Khmer Minimal Setup

\documentclass[a4paper,11pt]{article}
\makeatletter
\usepackage[top=2cm, left=1.5cm, bottom=1.5cm, right=1.5cm]{geometry}
\usepackage{amsmath,amssymb}
\usepackage{tcolorbox}
\usepackage{dsfont}
\usepackage[export]{adjustbox}
\usepackage{graphicx}
\usepackage{wrapfig}
\usepackage{pgf}
\usepackage{tikz}% graphic drawing
\usetikzlibrary{arrows}
\pagestyle{empty}
\usepackage{wasysym}
\usepackage{mathpazo}% change math font
\usepackage{enumitem}% change list environment like enumerate, itemize and description
\usepackage{multicol}% multi columns
\usepackage{mathpazo}% change math font
\usepackage{xcolor}
\usepackage[no-math]{fontspec}% font specfication
\newcommand{\heart}{\ensuremath\heartsuit}
\newcommand{\butt}{\rotatebox[origin=c]{180}{\heart}}
\newcommand*\circled[1]{\tikz[baseline=(char.base)]{
\node[shape=circle,draw,inner sep=2pt] (char) {#1};}}
%
\makeatletter %line1-8 make Khmer known
\def\@khmernum#1{\expandafter\@@khmernum\number#1\@nil} 
\def\@@khmernum#1{%
\ifx#1\@nil 
\else 
\char\numexpr#1+"17E0\relax 
\expandafter\@@khmernum\fi 
} 
\def\knum#1{\expandafter\@khmernum\csname c@#1\endcsname}
\def\khmernumeral#1{\@សំអុន@khmernum#1\@nil}
\AddEnumerateCounter{\knum}{\@knum}{}
\makeatother
\makeatletter
\newcommand*{\kalph}[1]{%
\expandafter\@kalph\csname c@#1\endcsname%
}
\newcommand*{\@kalph}[1]{%
\ifcase#1\or ក\or ខ\or គ\or ឃ\or ង\or ច\or ឆ\or ជ\or ឈ\or ញ\or ដ\or ឋ\or ឌ\or ឍ\or ណ\or ត\or ថ\or ទ\or ធ\or ន\or ប\or ផ\or ព\or ភ\or ម\or យ\or រ\or ល\or វ\or ស\or ហ\or ឡ\or អ%
\else\@ctrerr\fi%
}
\AddEnumerateCounter{\kalph}{\@kalph}{}
\makeatother

\SetEnumitemKey{I}{%
leftmargin=*,
label={\protect\tikz[baseline=-0.9ex]\protect\node[text=blue,fill=blue!10!white]{\en\Roman*.};},%
font=\small\sffamily\bfseries,%
labelsep=1ex,%
topsep=0pt}
%
\SetEnumitemKey{a}{%
leftmargin=*,%
label={\protect\tikz[baseline=-0.9ex]\protect\node[text=blue,fill=magenta!10!white]{\kalph*.};},%
font=\small\kbk\bfseries,%
labelsep=1ex,%
topsep=0pt}
%
\SetEnumitemKey{1}{leftmargin=*,%
label={\protect\tikz[baseline=-0.9ex]\protect\node[text=red,fill=cyan!20!white]{\knum*.};},%
font=\small\kbk\bfseries,%
labelsep=1ex,%
topsep=0pt}
%
\def\hard{\leavevmode\makebox[0pt][r]{\large\ensuremath{\star}\hspace{2em}}}
%
\def\hhard{\leavevmode\makebox[0pt][r]{\large\ensuremath{\star\star}\hspace{2em}}}
%
\everymath{\protect\displaystyle\protect\color{black}}
%
\pagecolor{cyan!1!white}
%
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{wasysym}
\makeatother
\setmainfont{Khmer OS Content}% set default font to Khmer OS
\newcommand{\ko}{\fontspec[Script=Khmer]{Khmer OS}\selectfont}
\newcommand{\kml}{\fontspec[Script=Khmer]{Khmer OS Muol Light}\selectfont}
\newcommand{\kos}{\fontspec[Script=Khmer]{Khmer OS System}\selectfont}
\newcommand{\kb}{\fontspec[Script=Khmer]{Khmer OS Battambang}\selectfont}
\newcommand{\kbk}{\fontspec[Script=Khmer]{Khmer OS Bokor}\selectfont}
\newcommand{\en}{\fontspec{Liberation Serif}\selectfont}

\usepackage{fancyhdr}
\pagestyle{fancy}
\fancyhf{}
\lhead{\kbk វិទ្យាល័យរស្មីសោភ័ណ}
\chead{\kbk គណិតវិទ្យាត្រៀមប្រឡងចូលសាលាតិចណូ}
\rhead{\kbk វិញ្ញាសាត្រៀមប្រឡង}
\lfoot{\kbk ដោយ៖ លី ឆាយធាន}
\cfoot{\en\thepage}
\rfoot{\en Tel: 070 676 566}
\renewcommand{\headrulewidth}{0.7pt}
\renewcommand{\footrulewidth}{0.7pt}
\headheight = 25pt
\headsep = 5pt
\footskip = 25pt

\begin{document}

\vspace{0.15cm}
\begin{center}
{\kml \color{blue}{វិញ្ញាសាទី១}}
\end{center}

\vspace{-0.3cm}
\begin{enumerate}[I]
    \item គណនាតម្លៃនៃផលបូក $S = \sum_{k=1}^{13} \frac{1}{\cos \left( \frac{\pi}{3} + \frac{k\pi}{4} \right) \cos \left(\frac{\pi}{12} + \frac{k\pi}{4} \right)}$
    \item ចូររកអនុគមន៍ $f(x)$ ដែលផ្ទៀងផ្ទាត់សមីការ $f(x) = 3 + \int_{0}^{x} \frac{t}{f(t)}dt$
    \item គណនាអាំងតេក្រាល $I = \int_{0}^{1} \frac{xe^{x}}{(1+x)^2}dx$
    \item គណនាតម្លៃធំបំផុតនៃ $f(x) = \int_{0}^{\pi} t \sin(\pi x+t)dt$
    \item គេអោយផលបូក $S_{n} = \frac{3}{(3+1)(3^{2}+1)} + \frac{3^{2}}{(3^2-1)(3^{3}-1)} + ... + \frac{3^{n}}{(3^{n}+1)(3^{n+1}+1)}$ និង $S = \lim_{n \to +\infty} S_{n}$។ គណនាតម្លៃនៃ $S$។
    \item គណនាតម្លៃតូចបំផុត $y_{\text{min}}$ និងតម្លៃធំបំផុត $y_\text{max}$ នៃអនុគមន៍ $y = \frac{2\cos^{2}(x) + 3\sin(2x)+7}{2\cos^{2}(x) + \sin(2x) + 1}$។
    \item គេយក $f(x) = \int_{0}^{x^{2}} \frac{e^{t} -1}{t}dt$ ដែល $x > 0$ ជាអនុគមន៍និង $f'(x)$ ជាដេរីវេនៃ $f(x)$។ គណនា $f'(x)$។
    \item គណនា $f' \left(\frac{\sqrt{\pi}}{2} \right)$ ដោយដឹងថា $f(x) = \frac{2}{\sqrt{\pi}\sin^{3}(x^{2})}$។
    \item គេមានអនុគមន៍ $f$ កំណត់លើ $\mathds{R}$ ដោយ $f(x) = xe^{x}$។ គ្រប់ $n \in \mathds{N}$ ជាដេរីវេទី $n$ នៃ $f(x)$។\\ ចូរគណនា $S_{n} = \sum_{k=1}^{n} f^{k}(0)$។
    \item គណនាតម្លៃពិតប្រាកដនៃកន្សោម $S = \cos \left( \frac{2\pi}{7} \right) + \cos \left(  \frac{4\pi}{7}\right) + \cos \left(  \frac{6\pi}{7} \right)$។
    \item គណនាអាំងតេក្រាលនៃ $I = \int_{\frac{\pi}{6}}^{\frac{\pi}{3}} x \left( \tan(x) + \cot(x) \right)^{2}dx$។
    \item គណនាលីមីតនៃ $S$ ចំពោះ $n \to \infty$ ដែល $S = \sum_{k=1}^{n} \frac{2^{k}}{(2^{k}+ 3)(2^{k+1}+3)}$។
    \item គណនាអាំងតេក្រាលនៃ $I = \int_{1}^{\sqrt{2}} \frac{x^{2} + 18}{x^{4} \sqrt{x^{2}+ 3}}dx$។
    \item បើគេមានអនុគមន៍ $f(x) = \frac{2}{\sqrt{\pi}}\int_{0}^{x} e^{t^{2}}dt$ ហើយ $z = \sqrt{2\ln(2)-\ln(\sqrt\pi)}$។\\
    ចូរគណនាតម្លៃនៃ $f'(z)$។
\end{enumerate}
\end{document}