% =============================================================================
% VMRC Mentor Proposal — SAMPLE (2-column HTML-matched, no frame, no cut-off)
% =============================================================================
% -------------------------- INSTRUCTIONS FOR APPLICANTS -----------------------
% • Compile with a modern TeX distribution (TeX Live 2021+). pdflatex works.
% • Submit BOTH your PDF and this .tex source (VMRC prefers LaTeX).
% • Font & size: Times-like via newtxtext/newtxmath; 10pt document class.
% • Two-column look: left column (labels) right-aligned; right column (content)
%   left-aligned—this mirrors the VMRC website sample.
% • Replace the placeholder text ONLY (e.g., dates, mentor info, title, topic,
%   Background, Abstract, Outline, References). Do not alter the layout macros.
% • Divider lines: the subtle fade uses TikZ shading. If your TeX install lacks
%   TikZ shadings, a solid rule still appears (graceful fallback).
% • Page break: we use \tcbbreak (inside the tcolorbox) before the Outline section.
%   If your content is shorter/longer, you MAY delete that single \tcbbreak line.
% • Left label column width is set in \newcolumntype{L}{...p{1.35in}} (below).
%   You MAY adjust 1.35in to fit longer labels (e.g., 1.5in) if needed.
% • Bullet indentation under “Outline/Timeline” is controlled in the itemize
%   options (leftmargin=2.6em, labelsep=.8em). You MAY tweak those two values.
% • References numbering and alignment are set to match the website look; don’t
%   change the enumerate options unless spacing truly needs tuning.
% • Keep the \sectiondivider macro as-is to preserve the faded divider style.
% ------------------------------------------------------------------------------

\documentclass[10pt]{article}

\usepackage[margin=1in]{geometry}
\usepackage{amsmath}                 % load amsmath first if you need it
\usepackage{newtxtext,newtxmath}     % Times-like text + math (includes AMS symbols)
% \usepackage{amssymb}               % ← not needed with newtxmath
\usepackage{enumitem}
\usepackage{xcolor}
\usepackage{hyperref}
\usepackage{array}
\usepackage{tabularx}
\usepackage{microtype}
\usepackage[most]{tcolorbox}
\usepackage{ragged2e}           % better ragged alignment
\usepackage{tikz}               % for the gradient overlay
\usetikzlibrary{shadings}

\definecolor{Muted}{HTML}{5B6470}

% Comfortable paragraph spacing, like the site
\setlength{\parskip}{6pt}
\setlength{\parindent}{0pt}
\setlist[itemize]{topsep=4pt,itemsep=8pt,parsep=0pt}
\raggedbottom

% Column helpers
% NOTE: You may widen 1.35in to accommodate longer labels if absolutely needed.
\newcolumntype{L}{>{\RaggedLeft\arraybackslash\scshape}p{1.35in}}
\newcolumntype{Y}{>{\RaggedRight\arraybackslash}X}

% Spacing knobs for rows
\newcommand{\VMRCrowsep}{10pt}
\newcommand{\VMRCrowsepMentor}{14pt}
\newcommand{\VMRCmentorpad}{\vspace{10pt}}

% Small gap to separate consecutive tabularx blocks (no divider)
\newcommand{\VMRCblockgap}{\vspace{6pt}}

% Date range helpers (raised en-dash rule)
\newcommand{\VMRCdashraise}{0.72ex}
\newcommand{\solidendash}{\rule[\VMRCdashraise]{0.9em}{0.5pt}}
\newcommand{\daterange}[2]{#1~\solidendash~#2}

% --------- Divider (solid rule + gradient overlay for HTML-like fade) ---------
% Do not modify these unless you need a darker/lighter fade.
\newcommand{\VMRCdivht}{1.2pt}     % thickness of the gradient band
\newcommand{\VMRCdivmid}{black!26} % center intensity of the fade

\newcommand{\sectiondivider}{%
  \par\vspace{10pt}%
  % Solid fallback that always shows up
  \noindent{\color{black!26}\rule{\linewidth}{0.6pt}}%
  % Thin gradient band on top (if shadings render)
  \noindent
  \begin{tikzpicture}
    \path[use as bounding box] (0,0) rectangle (\linewidth,\VMRCdivht);
    \shade[shading=axis, left color=white, middle color=\VMRCdivmid, right color=white]
      (0,0) rectangle (\linewidth,\VMRCdivht);
  \end{tikzpicture}%
  \vspace{8pt}\par}

% Card container — frame removed (no left/right borders)
\tcbset{
  vmrcbox/.style={
    enhanced, breakable,
    colback=white, colframe=white, boxrule=0pt, arc=0pt,
    left=10pt,right=10pt,top=10pt,bottom=10pt, overlay={}
  }
}

\begin{document}
\begin{tcolorbox}[vmrcbox]

  % ============================== TITLE BLOCK ================================
  % ACTION: Replace the title/subtitle text only. Do not change centering/format.
  \begin{center}
    {\large\bfseries Research Proposal — Virtual Math Research Circle}\\
    {\small\color{Muted} Sample for mentor submissions}
  \end{center}
  \vspace{8pt}

  % ============================ METADATA + BACKGROUND ========================
  % ACTION: Replace values on the right (Y column). Keep label wording on the left.
  \begin{tabularx}{\linewidth}{@{} L Y @{}}
    Session: &
      Session 2: \daterange{Jul 18, 2021}{Aug 6, 2021} \\[\VMRCrowsep]

    Mentor: &
      \begin{tabular}[t]{@{}l@{}}
        Sample Mentor \\
        Postdoctoral Researcher \\
        Department of Mathematics \\
        Sample University\VMRCmentorpad
      \end{tabular}\\[\VMRCrowsepMentor]

    Project title: & \emph{Probability and Geometric Series} \\[\VMRCrowsep]
    Topic area:    & Probability Theory \\[\VMRCrowsep]

    Background: &
      This research project requires little background knowledge other than basic
      algebra and arithmetic. All required skills needed to approach the problem
      will be taught by the instructor. This project serves as an excellent
      introduction to \emph{probability theory}, \emph{counting methods},
      \emph{sequences}, \emph{series}, and \emph{mathematical proofs}. During
      the experimental stage, we will implement computational tools such as
      \emph{Excel}, \emph{Python}, and/or \emph{R Studio}. During the
      development of the final presentation, we will use tools such as
      \emph{PowerPoint}, \emph{Beamer}, \emph{ShareLaTeX}, and/or \emph{TeXShop}. \\
  \end{tabularx}

  \sectiondivider

  % ================================= ABSTRACT ================================
  % ACTION: Replace the abstract body only (right column). Keep "Abstract:" label.
  \begin{tabularx}{\linewidth}{@{} L Y @{}}
    Abstract: &
      A \emph{fair coin} has equal probability of obtaining heads or tails. Say you
      want to make an equal-probability decision between two outcomes, such as which
      team in a sports game gets to go first. You can accomplish this using the fair
      coin by assigning heads to one outcome and tails to the other. But what if you
      have to make an equal-probability decision between \emph{three} outcomes? Is
      it possible to use a fair coin in some way to accomplish this?

      \par\medskip
      We will develop an algorithm using a fair coin to select from three mutually
      disjoint events with equal probability. We then hope to extend this result to
      the case of $n$ pairwise disjoint events, developing an algorithm so that,
      using a (two-sided) fair coin, each event has probability $1/n$.

      \par\medskip
      If this can be achieved, we seek to generalize further by considering the case
      where the coin is no longer fair; that is, for some $0<p<1$, we have
      \[
        P(H)=p,\qquad P(T)=1-p.
      \]
      Under this modification, is it still possible to choose from among $n$
      pairwise disjoint events so that each would occur with equal probability? \\
  \end{tabularx}

  % ---- Add a bit of breathing room between Abstract and Possible extension ---
  \VMRCblockgap

  % ============================ POSSIBLE EXTENSION ===========================
  % ACTION: Replace the text after "Possible extension:" with your own.
  \begin{tabularx}{\linewidth}{@{} L Y @{}}
    Possible extension: &
      Once the iteration algorithm is established, let $N$ denote the random
      variable associated to the number of iterations needed to produce the desired
      outcome of equal probabilities. Can the expected value $E(N)$, variance $V(N)$,
      and standard deviation $STD(N)$ of $N$ be computed? \\
  \end{tabularx}

  \sectiondivider

\tcbbreak % ACTION: You may delete this line if you want to avoid a page break here.

  % ============================ OUTLINE / TIMELINE ===========================
  % NOTE: Divider above is intentional; keep it for visual consistency.
    \sectiondivider
\begin{tabularx}{\linewidth}{@{} L Y @{}}
  Outline/Timeline: &
    This is a general (tentative) outline of how the research project will
    progress. We will adjust the schedule as needed throughout the 3-week period.

    % ACTION: You may adjust leftmargin (indent) and labelsep (gap after bullet)
    %         to fit your content, but keep the overall structure.
    \begin{itemize}[leftmargin=2.6em, labelsep=.8em, itemsep=8pt, topsep=6pt]
      \item \textbf{Week 1 (Background \& Examples).}
        The first week will be more instruction-based. We will begin with a brief
        overview of probability theory using references [1, 2]. The students will
        be introduced to basic notions such as \emph{probability functions},
        \emph{sampling with/without replacement}, \emph{mutual independence},
        \emph{conditional probability}, and \emph{Bayes’ Rule}.

        \par\medskip
        We will introduce the basic notions of a rigorous mathematical proof,
        including \emph{proof by exhaustion}, \emph{proof by induction}, and
        \emph{proof by contradiction}. The students will be given several examples
        and assignments to help strengthen their skills in proof writing.

      \item \textbf{Week 2 (Experimentation \& Main Problem).}
        The second week will be based heavily on experimentation. The goal during
        this stage is to determine an algorithm (‘game’) between three people,
        which, using a coin, yields a 1/3 probability of winning. All experiments
        will be implemented via \emph{Excel}, \emph{Python}, and/or \emph{R Studio}.

        \par\medskip
        Once a method has been found with strong empirical evidence of success,
        the next stage is to prove the assertion rigorously. We will first attempt
        to prove the base case $n=3$ and then generalize to the case $n\ge 3$.
        We will then extend, if possible, to the case where the coin is no longer
        ‘fair’ (i.e., $P(H)=p$ and $P(T)=1-p$ for some $0<p<1$).

      \item \textbf{Week 3 (Finalizing Results \& Presentation).}
        The first half of week three will be spent finalizing the empirical
        results of the experiments and the mathematical proofs.

        \par\medskip
        The second half of the final week will be spent creating the final
        presentation. We will decide whether to use \emph{PowerPoint} or the more
        sophisticated \emph{Beamer} presentation using \emph{LaTeX}. This will
        depend on the comfort levels of the students with \emph{LaTeX} typesetting.
    \end{itemize} \\
\end{tabularx}


% ================================= REFERENCES =============================
% ACTION: Replace these items with your project’s references (use the same format).
\sectiondivider
\begin{tabularx}{\linewidth}{@{} L Y @{}}
  References: &
    \begin{enumerate}[label={[}\arabic*{]}, leftmargin=*, labelsep=.5em,
      topsep=0pt, partopsep=0pt, parsep=0pt, itemsep=3pt,
      before=\vspace{-\parskip}\vspace{-0.5\baselineskip}] % nudge up to align with "References:"
      \item G. Casella, R. Berger, \emph{Statistical Inference}, 2nd ed., Duxbury Advanced Series, 2001.
      \item J. Rice, \emph{Mathematical Statistics and Data Analysis}, 3rd ed., Duxbury Advanced Series, 2006.
      \item J. Stewart, \emph{Calculus: Early Transcendentals}, 8th ed., Cengage Learning, 2016.
    \end{enumerate} \\
\end{tabularx}

  
\end{tcolorbox}
\end{document}
