) rectangle (7,7) ;
          \draw[colLogo,line width=1pt,name path=d] (0,0) -- (7,1.66667) ;
          \foreach \n/\l in {1/1,2/2,3/3,4/4,5/5,6/6}
          {
            \path[name path=\n] (\n,0)  -- (\n,6) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }
          \draw[colLogo,line width=1pt,name path=d] (0,0) -- (7,2.8) ;
          \foreach \n/\l in {1/3,2/6}
          {
            \path[name path=\n] (0,\n)  -- (6,\n) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }
          \foreach \n/\l in {1/1,2/2,3/4,4/5}
          {
            \path[name path=\n] (\n,0)  -- (\n,7) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }
          \draw[colLogo,line width=1pt,name path=d] (0,0) -- (7,5.25) ;
          \foreach \n/\l in {1/2,2/4}
          {
            \path[name path=\n] (0,\n)  -- (6,\n) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }
          \foreach \n/\l in {1/1,2/3,3/5,4/6}
          {
            \path[name path=\n] (\n,0)  -- (\n,7) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }
          \draw[colLogo,line width=1pt,name path=d] (0,0) -- (7,9.3333) ;
          \foreach \n/\l in {1/1,2/3,3/5}
          {
            \path[name path=\n] (0,\n)  -- (6,\n) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }
          \foreach \n/\l in {1/2,2/4,3/6}
          {
            \path[name path=\n] (\n,0)  -- (\n,7) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }

          \draw[colLogo,line width=1pt,name path=d] (0,0) -- (7,17.5) ;
          \foreach \n/\l in {1/1,2/2,3/4,4/5}
          {
            \path[name path=\n] (0,\n)  -- (6,\n) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }
          \foreach \n/\l in {1/3,2/6}
          {
            \path[name path=\n] (\n,0)  -- (\n,7) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }
          \draw[colLogo,line width=1pt,name path=d] (0,0) -- (1.66667,7) ;
          \foreach \n/\l in {1/1,2/2,3/3,4/4,5/5,6/6}
          {
            \path[name path=\n] (0,\n)  -- (6,\n) ;
            \fill [name intersections={of=d and \n,by=x}]
            (x) circle (1pt) node[right] {$M_\l$};
          }
         \end{tikzpicture}
         \caption{Motif $CHVHCHVH$}
         \label{fig_olymp_19BfC}
       \end{figure}
     \item Pour que le mot associé soit périodique, il faut qu'il
       passe par un point de coordonnées entières. Soit $
       \coord(a,b)$. La droite a donc pour équation $ y =
       \frac{b}{a}x$.

       Réciproquement, une droite a pour équation $y = \frac{b}{a}x$
       avec $a$ et $b$ des entiers, alors la droite passe au point de
       coordonnées $\coord(a,b)$ et donc le motif est périodique.
     \item La droite d'équation $y = \frac{1}{p}x$ donne un motif
       périodique de période $p$.
     \item Les points sur la droite $y = \sqrt{2}x$ sont soit des
       $V$ soit des $C$. Mais comme $\sqrt{2}$ n'est pas rationnel,
       la droite ne passe jamais par un point à coordonnées
       entières. C'est donc forcément des $V$.

       Les points associés à $H$ sont les points $\coord(x,y)$ pour
       lesquels il existe un entier $m$ tel que $y = m$, $x =
       \frac{m}{\sqrt{2}}$ et $\frac{m}{\sqrt{2}} < n$. Leur nombre
       est égal au plus grand entier inférieur à $n \sqrt{2}$. Cela
       donne $n \sqrt{2}-1 \leq F_W(n) \leq n \sqrt{2}$. Donc la
       limite de $\frac{F_W(n)}{n}$ existe et vaut $\sqrt{2}$.
     \end{enumerate}
  \end{Corrige}
\end{Enonce}

\begin{Enonce}[Additionnons des points]{}%
  \def\courbe{\begin{scope}[draw=colLogo,line width=1pt]
        \draw (0,0.0)--(0.02,0.0594)--(0.04,0.0865)--(0.06,0.1089)--
        (0.08,0.1292)--(0.1,0.1483)--(0.1200,0.1666)--(0.14,0.1845)--
        (0.16,0.2020)--(0.18,0.2194)--(0.1999,0.2366)--(0.2199,0.2538)--
        (0.2399,0.2709)--(0.2599,0.2880)--(0.2799,0.3052)--(0.3,0.3224)--
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\end{scope}
}%
  \Partie{Somme de carrés}
Un nombre entier $n$ étant donné, on cherche dans cette partie à
estimer
\[
S(n) = 1^2+2^2+3^2+ \cdots + (n-1)^2+ n^2.
\]

Sur la la figure \ref{fig_olymp_19Ba}, on a représenté une pyramide de
boulets de pierre, à base carrée, avec $6$ étages qui contiennent
$6^2$, $5^2$,\ldots, $1$ boulets. Le nombre de boulets qui la
composent est $S(6) = 91$.

\begin{figure}[!h]
  \centering
  \begin{asy} % version asymptote.
import graph ; 
import math ; 
import geometry ;
usepackage("calrsfs") ;  
import interpolate ;
import animate ;
import three ;
unitsize(0.75cm,0.75cm) ; 

pen colLogo = rgb(51/256, 183/256, 100/256) ;
currentprojection=perspective(
camera=(11.9965924074638,4.63653988312078,2.00503580390163),
up=(-0.00338670140379748,0.000626475987205595,0.035109145972391),
target=(2.72080636029237,2.22712739642788,1.15326786596917),
zoom=1,
angle=35.6670390835825,
autoadjust=false);


real x = 0 ;
real y = 0 ;
real z = 0 ;
real R = 0.5 ;

for(int k = 6 ; k > 0 ; k = k -1)
  {
  write("Plaque ", 6-k+1) ;
  for(int  j = 0 ; j < k ; ++j)
     {
     write("ligne ", j+1) ;
     for(int i = 0 ; i < k ; ++i)
       {
          x = i + (6-k) * R ;
          y = j + (6-k) * R ;
          z = (6-k)*sqrt(2)/2 ;
string mot = "("+string(x)+";"+string(y)+";"+string(z)+")"+" -> ("+string(x-0.375*y)+";"+string(z+0.375*y)+")" ;
          write(mot) ;
          draw(shift((x,y,z))*scale3(R)*unitsphere,colLogo) ;
      }
   }
}

\end{asy} 

% \begin{tikzpicture}[y=0.80pt, x=0.80pt, yscale=-1.000000,
%   xscale=1.000000, inner sep=0pt, outer sep=0pt]
%   % version réalisée à partir du code asymptote, puis avec inkscape
%   % exporté vers tikz.
%   % Une version native ne fonctionne pas, support de la 3d vraiment
%   % minimaliste en TiKz, de plus avec un rendu correct, l'affichage
%   % est très long.
%   \definecolor{cfbfafa}{RGB}{251,250,250}
% \definecolor{cf15858}{RGB}{241,88,88}
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% \definecolor{caf0505}{RGB}{175,5,5}
% \definecolor{c8f0101}{RGB}{143,1,1}
% \definecolor{c6f0000}{RGB}{111,0,0}
% \definecolor{c500001}{RGB}{80,0,1}
% \definecolor{c1d0202}{RGB}{29,2,2}

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% \end{scope}

% \end{tikzpicture}

  \caption{Pyramide de boulets}
  \label{fig_olymp_19Ba}
\end{figure}

\begin{enumerate}
\item Dans le cas $n=5$, le puzzle présenté à la figure \ref{fig_olymp_19Bb} permet d’estimer
  $S(5)$. Comment ? Quelle valeur obtient-on par cette méthode ?
  \begin{figure}[!h]
    \centering
    \begin{tikzpicture}[scale=0.5]
    \draw[lightgray](-1,-1) grid (16,12) ;
    \draw[colLogoFort,line width=1pt] (0,0) rectangle (5,5) ;
    \draw[colLogoFort,line width=1pt,shift={(0,6)}] (0,0) rectangle (5,5) node at (2.5,2.5) {$5^2$} ;
    \draw[colLogoFort,line width=1pt,shift={(5,0)}] (0,0) rectangle (4,4) ;
    \draw[colLogoFort,line width=1pt,shift={(5,7)}] (0,0) rectangle (4,4) node at (2,2) {$4^2$};
    \draw[colLogoFort,line width=1pt,shift={(9,0)}] (0,0) rectangle (3,3) ;
    \draw[colLogoFort,line width=1pt,shift={(9,8)}] (0,0) rectangle (3,3) node at (1.5,1.5) {$3^2$};
    \draw[colLogoFort,line width=1pt,shift={(12,0)}] (0,0) rectangle (2,2) ;
    \draw[colLogoFort,line width=1pt,shift={(12,9)}] (0,0) rectangle (2,2) node at (1,1) {$2^2$};
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    \draw[colLogoFort,line width=1pt,shift={(14,10)}] (0,0) rectangle (1,1) node at (.5,.5) {$1^2$};
    \draw[colLogoFort,line width=1pt,shift={(0,5)}] (0,0) rectangle (5,1) node at (2.5,0.5) {$5$} ;
    \draw[colLogoFort,line width=1pt,shift={(10,3)},rotate=90] (0,0) rectangle (5,1) node at
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    \draw[colLogoFort,line width=1pt,shift={(12,3)},rotate=90] (0,0) rectangle (5,1) node at
    (2.5,0.5) {$5$} ;
    \draw[colLogoFort,line width=1pt,shift={(15,5)},rotate=90] (0,0) rectangle (5,1) node at
    (2.5,0.5) {$5$} ;
    \draw[colLogoFort,line width=1pt,shift={(5,6)},rotate=0] (0,0) rectangle (4,1) node at
    (2,0.5) {$4$} ;
    \draw[colLogoFort,line width=1pt,shift={(13,5)},rotate=90] (0,0) rectangle (4,1) node at
    (2,0.5) {$4$} ;
    \draw[colLogoFort,line width=1pt,shift={(14,5)},rotate=90] (0,0) rectangle (4,1) node at
    (2,0.5) {$4$} ;
    \draw[colLogoFort,line width=1pt,shift={(15,1)},rotate=90] (0,0) rectangle (4,1) node at
    (2,0.5) {$4$} ;
    \draw[colLogoFort,line width=1pt,shift={(6,5)},rotate=0] (0,0) rectangle (3,1) node at
    (1.5,0.5) {$3$} ;
    \draw[colLogoFort,line width=1pt,shift={(13,2)},rotate=90] (0,0) rectangle (3,1) node at
    (1.5,0.5) {$3$} ;
    \draw[colLogoFort,line width=1pt,shift={(14,2)},rotate=90] (0,0) rectangle (3,1) node at
    (1.5,0.5) {$3$} ;
    \draw[colLogoFort,line width=1pt,shift={(6,4)},rotate=0] (0,0) rectangle (2,1) node at
    (1,0.5) {$2$} ;
    \draw[colLogoFort,line width=1pt,shift={(6,4)},rotate=90] (0,0) rectangle (2,1) node at
    (1,0.5) {$2$} ;
    \draw[colLogoFort,line width=1pt,shift={(8,4)},rotate=0] (0,0) rectangle (1,1) node at
    (0.5,0.5) {$1$} ;
  \end{tikzpicture}
    \caption{Puzzle}
    \label{fig_olymp_19Bb}
  \end{figure}
\item Les exemples précédents inspirent la formule
  \[
    S(n) = \frac{1}{6}n(n+1)(2n+1).
  \]
  \begin{enumerate}
  \item Montrer que l’entier $n(n+1)(2n+1)$ est bien un multiple de $2$ et de $3$.
  \item Si on suppose que pour un certain $n$, sur lequel on ne fait
    aucune autre hypothèse, $S(n)=  \frac{1}{6}n(n+1)(2n+1)$, quelle
    formule obtient-on pour $S(n)+ (n+1)^2$. On peut 
    donc décider que la formule de $S(n)$ est vraie pour tout entier $n$.
  \end{enumerate}
\item Montrer que $S(24)$, somme des $24$ premiers carrés, est un carré.
\end{enumerate}

\Partie{Additionner sur une courbe}
On considère l’ensemble $(E)$ des points du plan dont les coordonnées
$\coord(x,y)$ satisfont la relation :
\[
y^2 = \frac{1}{6}x(x+1)(2x+1).
\]
\begin{enumerate}[resume]
\item Un peu d’exploration
  \begin{enumerate}
  \item Les points de coordonnées $\coord(1,1)$ et $\coord(24,70)$ appartiennent-ils à cet
    ensemble ?  Fournir deux autres exemples.  
  \item Les points de coordonnées $\coord(-2,1)$ et
    $\coord(-\frac{1}{4},3)$ appartiennent-ils à $(E)$ ?
  \item Quelles relations
    doit vérifier le réel $x$ pour qu’il y ait un point d’abscisse dans
    l’ensemble (E) ?
    Dans ce cas, combien y a-t-il de points $x$ d’abscisse dans $(E)$ ?
  \end{enumerate}
\item  La forme de la courbe
  \begin{figure}[!p]
    \centering
 
    \begin{tikzpicture}[yscale=2]
      \draw [clip] (-3,-2) rectangle (3,2) ;
      \draw [lightgray] (-3,-2) grid[xstep=1,ystep=0.5] (3,2) ;
      \foreach \x in {-2,-1,...,2}
      {
        \node[below] at (\x,0) {\x} ;
      }
      \foreach \y in {-1.5,-1,...,1.5}
      {
        \node[left] at (0,\y) {\y} ;
      }
      \draw[black,->,>=stealth,line width=0.8pt] (-3,0) -- (3,0) ;
      \draw[black,->,>=stealth,line width=0.8pt] (0,-2) -- (0,2) ;
      \node[label=above right:$A$,circle,fill=colLogo,inner sep=1pt] at (1,-1) {} ;
      \node[label=above left:$B$,circle,fill=colLogo,inner sep=1pt] at (-0.9,0.1095) {} ;
      % Les données sont calculées à l'éxtérieur de LaTeX
      \courbe
        \end{tikzpicture}\hfill
      \begin{tikzpicture}[yscale=2]
      \draw [clip] (-3,-2) rectangle (3,2) ;
      \draw [lightgray] (-3,-2) grid[xstep=1,ystep=0.5] (3,2) ;
      \foreach \x in {-2,-1,...,2}
      {
        \node[below] at (\x,0) {\x} ;
      }
      \foreach \y in {-1.5,-1,...,1.5}
      {
        \node[left] at (0,\y) {\y} ;
      }
      \draw[black,->,>=stealth,line width=0.8pt] (-3,0) -- (3,0) ;
      \draw[black,->,>=stealth,line width=0.8pt] (0,-2) -- (0,2) ;
      \node[label=above left:$A$,circle,fill=colLogo,inner sep=1pt] at (-0.9,0.1095) {} ;
      % Les données sont calculées à l'éxtérieur de LaTeX
       \courbe
    \end{tikzpicture}

    \vspace*{1ex}
    \begin{tikzpicture}[yscale=2]
      \draw [clip] (-3,-2) rectangle (3,2) ;
      \draw [lightgray] (-3,-2) grid[xstep=1,ystep=0.5] (3,2) ;
      \foreach \x in {-2,-1,...,2}
      {
        \node[below] at (\x,0) {\x} ;
      }
      \foreach \y in {-1.5,-1,...,1.5}
      {
        \node[left] at (0,\y) {\y} ;
      }
      \draw[black,->,>=stealth,line width=0.8pt] (-3,0) -- (3,0) ;
      \draw[black,->,>=stealth,line width=0.8pt] (0,-2) -- (0,2) ;
      \node[label=above left:$A$,circle,fill=colLogo,inner sep=1pt] at (-0.9,0.1095) {} ;
      % Les données sont calculées à l'éxtérieur de LaTeX
      \courbe
    \end{tikzpicture}

    % version asymptote
    % \begin{asy}
% import graph ; 
% import math ; 
% import geometry ;
% usepackage("calrsfs") ;  
% import interpolate ;
% settings.outformat="pdf" ;
% import graph ; 
% import math ; 
% import geometry ;
% usepackage("calrsfs") ;  
% import interpolate ;
% import animate ;


% picture A ;
% unitsize(A,1cm,2cm) ; 

% real xmin =-3 , xmax =3  ;
% real ymin =-2 , ymax =2  ; 
% real xStep =1, xsubStep = xStep/1 ;
% real yStep =0.5, ysubStep = yStep/1 ;

% pen colLogo=rgb( 51/256, 183/256, 100/256) ;

% // Si il y a un graphe de fonction, le mettre avant cette ligne
% // Tracer de la courbe representant la fonction f, avec
% // une couleur mixant, 75% de bleu pur et 25% de noir.
% real f(real x){return sqrt(x*(x+1)*(2*x+1)/6);}
% real g(real x){return -f(x);}
% // draw doit être appelé avant limits avec Crop.
% real e = 0 ;
% // pourquoi il ne prend pas le f dans le graph ?
% path Cf1 = graph(f,0,xmax,n=250) ;
% path Cf2 = graph(g,0,xmax,n=250) ;
% path Cf3 = graph(f,-1,-0.5,n=250) ;
% path Cf4 = graph(g,-1,-0.5,n=250) ;
% draw(A,Cf1^^Cf2^^Cf3^^Cf4,linewidth(1bp)+colLogo) ;

% limits(A,(xmin,ymin),(xmax,ymax),Crop) ;

% point pA=(1,g(1)) ;
% point pB=(-0.9,f(-0.9)) ;

% label(A,"$ A $",pA,NE) ;
% label(A,"$ B $",pB,NW) ;
% dot(A,pA--pB,3bp+colLogo) ;
% pen styloOli = 0.5bp+grey ;

% xaxis(A,BottomTop,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,//
% 		  Step=xStep,//
% 		  step=xsubStep,//
% 		  pTick=styloOli,
% 		  ptick=styloOli,
% 		  Size=2mm,
% 		  size=0mm)) ;
% yaxis(A,LeftRight,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,
% 		  Step=yStep,
% 		  step=ysubStep,
% 		  pTick=styloOli,
% 		  ptick=styloOli,Size=2mm,
% 		  size=0mm)); 

% xequals(A,L="",//
% 	0,//
% 	ymin,//
% 	ymax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=yStep,step=ysubStep)
% 	) ;
% yequals(A,L="",//
% 	0,//
% 	xmin,//
% 	xmax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=xStep,step=xsubStep,Size=2mm,size=0mm)
% 	);

% // label(A,"$x$",(xmax,0),E,black) ;
% // label(A,"$y$",(0,ymax),N,black) ;

% picture B ;
% unitsize(B,1cm,2cm) ; 
% draw(B,Cf1^^Cf2^^Cf3^^Cf4,linewidth(1bp)+colLogo) ;

% limits(B,(xmin,ymin),(xmax,ymax),Crop) ;


% point pA=(-0.9,f(-0.9)) ;

% label(B,"$ B $",pB,NW) ;
% dot(B,pB,3bp+colLogo) ;
% pen styloOli = 0.5bp+grey ;

% xaxis(B,BottomTop,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,//
% 		  Step=xStep,//
% 		  step=xsubStep,//
% 		  pTick=styloOli,
% 		  ptick=styloOli,
% 		  Size=2mm,
% 		  size=0mm)) ;
% yaxis(B,LeftRight,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,
% 		  Step=yStep,
% 		  step=ysubStep,
% 		  pTick=styloOli,
% 		  ptick=styloOli,Size=2mm,
% 		  size=0mm)); 

% xequals(B,L="",//
% 	0,//
% 	ymin,//
% 	ymax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=yStep,step=ysubStep)
% 	) ;
% yequals(B,L="",//
% 	0,//
% 	xmin,//
% 	xmax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=xStep,step=xsubStep,Size=2mm,size=0mm)
% 	);

% // label(B,"$x$",(xmax,0),E,black) ;
% // label(B,"$y$",(0,ymax),N,black) ;

% picture B ;
% unitsize(B,1cm,2cm) ; 
% draw(B,Cf1^^Cf2^^Cf3^^Cf4,linewidth(1bp)+colLogo) ;

% limits(B,(xmin,ymin),(xmax,ymax),Crop) ;


% point pA=(-0.9,f(-0.9)) ;

% label(B,"$ B $",pB,NW) ;
% dot(B,pB,3bp+colLogo) ;
% pen styloOli = 0.5bp+grey ;

% xaxis(B,BottomTop,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,//
% 		  Step=xStep,//
% 		  step=xsubStep,//
% 		  pTick=styloOli,
% 		  ptick=styloOli,
% 		  Size=2mm,
% 		  size=0mm)) ;
% yaxis(B,LeftRight,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,
% 		  Step=yStep,
% 		  step=ysubStep,
% 		  pTick=styloOli,
% 		  ptick=styloOli,Size=2mm,
% 		  size=0mm)); 

% xequals(B,L="",//
% 	0,//
% 	ymin,//
% 	ymax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=yStep,step=ysubStep)
% 	) ;
% yequals(B,L="",//
% 	0,//
% 	xmin,//
% 	xmax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=xStep,step=xsubStep,Size=2mm,size=0mm)
% 	);

% // label(B,"$x$",(xmax,0),E,black) ;
% // label(B,"$y$",(0,ymax),N,black) ;
% picture C ;
% unitsize(C,1cm,2cm) ; 
% draw(C,Cf1^^Cf2^^Cf3^^Cf4,linewidth(1bp)+colLogo) ;

% limits(C,(xmin,ymin),(xmax,ymax),Crop) ;


% point pA=(-0.9,f(-0.9)) ;

% label(C,"$ B $",pB,NW) ;
% dot(C,pB,3bp+colLogo) ;
% pen styloOli = 0.5bp+grey ;

% xaxis(C,BottomTop,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,//
% 		  Step=xStep,//
% 		  step=xsubStep,//
% 		  pTick=styloOli,
% 		  ptick=styloOli,
% 		  Size=2mm,
% 		  size=0mm)) ;
% yaxis(C,LeftRight,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,
% 		  Step=yStep,
% 		  step=ysubStep,
% 		  pTick=styloOli,
% 		  ptick=styloOli,Size=2mm,
% 		  size=0mm)); 

% xequals(C,L="",//
% 	0,//
% 	ymin,//
% 	ymax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=yStep,step=ysubStep)
% 	) ;
% yequals(C,L="",//
% 	0,//
% 	xmin,//
% 	xmax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=xStep,step=xsubStep,Size=2mm,size=0mm)
% 	);

% // label(C,"$x$",(xmax,0),E,black) ;
% // label(C,"$y$",(0,ymax),N,black) ;



% add(A.fit(),(0,0),NW) ;
% add(B.fit(),(0,0),NE) ;
% add(C.fit(),(0,0),S) ;
% \end{asy}      
    \caption{Annexe de l'exerice Additionnons des points }
    \label{fig_olymp_19Bc}
  \end{figure}
  \begin{enumerate}
  \item Sur l’annexe qui servira aux tracés, on a représenté
    l’ensemble $(E)$ qu’on appellera dorénavant courbe $(E)$. On peut
    identifier des « branches infinies ».

    Comment expliquer que le
    quotient $ \frac{3y^2}{x^3}$ s’approche de $1$ lorsque l’abscisse $x$ du point de
    coordonnées $\coord(x,y)$ de la courbe devient très grande ? On a
    alors $y \approx \frac{x\sqrt(x)}{3}$..
  \item Expliquer pourquoi la courbe (E) présente une partie
    « fermée ».
  \end{enumerate}
\item Somme de deux points
  \begin{enumerate}
  \item Sur la feuille annexe, à rendre avec la copie, on a représenté
    la courbe $(E)$. Pour tout couple $\coord(A, B)$ de points
    distincts de la courbe $(E)$, on trace la droite $(AB)$.
    Quand elle recoupe la courbe en un troisième point $D$, on note $A
    \oplus B$ le symétrique de $D$ par rapport à l’axe des abscisses.
    Représenter le point $A \oplus B$ sur la figure 1. 
  \item Pour tout point $A$ de la courbe $(E)$, on trace la
    tangente en $A$ à la courbe $(E)$. Quand elle recoupe la
    courbe $(E)$ en un point $D$, on note $A \oplus B$ le symétrique
    de $D$ par rapport à l’axe des abscisses. Représenter le point  $A
   \oplus A$ sur la figure 2.
 \item On note $2A = A \oplus A$, $3A = A \oplus 2A$ ,
   etc. Représenter le point $3A$ sur la figure 3.  
  \end{enumerate}
\item Un système de codage
La correspondance entre Alice et Bruno est protégée par une clef : les
quatre premières décimales de l’abscisse du point $abM$, obtenu de la
manière suivante : 
\begin{itemize}
\item Ils choisissent ensemble un point $M$ sur la courbe $(E)$ ;
\item Alice choisit un entier $a$ et ne donne à Bruno que les coordonnées
  du point $aM$ ; 
\item Bruno choisit un entier $b$ et ne donne à Alice que les
  coordonnées du point $bM$ ;
\end{itemize}
Dans ce processus, ils choisissent le point $M$ de coordonnées
$\coord(1, 1)$. Alice donne à Bruno les coordonnées
$\coord(0.02083333,0.06076389)$. Bruno donne à Alice les coordonnées
$\coord(0.02908309,-0.07265188)$. En s’aidant du tableau
\ref{tab_olymp_19Ba}, indiquer quelle est la clef.

\begin{table}[!h]
  \centering
  \begin{tabular}{*{3}{>{$}c<{$}}}
    \toprule
    k &  \multicolumn{1}{m{20ex}}{\centering Abscisse du point $M$} 
    & \multicolumn{1}{m{20ex}}{\centering Abscisse du point $kM$} \\
    \midrule
    1 & 1 & 1 \\
    2 & 0,02083333 & 0,06076389 \\
    3 & 5,63265731 & -8,73904883 \\
    4 & 1,06401114 & 1,07001139 \\
    5 & 0,02908309 & -0,07265188 \\
    6 & 3,93363878 & 5,35550512 \\
    7 & 0,651426 & -0,64256859 \\
    8 & 0,00115486 & 0,01389763 \\
    9 & 107,8640 & -651,271826 \\
    10 & 26,5927 & 81,40364156 \\
    \bottomrule
  \end{tabular}
  \caption{ }
  \label{tab_olymp_19Ba}
\end{table}
\end{enumerate}


\begin{Corrige}
\def\intervalA{\interval[scaled]{-1}{-\frac{1}{2}}}
\def\intervalB{\interval[open right]{0}{+\infty}}
   \def\courbe{\begin{scope}[draw=colLogo,line width=1pt]
        \draw (0,0.0)--(0.02,0.0594)--(0.04,0.0865)--(0.06,0.1089)--
        (0.08,0.1292)--(0.1,0.1483)--(0.1200,0.1666)--(0.14,0.1845)--
        (0.16,0.2020)--(0.18,0.2194)--(0.1999,0.2366)--(0.2199,0.2538)--
        (0.2399,0.2709)--(0.2599,0.2880)--(0.2799,0.3052)--(0.3,0.3224)--
        (0.32,0.3397)--(0.34,0.3571)--(0.3600,0.3746)--(0.3800,0.3922)--
        (0.4000,0.4098)--(0.4200,0.4276)--(0.4400,0.4455)--(0.4600,0.4635)--
        (0.4800,0.4817)--(0.5000,0.5)--(0.5200,0.5183)--(0.5400,0.5369)--
        (0.5600,0.5555)--(0.5800,0.5743)--(0.6000,0.5932)--(0.6200,0.6123)--
        (0.6400,0.6315)--(0.6600,0.6508)--(0.6800,0.6703)--(0.7000,0.6899)--
        (0.7200,0.7096)--(0.7400,0.7295)--(0.7600,0.7495)--(0.7800,0.7696)--
        (0.8000,0.7899)--(0.8200,0.8103)--(0.8400,0.8308)--(0.8600,0.8515)--
        (0.8800,0.8723)--(0.9000,0.8933)--(0.9200,0.9143)--(0.9400,0.9355)--
        (0.9600,0.9569)--(0.9800,0.9783)--(1.0000,1.0000)--(1.0200,1.0217)--
        (1.0400,1.0435)--(1.0600,1.0655)--(1.0800,1.0877)--(1.1000,1.1099)--
        (1.1200,1.1323)--(1.1400,1.1548)--(1.1600,1.1774)--(1.1800,1.2002)--
        (1.2000,1.2231)--(1.2200,1.2461)--(1.2400,1.2692)--(1.2600,1.2925)--
        (1.2800,1.3158)--(1.3000,1.3394)--(1.3200,1.3630)--(1.3400,1.3867)--
        (1.3600,1.4106)--(1.3800,1.4346)--(1.4000,1.4587)--(1.4200,1.4830)--
        (1.4400,1.5073)--(1.4600,1.5318)--(1.4800,1.5564)--(1.5000,1.5811)--
        (1.5200,1.6059)--(1.5400,1.6309)--(1.5600,1.6559)--(1.5800,1.6811)--
        (1.6000,1.7064)--(1.6200,1.7318)--(1.6400,1.7573)--(1.6600,1.7830)--
        (1.6800,1.8087)--(1.7000,1.8346)--(1.7200,1.8606)--(1.7400,1.8867)--
        (1.7600,1.9129)--(1.7800,1.9392)--(1.8000,1.9657)--(1.8200,1.9922)--
        (1.8400,2.0189)--(1.8600,2.0456)--(1.8800,2.0725)--(1.9000,2.0995)--
        (1.9200,2.1266)--(1.9400,2.1538)--(1.9600,2.1811)--(1.9800,2.2085);

        \draw[yscale=-1] (0,0.0)--(0.02,0.0594)--(0.04,0.0865)--(0.06,0.1089)--
         (0.08,0.1292)--(0.1,0.1483)--(0.1200,0.1666)--(0.14,0.1845)--
         (0.16,0.2020)--(0.18,0.2194)--(0.1999,0.2366)--(0.2199,0.2538)--
         (0.2399,0.2709)--(0.2599,0.2880)--(0.2799,0.3052)--(0.3,0.3224)--
         (0.32,0.3397)--(0.34,0.3571)--(0.3600,0.3746)--(0.3800,0.3922)--
         (0.4000,0.4098)--(0.4200,0.4276)--(0.4400,0.4455)--(0.4600,0.4635)--
         (0.4800,0.4817)--(0.5000,0.5)--(0.5200,0.5183)--(0.5400,0.5369)--
         (0.5600,0.5555)--(0.5800,0.5743)--(0.6000,0.5932)--(0.6200,0.6123)--
         (0.6400,0.6315)--(0.6600,0.6508)--(0.6800,0.6703)--(0.7000,0.6899)--
         (0.7200,0.7096)--(0.7400,0.7295)--(0.7600,0.7495)--(0.7800,0.7696)--
         (0.8000,0.7899)--(0.8200,0.8103)--(0.8400,0.8308)--(0.8600,0.8515)--
         (0.8800,0.8723)--(0.9000,0.8933)--(0.9200,0.9143)--(0.9400,0.9355)--
         (0.9600,0.9569)--(0.9800,0.9783)--(1.0000,1.0000)--(1.0200,1.0217)--
         (1.0400,1.0435)--(1.0600,1.0655)--(1.0800,1.0877)--(1.1000,1.1099)--
         (1.1200,1.1323)--(1.1400,1.1548)--(1.1600,1.1774)--(1.1800,1.2002)--
         (1.2000,1.2231)--(1.2200,1.2461)--(1.2400,1.2692)--(1.2600,1.2925)--
         (1.2800,1.3158)--(1.3000,1.3394)--(1.3200,1.3630)--(1.3400,1.3867)--
         (1.3600,1.4106)--(1.3800,1.4346)--(1.4000,1.4587)--(1.4200,1.4830)--
         (1.4400,1.5073)--(1.4600,1.5318)--(1.4800,1.5564)--(1.5000,1.5811)--
         (1.5200,1.6059)--(1.5400,1.6309)--(1.5600,1.6559)--(1.5800,1.6811)--
         (1.6000,1.7064)--(1.6200,1.7318)--(1.6400,1.7573)--(1.6600,1.7830)--
         (1.6800,1.8087)--(1.7000,1.8346)--(1.7200,1.8606)--(1.7400,1.8867)--
         (1.7600,1.9129)--(1.7800,1.9392)--(1.8000,1.9657)--(1.8200,1.9922)--
         (1.8400,2.0189)--(1.8600,2.0456)--(1.8800,2.0725)--(1.9000,2.0995)--
         (1.9200,2.1266)--(1.9400,2.1538)--(1.9600,2.1811)--(1.9800,2.2085);

         \draw(-1,0.0)--(-0.995,0.0286)--(-0.99,0.0402)--(-0.985,0.0488)--
         (-0.98,0.0560)--(-0.975,0.0621)--(-0.97,0.0675)--(-0.965,0.0723)--
         (-0.96,0.0767)--(-0.955,0.0807)--(-0.95,0.0844)--(-0.945,0.0878)--
         (-0.94,0.0909)--(-0.9349,0.0938)--(-0.9299,0.0965)--(-0.9249,0.0991)--
         (-0.9199,0.1015)--(-0.9149,0.1037)--(-0.9099,0.1057)--(-0.9049,0.1077)--
         (-0.8999,0.1095)--(-0.8949,0.1112)--(-0.8899,0.1128)--(-0.8849,0.1142)--
         (-0.8799,0.1156)--(-0.8749,0.1169)--(-0.8699,0.1181)--(-0.8649,0.1191)--
         (-0.8599,0.1201)--(-0.8549,0.1211)--(-0.8499,0.1219)--(-0.8449,0.1227)--
         (-0.8399,0.1234)--(-0.8349,0.1240)--(-0.8299,0.1245)--(-0.8249,0.1250)--
         (-0.8199,0.1254)--(-0.8149,0.1258)--(-0.8099,0.1261)--(-0.8049,0.1263)--
         (-0.7999,0.1264)--(-0.7949,0.1265)--(-0.7899,0.1266)--(-0.7849,0.1266)--
         (-0.7799,0.1265)--(-0.7749,0.1264)--(-0.7699,0.1262)--(-0.7649,0.1260)--
         (-0.7599,0.1257)--(-0.7549,0.1253)--(-0.7499,0.1249)--(-0.7449,0.1245)--
         (-0.7399,0.1240)--(-0.7349,0.1235)--(-0.7299,0.1229)--(-0.7249,0.1222)--
         (-0.7199,0.1215)--(-0.7149,0.1208)--(-0.7099,0.1200)--(-0.7049,0.1192)--
         (-0.6999,0.1183)--(-0.6949,0.1173)--(-0.6899,0.1163)--(-0.6849,0.1153)--
         (-0.6799,0.1142)--(-0.6749,0.1131)--(-0.6699,0.1119)--(-0.6649,0.1106)--
         (-0.6599,0.1093)--(-0.6549,0.1080)--(-0.6499,0.1066)--(-0.6449,0.1052)--
         (-0.6399,0.1036)--(-0.6349,0.1021)--(-0.6299,0.1005)--(-0.6249,0.0988)--
         (-0.6199,0.0970)--(-0.6149,0.0952)--(-0.6099,0.0933)--(-0.6049,0.0914)--
         (-0.5999,0.0894)--(-0.5949,0.0873)--(-0.5899,0.0851)--(-0.5849,0.0829)--
         (-0.5799,0.0805)--(-0.5749,0.0781)--(-0.5699,0.0756)--(-0.5649,0.0729)--
         (-0.5599,0.0701)--(-0.5549,0.0672)--(-0.5499,0.0642)--(-0.5449,0.0609)--
         (-0.5399,0.0575)--(-0.5349,0.0538)--(-0.5299,0.0499)--(-0.5249,0.0455)--
         (-0.5199,0.0407)--(-0.5149,0.0353)--(-0.5099,0.0288)--(-0.5049,0.0204)--
         (-0.5,0);
         \draw[yscale=-1](-1,0.0)--(-0.995,0.0286)--(-0.99,0.0402)--(-0.985,0.0488)--
         (-0.98,0.0560)--(-0.975,0.0621)--(-0.97,0.0675)--(-0.965,0.0723)--
         (-0.96,0.0767)--(-0.955,0.0807)--(-0.95,0.0844)--(-0.945,0.0878)--
         (-0.94,0.0909)--(-0.9349,0.0938)--(-0.9299,0.0965)--(-0.9249,0.0991)--
         (-0.9199,0.1015)--(-0.9149,0.1037)--(-0.9099,0.1057)--(-0.9049,0.1077)--
         (-0.8999,0.1095)--(-0.8949,0.1112)--(-0.8899,0.1128)--(-0.8849,0.1142)--
         (-0.8799,0.1156)--(-0.8749,0.1169)--(-0.8699,0.1181)--(-0.8649,0.1191)--
         (-0.8599,0.1201)--(-0.8549,0.1211)--(-0.8499,0.1219)--(-0.8449,0.1227)--
         (-0.8399,0.1234)--(-0.8349,0.1240)--(-0.8299,0.1245)--(-0.8249,0.1250)--
         (-0.8199,0.1254)--(-0.8149,0.1258)--(-0.8099,0.1261)--(-0.8049,0.1263)--
         (-0.7999,0.1264)--(-0.7949,0.1265)--(-0.7899,0.1266)--(-0.7849,0.1266)--
         (-0.7799,0.1265)--(-0.7749,0.1264)--(-0.7699,0.1262)--(-0.7649,0.1260)--
         (-0.7599,0.1257)--(-0.7549,0.1253)--(-0.7499,0.1249)--(-0.7449,0.1245)--
         (-0.7399,0.1240)--(-0.7349,0.1235)--(-0.7299,0.1229)--(-0.7249,0.1222)--
         (-0.7199,0.1215)--(-0.7149,0.1208)--(-0.7099,0.1200)--(-0.7049,0.1192)--
         (-0.6999,0.1183)--(-0.6949,0.1173)--(-0.6899,0.1163)--(-0.6849,0.1153)--
         (-0.6799,0.1142)--(-0.6749,0.1131)--(-0.6699,0.1119)--(-0.6649,0.1106)--
         (-0.6599,0.1093)--(-0.6549,0.1080)--(-0.6499,0.1066)--(-0.6449,0.1052)--
         (-0.6399,0.1036)--(-0.6349,0.1021)--(-0.6299,0.1005)--(-0.6249,0.0988)--
         (-0.6199,0.0970)--(-0.6149,0.0952)--(-0.6099,0.0933)--(-0.6049,0.0914)--
         (-0.5999,0.0894)--(-0.5949,0.0873)--(-0.5899,0.0851)--(-0.5849,0.0829)--
         (-0.5799,0.0805)--(-0.5749,0.0781)--(-0.5699,0.0756)--(-0.5649,0.0729)--
         (-0.5599,0.0701)--(-0.5549,0.0672)--(-0.5499,0.0642)--(-0.5449,0.0609)--
         (-0.5399,0.0575)--(-0.5349,0.0538)--(-0.5299,0.0499)--(-0.5249,0.0455)--
         (-0.5199,0.0407)--(-0.5149,0.0353)--(-0.5099,0.0288)--(-0.5049,0.0204)--
         (-0.5,0);
\end{scope}
}

  \Partie{Somme de carrés}
\begin{enumerate}
\item L'aire du rectangle est $(5+4+3+2+1) \times (2 \times 5
  +1)$. Mais c'est aussi la somme des différents rectangles du puzzle. 
  \begin{align*}
    (5+4+3+2+1) \times (2 \times 5 + 1) &= 2 \times 5^2+2 \times 4^2+2
                                          \times 3^2+2 \times 2^2+2
                                          \times 1^2 + 5
                                          \times 5 + 4 \times 4 + 3
                                          \times 3 + 2 \times 2  1 \\
                                        &= 3 \times 5^2+3 \times 4^2+2
                                          \times 3^2+3 \times 2^2+3
                                          \times 1^2 
  \end{align*}
  Soit :
  \[
    S(5) = \frac{1}{3} (5+4+3+2+1) \times (2 \times 5 +1) =
    \frac{1}{3} \times 15 \times 11 = 55 
  \]
\item
  \begin{enumerate}
  \item $N$ ou $n+1$ est un multiple de $2$. L'un des trois nombres
    $n$, $n+1$, ou $2n+1$ est un multiple de $3$. Si ni $n$, ni $n+1$
    ne sont des multiples de $3$, alors $n+2$ est un multiple de $3$,
    soit $n+2 = 3k$, donc $n = 3k -2$, et donc $2(3k-2) +1 = 6k - 3$,
    donc, $2k-1$ est multiple de $3$.
  \item On a :
    \begin{align*}
      S(n) + (n+1)^2 &= \frac{1}{6} n(n+1)(2n+1) + (n+1)^2\\
                     & = \frac{1}{6}(n+1) (2n^2 + n + 6n + 6) \\
    \end{align*}
    Or $(n+2)(2n+3) = 2n^2 + 7n +6$, on en déduit que
    \[
      S(n+1) = \frac{1}{6}(n+1)(n+2)(2n+3)
    \]
  \end{enumerate}
\item $S(24) = \frac{1}{6} \times 24 \times 25 \times 49 = 4 \times 25
  \times 49 =  \num{4900} = 70^2$.
\end{enumerate}

\Partie{Additionner sur une courbe}
\begin{enumerate}[resume]
\item
  \begin{enumerate}
  \item $S(1) = 1$, donc $ \coord(1,1)$ appartient à
    $(E)$. $\coord(24,70)$ appartient à $(E)$ d'après la question
    précédente. $0$ et $-1$ appartiennent à $(E)$. Pour $x = 2$, on
    $S(2) = 5$, donc $\left(2\,;\,\sqrt{5}\right)$ et
    $\left(2\,;\,-\sqrt{5}\right)$ appartien à $(E)$.
  \item $\coord(-2,1)$ n'appartient pas à $(E)$, car $S(-2) = -1$, et
    il n'y a pas de nombre $y$ donc le carré vaut $-1$.

    De même $S \left(-\frac{1}{4}\right)$ est négatif.
  \item L'étude du signe de $S(x)$ donne :
    
    \begin{center}
      \begin{tikzpicture}
        \tkzTabInit[espcl=1.5]{$x$ /1, $x$ /1, $x+1$ /1, $2x+1$ /1,
          $S(x)$ /2}
        {$-\infty$,$-1$,$-\frac{1}{2}$,$0$,$+\infty$} % Valeurs de x, séparés par des virgules
        \tkzTabLine{,-,t,-,t,-,z,+,} % signe, commence par une virgule et finis par une virgule,
        \tkzTabLine{,-,z,+,t,+,t,+,} \tkzTabLine{,-,t,-,z,+,t,+,}
        \tkzTabLine{,-,z,+,z,-,z,+,}
      \end{tikzpicture}
    \end{center}
    On en déduit que pour $x \in \intervalA \cup \intervalB$, les
    points $ \left(x\,;\,\sqrt{S(x)}\right)$ et
    $ \left(x\,;\,-\sqrt{S(x)}\right)$ sont éléments de $(E)$.
    
    Il y a deux points pour chaque valeur de $x$ éventuellement
    double.
  \end{enumerate}
\item
  \begin{enumerate}
  \item Quand $x$ est grand, $x+1 \approx x$ et $2x+1 \approx 2x$,
    donc $ \frac{3y^2}{x^3} \approx \frac{6x^3}{6x^3} = 1$.
  \item La courbe est formée de deux parties symétriques sur
    l'intervalle $\intervalA$. La partie supérieurs, est croissante
    puis décroissante, on en déduit que la courbe $(E)$ présente une
    partie fermée.
  \end{enumerate} 
\item Somme de deux points
  \begin{enumerate}
  \item Voir la figure \ref{fig_olymp_19BcC}.
  \item Voir la figure \ref{fig_olymp_19BcC}.
  \item Voir la figure \ref{fig_olymp_19BcC}.
  \end{enumerate}
  \begin{figure}[!p]
    \centering  
    \begin{tikzpicture}[yscale=4,xscale=2]
      \draw [clip] (-1.5,-1.25) rectangle (1.5,1.25) ;
      \draw [lightgray] (-3,-2) grid[xstep=0.5,ystep=0.25] (3,2) ;
      \foreach \x in {-2,-1,...,2}
      {
        \node[below] at (\x,0) {\x} ;
      }
      \foreach \y in {-1.5,-1,...,1.5}
      {
        \node[left] at (0,\y) {\y} ;
      }
      \draw[black,->,>=stealth,line width=0.8pt] (-3,0) -- (3,0) ;
      \draw[black,->,>=stealth,line width=0.8pt] (0,-2) -- (0,2) ;
      \node[label=above right:$A$,circle,name=A,fill=colLogo,inner sep=1.5pt] at (1,-1) {} ;
      \node[label=left:$B$,circle,name=B,fill=colLogo,inner sep=1.5pt] at (-0.9,0.1095) {} ;
      % Les données sont calculées à l'éxtérieur de LaTeX
      \draw[colLogoFort,line width=1pt,name path=d] (-3,1.3358) --
      (3,-2.1679) ;
      \node[label=below:$D$,circle,name=D,fill=colLogo,inner sep=1.5pt]
      at (-0.5769,-0.0791) {} ;
      \node[label=above:$A \oplus B$,circle,name=D1,fill=colLogo,inner
      sep=1.5pt] at (-0.5769,0.0791) {} ;
      \courbe          
    \end{tikzpicture}\hfill
    \begin{tikzpicture}[yscale=4,xscale=2]
      \draw [clip] (-1.5,-1.25) rectangle (1.5,1.25) ;
      \draw [lightgray] (-3,-2) grid[xstep=0.5,ystep=0.25] (3,2) ;
      \foreach \x in {-2,-1,...,2}
      {
        \node[below] at (\x,0) {\x} ;
      }
      \foreach \y in {-1.5,-1,...,1.5}
      {
        \node[left] at (0,\y) {\y} ;
      }
      \draw[black,->,>=stealth,line width=0.8pt] (-3,0) -- (3,0) ;
      \draw[black,->,>=stealth,line width=0.8pt] (0,-2) -- (0,2) ;
      \node[label=above left:$A$,circle,fill=colLogo,inner sep=1.5pt]
      at (-0.9,0.1095) {} ;
      \draw[colLogoFort,line width=1pt] (-1.5,-0.1004)--(1.5,0.9493)
      ;
      \node[label=above :$D$,circle,fill=colLogo,inner sep=1.5pt]
      at (0.6673,0.6580) {} ;
      \node[label=below :$A\oplus A$,circle,fill=colLogo,inner sep=1.5pt]
      at (0.6673,-0.6580) {} ;

      % Les données sont calculées à l'éxtérieur de LaTeX
      \courbe
    \end{tikzpicture}

   \vspace*{1ex}
    \begin{tikzpicture}[yscale=4,xscale=2]
      \draw [clip] (-1.5,-1.25) rectangle (1.5,1.25) ;
      \draw [lightgray] (-3,-2) grid[xstep=0.5,ystep=0.25] (3,2) ;
      \foreach \x in {-2,-1,...,2}
      {
        \node[below] at (\x,0) {\x} ;
      }
      \foreach \y in {-1.5,-1,...,1.5}
      {
        \node[left] at (0,\y) {\y} ;
      }
      \draw[black,->,>=stealth,line width=0.8pt] (-3,0) -- (3,0) ;
      \draw[black,->,>=stealth,line width=0.8pt] (0,-2) -- (0,2) ;
      \node[label=above left:$A$,circle,fill=colLogo,inner sep=1.5pt]
      at (-0.9,0.1095) {} ;
      \draw[colLogoFort,line width=1pt] (-1.5,-0.1004)--(1.5,0.9493)
      ;
      \node[label=above :$D$,circle,fill=colLogo,inner sep=1.5pt]
      at (0.6673,0.6580) {} ;
      \node[label=below :$A\oplus A$,circle,fill=colLogo,inner sep=1.5pt]
      at (0.6673,-0.6580) {} ;
      \draw[colLogoFort,line width=1pt] (-1.5,0.4034)--(1.5,-1.0658) ;
      \node[label=below :$3A$,circle,fill=colLogo,inner sep=1.5pt]
      at (-0.5479,-0.0629) {} ;

      % Les données sont calculées à l'éxtérieur de LaTeX
      \courbe
    \end{tikzpicture}
    %     version asymptote
    % \begin{asy}
% import graph ; 
% import math ; 
% import geometry ;
% usepackage("calrsfs") ;  
% import interpolate ;
% settings.outformat="pdf" ;
% import graph ; 
% import math ; 
% import geometry ;
% usepackage("calrsfs") ;  
% import interpolate ;
% import animate ;


% picture A ;
% unitsize(A,2cm,4cm) ; 

% real xmin =-1.5 , xmax =1.5  ;
% real ymin =-1.25 , ymax =1.25  ; 
% real xStep =1, xsubStep = xStep/2 ;
% real yStep =0.5, ysubStep = yStep/2 ;

% pen colLogo=rgb( 51/256, 183/256, 100/256) ;

% // Si il y a un graphe de fonction, le mettre avant cette ligne
% // Tracer de la courbe representant la fonction f, avec
% // une couleur mixant, 75% de bleu pur et 25% de noir.
% real f(real x){return sqrt(x*(x+1)*(2*x+1)/6);}
% real g(real x){return -f(x);}
% // draw doit être appelé avant limits avec Crop.
% real e = 0 ;
% // pourquoi il ne prend pas le f dans le graph ?
% path Cf1 = graph(f,0,xmax,n=250) ;
% path Cf2 = graph(g,0,xmax,n=250) ;
% path Cf3 = graph(f,-1,-0.5,n=250) ;
% path Cf4 = graph(g,-1,-0.5,n=250) ;
% draw(A,Cf1^^Cf2^^Cf3^^Cf4,linewidth(1bp)+colLogo) ;

% limits(A,(xmin,ymin),(xmax,ymax),Crop) ;

% point pA=(1,g(1)) ;
% point pB=(-0.9,f(-0.9)) ;

% label(A,"$ A $",pA,NE) ;
% label(A,"$ B $",pB,NW) ;
% dot(A,pA--pB,3bp+colLogo) ;
% line d = line(pA,pB) ;
% draw(A,d,1bp+colLogo) ;

% point[] pM = intersectionpoints(d,Cf4) ;

% point pAB = reflect(line((0,0),(1,0)))*pM[0] ;
% write(pM[0].x,pM[0].y) ;
% label(A,"$ D $",pM[0],S) ;
% dot(A,pAB,3bp+colLogo) ;
% label(A,"$ A \oplus B $",pAB,NE) ;
% dot(A,pAB,3bp+colLogo) ;

% pen styloOli = 0.5bp+grey ;

% xaxis(A,BottomTop,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,//
% 		  Step=xStep,//
% 		  step=xsubStep,//
% 		  pTick=styloOli,
% 		  ptick=styloOli,
% 		  Size=2mm,
% 		  size=0mm)) ;
% yaxis(A,LeftRight,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,
% 		  Step=yStep,
% 		  step=ysubStep,
% 		  pTick=styloOli,
% 		  ptick=styloOli,Size=2mm,
% 		  size=0mm)); 

% xequals(A,L="",//
% 	0,//
% 	ymin,//
% 	ymax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=yStep,step=ysubStep)
% 	) ;
% yequals(A,L="",//
% 	0,//
% 	xmin,//
% 	xmax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=xStep,step=xsubStep,Size=2mm,size=0mm)
% 	);

% // label(A,"$x$",(xmax,0),E,black) ;
% // label(A,"$y$",(0,ymax),N,black) ;

% picture B ;
% unitsize(B,2cm,4cm) ; 
% draw(B,Cf1^^Cf2^^Cf3^^Cf4,linewidth(1bp)+colLogo) ;

% limits(B,(xmin,ymin),(xmax,ymax),Crop) ;


% point pA=(-0.9,f(-0.9)) ;

% label(B,"$ B $",pB,NW) ;
% dot(B,pB,3bp+colLogo) ;
% pen styloOli = 0.5bp+grey ;

% xaxis(B,BottomTop,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,//
% 		  Step=xStep,//
% 		  step=xsubStep,//
% 		  pTick=styloOli,
% 		  ptick=styloOli,
% 		  Size=2mm,
% 		  size=0mm)) ;
% yaxis(B,LeftRight,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,
% 		  Step=yStep,
% 		  step=ysubStep,
% 		  pTick=styloOli,
% 		  ptick=styloOli,Size=2mm,
% 		  size=0mm)); 

% xequals(B,L="",//
% 	0,//
% 	ymin,//
% 	ymax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=yStep,step=ysubStep)
% 	) ;
% yequals(B,L="",//
% 	0,//
% 	xmin,//
% 	xmax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=xStep,step=xsubStep,Size=2mm,size=0mm)
% 	);

% // label(B,"$x$",(xmax,0),E,black) ;
% // label(B,"$y$",(0,ymax),N,black) ;

% picture B ;
% unitsize(B,2cm,4cm) ; 
% draw(B,Cf1^^Cf2^^Cf3^^Cf4,linewidth(1bp)+colLogo) ;

% // Tangente au point d'abscisse a
% // on calcule dy en fonction de a et de dx
% // la tangente est 
% real a = -0.9;
% real dx = 1e-9 ;
% real dy = (f(a+dx)-f(a-dx))/(2*dx) ;
% real h =  3;// longueur de la tangente
% // On trace la droite
% line tangente = line((a-h,f(a)-h*dy),(a+h,f(a)+h*dy)) ;
% draw(B,tangente,linewidth(1bp)+colLogo) ;

% point[] pN = intersectionpoints(tangente,Cf1) ;
% dot(B,pN[0],3bp+colLogo) ;
% label(B,"$ D $",pN[0],NW) ;
% point pAB = reflect(line((0,0),(1,0)))*pN[0] ;
% label(B,"$ A \oplus A $",pAB,E) ;
% dot(B,pAB,3bp+colLogo) ;
% limits(B,(xmin,ymin),(xmax,ymax),Crop) ;


% point pA=(-0.9,f(-0.9)) ;

% label(B,"$ A $",pA,NW) ;
% dot(B,pA,3bp+colLogo) ;
% pen styloOli = 0.5bp+grey ;

% xaxis(B,BottomTop,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,//
% 		  Step=xStep,//
% 		  step=xsubStep,//
% 		  pTick=styloOli,
% 		  ptick=styloOli,
% 		  Size=2mm,
% 		  size=0mm)) ;
% yaxis(B,LeftRight,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,
% 		  Step=yStep,
% 		  step=ysubStep,
% 		  pTick=styloOli,
% 		  ptick=styloOli,Size=2mm,
% 		  size=0mm)); 

% xequals(B,L="",//
% 	0,//
% 	ymin,//
% 	ymax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=yStep,step=ysubStep)
% 	) ;
% yequals(B,L="",//
% 	0,//
% 	xmin,//
% 	xmax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=xStep,step=xsubStep,Size=2mm,size=0mm)
% 	);

% // label(B,"$x$",(xmax,0),E,black) ;
% // label(B,"$y$",(0,ymax),N,black) ;
% picture C ;
% unitsize(C,2cm,4cm) ; 
% draw(C,Cf1^^Cf2^^Cf3^^Cf4,linewidth(1bp)+colLogo) ;

% draw(C,tangente,linewidth(1bp)+colLogo) ;

% label(C,"$ A $",pA,NW) ;
% dot(C,pA,3bp+colLogo) ;
% point[] pN = intersectionpoints(tangente,Cf1) ;
% dot(C,pN[0],3bp+colLogo) ;
% label(C,"$D$",pN[0],NW) ;
% point pE = reflect(line((0,0),(1,0)))*pN[0] ;
% dot(C,pE,3bp+colLogo) ;
% label(C,"$ A \oplus A $",pE,E) ;

% line d2 = line(pA,pE) ;
% draw(C,d2,1bp+colLogo) ;

% point pP[] = intersectionpoints(d2,Cf4) ;
% dot(C,pP[0],3bp+colLogo) ;
% label(C,"$ 3A $",pP[0],S) ;


% limits(C,(xmin,ymin),(xmax,ymax),Crop) ;



% pen styloOli = 0.5bp+grey ;

% xaxis(C,BottomTop,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,//
% 		  Step=xStep,//
% 		  step=xsubStep,//
% 		  pTick=styloOli,
% 		  ptick=styloOli,
% 		  Size=2mm,
% 		  size=0mm)) ;
% yaxis(C,LeftRight,p=0.5bp+grey,//
%       ticks=Ticks("%",
% 		  extend=true,
% 		  Step=yStep,
% 		  step=ysubStep,
% 		  pTick=styloOli,
% 		  ptick=styloOli,Size=2mm,
% 		  size=0mm)); 

% xequals(C,L="",//
% 	0,//
% 	ymin,//
% 	ymax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=yStep,step=ysubStep)
% 	) ;
% yequals(C,L="",//
% 	0,//
% 	xmin,//
% 	xmax,//
% 	1bp+black,//
% 	Arrow(TeXHead,1bp),//
% 	ticks=Ticks(NoZero,Step=xStep,step=xsubStep,Size=2mm,size=0mm)
% 	);

% // label(C,"$x$",(xmax,0),E,black) ;
% // label(C,"$y$",(0,ymax),N,black) ;



% add(A.fit(),(0,0),NW) ;
% add(B.fit(),(0,0),NE) ;
% add(C.fit(),(0,0),S) ;


% \end{asy}      
    \caption{Construction de $A \oplus B$, $A \oplus A$ et $3A$.}
    \label{fig_olymp_19BcC}
  \end{figure}
\item Alice donne à Bruno les coordonnées $\coord(0.02083333,
  0.06076389)$, qui sont les coordonnées de $2M$. Bruno donne à Alice
  les coordonnées $\coord(0.02908309, -0.07265188)$, qui sont les coordonnées
  de $5M$. La clef est donc constituée des quatre premières décimales de
  l’abscisse de $10M$ : $5927$  
\end{enumerate}
\end{Corrige}

\end{Enonce}

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