\documentclass[a4paper,12pt]{article}
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\usepackage[utf8]{inputenc}
\usepackage[french]{babel}
\usepackage{amsmath, amssymb}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{multicol}
\usepackage{enumitem}
\usepackage{mathrsfs}
\usepackage{tcolorbox,varwidth}
\usepackage{fontspec}

\usepackage{colortbl}
\usepackage{libertinus}

\tcbuselibrary{skins,breakable}
\usepackage{tikz}
\usetikzlibrary{calc}
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\usepackage{tabularx, array}
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% --- Arabic support ---
\usepackage{fontspec} % XeLaTeX or LuaLaTeX required
\usepackage{polyglossia}

% Set Arabic as main language
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\setotherlanguage{english}

% Arabic font configuration
\newfontfamily\arabicfont[Script=Arabic,Scale=1.2]{Amiri} % Amiri is a great default choice
\newfontfamily\arabicfontsf[Script=Arabic]{Amiri}
\newfontfamily\arabicfonttt[Script=Arabic]{Amiri}
% --- Basic Settings ---

% --- Basic document info ---
\def\professor{ ذ. عبد الله العمراني}
\def\academicyear{}
\def\examtitle{ فرض منزلي  02 د 01}
\def\schoolname{\textbf{ثانوية :} الخوارزمي التأهيلية}
\def\classname{السنة الدراسية: 2025/2026 }
%\def\subject{Mathématiques}
\def\province{المديرية الإقليمية بوهودة تاونات}
\def\subject{جذع مشترك علوم}
\def\duration{2h}
\def\secondtitle{\small(الحسابيات والحساب المتجهي)}%\LR{(Arithmétiques \& Calcul vectoriel)}
\def\bottommsg{بالتوفيق وحظ سعيد.}
\def\logo{\includegraphics[width=\linewidth]{images/logo-men.png}}
\def\wsite{\href{https://www.mosaid.xyz}{\textcolor{magenta}{\texttt{www.mosaid.xyz}}}}
\def\ddate{\hfill \number\day/\number\month/\number\year~~}
\def\bottommsg{Bonne chance!}
\setstretch{1.3}
\everymath{\displaystyle}

% --- Exercise Theme 1 ---
% Exercise Theme 1: TikZ shadow title
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  \if\relax\detokenize{#2}\relax
    \def\fulltitle{#1 تمرين }%
  \else
    \def\fulltitle{Exercice #1~#2}%
  \fi
  \exothemeone{\fulltitle}%
  \noindent #3%
  \vspace{0.2cm}%
}

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% Head Style 8: Modern tabular header
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    &
     ~~~~\textbf{\examtitle} \newline \textbf{\subject}
    &
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\begin{document}

% Two-column layout
\setlength{\columnsep}{0.5cm}
\setlength{\columnseprule}{2pt}
\renewcommand{\columnseprulecolor}{\color{black}}
\begin{multicols}{2}[\raggedcolumns]
\vspace*{-\topskip}

% Exercise 1
\printexo{1}{}{
\hspace*{1cm}أتمم باستعمال
~$\in$~
أو
~$\not\in$~
  \vspace*{-0.5cm}
\begin{flalign*}
&~\sqrt2\cdots\cdots\mathbb{R}~~;~~
~\pi\cdots\cdots\mathbb{D}~~;~~
  ~-13\cdots\cdots\mathbb{N}\\&
~\frac{4}{3}\cdots\cdots\mathbb{Q}~\quad  ;\quad
~\frac{4}{3}\cdots\cdots\mathbb{D}~~;~~
  ~1,5\cdots\cdots\mathbb{N}\\&
~-1\cdots\cdots\mathbb{R^+}~~;~~
~2\cdots\cdots\mathbb{R^-}~~;~~
~\frac{1}{4}\cdots\cdots\mathbb{R^*_+}
\end{flalign*}
}
  \vspace*{-0.75cm}
% Exercise 2
\printexo{2}{}{
\hspace*{1cm}بسط ما يلي:
\begin{flalign*}
  ~A&=\frac{1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}}{1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}}~\\
  ~B&=\frac{\frac{1}{x}}{\frac{1}{x}-\frac{1}{y}}-\frac{\frac{1}{y}}{\frac{1}{x}+\frac{1}{y}}~
\quad ~x-y\neq 0\quad ,\quad x+y\neq 0~\\
  ~C&=\frac{1}{x-1}-\frac{1-\frac{1}{x-2}}{1+\frac{1}{x-2}}~
\quad ~x\ne 1\quad ,x\ne 2~\\
  ~D&=\frac{\frac{x-1}{x+1}-\frac{x+1}{x-1}}{\frac{1}{x+1}+\frac{1}{x-1}}~
\quad ~x\ne 0 \quad , x^2\ne 1~
\end{flalign*}
}

% Exercise 3
\printexo{3}{}{\\
1.) بين أن :
~$\frac{4+44+444+4444}{5+55+555+5555}\in \mathbb{D}$~\\[0.3cm]
2.) ليكن
~$x$~
و
~$y$~
عددين حيث\\[0.3cm]
~$x=0.272727\cdots$~
و
~$y=3.66666\cdots$~\\[0.3cm]
بين أن
~$x^{2025} \times y^{2025}\in\mathbb{N}$\\[0.3cm]
3.) ليكن
~$n$~
عددا صحيحا طبيعيا،
  بين أن\\
 أ ~$-$~
    ~$\frac{11^{n+1}+11}{11^n+1}\in\mathbb{N}$~
 ب ~$-$~
    ~$\frac{(25^{n+1}-25^n)^2}{(5^{2n+1}+5^{2n})^2}\in\mathbb{N}$~
}

% Exercise 4
\printexo{4}{}{\\
1.) ليكن
~$a\in\mathbb{R^*_+}$~
حيث
~$a+\frac{1}{a}=7$~\\[0.3cm]
أحسب
~$
a^2 + \frac{1}{a^2} $~~;~~
~$ a^3 + \frac{1}{a^3}$~~;~~
~$
a^4 + \frac{1}{a^4}
$~\\[0.3cm]
2.) بين أن
~$
(499999)^2 + 999999 = 25 \times 10^{10}
$~
}
\columnbreak
% Exercise 5
\printexo{5}{}{
1.) عمل ما يلي\\[0.3cm]
\hspace*{0.5cm}
أ)
~$x^8+x+1$~\quad \quad
ب)
 ~$x^7-1$~\quad \quad
ج)
~$x^6-1$~\\[0.3cm]
2.)
باستعمال نتيجة السؤال (ج) أحسب\\[0.3cm]
~$A=1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{32}$~
;
~$B=1+\frac{2}{3}+\frac{4}{9}+\cdots+\frac{32}{243}$~\\[0.3cm]
3.)
باستعمال نتيجة السؤال (ب) أحسب\\[0.3cm]
\hspace*{1cm}
~$A+\frac{1}{64}$~
~~و~~
~$B+\frac{64}{729}$~
}

% Exercise 6
%\printexo{6}{}{\\
%قارن\\
%~$7$~
%و
%~$\sqrt3\times\sqrt{15}$~ ;~
%~$-4\sqrt2$~
%و
%~$-2\sqrt7$
%~;~
%~$10\sqrt{51}$~
%و
%~$70+\sqrt2$~
%}

% Exercise 7
\printexo{6}{}{\\
أكتب بدون رمز القيمة المطلقة\\
~$A=|\pi-\sqrt2|$~~;~~
~$B=|2\sqrt3-3\sqrt2|$~~;~~
~$C=\sqrt{(1-\sqrt3)^2}$\\[0.3cm]
~$D=\sqrt{(\sqrt3-\sqrt2)^2}+\sqrt{(1-\sqrt2)^2}+4|5-\sqrt3|$~\\[0.3cm]
~$E=\sqrt{(3-2\sqrt2)^2}+\sqrt{(7-5\sqrt2)^2}+3|2-\sqrt2|$~
}

% Exercise 8
\printexo{7}{}{\\
\noindent 1.) حدد قيمة
~$x$~
:
~$|2x+1|=|-4x+7|$~\\
2.) حدد
~$y$~
حيث
~$\Big||3y-2|-7\Big|=1$~\\
3.) حدد المجالات التي ينتمي إليها
~$x$~
في الحالات التالية\\
   1.~ ~$-5<x\le3$
  \quad \quad ;\quad 2.~ ~$|x-2|<1$~\\
3. ~$x<7$~ أو ~$x\ge10$~
  \quad \quad ;\quad 4.~  ~$|x+1|\ge2$~\\
5. ~$-2<x<1$~ أو ~$x\ne 0$~
}

% Exercise 9
\printexo{8}{}{
ليكن
~$x$~
و
~$y$~
عددين حقيقيين حيث\\[0.3cm]
~$-3\le x\le 2$~
و
~$-7\le y \le 1$~
\\
 أطر :
~$x+2y$~
و
~$2x-y$~
و
~$-5x+3y-8$~
و
~$xy$~
}

% Exercise 10
\printexo{9}{}{
ليكن
~$x$~
و
~$y$~
عددين حقيقيين حيث\\[0.3cm]
~$x-y=6$~
و
~$y\le -1 $~
و
~$x\ge -2 $~
\\
1.)
أحسب
~$A=\sqrt{(x+2)^2}+\sqrt{(y+1)^2}$~\\
2.)
بين أن
~$x\le 5$~
و
~$y\ge-8$~\\
3.)
حدد قيمة
~$B=|x+y-4|+|x+y+10|$~
}

\end{multicols}


\end{document}