From 678bff430cf06a06be9c7629c0569aa637ed7e75 Mon Sep 17 00:00:00 2001 From: Mark Date: Sun, 15 Feb 2026 10:32:23 -0800 Subject: [PATCH] Generating Functions edits --- src/Advanced/Generating Functions/main.tex | 3 +- .../parts/01 fibonacci.tex | 8 +-- .../Generating Functions/parts/02 dice.tex | 2 +- .../Generating Functions/parts/03 coins.tex | 2 +- .../Generating Functions/parts/04 bonus.tex | 57 +++++++++++++++++++ 5 files changed, 65 insertions(+), 7 deletions(-) create mode 100755 src/Advanced/Generating Functions/parts/04 bonus.tex diff --git a/src/Advanced/Generating Functions/main.tex b/src/Advanced/Generating Functions/main.tex index a670fd7..8066ddb 100755 --- a/src/Advanced/Generating Functions/main.tex +++ b/src/Advanced/Generating Functions/main.tex @@ -1,7 +1,7 @@ % use [nosolutions] flag to hide solutions. % use [solutions] flag to show solutions. \documentclass[ - solutions, + %solutions, singlenumbering ]{../../../lib/tex/handout} \usepackage{../../../lib/tex/macros} @@ -19,4 +19,5 @@ \input{parts/01 fibonacci.tex} \input{parts/02 dice.tex} \input{parts/03 coins.tex} + \input{parts/04 bonus.tex} \end{document} \ No newline at end of file diff --git a/src/Advanced/Generating Functions/parts/01 fibonacci.tex b/src/Advanced/Generating Functions/parts/01 fibonacci.tex index 3c99478..bbaf986 100755 --- a/src/Advanced/Generating Functions/parts/01 fibonacci.tex +++ b/src/Advanced/Generating Functions/parts/01 fibonacci.tex @@ -77,7 +77,7 @@ A \textit{rational function} $f$ is a function that can be written as a quotient That is, $f(x) = \frac{p(x)}{q(x)}$ where $p$ and $q$ are polynomials. \problem{} -Solve the equation from \ref for $F(x)$, expressing it as a rational function. +Solve the equation from \ref{fibo} for $F(x)$, expressing it as a rational function. \begin{solution} \begin{align*} @@ -99,8 +99,8 @@ Solve the equation from \ref for $F(x)$, expressing it as a rational funct \definition{} -\textit{Partial fraction decomposition} is an algebreic technique that works as follows: \par -If $p(x)$ is a polynomial and $a$ and $b$ are constants, +\textit{Partial fraction decomposition} is an algebraic technique that works as follows: \par +If $p(x)$ is a polynomial of degree 1 and $a$ and $b$ are constants, we can rewrite the rational function $\frac{p(x)}{(x-a)(x-b)}$ as follows: \begin{equation*} \frac{p(x)}{(x-a)(x-b)} = \frac{c}{x-a} + \frac{d}{x-b} @@ -131,7 +131,7 @@ find a closed-form expression for its coefficients using partial fraction decomp \problem{} Using problems from the introduction and \ref{pfd}, find an expression -for the coefficients of $F(x)$ (and this, for the Fibonacci numbers). +for the coefficients of $F(x)$ (and thus, for the Fibonacci numbers). \begin{solution} diff --git a/src/Advanced/Generating Functions/parts/02 dice.tex b/src/Advanced/Generating Functions/parts/02 dice.tex index f893350..d0809db 100755 --- a/src/Advanced/Generating Functions/parts/02 dice.tex +++ b/src/Advanced/Generating Functions/parts/02 dice.tex @@ -76,7 +76,7 @@ the probability that the sum of the two dice is $k$. \problem{} Using generating functions, find two six-sided dice whose sum has the same -distribution as the sum of two standard six-sided dice? \par +distribution as the sum of two standard six-sided dice. \par That is, for any integer $k$, the number if ways that the sum of the two nonstandard dice rolls as $k$ is equal to the number of ways the sum of diff --git a/src/Advanced/Generating Functions/parts/03 coins.tex b/src/Advanced/Generating Functions/parts/03 coins.tex index 6ab2e22..e5426ef 100755 --- a/src/Advanced/Generating Functions/parts/03 coins.tex +++ b/src/Advanced/Generating Functions/parts/03 coins.tex @@ -9,7 +9,7 @@ using pennies, nickels, dimes, quarters and half-dollars?} \vspace{2mm} Most ways of solving this involve awkward brute-force -approache that don't reveal anything interesting about the problem: +approaches that don't reveal anything interesting about the problem: how can we change our answer if we want to make change for \$0.51, or \$1.05, or some other quantity? diff --git a/src/Advanced/Generating Functions/parts/04 bonus.tex b/src/Advanced/Generating Functions/parts/04 bonus.tex new file mode 100755 index 0000000..a59aba1 --- /dev/null +++ b/src/Advanced/Generating Functions/parts/04 bonus.tex @@ -0,0 +1,57 @@ +\section{Extra Problems} + + +\problem{USAMO 1996 Problem 6} +Determine (with proof) whether there is a subset $X$ of +the nonnegative integers with the following property: for any nonnegative integer $n$ there is exactly +one solution of $a + 2b = n$ with $a, b \in X$. +(The original USAMO question asked about all integers, not just nonnegative - this is harder, +but still approachable with generating functions.) + + +\vfill + +\problem{IMO Shortlist 1998} +Let $a_0, a_1, ...$ be an increasing sequence of nonnegative integers +such that every nonnegative integer can be +expressed uniquely in the form $a_i + 2a_j + 4a_k$, +where $i, j, k$ are not necessarily distinct. + +Determine $a_1998$. + + +\vfill + +\problem{USAMO 1986 Problem 5} +By a partition $\pi$ of an integer $n \geq 1$, we mean here a +representation of $n$ as a sum of one or more positive integers where the summands must be put in +nondecreasing order. (e.g., if $n = 4$, then the partitions $\pi$ are +$1 + 1 + 1 + 1$, $1 + 1 + 2$, $1 + 3, 2 + 2$, and $4$). + + +For any partition $\pi$, define $A(\pi)$ to be the number of ones which appear in $\pi$, and define $B(\pi)$ +to be the number of distinct integers which appear in $\pi$ (e.g, if $n = 13$ and $\pi$ is the partition +$1 + 1 + 2 + 2 + 2 + 5$, then $A(\pi) = 2$ and $B(\pi) = 3$). + +Show that for any fixed $n$, the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of +$B(\pi)$ over all partitions of $\pi$ of $n$. + +\vfill + +\problem{USAMO 2017 Problem 2} +Let $m_1, m_2, ..., m_n$ be a collection of $n$ distinct positive +integers. For any sequence of integers $A = (a_1, ..., a_n)$ and any permutation $w = w_1, ..., w_n$ of +$m_1, ..., m_n$, define an $A$-inversion of $w$ to be a pair of entries $w_i, w_j$ with $i < j$ for which one of the +following conditions holds: +\begin{itemize} + \item $ai \geq wi > wj$ + \item $wj > ai \geq wi$ + \item $wi > wj > ai$ +\end{itemize} + +Show that for any two sequences of integers $A = (a_1, ..., a_n)$ and $B = (b_1, ..., b_n)$ and for any +positive integer $k$, the number of permutations of $m_1, ..., m_n$ having exactly $k$ $A$-inversions is equal +to the number of permutations of $m_1, ..., m_n$ having exactly $k$ $B$-inversions. +(The original USAMO problem allowed the numbers $m_1, ..., m_n$ to not necessarily be distinct.) + +\vfill \ No newline at end of file