Edits

Advanced/Intro to Proofs/main.tex

@@ -1,7 +1,7 @@
 % use [nosolutions] flag to hide solutions.
 % use [solutions] flag to show solutions.
 \documentclass[
-	solutions,
+	nosolutions,
 	singlenumbering
 ]{../../resources/ormc_handout}
 \usepackage{../../resources/macros}
@@ -18,7 +18,7 @@
 	\maketitle
 
 
-
+	\section{}
 
 	\problem{}
 	We say an integer $x$ is \textit{even} if $x = 2k$ for some $k \in \mathbb{Z}$.
@@ -79,7 +79,7 @@
 
 
 	\problem{}
-	Let $X = \{x \in \mathbb{Z} ~\bigl|~ n \geq 2 \}$. For $k \geq 2$, degine $X_k = \{kx ~\bigl|~ x \in X \}$. \par
+	Let $X = \{x \in \mathbb{Z} ~\bigl|~ x \geq 2 \}$. For $k \geq 2$, define $X_k = \{kx ~\bigl|~ x \in X \}$. \par
 	What is $X - (X_2 \cup X_3 \cup X_4 \cup ...)$? Prove your claim.
 
 	\vfill
@@ -91,8 +91,6 @@
 
 
 
-
-
 	\problem{}
 	Show that there are infinitely may primes. \par
 	You may use the fact that every integer has a prime factorization.
@@ -185,12 +183,14 @@
 
 
 
+	\theorem{The Division Algorithm}<divalgo>
+	Given two integers $a, b$, we can find two integers $q, r$, where $0 \leq r < b$ and $a = qb + r$. \par
+	In other words, we can divide $a$ by $b$ to get $q$ remainder $r$.
 
 	\problem{}
 	Let $x, y \in \mathbb{N}$ be natural numbers.
-	Consider the set $S = \{ax + by ~\bigl|~ a, b \in \mathbb{Z}, ax + by = 0\}$. \par
+	Consider the set $S = \{ax + by ~\bigl|~ a, b \in \mathbb{Z}, ax + by > 0\}$. \par
 	The well-ordering principle states that every nonempty subset of the natural numbers has a least element.
-	You many also need the division algorithm.
 
 	\vspace{4mm}
 	\begin{itemize}[itemsep=4mm]
@@ -252,7 +252,7 @@
 		$$
 			(a + b)^N = \sum_{k=0}^N \binom{N}{k}a^kb^{N-k}
 		$$
-		Now, show that this formula also works for $N = N + 1$.
+		Now, show that this formula also works for $N + 1$.
 
 		\item Conclude that this formula works for all $a, b \in \mathbb{R}$ and $n \in \mathbb{Z}^+$\hspace{-0.5ex}.
 	\end{itemize}
@@ -284,4 +284,57 @@
 	\vfill
 	\pagebreak
 
+	\problem{}
+	Show that there exist two positive irrational numbers $a$ and $b$ so that $a^b$ is rational.
+
+	% Solution: a = b = root 2, check all cases
+	% if irrational,a=rt, s = rt^rt, which is rational
+
+	\vfill
+
+	\problem{}
+	Show that the following holds for any planar graph:
+	$$
+	\text{vertices} - \text{edges} + \text{faces} = 2
+	$$
+	\hint{If you don't know what these words mean, ask an instructor.}
+
+
+	\vfill
+	\pagebreak
+
+	\problem{}
+	Consider a rectangular chocolate bar of arbitrary size. \par
+	What is the minimum number of breaks you need to make to
+	seperate all its pieces?
+
+	\begin{solution}
+		number of squares minus one, simple proof by induction.
+	\end{solution}
+
+	\vfill
+
+	\problem{}
+	Four travellers are on a plane, each moving along a straight line at an arbitrary constant speed. \par
+	No two of their paths are parallel, and no three intersect at the same point. \par
+	We know that traveller A has met traveler B, C, and D, and that B has met C and D (and A). \par
+	Show that C and D must also have met.
+
+	\begin{solution}
+		When a body travels at a constant speed, its graph with respect to time is a straight line. \par
+		So, we add time axis in the third dimension, perpendicular to our plane. \par
+		Naturally, the projection of each of these onto the plane corresponds to a road.
+
+		Now, note that two intersecting lines define a plane and use the conditions in the problem to show that no two lines are parallel.
+	\end{solution}
+
+
+	\vfill
+
+
+	\problem{}
+	Say we have an $n$-gon with non-intersecting edges. \par
+	What is the size of the smallet set of vertices that can \say{see} every point inside the polygon?
+	\vfill
+	\pagebreak
 \end{document}