handouts/Advanced/Estimathon/main.tex

% use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions.
\documentclass[
	solutions,
	singlenumbering,
	nopagenumber
]{../../resources/ormc_handout}
\usepackage{../../resources/macros}

\uptitlel{Advanced 2}
\uptitler{Winter 2023}
\title{Estimathon}
\subtitle{Prepared by \githref{Mark} on \today{}}


\begin{document}


	\maketitle

	\section{Rules}

	Your team will have 45 minutes to work on 16 estimation problems.
	The answer to each problem is a positive real number. Your team will
	submit intervals for each problem, which (ideally) contain the specified quantity.


	\vspace{2mm}

	An interval is \say{good} if it contains the true value. After the end of the game,
	your team's score will be calculated as follows:

	\begin{equation*}
		\Biggl(10 +\sum_\text{good intervals}\biggl\lfloor\frac{\text{max}}{\text{min}}\biggr\rfloor\Biggr)
		\times 2^{16 ~-~ \text{number of good intervals}}
	\end{equation*}


	For every problem you miss or leave blank, your score doubles. \par
	Your job is to \textbf{minimize} your score.


	\vspace{8mm}

	Every team will get 20 answer sheets. You may use one of these sheets to submit an interval at any time.
	Make sure you write your team name, problem number, and interval (min and max) every time you submit.

	\vspace{2mm}

	There are 16 problems, but you are given 20 answer sheets. You may re-submit your solution to any problem
	(as long as you have sheets remaining). Your latest answer will be kept.

	\vspace{2mm}

	Your interval may not use any mathematical operations except for scientific notation \par
	(for example, $[2 \times 10^2, 3 \times 10^2]$)



	\vfill
	\pagebreak

	\section{Problems}

	\problem{}
	What is the highest posted speed limit in the United States?

	\begin{solution}
		$85$ mi/hr
	\end{solution}

	\problem{}
	How many words are in Isaac Asimov's \textit{Foundation} trilogy?

	\begin{solution}
		About 250,000
	\end{solution}


	\problem{}
	How much horsepower can the average horse produce, disregarding fatigue?

	\begin{solution}
		About 15HP, as measured in 1925.
	\end{solution}

	\problem{}
	What is $\sqrt[100]{2} - 1$?

	\begin{solution}
		$0.06956$
	\end{solution}


	%\problem{}
	%What is the approximate speed of the magnetic north pole's drift? (in km/year)
	%\begin{solution}
	%	60km/yr
	%\end{solution}

	\problem{}
	What was the stock price of Apple on $2023-01-10$?
	\begin{solution}
		$\$186.19$
	\end{solution}

	\problem{}
	How many distinct (non-isomorphic) groups are there on $60$ elements?

	\begin{solution}
		13
	\end{solution}


	\problem{}
	How many undergraduates were enrolled at UCLA in the Fall of 2021?

	\begin{solution}
		32,121
	\end{solution}


\makeatletter
\if@solutions
	\vfill
	\pagebreak
\fi
\makeatother


	%\problem{}
	%How many different creatures are there in \textit{Dwarf Fortress}?

	%\begin{solution}
	%	500 (estimate, no way I'm counting them all)
	%\end{solution}

	\problem{}
	Find the smallest $k > 10$ where
	$
		\sqrt{
			\frac{k!(k+1)!}{2}
		}
	$
	is an integer

	\begin{solution}
		$\frac{k!(k+1!)}{2} = (k!)^2 \times \frac{k+1}{2}$, so $\frac{k+1}{2}$ must be a perfect square. \par
		If $k > 10$, $\frac{k+1}{2} > \frac{11}{2} > 4$. 9 is the next smallest perfect square, so $\frac{k+1}{2} = 9$ and $k =17$.
	\end{solution}


	\problem{}
	How many hours of podcasts has Mark listened to in 3.5 years of driving to UCLA?

	\begin{solution}
		831.4 hours
	\end{solution}


	\problem{}
	For how many positive integers $n$ less than $10,000$ is $2^n - n^2$ divisible by $7$?

	\begin{solution}
		2858
	\end{solution}

	%\problem{}
	%How many Serbian dinars can you exchange for $\$32.53$?
	%\begin{solution}
	%	3,473.02
	%\end{solution}

	\problem{}
	How many lines of code were in the Linux repository in 2022?

	\begin{solution}
		About 27.8 million
	\end{solution}


	\problem{}
	Suppose you drop 16 needles of length 5 on ruled paper with distance 8. \par
	What is the probability that three, four, or five needles cross a line?

	\begin{solution}
		0.316
	\end{solution}


	\problem{}
	How many officially-recognized time zones are there?

	\begin{solution}
		Oddly enough, 38
	\end{solution}


	\problem{}
	What is the smallest number ending in 34, divisible by 34, with a sum of digits equal to 34?
	\begin{solution}
		198934
	\end{solution}


\makeatletter
\if@solutions
	\vfill
	\pagebreak
\fi
\makeatother


	\problem{}
	How many distinct typewriter models have been produced by \textit{Smith Corona} since 1886?

	\begin{solution}
		106
	\end{solution}

	%\problem{}
	%How many people live on Antarctica during the winter?

	%\begin{solution}
	%	About 1100; rises to about 5000 in Summer.
	%\end{solution}


	\problem{}
	What is the standard deviation of the above solutions?

	\begin{solution}
		$7.421 \times 10^6$
	\end{solution}

\end{document}