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\def \RM{\mathbb{R}}%        corps des reels
\def \NM{\mathbb{N}}%        entiers naturels
\def \ZM{\mathbb{Z}}%        entiers relatifs
\def \CM{\mathbb{C}}%        nombres complexes
\def \QM{\mathbb{Q}}%        nombres rationnels
 
 
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%\title{Pr{\'e}paration au CAPES de math{\'e}matiques.}
 
%\author{Test d'Alg{\`e}bre du mercredi 6 septembre 2006}

%\address{E.F.~: Institut Fourier, BP 74, 38402 St Martin d'H{\`e}res Cedex, France.}                           
                          
 
%\email{emmanuel.ferrand@ujf-grenoble.fr}
\parindent=0cm
%\subjclass{53D10}
%\keywords{contact topology, Morse theory}
%\date{First version}
 
\begin{document}

%\maketitle

{\bf Pr{\'e}pa CAPES UPMC 2006} \hfill Emmanuel FERRAND, Laurent KOELBLEN.
\bigskip

\centerline{\bf TD d'Alg{\`e}bre du mercredi 13 septembre 2006}

\bigskip


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\section{} Soit $\theta$ un nombre r{\'e}el et $n$ un entier. Calculer
$\sum_{k=0}^{k=n} cos(k\theta)$.

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\section{} Donner une condition n{\'e}cessaire et suffisante sur les
complexes 
$z_1,\dots,z_n$ pour que l'on ait $|z_1+\dots +z_n|=|z_1|+ \dots +|z_n|$


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\section{} R{\'e}soudre $z^3=1+i$.


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\section{} R{\'e}soudre $z^2-(3+i)z+4+3i=0$.

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\section{} Résoudre le système $b+c+1=0, b^2+c^2+1=0$.

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\section{} Determiner les affixes des 2 points d'intersection des
deux cercles de rayon 1 et de centres $0$ et $-1$. 

\medskip

Pour quelles valeurs de $n$ l'équation $(x+1)^n=x^{n+1}+1$
poss\`ede-t-elle une racine multiple~?

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\section{} Si $a,b$ sont les affixes de deux sommets d'un carré,
déterminer les affixes des deux autres sommets.

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\section{} Soient $a,b,c$ trois complexes distincts.
Montrer que les 4 propositions suivantes sont
équivalentes~:

\begin{itemize}  

\item $a,b,c$ sont les affixes des sommets d'un triangle équilatéral.


\item $a^2+b^2+c^2=ab+bc+ac$


\item $j$ ou $j^2$ est racine de $az^2+bz+c$

\item $\frac{1}{a-b}+\frac{1}{b-c}+\frac{1}{c-a}=0$

\end{itemize}


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\section{}

Déterminer les complexes $z$ tels que le triangle $z,z^2,z^3$ soit
rectangle en $z$.

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\section{} Dessiner la courbe $x^2+y^2-2ax=0$ ainsi que sont image par
$z \mapsto \frac{z-1}{z+1}$.



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\section{} On consid{\`e}re la suite $(u_n)$ d{\'e}finie par $u_0=a \in \CM$, et
$u_{n+1}=\frac{3u_n+1}{u_n+3}$.
Montrer que tous ses termes sont
situ{\'e}s sur un m{\^e}me cercle. Quelle est la limite de cette suite.

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\end{document}