\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsfonts,amsmath,amssymb,amsthm}

\PassOptionsToPackage{obeyspaces}{url}
\usepackage[colorlinks=true,citecolor=blue,urlcolor=blue,linkcolor=blue,bookmarksopen=true]{hyperref}
\usepackage{tikz}

\usepackage{enumitem}
\setlist{itemsep=0.5em}

%% header definitions
\def\course{MATH/CMSC H395}
\def\assign{HW \#4}
\def\doclink{\url{https://mathematicaster.org/teaching/discrete2026/hw/hw4.tex}}
\def\due{Sep 28}

%% environments
% theorem style
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{defn}[theorem]{Definition}

% definition style
\theoremstyle{definition}
\newtheorem{problem}{Problem}

% custom
\newenvironment{solution}{\noindent\textbf{Solution.}}{\qed}

%% custom commands
\newcommand*{\arXiv}[1]{\href{http://arxiv.org/pdf/#1}{arXiv:#1}}       % arXiv link
\newcommand*{\eqdef}{\stackrel{\mbox{\normalfont\tiny{def}}}{=}}        % definition by equality
\newcommand*{\abs}[1]{\lvert #1\rvert}     % absolute values, cardinality
\newcommand*{\eps}{\varepsilon}            % prettier epsilon
\newcommand*{\R}{\mathbb{R}}               % real numbers
\newcommand*{\C}{\mathbb{C}}               % complex numbers
\newcommand*{\Q}{\mathbb{Q}}               % rational numbers
\newcommand*{\Z}{\mathbb{Z}}               % integers
\newcommand*{\N}{\mathbb{N}}               % natural numbers
\newcommand*{\F}{\mathbb{F}}               % field
\newcommand*{\family}{\mathcal{F}}         % family
\newcommand*{\GL}{\mathbf{GL}}             % general linear group
\newcommand*{\what}[1]{\widehat{#1}}       % wider hat
\newcommand*{\wbar}[1]{\overline{#1}}      % wider bar
\newcommand*{\mcal}[1]{\mathcal{#1}}       % mathcal
\newcommand*{\mbf}[1]{\mathbf{#1}}         % mathbf
\newcommand*{\mbb}[1]{\mathbb{#1}}         % mathbb

% operators
\let\Pr\relax\DeclareMathOperator*{\Pr}{\mathbf{Pr}}       % probability
\DeclareMathOperator*{\E}{\mathbb{E}}          % expectation
\DeclareMathOperator*{\Var}{\mathbf{Var}}      % variance
\DeclareMathOperator*{\Span}{span}             % span
\DeclareMathOperator{\tr}{tr}                  % trace
\DeclareMathOperator{\rank}{rank}              % rank
\DeclareMathOperator{\supp}{supp}              % support
\DeclareMathOperator{\diam}{diam}



\begin{document}
\pagestyle{empty}

\noindent
\begin{minipage}{.33\textwidth}
	\flushleft {\bf \course}
\end{minipage}
\begin{minipage}{.33\textwidth}
	\centering {\bf \assign}
\end{minipage}
\begin{minipage}{.33\textwidth}
	\flushright {\bf \due}
\end{minipage}
\vskip3pt
\hrule\hrule\hrule
\vskip5pt

\noindent {\footnotesize This document is from \doclink}
\vskip10pt

\noindent {\small
    Solutions must be typeset in \LaTeX.
    Collaboration is allowed, but must be fully acknowledged.
    Any collaboration should involve only discussion; all writing \textbf{must} be done individually.
    Use of generative AI is \textbf{forbidden}.
    You are free to use the result from any problem on this (or previous) assignment as a part of your solution to a different problem even if you have not solved the former problem.
}

\vskip20pt

\begin{problem}
    $61$ points are placed in a disk of radius $4$.
    Prove that some pair of points are at most $\sqrt 2$ units apart.
\end{problem}


\begin{problem}
    $n+1$ integers are chosen from the set $[2n]$.
    Prove that one of these integers divides another.

    Hint: Every positive integer can be written as $2^ab$ where $a\geq 0$ and $b$ is odd.
\end{problem}


\begin{problem}
    Let $G$ be a (simple) graph on at least $2$ vertices.
    Show that $G$ has some pair of vertices of the same degree.
\end{problem}


\begin{problem}
    Fix $n\geq 3$ and suppose that $S\subseteq[n]$ with $\abs S\geq 2\log_2 n$.
    Show that there are two distinct subsets $A,B\subseteq S$ with $\sum A=\sum B$.
\end{problem}




\end{document}
