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11
Advanced/Cryptography/main.tex
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@ -5,18 +5,17 @@
singlenumbering singlenumbering
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{multicol} \usepackage{multicol}
\uptitlel{Advanced 1A}
\uptitler{Spring 2022}
\title{Intro to Cryptography}
\subtitle{Prepared by Mark on \today{}}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 1A>
<Spring 2022>
{Intro to Cryptography}
{Prepared by Mark on \today{}}
\input{parts/part 1} \input{parts/part 1}
\input{parts/part 2} \input{parts/part 2}

15
Advanced/DFAs/main.tex
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@ -6,16 +6,17 @@
\input{tikxset.tex} \input{tikxset.tex}
\uptitlel{Advanced 2}
\uptitler{Winter 2022}
\title{Finite Automata}
\subtitle{Prepared by Mark and Nikita on \today{}}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 2>
<Winter 2022>
{Finite Automata}
{Prepared by Mark and Nikita on \today}
\input{parts/0 DFA.tex}
\input{parts/0 DFA.tex} \input{parts/1 regular.tex}
\input{parts/1 regular.tex}
\end{document} \end{document}

15
Advanced/Definable Sets/main.tex
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@ -8,15 +8,14 @@
% for \coloneqq, a centered := % for \coloneqq, a centered :=
\usepackage{mathtools} \usepackage{mathtools}
\begin{document} \uptitlel{Advanced 2}
\maketitle \uptitler{Spring 2023}
<Advanced 2> \title{Definable Sets}
<Spring 2023> \subtitle{Prepared by Mark on \today{}}
{Definable Sets}
{
Prepared by Mark on \today
}
\begin{document}
\maketitle
\input{parts/0 logic.tex} \input{parts/0 logic.tex}
\input{parts/1 structures.tex} \input{parts/1 structures.tex}

19
Advanced/Error-Correcting Codes/main.tex
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@ -1,22 +1,23 @@
% use [nosolutions] flag to hide solutions. % use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions. % use [solutions] flag to show solutions.
\documentclass[ \documentclass[
solutions solutions,
singlenumbering
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
\usepackage{tikz} \usepackage{tikz}
\uptitlel{Advanced 2}
\uptitler{Fall 2022}
\title{Error-Correcting Codes}
\subtitle{
Based on a handout by Yingkun Li \\
Revised by Mark on \today
}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 2>
<Fall 2022>
{Error-Correcting Codes}
{
Based on a handout by Yingkun Li \\
Revised by Mark on \today
}
\input{parts/00 detection} \input{parts/00 detection}
\input{parts/01 correction} \input{parts/01 correction}

22
Advanced/Euler's Number/main.tex
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@ -1,13 +1,10 @@
% use [nosolutions] flag to hide solutions. % use [nosolutions] flag to hide solutions.
% use [solutions] flag to show solutions. % use [solutions] flag to show solutions.
\documentclass[ \documentclass[
solutions,
singlenumbering, singlenumbering,
nosolutions
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{multicol} \usepackage{multicol}
\usepackage{tikz} \usepackage{tikz}
\usepackage{graphicx} \usepackage{graphicx}
@ -17,20 +14,19 @@
\newcommand{\qgt}{\stackrel{?}{>}} \newcommand{\qgt}{\stackrel{?}{>}}
\newcommand{\qlt}{\stackrel{?}{<}} \newcommand{\qlt}{\stackrel{?}{<}}
\uptitlel{Advanced 2}
\uptitler{Fall 2022}
\title{Euler's Number}
\subtitle{
By Oleg Gleizer and Olga Radko. \\
Prepared by Mark on \today
}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 2>
<Fall 2022>
{Euler's Number}
{
By Oleg Gleizer and Olga Radko. \\
Prepared by Mark on \today
}
\paragraph{}
The goal of this mini-course is to construct Euler's number, one of the most important constants in mathematics, physics, economics, and finance. Make sure you fully understand all definitions before trying to solve problems that use them. The goal of this mini-course is to construct Euler's number, one of the most important constants in mathematics, physics, economics, and finance. Make sure you fully understand all definitions before trying to solve problems that use them.
\section{Compound Interest} \section{Compound Interest}

12
Advanced/Graph Algorithms/main.tex
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@ -4,17 +4,17 @@
solutions solutions
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
\input{tikxset} \input{tikxset}
\uptitlel{Advanced 2}
\uptitler{Fall 2022}
\title{Algorithms on Graphs: Flow}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 2>
<Fall 2022>
{Algorithms on Graphs: Flow}
{Prepared by Mark on \today}
\input{parts/00 review} \input{parts/00 review}
\input{parts/01 flow} \input{parts/01 flow}

12
Advanced/Group Theory/main.tex
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@ -6,16 +6,14 @@
\usepackage{tikz} \usepackage{tikz}
\uptitlel{Advanced 2}
\uptitler{Fall 2022}
\title{Group Theory}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 2>
<Fall 2022>
{Group Theory}
{
Prepared by Mark on \today
}
\input{parts/00 review} \input{parts/00 review}
\input{parts/01 groups} \input{parts/01 groups}

12
Advanced/Lambda Calculus/main.tex
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@ -29,16 +29,14 @@
% Notice that I a b = 1 a b! % Notice that I a b = 1 a b!
\uptitlel{Advanced 2}
\uptitler{Fall 2022}
\title{Lambda Calculus}
\subtitle{Prepared by Mark on \today{}}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 2>
<Fall 2022>
{Lambda Calculus}
{
Prepared by Mark on \today
}
\begin{minipage}{8cm} \begin{minipage}{8cm}
Beware of the Turing tar-pit in which everything is possible but nothing of interest is easy. Beware of the Turing tar-pit in which everything is possible but nothing of interest is easy.

16
Advanced/Lattices/main.tex
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@ -6,23 +6,21 @@
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
\usepackage{ifthen} \usepackage{ifthen}
%\usepackage{lua-visual-debug} %\usepackage{lua-visual-debug}
\renewcommand{\arraystretch}{1.2} \renewcommand{\arraystretch}{1.2}
\uptitlel{Advanced 2}
\uptitler{Spring 2023}
\title{Lattices}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 2>
<Spring 2023>
{Lattices}
{
Prepared by Mark on \today \\
}
\input{parts/0 intro.tex} \input{parts/0 intro.tex}
\input{parts/1 minkowski.tex} \input{parts/1 minkowski.tex}
\input{parts/2 orchard.tex} \input{parts/2 orchard.tex}
\end{document} \end{document}

15
Advanced/Linear Algebra 101/main.tex
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@ -2,7 +2,7 @@
% use [solutions] flag to show solutions. % use [solutions] flag to show solutions.
\documentclass[ \documentclass[
solutions, solutions,
nowarning, hidewarning,
%singlenumbering %singlenumbering
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
@ -20,16 +20,15 @@
} }
\input{tikzset} \input{tikzset}
\uptitlel{Advanced 2}
\uptitler{Spring 2023}
\title{Linear Algebra 101}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 2>
<Spring 2023>
{Linear Algebra 101}
{
Prepared by Mark on \today \\
}
\input{parts/0 notation} \input{parts/0 notation}
\input{parts/1 vectors} \input{parts/1 vectors}

17
Advanced/Mock a Mockingbird/main.tex
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@ -63,18 +63,21 @@
\newcommand{\qed}{\(\blacksquare\)} \newcommand{\qed}{\(\blacksquare\)}
\uptitlel{Advanced 2}
\uptitler{Spring 2023}
\title{To Mock a Mockingbird}
\subtitle{
Prepared by Mark on \today \\
Based on a book of the same name.
}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 2>
<Spring 2023>
{To Mock a Mockingbird}
{
Prepared by Mark on \today \\
Based on a book of the same name.
}
\input{parts/00 intro} \input{parts/00 intro}
\input{parts/01 tmam} \input{parts/01 tmam}
\input{parts/02 kestrel} \input{parts/02 kestrel}
\end{document} \end{document}

5
Advanced/Origami/main.tex
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@ -7,6 +7,11 @@
\graphicspath{ {./images/} } \graphicspath{ {./images/} }
\uptitlel{Advanced 2}
\uptitler{Winter 2022}
\title{Origami}
\subtitle{Prepared by everyone on \today}
\begin{document} \begin{document}
\maketitle \maketitle

12
Advanced/Pidgeonhole Problems/main.tex
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@ -8,16 +8,14 @@
\usepackage{amssymb} \usepackage{amssymb}
\usepackage{tikz} \usepackage{tikz}
\uptitlel{Advanced 2}
\uptitler{Winter 2022}
\title{Pidgeonhole Problems}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
<Advanced 2>
<Winter 2023>
{Pidgeonhole Problems}
{Prepared by Mark on \today}
\vspace{3ex}
\problem{} \problem{}

16
Intermediate/An Introduction to Graph Theory/main.tex
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@ -8,17 +8,17 @@
\usepackage{tkz-graph} \usepackage{tkz-graph}
\uptitlel{Intermediate 2}
\uptitler{ORMC Summer Sessions}
\title{An Introduction to Graph Theory}
\subtitle{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\begin{document} \begin{document}
\maketitle \maketitle
<Intermediate 2>
<ORMC Summer Sessions>
{An Introduction to Graph Theory}
{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\section{Graphs} \section{Graphs}

10
Intermediate/Combinatorics/main.tex
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@ -14,14 +14,14 @@
} }
} }
\uptitlel{Intermediate 2}
\uptitler{ORMC Summer Sessions}
\title{Combinatorics}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
<Intermediate 2>
<ORMC Summer Sessions>
{Combinatorics}
{Prepared by Mark on \today}
\section{Getting started} \section{Getting started}
@ -241,7 +241,7 @@
\problem{} \problem{}
A stressed-out student consumes at least one espresso every day of a particular year, drinking $500$ overall. Prove that on some consecutive sequence of whole days the student drinks exactly $100$ espressos. A stressed-out student consumes at least one espresso every day of a particular year, drinking $500$ overall. Prove that on some consecutive sequence of whole days the student drinks exactly $100$ espressos.
\note<Warning>{This problem is significantly harder than anything else in the handout.} \note[Warning]{This problem is significantly harder than anything else in the handout.}
\vfill \vfill

27
Intermediate/Instant Insanity/main.tex
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@ -8,26 +8,27 @@
\usepackage{tkz-graph} \usepackage{tkz-graph}
\uptitlel{Intermediate 2}
\uptitler{ORMC Summer Sessions}
\title{Graph Theory and Instant Insanity}
\subtitle{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\begin{document} \begin{document}
\maketitle \maketitle
<Intermediate 2>
<ORMC Summer Sessions>
{Graph Theory and Instant Insanity}
{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\section{Instant Insanity} \section{Instant Insanity}
The puzzle you have in front of you is called {\it Instant Insanity}. The puzzle you have in front of you is called {\it Instant Insanity}.
It consists of four cubes, with faces colored with four colors: It consists of four cubes, with faces colored with four colors:
red, blue, green, and white. The objective is to put the cubes in a row red, blue, green, and white. The objective is to put the cubes in a row
so that each side, front, back, upper, and lower, so that each side, front, back, upper, and lower,
of the row shows each of the four colors. \\ of the row shows each of the four colors. \\
\begin{center} \begin{center}
\includegraphics[width=2.2in] \includegraphics[width=2.2in]

15
Intermediate/Newton's Laws/main.tex
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@ -5,16 +5,17 @@
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
\uptitlel{Intermediate 2}
\uptitler{ORMC Summer Sessions}
\title{Newton's Laws of Motion}
\subtitle{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\begin{document} \begin{document}
\maketitle \maketitle
<Intermediate 2>
<ORMC Summer Sessions>
{Newton's Laws of Motion}
{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\section{Newton's First Law} \section{Newton's First Law}
If the net force acting on an object is zero, the velocity of that object does not change. \\ If the net force acting on an object is zero, the velocity of that object does not change. \\

551
Intermediate/Probability/main.tex
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@ -11,331 +11,334 @@
\ensuremath{\P{\text{#1}}} \ensuremath{\P{\text{#1}}}
} }
\uptitlel{Intermediate 2}
\uptitler{ORMC Summer Sessions}
\title{Probability}
\subtitle{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\begin{document} \begin{document}
\maketitle \maketitle
<Intermediate 2>
<ORMC Summer Sessions>
{Probability}
{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\problem{}
A Zoo manager thinks of a way
to set up a new pavilion.
He has 7 different plants
and 12 different animals. \\
\begin{itemize} \problem{}
\item How many ways are there to choose two animals and three plants? A Zoo manager thinks of a way
to set up a new pavilion.
He has 7 different plants
and 12 different animals. \\
\begin{itemize}
\item How many ways are there to choose two animals and three plants?
\vfill
\item The manager finds that he only has 5 exhibits available. How many different sets of animals can he choose, if only one can be in each exhibit time?
\vfill
\end{itemize}
\problem{}
How many different 7-symbol license plates are possible if the first three symbols are letters and the remaining four are digits 0-9? \\
\hint{Symbols can repeat, but letters must be uppercase}
\vfill \vfill
\item The manager finds that he only has 5 exhibits available. How many different sets of animals can he choose, if only one can be in each exhibit time?
\pagebreak
\problem{}
There are two plates on a table.
One plate has 10 identical candies,
the other has 8 different fruits. \\
\begin{itemize}
\item How many ways are there to choose one candy?
\vfill
\item How many ways are there to choose seven candies?
\vfill
\item How many ways are there to choose five fruits?
\vfill
\item How many ways are there to choose three candies and six fruits?
\vfill
\item Gregory chooses two fruits and two candies, and lines up the four objects on the table. In how many ways can he do it?
\vfill
\item Gregory chooses five fruits and seven candies, and lines up the twelve objects on the table. In how many ways can he do it?
\vfill
\end{itemize}
\pagebreak
\vspace{190pt}
\pagebreak
\begin{center}
\section{Probability}
\end{center}
\vspace{10pt}
A \textit{probability}, also known as a \textit{chance}, is a number showing how likely some event is to happen. Let us call the event $X$. Then the probability of $X$ taking place is
$$
\P{X} = \frac{
\text{The number of the outcomes such that $X$ happens.}
}{
\text{The number of all the possible outcomes.}
}
$$
Note that by definition, $0 \leq P(X) \leq 1$. \\
In some of the following problems, we will be flipping a coin. Let us use $H$ to represent the event of the coin landing heads, and $T$, the event of the coin landing tails.
\problem{}
Compute each of the following:
\begin{enumerate}
\item \Pt{Rolling a six-sided die and getting 2}
\item \Pt{Flipping a coin twice and getting the sequence HH}
\item \Pt{Flipping a coin twice and getting one head and one tail in any order}
\item \Pt{Rolling two six-sided dice and getting a sum of 5}
\end{enumerate}
\vfill \vfill
\end{itemize} \pagebreak
\problem{} Some parts of the previous problem involve repeated trials: two dice, or two coins. You may have solved these by listing out all the possible outcomes. Though this simple approach works for small problems, it isn't particularly useful for larger ones: ten coin flips create 1024 possible outcomes, and ten dice rolls, 60466174.
How many different 7-symbol license plates are possible if the first three symbols are letters and the remaining four are digits 0-9? \\
\hint{Symbols can repeat, but letters must be uppercase}
\vfill
\pagebreak \medskip
\problem{} A better way to think about repeated trials is as a ``tree,'' where each outcome represents a path. The following tree represents two coin flips, and the four paths down it (from left to right) correspond to the four possible outcomes: HH, HT, TH, TT.
There are two plates on a table.
One plate has 10 identical candies,
the other has 8 different fruits. \\
\begin{itemize} % Ugly hack
\item How many ways are there to choose one candy? \tikzstyle{level 1}=[level distance=3.5cm, sibling distance=3.5cm]
\vfill \tikzstyle{level 2}=[level distance=3.5cm, sibling distance=2cm]
\item How many ways are there to choose seven candies? \tikzstyle{split} = [text width=1em, text centered]
\vfill \tikzstyle{tsplit} = [text width=0, text centered]
\item How many ways are there to choose five fruits? \tikzstyle{end} = [minimum width=3pt, inner sep=0pt]
\vfill
\item How many ways are there to choose three candies and six fruits?
\vfill
\item Gregory chooses two fruits and two candies, and lines up the four objects on the table. In how many ways can he do it?
\vfill
\item Gregory chooses five fruits and seven candies, and lines up the twelve objects on the table. In how many ways can he do it?
\vfill
\end{itemize}
\begin{center}
\pagebreak \begin{tikzpicture}[grow=right]
\node[tsplit] {}
\vspace{190pt}
\pagebreak
\begin{center}
\section{Probability}
\end{center}
\vspace{10pt}
A \textit{probability}, also known as a \textit{chance}, is a number showing how likely some event is to happen. Let us call the event $X$. Then the probability of $X$ taking place is
$$
\P{X} = \frac{
\text{The number of the outcomes such that $X$ happens.}
}{
\text{The number of all the possible outcomes.}
}
$$
Note that by definition, $0 \leq P(X) \leq 1$. \\
In some of the following problems, we will be flipping a coin. Let us use $H$ to represent the event of the coin landing heads, and $T$, the event of the coin landing tails.
\problem{}
Compute each of the following:
\begin{enumerate}
\item \Pt{Rolling a six-sided die and getting 2}
\item \Pt{Flipping a coin twice and getting the sequence HH}
\item \Pt{Flipping a coin twice and getting one head and one tail in any order}
\item \Pt{Rolling two six-sided dice and getting a sum of 5}
\end{enumerate}
\vfill
\pagebreak
Some parts of the previous problem involve repeated trials: two dice, or two coins. You may have solved these by listing out all the possible outcomes. Though this simple approach works for small problems, it isn't particularly useful for larger ones: ten coin flips create 1024 possible outcomes, and ten dice rolls, 60466174.
\medskip
A better way to think about repeated trials is as a ``tree,'' where each outcome represents a path. The following tree represents two coin flips, and the four paths down it (from left to right) correspond to the four possible outcomes: HH, HT, TH, TT.
% Ugly hack
\tikzstyle{level 1}=[level distance=3.5cm, sibling distance=3.5cm]
\tikzstyle{level 2}=[level distance=3.5cm, sibling distance=2cm]
\tikzstyle{split} = [text width=1em, text centered]
\tikzstyle{tsplit} = [text width=0, text centered]
\tikzstyle{end} = [minimum width=3pt, inner sep=0pt]
\begin{center}
\begin{tikzpicture}[grow=right]
\node[tsplit] {}
child {
node[split] {T}
child {
node[end, label=right:{T\ \ \ (TT)}] {}
edge from parent
}
child {
node[end, label=right:{H\ \ \ (TH)}] {}
edge from parent
}
edge from parent
}
child {
node[split] {H}
child { child {
node[end, label=right:{T\ \ \ (HT)}] {} node[split] {T}
child {
node[end, label=right:{T\ \ \ (TT)}] {}
edge from parent
}
child {
node[end, label=right:{H\ \ \ (TH)}] {}
edge from parent
}
edge from parent edge from parent
} }
child {
node[end, label=right:{H\ \ \ (HH)}] {}
edge from parent
}
edge from parent
};
\end{tikzpicture}
\end{center}
If we label each edge with the probability of each event, we can calculate the probability of each outcome by multiplying the edges we passed:
\begin{center}
\begin{tikzpicture}[grow=right]
\node[tsplit] {}
child {
node[split] {T}
child {
node[end, label=right:{T\ \ \ ({TT, $\frac{1}{4}$})}] {}
edge from parent
node[below] {$\frac{1}{2}$}
}
child {
node[end, label=right:{H\ \ \ ({TH, $\frac{1}{4}$})}] {}
edge from parent
node[below] {$\frac{1}{2}$}
}
edge from parent
node[below] {$\frac{1}{2}$}
}
child {
node[split] {H}
child { child {
node[end, label=right:{T\ \ \ ({HT, $\frac{1}{4}$})}] {} node[split] {H}
edge from parent
node[below] {$\frac{1}{2}$}
}
child { child {
node[end, label=right:{H\ \ \ ({HH, $\frac{1}{4}$})}] {} node[end, label=right:{T\ \ \ (HT)}] {}
edge from parent
}
child {
node[end, label=right:{H\ \ \ (HH)}] {}
edge from parent
}
edge from parent
};
\end{tikzpicture}
\end{center}
If we label each edge with the probability of each event, we can calculate the probability of each outcome by multiplying the edges we passed:
\begin{center}
\begin{tikzpicture}[grow=right]
\node[tsplit] {}
child {
node[split] {T}
child {
node[end, label=right:{T\ \ \ ({TT, $\frac{1}{4}$})}] {}
edge from parent
node[below] {$\frac{1}{2}$}
}
child {
node[end, label=right:{H\ \ \ ({TH, $\frac{1}{4}$})}] {}
edge from parent
node[below] {$\frac{1}{2}$}
}
edge from parent edge from parent
node[below] {$\frac{1}{2}$} node[below] {$\frac{1}{2}$}
} }
edge from parent child {
node[below] {$\frac{1}{2}$} node[split] {H}
}; child {
\end{tikzpicture} node[end, label=right:{T\ \ \ ({HT, $\frac{1}{4}$})}] {}
\end{center} edge from parent
node[below] {$\frac{1}{2}$}
}
child {
node[end, label=right:{H\ \ \ ({HH, $\frac{1}{4}$})}] {}
edge from parent
node[below] {$\frac{1}{2}$}
}
edge from parent
node[below] {$\frac{1}{2}$}
};
\end{tikzpicture}
\end{center}
We can formalize this idea as follows: We can formalize this idea as follows:
\proposition{} \proposition{}
If we have two independent events $A$ and $B$, then $\Pt{A and B} = \P{A} \times \P{B}$. \\ If we have two independent events $A$ and $B$, then $\Pt{A and B} = \P{A} \times \P{B}$. \\
Usually we write $\Pt{A and B}$ as $\P{A \cap B}$. \\ Usually we write $\Pt{A and B}$ as $\P{A \cap B}$. \\
\vfill \vfill
Here's another important thought: Here's another important thought:
\proposition{} \proposition{}
If the probability of event $A$ happening is $\P{A}$, the probability of $A$ \textit{not} happening is $1 - \P{A}$ If the probability of event $A$ happening is $\P{A}$, the probability of $A$ \textit{not} happening is $1 - \P{A}$
\pagebreak \pagebreak
\problem{} \problem{}
There are three cans of white paint and two cans of black paint in a dark storage room. You take two cans out without looking. What is the probability that you'll choose two cans of the same color? There are three cans of white paint and two cans of black paint in a dark storage room. You take two cans out without looking. What is the probability that you'll choose two cans of the same color?
\vfill \vfill
\problem{} \problem{}
Hospital records show that of patients Hospital records show that of patients
suffering from a certain disease, suffering from a certain disease,
75\% die of it. What is the probability 75\% die of it. What is the probability
that of 5 randomly selected patients, that of 5 randomly selected patients,
4 will recover? \\ 4 will recover? \\
\hint{What is the probability of a patient recovering?} \hint{What is the probability of a patient recovering?}
\vfill \vfill
\problem{} \problem{}
When Oleg calls his daughter Anya, When Oleg calls his daughter Anya,
the chance of the call getting through is 60\%. the chance of the call getting through is 60\%.
How likely is it to have at least one connection How likely is it to have at least one connection
in four calls? in four calls?
\vfill \vfill
\problem{} \problem{}
The chance of a runner to improve The chance of a runner to improve
his own personal record in a race is $p$. his own personal record in a race is $p$.
What is the probability that his record will improve after 3 races? What is the probability that his record will improve after 3 races?
\vfill \vfill
\newpage \newpage
\problem{} \problem{}
You toss a pair of fair dice five times. You toss a pair of fair dice five times.
What is the probability that you get a sum of ten exactly two times? What is the probability that you get a sum of ten exactly two times?
\vfill \vfill
\problem{} \problem{}
You toss a pair of fair dice five times. You toss a pair of fair dice five times.
What is the probability that you get ten What is the probability that you get ten
at least twice? at least twice?
\vfill \vfill
\problem{} \problem{}
A fair coin is tossed 4 times. What is the chance of getting more heads than tails? A fair coin is tossed 4 times. What is the chance of getting more heads than tails?
\vfill \vfill
\problem{} \problem{}
A pharmaceutical study shows that a new drug causes negative side effects in 3 of every 100 patients. A pharmaceutical study shows that a new drug causes negative side effects in 3 of every 100 patients.
To check the number, a researcher chooses 5 random people to survey. To check the number, a researcher chooses 5 random people to survey.
Assuming the study is accurate, what is the probabilty of the following? \\ Assuming the study is accurate, what is the probabilty of the following? \\
\begin{enumerate} \begin{enumerate}
\item None of the five patients experience side effects. \item None of the five patients experience side effects.
\item At least two experience side effects. \item At least two experience side effects.
\end{enumerate} \end{enumerate}
\vfill \vfill
\pagebreak \pagebreak
\problem{} \problem{}
You pick up a natural number (positive integer) You pick up a natural number (positive integer)
at random. What is the probability at random. What is the probability
that the number is divisible by either two that the number is divisible by either two
or three? or three?
\vfill \vfill
\problem{} \problem{}
Three players are tossing a fair coin. Three players are tossing a fair coin.
The first to have a HEAD wins. The first to have a HEAD wins.
What are the players' chances of winning? What are the players' chances of winning?
\vfill \vfill
\section{Harder Probabilities} \section{Harder Probabilities}
\problem{} \problem{}
Oleg wrote ten letters to Math Circle parents Oleg wrote ten letters to Math Circle parents
and addressed the ten envelopes. However, he and addressed the ten envelopes. However, he
left the final stages of mailing to a careless clerk who left the final stages of mailing to a careless clerk who
didn't pay attention, inserting the letters didn't pay attention, inserting the letters
into the envelopes at random. into the envelopes at random.
(However, he did manage to fit exactly (However, he did manage to fit exactly
one letter in each envelope.) one letter in each envelope.)
What is the probability that exactly nine of What is the probability that exactly nine of
the ten letters is correctly addressed? the ten letters is correctly addressed?
\vfill \vfill
\problem{} \problem{}
On a sold-out flight, the first person to On a sold-out flight, the first person to
board the plane forgot which seat was his board the plane forgot which seat was his
and chose a random seat. Subsequent passengers and chose a random seat. Subsequent passengers
took their assigned seat if available, or a took their assigned seat if available, or a
randomly chosen seat if not. When the last randomly chosen seat if not. When the last
person boards, there is only one seat left. person boards, there is only one seat left.
What is the probability that this was the What is the probability that this was the
seat assigned to the last passenger? seat assigned to the last passenger?
\vfill \vfill
\pagebreak \pagebreak
\problem{} \problem{}
Your new neighbor has two children. Your new neighbor has two children.
Assuming that it is equally likely Assuming that it is equally likely
for a child to be a boy or a girl, for a child to be a boy or a girl,
what is the probability that both what is the probability that both
of the neighbor's children are girls? of the neighbor's children are girls?
Does the probability change if you Does the probability change if you
discover that one of the children discover that one of the children
is indeed a girl? If so, how? is indeed a girl? If so, how?
\vfill \vfill
\problem{} \problem{}
A bag contains a marble which is either A bag contains a marble which is either
black or white --- but we don't know which. black or white --- but we don't know which.
We put a white marble into the bag and shake it. We put a white marble into the bag and shake it.
We then draw out a white marble. We then draw out a white marble.
What is the probability that the marble What is the probability that the marble
left inside the bag is also white? left inside the bag is also white?
\vfill \vfill
\problem{The Monty Hall Problem} \problem{The Monty Hall Problem}
You are a contestant on a certain game show. You are a contestant on a certain game show.
There are three doors. There are three doors.
Behind one door is a brand-new car. Behind one door is a brand-new car.
Behind the other two doors are goats. Behind the other two doors are goats.
You are invited to choose one of the doors. You are invited to choose one of the doors.
Before opening the selected door, the game show host Before opening the selected door, the game show host
opens one of the other two doors, revealing a goat. opens one of the other two doors, revealing a goat.
You can now either keep your (original) You can now either keep your (original)
choice, or switch to the other unopened door. choice, or switch to the other unopened door.
Which choice gives you a better chance of winning Which choice gives you a better chance of winning
the car? Does it matter? Explain your answer. the car? Does it matter? Explain your answer.
\vfill \vfill
\end{document} \end{document}

14
Intermediate/Slide Rules/main.tex
View File

@ -6,6 +6,7 @@
\usepackage{pdfpages} \usepackage{pdfpages}
\usepackage{sliderule} \usepackage{sliderule}
\usepackage{changepage}
% Args: % Args:
% x, top scale y, label % x, top scale y, label
@ -22,15 +23,16 @@
node [below] {#3}; node [below] {#3};
} }
\uptitlel{Intermediate 2}
\uptitler{Summer 2022}
\title{Slide Rules}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
<Intermediate 2>
<Summer 2022>
{Slide Rules}
{
Prepared by Mark on \today
}
\begin{center} \begin{center}
\begin{minipage}{6cm} \begin{minipage}{6cm}

31
Intermediate/Vectors 1/main.tex
View File

@ -3,23 +3,18 @@
\documentclass[solutions]{../../resources/ormc_handout} \documentclass[solutions]{../../resources/ormc_handout}
\usepackage{adjustbox} \usepackage{adjustbox}
\uptitlel{Intermediate 2}
\uptitler{ORMC Summer Sessions}
\title{Vectors 1}
\subtitle{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\begin{document} \begin{document}
\begin{adjustbox}{minipage=0.7\textwidth, margin=0pt \smallskipamount,center} \maketitle
\begin{center}
\textsc{Intermediate 2 \hfill ORMC Summer Sessions} \\
\rule{\linewidth}{0.2mm}\\
\huge
Vectors 1\\
\normalsize
\vspace{1ex}
Prepared by Mark on \today. \\
Based on a handout by Oleg Gleizer.
\rule{\linewidth}{0.2mm}\\
\end{center}
\end{adjustbox}
\section{Warm-Up} \section{Warm-Up}
@ -113,11 +108,11 @@
In other words, two vectors are equivalent if they have the same length and direction. If this is the case, we write $v = w$. In other words, two vectors are equivalent if they have the same length and direction. If this is the case, we write $v = w$.
\note<Note 1>{ \note[Note 1]{
Convince yourself that this is true. Why are these two definitions of vector equivalence interchangeable? Convince yourself that this is true. Why are these two definitions of vector equivalence interchangeable?
} }
\note<Note 2>{ \note[Note 2]{
A vector is characterized by its direction and length. One cannot make a formal definition out of this observation, because a ``direction'' is formally defined in terms of a vector. A vector is characterized by its direction and length. One cannot make a formal definition out of this observation, because a ``direction'' is formally defined in terms of a vector.
} }
@ -450,7 +445,7 @@
\vfill \vfill
\note<Note>{ \note[Note]{
With the tools we have thus far, we can multiply vectors by any rational number using only a compass and a ruler. Multiplying a vector by an irrational number is a bit more tricky, but it is doable... With the tools we have thus far, we can multiply vectors by any rational number using only a compass and a ruler. Multiplying a vector by an irrational number is a bit more tricky, but it is doable...
} }

17
Intermediate/Vectors 2/main.tex
View File

@ -3,18 +3,17 @@
\documentclass[solutions]{../../resources/ormc_handout} \documentclass[solutions]{../../resources/ormc_handout}
\uptitlel{Intermediate 2}
\uptitler{ORMC Summer Sessions}
\title{Vectors 2}
\subtitle{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\begin{document} \begin{document}
\maketitle \maketitle
<Intermediate 2>
<ORMC Summer Sessions>
{Vectors 2}
{
Prepared by Mark on \today \\
Based on a handout by Oleg Gleizer
}
\section{Review} \section{Review}

20
Misc/Moscow Puzzles/main.tex
View File

@ -8,18 +8,20 @@
\usepackage{graphicx} \usepackage{graphicx}
\graphicspath{ {./images} } \graphicspath{ {./images} }
\uptitlel{Intermediate 2}
\uptitler{ORMC Summer Sessions}
\title{Warm - Up}
\subtitle{
Prepared by Mark on \today \\
\medskip
These problems were originally found in \\
Boris Kordemsky's \textit{The Moscow Puzzles}
}
\begin{document} \begin{document}
\maketitle \maketitle
<Intermediate 2>
<ORMC Summer Sessions>
{Warm - Up}
{
Prepared by Mark on \today \\
\medskip
These problems were originally found in \\
Boris Kordemsky's \textit{The Moscow Puzzles}
}
\subfile{problems/217} \subfile{problems/217}
\vfill \vfill

11
Misc/Warm-Ups/ast.tex
View File

@ -1,19 +1,20 @@
\documentclass[ \documentclass[
solutions, solutions,
nowarning, hidewarning,
singlenumbering, singlenumbering,
nopagenumber nopagenumber
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
\usepackage[linguistics]{forest} \usepackage[linguistics]{forest}
\title{Warm-Up: What's an AST?}
\subtitle{Prepared by Mark on \today. \\ Based on a true story.}
\begin{document} \begin{document}
\maketitle \maketitle
{Warm-Up: What's an AST?}
{Prepared by Mark on \today. \\ Based on a true story.}
\vspace{2mm}
Say you have a valid string of simple arithmetic that contains no unary operators (like $3!$ or $-4$) and no parenthesis: Say you have a valid string of simple arithmetic that contains no unary operators (like $3!$ or $-4$) and no parenthesis:
$$ $$

10
Misc/Warm-Ups/electician.tex
View File

@ -1,6 +1,6 @@
\documentclass[ \documentclass[
solutions, solutions,
nowarning, hidewarning,
singlenumbering, singlenumbering,
nopagenumber nopagenumber
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
@ -20,13 +20,15 @@
} }
} }
\title{The Electrician's Warm-Up}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
{The Electrician's Warm-Up}
{Prepared by Mark on \today}
\vspace{2mm}
Ivan the electician is working in an apartment. He has a box of switches, which come in three types: Ivan the electician is working in an apartment. He has a box of switches, which come in three types:
\begin{center} \begin{center}

11
Misc/Warm-Ups/median.tex
View File

@ -1,17 +1,20 @@
\documentclass[ \documentclass[
nosolutions, solutions,
warning, hidewarning,
singlenumbering, singlenumbering,
nopagenumber nopagenumber
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
\usepackage[linguistics]{forest} \usepackage[linguistics]{forest}
\title{Warm-Up: A Familiar Concept}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
{Warm-Up: A Familiar Concept}
{Prepared by Mark on \today.}
\problem{}<one> \problem{}<one>

10
Misc/Warm-Ups/raid.tex
View File

@ -1,19 +1,21 @@
\documentclass[ \documentclass[
solutions, solutions,
nowarning, hidewarning,
singlenumbering, singlenumbering,
nopagenumber nopagenumber
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
\usepackage{tikz} \usepackage{tikz}
\title{The Sysadmin's Warm-Up}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
{The Sysadmin's Warm-Up}
{Prepared by Mark on \today}
\vspace{2mm}
Most of you have seen a hard drive. Many have touched one, and a lucky few have poked around inside one. These devices have two interesting properties: Most of you have seen a hard drive. Many have touched one, and a lucky few have poked around inside one. These devices have two interesting properties:

9
Misc/Warm-Ups/regex.tex
View File

@ -1,6 +1,6 @@
\documentclass[ \documentclass[
solutions, solutions,
nowarning hidewarning,
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
@ -11,13 +11,14 @@
\sethlcolor{Light} \sethlcolor{Light}
\newcommand{\htexttt}[1]{\texttt{\hl{#1}}} \newcommand{\htexttt}[1]{\texttt{\hl{#1}}}
\title{The Regex Warm-Up}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
{The Regex Warm-Up}
{Prepared by Mark on \today}
\vspace{2mm}
Last time, we discussed Deterministic Finite Automata. One interesting application of these mathematical objects is found in computer science: Regular Expressions. \\ Last time, we discussed Deterministic Finite Automata. One interesting application of these mathematical objects is found in computer science: Regular Expressions. \\
(abbreviated \say{regex}, which is pronounced like \say{gif}) (abbreviated \say{regex}, which is pronounced like \say{gif})

11
Misc/Warm-Ups/sin.tex
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@ -1,19 +1,18 @@
\documentclass[ \documentclass[
nosolutions, solutions,
warning,
singlenumbering, singlenumbering,
nopagenumber nopagenumber
]{../../resources/ormc_handout} ]{../../resources/ormc_handout}
\usepackage[linguistics]{forest} \usepackage[linguistics]{forest}
\title{Warm-Up: Exact answers}
\subtitle{Prepared by Mark on \today}
\begin{document} \begin{document}
\maketitle \maketitle
{Warm-Up: Exact answers}
{Prepared by Mark on \today.}
\vspace{2mm}
Compute the exact value of $\sin(x^\circ)$ for as many integers $x \in [0, 90]$ as you can. Compute the exact value of $\sin(x^\circ)$ for as many integers $x \in [0, 90]$ as you can.

8
Problems/all.tex
View File

@ -1,6 +1,6 @@
\documentclass[ \documentclass[
solutions, solutions,
nowarning hidewarning,
]{../resources/ormc_handout} ]{../resources/ormc_handout}
@ -88,6 +88,12 @@
} }
\title{Mark's Problem Library}
\subtitle{
This document lists all problems in this library. \\
Use it to find problems or debug the source.
}
\begin{document} \begin{document}
\maketitle \maketitle
{Mark's Problem Library} {Mark's Problem Library}