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\title{Selected Problems in Discrete Mathematics and Analysis}
\author{Henry Shin}
\date{}

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\maketitle

\section*{Problems}

\begin{enumerate}

    \item For positive integers \(1 \leq k \leq n\), show that the number of nondecreasing sequences \(a_1 \leq a_2 \leq \cdots \leq a_k\) where \(a_1, a_2, \ldots, a_k\) are chosen from \(\{1, 2, \ldots, n\}\) is exactly \(\binom{n+k-1}{k}\).

    \item For a nonzero integer \(a\) and an integer \(b\), denote \(P_{a,b} = \{ an + b \mid n \text{ is an integer} \}\). Is there a finite set of integers \(a_1, b_1, \ldots, a_n, b_n\) with \( a_i \neq a_j\) for all \( i \neq j \) such that \(\mathbb{Z} = \bigcup_{i=1}^{n} P_{a_i, b_i}\)?

    \item Given a finite set \( S = \{1, 2, \ldots, n\} \) and a positive integer \( k \), determine the number of \( k \)-tuples \( (i_1, i_2, \ldots, i_k) \) such that \( \{i_1, i_2, \ldots, i_k\} \) is a subset of \( S \) and satisfies \( i_1 < i_2 < \cdots < i_k \) with the additional condition that \( i_2 \geq i_1 + 2 \), \( i_3 \geq i_2 + 2 \), \(\ldots\), \( i_k \geq i_{k-1} + 2 \).

    \item Determine the number of binary strings of length \( n \), where \( n \) is a positive integer, that do not contain the substring \( 1011 \).

    \item Determine the real value \( s \) such that the series
\[
\sum_{n=1}^{\infty} \left[(n+s)\log\left(1 + \frac{1}{n}\right) - 1\right]
\]
converges absolutely. Additionally, analyze the rate of convergence and investigate the limit to which the series converges. Bonus points if you can show that the series equals exactly \( 2 \log 2 + \frac{1}{2} \log \pi - 1 \) for the particular value of \( s\).

    \item Imagine a testing setting with 5000 bottles of wine, exactly one of which is poisoned, and a large number of prisoners available for testing. The goal is to identify the poisoned bottle by giving small amounts of wine to the prisoners.

    Here are the conditions:
    \begin{itemize}
        \item If a prisoner consumes 5 mg of the poisoned wine, they will become very sick.
        \item If a prisoner consumes 10 mg of the poisoned wine, they will die.
        \item Each bottle has an essentially unlimited supply of wine.
        \item The testing results take one hour to appear, and the poisoned bottle must be identified within one hour and ten minutes.
    \end{itemize}
    Given these conditions, what is the minimal number of prisoners needed to identify the poisoned wine bottle?
\end{enumerate}
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