User:Crazyvideogamez

AOPS Contributions

AMC

2019 AMC 10B Problems/Problem 18 Solution 5 (Combinatorics)
2020 AMC 10A Problems/Problem 24 Solution 12 (Number Theory)
2019 AMC 10B Problems/Problem 25 Solution 1 (Combinatorics)
2021 Fall AMC 12A Problems/Problem 20 Solution 2 (Number Theory)
2021 Fall AMC 12A Problems/Problem 23 Solution 2.5 (Algebra)

AIME

2011 AIME II Problems/Problem 8 Solution $1.\overline{6}$ (Complex Numbers)

Problems that I enjoyed solving

Problem

Let $k$ be a positive integer. Prove that there are exactly $k$ ordered pairs $(x, y)$ of nonnegative integers that satisfy any one of the following equations:

\begin{align*} x+3y &= 2k-1, \\ 3x + 5y &= 2k - 3, \\ \vdots \\ (2k-1)x + (2k+1)y &= 1. \end{align*}

(Source: Mandelbrot)

My Solution

First, allow us to prove that none of these equations share a solution. Let $i$ and $j$ be any integer from $1$ to $k$ . Then, we solve

\begin{align*} (2i - 1)x + (2i + 1)y = 2k - 2i + 1 &\implies (2i - 1)(x+1) + (2i + 1)y = 2k \\ (2j - 1)x + (2j + 1)y = 2k - 2j + 1 &\implies (2j - 1)(x+1) + (2j + 1)y = 2k \\ \text{Subtracting the }&\text{two equations, we get} \\ (2i - 2j)(x+1) + (2i - 2j) y = 0 &\implies x + y + 1 = 0 \implies x + y = -1 \end{align*}

This clearly has no solutions in nonnegative integers. Therefore, each solution each equation provides is guaranteed to be distinct from one another without any overcounting.

Next, let us represent the number of solutions to each equation as a generating function. The equation

\begin{equation} (2i - 1)(x+1) + (2i + 1)y = 2k \end{equation}

represents a sum of multiples of $2i-1$ and $2i+1$ equaling $2k$ . So, we can represent it as so:

\begin{align*} (m^{2i-1} + m^{2(2i-1)} + m^{3(2i-1)} \ldots)&(1 + m^{2i+1} + m^{2(2i+1)} \ldots) \\ \implies \frac{m^{2i-1}}{1-m^{2i-1}}\cdot&\frac{1}{1 - m^{2i+1}} \end{align*}

where the number of solutions is the coefficient of $m^{2k}$ .

Then, to find the total number of solutions, we sum each of the generating functions corresponding to each equation and derive the coefficient of $m^{2k}$ .

\begin{align*} \sum_{i=1}^{k} \frac{m^{2i-1}}{1-m^{2i-1}}\cdot \frac{1}{1 - m^{2i+1}} &= \sum_{i=1}^{k} m^{2i-1} \Bigl(\frac{1}{1-m^{2i-1}}\cdot\frac{1}{1 - m^{2i+1}}\Bigr) \\ &= \sum_{i=1}^{k} \frac{m^{2i-1}}{1-m^2} \Bigl(\frac{1}{1-m^{2i-1}} - \frac{m^2}{1 - m^{2i+1}}\Bigr) \\ &= \frac{1}{1-m^2} \sum_{i=1}^{k} \frac{m^{2i-1}}{1-m^{2i-1}} - \frac{m^{2i+1}}{1 - m^{2i+1}} \end{align*}

Let $n = 1/m$ . Then, we have

\begin{align*} \frac{1}{1-m^2} \sum_{i=1}^{k} \frac{m^{2i-1}}{1-m^{2i-1}} - \frac{m^{2i+1}}{1 - m^{2i+1}} &= \frac{1}{1-m^2} \sum_{i=1}^{k} \frac{1}{\frac{1}{m^{2i-1}}-1} - \frac{1}{\frac{1}{m^{2i+1}}-1} \\ &= \frac{1}{1-m^2} \sum_{i=1}^{k} \frac{1}{n^{2i-1}-1} - \frac{1}{n^{2i+1}-1} \end{align*}

Notice that

\begin{align*} \sum_{i=1}^{k} \frac{1}{n^{2i-1}-1} - \frac{1}{n^{2i+1}-1} &= \frac{1}{n - 1} - \frac{1}{n^3 - 1} + \frac{1}{n^3 - 1} - \frac{1}{n^5 - 1} + \cdots + \frac{1}{n^{2k-1} - 1} - \frac{1}{n^{2k+1} - 1} \\ &= \frac{1}{n-1} - \frac{1}{n^{2k+1}-1} \end{align*}

Via the factorization $n^{2k+1} - 1 = (n - 1)(n^{2k} + n^{2k-1} + \ldots + 1)$ , we obtain

\begin{align*} \frac{n^{2k} + n^{2k-1} + \ldots + 1 - 1}{n^{2k+1} - 1} &= \frac{n^{2k} + n^{2k-1} + \ldots + n}{n^{2k+1} - 1} \\ &= \frac{m^{2k+1} (n^{2k} + n^{2k-1} + \ldots + n)}{m^{2k+1}(n^{2k+1}-1)} \\ &= \frac{m + m^2 + \ldots + m^{2k}}{1 - m^{2k+1}} \end{align*}

Returning to our original expression, we have

\begin{align*} \frac{1}{1-m^2} \sum_{i=1}^{k} \frac{1}{n^{2i-1}-1} - \frac{1}{n^{2i+1}-1} &= \frac{1}{1-m^2} \cdot \frac{m + m^2 + \ldots + m^{2k}}{1 - m^{2k+1}} \end{align*}

Expanding the infinite geometric series, we see that

\begin{align*} \frac{1}{1-m^{2k+1}} &= 1 + m^{2k+1} + m^{2(2k+1)} + \ldots \\ \frac{1}{1-m^2} &= 1 + m^2 + m^4 + m^6 + \ldots \end{align*}

To find the coefficient of $m^{2k}$ , we can ignore all the powers of $m$ in $\frac{1}{1-m^{2k+1}}$ since they are all above $2k$ . Thus, we must only find the coefficient of $m^{2k}$ in the expression

\begin{equation} (m + m^2 + \cdots + m^{2k})(1 + m^2 + m^4 + m^6 + \cdots) \end{equation}

Each $m^{2k} \cdot 1, m^{2k - 2} \cdot m^2, m^{2k-4} \cdot m^4, \cdots, m^{2} \cdot m^{2k-2}$ (first power of $m$ originating from the first series, second power of $m$ originating from the infinite geometric series) contribute one factor of $m^{2k}$ . There are $k$ of these factors. Therefore, there are exactly $\boxed{k}$ ordered pairs of $(x, y)$ satisfying any of the equations listed in the problem.

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