\documentclass[a4paper,11pt]{article}
\usepackage{francois_meria}
\usepackage[dvips]{graphicx}
\usepackage[dvips]{epsfig}
\usepackage{psfig}
\usepackage{pst-eucl}
    \lhead{6\ieme~- \textit{Math�matiques}}
    \chead{}
    \rhead{\textit{Ann�e} 2005/2006}
    \rfoot{}
    \pagestyle{fancy}
  \renewcommand{\headrulewidth}{0.5pt}
\begin{document}
\begin{center}
    \textbf{DS \textit{correction} : \og Num�rations, nombres d�cimaux et g�om�trie \fg}\\
    {{\small\textit{Pour des raisons de photocopie, les figures des
    exercices $4$ et $7$ ne sont pas en vraies grandeurs}}}
\end{center}

\vskip 0.3cm \hrule\vspace{\baselineskip}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% DEBUT DES EXERCICES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{multicols}{2}
\begin{exercice} (4,5 points) - Les nombres demand�s, dans l'ordre, sont les suivants :
$153600$ ; $72050$ ; $4520000$ ; $125000000$ ; $709200$ ; $400000$
; $347675$ ; $16000523$ et $1489$.
\end{exercice}

\begin{exercice} (1 point) - Le nombre compris entre 570 et 600
est $593$.
\end{exercice}
\end{multicols}
\begin{exercice} (4 points) - On a :
\begin{multicols}{2}
\begin{questions}
    \item 1543 : $1\times 1000 + 5\times 100 + 4\times 10 +3$
    \item 25782 : $2\times 10000 + 5\times 1000 + 7\times 100 +
    8\times 10 +2$
    \item 10002 : $1\times 10000+2$
    \item 56278 : $5\times 10000+ 6\times 1000 + 2\times 100 + 7\times 10 +8$
    \item 000520 : $5\times100 + 2\times10$
    \item 008569 : $8\times1000+5\times100 +6\times 10 +9$
    \item 77777 : $7\times10000+7\times1000+7\times100+7\times10+7$
    \item 81 : $8\times10+1$
\end{questions}
\end{multicols}
\end{exercice}

\begin{exercice} (4,5 points)
\begin{multicols}{2}
\begin{questions}
    \item Les milieux des segments propos�s sont les points
    d'intersection entre la droite $(d)$ et les segments.
\begin{center}
\psset{linecolor=black, linewidth=.5pt, arrowsize=2pt 4}
\psset{unit=0.55cm} \pspicture*(0.0000,-0.5)(16.0000,5)
\uput{0.3000}[135.0000](1.2929,2.7071){$A$}
\uput{0.3000}[-45.0000](2.7071,1.2929){$B$}
\uput{0.3000}[100.0000](3.7395,3.4772){$C$}
\uput{0.3000}[280.0000](4.2605,0.5228){$D$}
\uput{0.3000}[45.0000](4.0858,3.4142){$E$}
\uput{0.3000}[-45.0000](6.9142,0.5858){$F$}
\uput{0.3000}[35.0000](8.7202,3.2045){$G$}
\uput{0.3000}[215.0000](5.2798,0.7955){$H$}
\uput{0.3000}[-135.0000](8.5050,2.4950){$I$}
\uput{0.3000}[-45.0000](9.4950,1.5050){$J$}
\uput{0.3000}[68.0000](11.5994,3.4835){$K$}
\uput{0.3000}[248.0000](10.4006,0.5165){$L$}
\uput{0.3000}[-35.0000](15.1789,0.4743){$M$}
\uput{0.3000}[165.0000](10.8211,3.5257){$N$}
\psline{|-|}(1.2929,2.7071)(2.7071,1.2929)
\psline{|-|}(3.7395,3.4772)(4.2605,0.5228)
\psline{|-|}(4.0858,3.4142)(6.9142,0.5858)
\psline{|-|}(8.7202,3.2045)(5.2798,0.7955)
\psline{|-|}(8.5050,2.4950)(9.4950,1.5050)
\psline{|-|}(11.5994,3.4835)(10.4006,0.5165)
\psline{|-|}(15.1789,0.4743)(10.8211,3.5257)

\psline(0,2)(15,2) \put(15,2.3){$(d)$}
\endpspicture
\end{center}
\begin{multicols}{2}
\begin{enumerate}[(a)]
    \item $AB =2$~cm
    \item $CD =3$~cm
    \item $EF =4$~cm
    \item $GH =4,2$~cm
    \item $IJ =1,5$~cm
    \item $KL =3,2$~cm
    \item $MN =5,4$~cm
\end{enumerate}
    \item On remarque que tous les milieux de
    ces segments sont align�s.
\end{multicols}
\end{questions}
\end{multicols}
\end{exercice}

\begin{multicols}{2}
\begin{exercice} (3 points)\\
\noindent Voici la figure demand�e avec $AB=5$~cm.
\begin{center}
\psset{linecolor=black, linewidth=.5pt, arrowsize=2pt 4}
\psset{unit=1.25cm} \pspicture*(-0.5000,2.0000)(4.5000,6.3000)
\pscircle(2.0000,4.0000){2.0000} \pscircle(1.0000,4.0000){1.0000}
\pscircle(3.0000,4.0000){1.0000}
\psline(0.0000,4.0000)(4.0000,4.0000)
\psline(0.8950,4.1050)(1.1050,3.8950)
\psline(1.1050,4.1050)(0.8950,3.8950)
\psline(2.8950,4.1050)(3.1050,3.8950)
\psline(3.1050,4.1050)(2.8950,3.8950) \psdots(0,4)
\uput[135](0,4){$A$} \psdots(4,4) \uput[45](4,4){$B$} \psdots(2,4)
\uput[45](2,4){$I$}
\endpspicture
\end{center}
\end{exercice}
\columnbreak
\begin{exercice} (4,5 points)
\noindent On a :\\
\begin{center}
\begin{tabular}{cccccc}
$B \ \in\ [AE]$ &    \qquad  \qquad & $B \ \in\ [AD]$ & \qquad  \qquad & $C \ \notin\ [ED]$ \\
$C \ \notin\ [AB)$ & \qquad  \qquad & $E \ \in\ [AD)$ & \qquad  \qquad & $E \ \in\ [AB)$ \\
$B \ \notin\ [ED)$ & \qquad  \qquad & $B \ \in\ (ED)$ & \qquad  \qquad & $B \ \in\ [AB]$ \\
\end{tabular}
\end{center}
\end{exercice}
\begin{exercice} (4,5 points) - Voici la construction demand�e. \end{exercice}
\end{multicols}
\begin{center}
\psset{unit=0.7cm}
    \pspicture(10,9)
    %\psgrid
    \pstGeonode[PointSymbol=none,PosAngle=-90](0,0){B}(10,0){U}
    \pstGeonode[PointSymbol=none,PointName=none](2,0){P_2}(9,0){P_1}
    \pstInterCC[PointSymbol=+,PosAngle=90]{B}{P_1}{U}{P_2}{T}{T'}
    \put(6.5,7.6){la droite $(TB)$} \put(5.5,8.5){la demi-droite $[UT)$}
    \put(4,0.5){le segment $[BU]$}
    \pstLineAB{B}{U}
    \pstLineAB[nodesepB=-2]{U}{T}
    \pstLineAB[nodesep=-1]{B}{T}
    \put(4.5,-0.5){10 cm} \put(2,4){9 cm} \put(8,4){8 cm}
    \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{P_1}{M_1}{100} \pstCurvAbsNode[PointName=none,PointSymbol=none]{B}{P_1}{M_2}{200}
    \pstArcOAB[linestyle=dashed]{B}{M_1}{M_2}
    \pstCurvAbsNode[PointName=none,PointSymbol=none]{U}{P_2}{N_1}{-200} \pstCurvAbsNode[PointName=none,PointSymbol=none]{U}{P_2}{N_2}{-100}
    \pstArcOAB[linestyle=dashed]{U}{N_1}{N_2}
    \pstGeonode[PointSymbol=+,PosAngle={150,90}](4.27,5){F}(5.17,8){E}
    \endpspicture
\end{center}

\end{document}

