From c4ff36ac275fec12a2813ffd247ae5a859d34c1b Mon Sep 17 00:00:00 2001 From: Mark Date: Sun, 26 Feb 2023 12:31:27 -0800 Subject: [PATCH] Cleaned up pidgeonhole handout --- Advanced/Pidgeonhole Problems/main.tex | 89 +++++++++++++++++++++----- 1 file changed, 72 insertions(+), 17 deletions(-) diff --git a/Advanced/Pidgeonhole Problems/main.tex b/Advanced/Pidgeonhole Problems/main.tex index 0a2edfe..f2b8d5d 100755 --- a/Advanced/Pidgeonhole Problems/main.tex +++ b/Advanced/Pidgeonhole Problems/main.tex @@ -11,8 +11,8 @@ \begin{document} \maketitle - - + + {Pidgeonhole Problems} {Prepared by Mark on \today} @@ -21,23 +21,28 @@ \problem{} + \note{Difficulty: Easy} Is it possible to cover an equilateral triangle with two smaller equilateral triangles? Why or why not? + \begin{solution} In order to completely cover an equilateral triangle, the two smaller triangles must cover all three vertices. Since the longest length of an equilateral triangle is one of its sides, a smaller triangle cannot cover more than one vertex. Therefore, we cannot completely cover the triangle with two smaller copies. \\ \textcolor{gray}{\textit{Bonus question:}} Can you cover a square with three smaller squares? \end{solution} + \vfill + \problem{} - You are given $n + 1$ integers. Prove that there exist at least two - of them such that their difference is divisible by $n$. + \note{Difficulty: Hard} + You are given $n + 1$ integers. \\ + Prove that at least two of them have a difference divisible by $n$. \begin{solution} - $n | (a-b) \iff a \equiv b \pmod{n}$ \\ + $n~|~(a-b) \iff a \equiv b \pmod{n}$ \\ Let $i_0 ... i_{n+1}$ be our set of integers. If we pick $i_0 ... i_{n+1}$ so that no two have a difference divisible by $n$, we must have $i_0 \not\equiv i_k \pmod{n}$ for all $1 \leq k \leq n+1$. There are $n$ such $i_k$, and there are $n$ equivalence classes mod $n$. \\ @@ -46,11 +51,15 @@ In either case, we can find $a, b$ so that $a \equiv b \pmod{n}$, which implies that $n$ divides $a-b$. \end{solution} + \vfill + \pagebreak + \problem{} - You are given an $8 \times 8$ chess board with a pair of opposite corner squares cut off. You are further given a set of dominoes each equal in size to a pair of the board squares with a common side. Is it possible to tile the board with the dominoes in such a way that all the board squares are covered while the dominoes neither overlap nor stick out? + \note{Difficulty: Easy} + You have an $8 \times 8$ chess board with two opposing corner squares cut off. You also have a set of dominoes, each of which is the size of two squares. Is it possible to completely cover the the board with dominos, so that none overlap nor stick out? \begin{solution} A domino covers two adjacent squares. Adjacent squares have different colors. \\ @@ -60,10 +69,12 @@ Since each domino must cover two colors, you cannot cover the modified board. \end{solution} + \vfill \problem{} + \note{Difficulty: Easy} The ocean covers more than a half of the Earth's surface. Prove that the ocean has at least one pair of antipodal points. \begin{solution} @@ -72,10 +83,12 @@ \textcolor{gray}{\textit{Note:}} This solution isn't very convincing. However, it is unlikely that the students know enough to provide a fully rigorous proof. \end{solution} + \vfill \problem{} + \note{Difficulty: Normal} There are $n > 1$ people at a party. Prove that among them there are at least two people who have the same number of acquaintances at the gathering. (We assume that if A knows B, then B also knows A) \begin{solution} @@ -85,11 +98,14 @@ (Remember, ``knowing'' must be mutual.) \end{solution} + \vfill + \pagebreak \problem{} - Among any five points with integer coordinates in the plane, there exist two such that the center of the line segment that connects them has integer coordinates as well. + \note{Difficulty: Normal} + Pick five points in $\mathbb{R}^2$ with integral coordinates. Show that two of these form a line segment that has an integral midpoint. \begin{solution} Let $e, o$ represent even and odd integers. \\ @@ -102,11 +118,14 @@ \end{solution} + \vfill + \problem{} - Prove that if every point on a straight line is painted either black or white, then there exist three points of the same color such that one is the midpoint of the line segment formed by the other two. + \note{Difficulty: Normal} + Every point on a line is painted black or white. Show that there exist three points of the same color where one is the midpoint of the line segment formed by the other two. \begin{solution} This is a proof by contradiction. We will try to construct a set of points where three points have such an arrangement. \\ @@ -172,48 +191,64 @@ \end{solution} + \vfill + \problem{} - All the points in the plane are painted with either one of two colors. Prove that there exist two points in the plane that have the same color and are located exactly one foot away from each other. + \note{Difficulty: Easy} + Every point on a plane is painted black or white. Show that there exist two points in the plane that have the same color and are located exactly one foot away from each other. \begin{solution} Pick three points that form an equilateral triangle with side length 1. \end{solution} + \vfill + \pagebreak + \problem{} - Each point of a circumference is colored either black or white. Prove that there exist three equally spaced points of the same color. + \note{Difficulty: Normal} + Each point on a circle is colored either black or white. Prove that there exist three equally spaced points of the same color. \begin{solution} This problem is exactly the same as \ref{line_threecolor}. \end{solution} + \vfill \problem{} + \note{Difficulty: Hard} Let n be an integer not divisible by $2$ and $5$. Show that n has a multiple consisting entirely of ones. + \vfill + \problem{} + \note{Difficulty: Brutal} Prove that for any $n > 1$, there exists an integer made of only sevens and zeros that is divisible by $n$. + \vfill \problem{} - You choose $n + 1$ integers between $1$ and $2n$. Show that you must select two co-prime numbers. + \note{Difficulty: Hard} + Choose $n + 1$ integers between $1$ and $2n$. Show that at least two of these are co-prime. + \vfill \problem{} - You choose $n + 1$ integers between $1$ and $2n$. Show you must select two numbers $a$ and $b$ such that $a$ divides $b$. + \note{Difficulty: Hard} + Choose $n + 1$ integers between $1$ and $2n$. Show that you must select two numbers $a$ and $b$ such that $a$ divides $b$. \begin{solution} Split the the set $\{1, ..., 2n\}$ into classes defined by each integer's greatest odd divisor. There will be $n$ classes since there are $\frac{k}{2}$ odd numbers between $1$ and $n$. Because we pick $n + 1$ numbers, at least two will come from the same class---they will be divisible. \\ @@ -228,23 +263,33 @@ \end{itemize} \end{solution} + \vfill + \pagebreak \problem{} - Prove that it is always possible to choose a subset of the set of integral numbers $a_1, a_2, ... , a_n$ so that the sum of the numbers in the subset is divisible by $n$. + \note{Difficulty: Hard} + Show that it is always possible to choose a subset of the set of integers $a_1, a_2, ... , a_n$ so that the sum of the numbers in the subset is divisible by $n$. + + \vfill \problem{} - Prove that there exists a positive integer divisible by $2013$ that has $2014$ as its last for digits. + \note{Difficulty: Hard} + Show that there exists a positive integer divisible by $2013$ that has $2014$ as its last four digits. + + \vfill \problem{} - Let $n$ be an odd number. Let $a_1, a_2, ... , a_n$ be a permutation of the numbers $1, 2, ... , n$. Prove that the product $(a_1 - 1) \times (a_2 - 2) \times ... \times (a_n - n)$ is an even number. + \note{Difficulty: Normal} + Let $n$ be an odd number. Let $a_1, a_2, ... , a_n$ be a permutation of the numbers $1, 2, ... , n$. \\ + Show that $(a_1 - 1) \times (a_2 - 2) \times ... \times (a_n - n)$ is even. \begin{solution} If $n$ is odd, there will be $m$ even and $m + 1$ odd numbers between $1$ and $n$. \\ @@ -253,11 +298,15 @@ The difference of two odd numbers is even, so the product above will have at least one factor of two. \end{solution} + \vfill + \pagebreak + \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. + \note{Difficulty: Hard} + A stressed-out student consumes at least one espresso every day of a particular year, drinking $500$ overall. Show the student drinks exactly $100$ espressos on some consecutive sequence of days. \begin{solution} Rearrange the problem. Don't think about days, think about espressos. Consider the following picture: @@ -315,14 +364,20 @@ \end{solution} + \vfill \problem{} - Prove that at a party with ten or more people, there are either three mutual acquaintances or four mutual strangers. + \note{Difficulty: Hard} + Show that there are either three mutual acquaintances or four mutual strangers at a party with ten or more people. + \vfill \problem{} + \note{Difficulty: Brutal} Given a table with a marked point, $O$, and with $2013$ properly working watches put down on the table, prove that there exists a moment in time when the sum of the distances from $O$ to the watches' centers is less than the sum of the distances from $O$ to the tips of the watches' minute hands. + + \vfill \end{document}