%% please leave the formatting untouched and modify just the
%% version name and the statements.
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% headers and macros section
\documentclass[12pt,a4paper]{article}

%\usepackage[latin1]{inputenc}
%\usepackage[T1]{fontenc}
%\usepackage[frenchb]{babel}

\font\fonttit=cmr12 scaled \magstep 3
\font\fontimo=cmr12 scaled \magstep 1
\font\fontversion=cmr12 scaled \magstep 1
\def\problem#1{\vskip 0.7 true cm\noindent{\bf #1.\ }}
\def\version#1{\rightline{Version: \fontversion#1}}
\oddsidemargin=0pt
\textwidth=21cm
\addtolength{\textwidth}{-2in}
\setlength{\textheight}{297mm}
\addtolength{\textheight}{-1.5in}
\topmargin= -.5in
\headheight=0pt
\headsep=0pt
%\addtolength{\textheight}{1in}
%\addtolength{\textheight}{1in}
\pagestyle{empty}
\begin{document}
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% please put the name of your language in english here:
\version{French}

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\vskip %1.5
0.4 true cm {\fonttit{\centerline{Premier Jour}}} \vskip .5 true
cm {\fontimo{\centerline{Olympiades d'Iran - 5 mai 1999}}} \vskip
0.3 true cm


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% those are the problems:

\problem{1}%
Existe-t-il un entier qui soit une puissance de $2$ et tel que
l'on puisse obtenir une autre puissance de $2$ par une permutation
de ses chiffres d\'ecimaux ?


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\problem{2}%
Soit $ABC$ un triangle dont les angles $\hat{B}$ et $\hat{C}$
mesurent plus de $45$ degr\'es. On construit les triangles
isoc\`ele-rectangles $CAM$ et $BAN$ à l'ext\'erieur du triangle
$ABC$ de telle sorte que les angles droits sont $\widehat{CAM}$ et
$\widehat{BAN}$. On construit \'egalement le triangle
isoc\`ele-rectangle $BPC$ à l'int\'erieur de $ABC$ de telle sorte
que son angle droit est $\widehat{BPC}$. Montrer que le triangle
$MNP$ est \'egalement isoc\`ele-rectangle.

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\problem{3}%
On a un r\'eseau $100\times 100$ avec un arbre plant\'e sur chacun
des $10000$ points. Combien peut-on couper d'arbres  au maximum de
telle mani\`ere que de chaque endroit o\`u un arbre a \'et\'e
coup\'e, on ne puisse constater la coupe d'aucun autre arbre. (En
d'autres termes, toute ligne joignant deux sites o\`u les arbres
ont \'et\'e coup\'es doit contenir au moins un arbre.)



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\vskip .5 true cm
\hrule width 4 true cm
\vskip .3 true cm
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% change here please, this is almost the end:

\noindent %Chaque probl\`eme vaut 7 points.
\par \noindent Temps
accord\'e: 4 heures.

% please put the name of your language in english here:
\version{French}

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\vskip %1.5
0.4 true cm { \fonttit {\centerline{Deuxi\`eme Jour}}} \vskip .5
true cm {\fontimo{\centerline{Olympiades d'Iran - 6 mai 1999}}}
\vskip .3 true cm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% those are the problems:

\problem{4}%
Trouver tous les entiers naturels $m$ pouvant s'\'ecrire:
$$m=\frac1{a_1}+\frac2{a_2}+\dots+\frac{1378}{a_{1378}}$$ o\`u
$a_1,\dots a_{1378}$ sont des entiers naturels.

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\problem{5}%
Soit un triangle $ABC$; $P,Q,R$ trois points situ\'es
respectivement sur les c\^ot\'es $AB,AC$ et $BC$. $A',B',C'$ trois
points situ\'es respectivement sur les droites $(PQ),(PR)$ et
$(QR)$ de telle sorte que $(AB)\parallel (A'B')$, $(AC)\parallel
(A'C')$ et $(BC)\parallel (B'C')$. Montrer que
$$\frac{AB}{A'B'}=\frac{\textbf{aire}(ABC)}{\textbf{aire}(PQR)}.$$

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\problem{6}%
$A_1,A_2,\dots A_n$ sont $n$ points distincts du plan. On colorie
en rouge le milieu de chaque segment $A_i A_j$, $i\ne j$. Trouver
le nombre minimum de points rouges.





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\vskip 2 true cm
\hrule width 4 true cm
\vskip .3 true cm
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% change here please, this is almost the end:

\noindent %Chaque probl\`eme vaut 7 points.
\par
\noindent Temps accord\'e: 4 heures.









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% final stuff
\end{document}
