User:Crazyvideogamez

AOPS Contributions

AMC

1. 2019 AMC 10B Problems/Problem 18 Solution 5 (Combinatorics)
2. 2020 AMC 10A Problems/Problem 24 Solution 12 (Number Theory)
3. 2019 AMC 10B Problems/Problem 25 Solution 1 (Combinatorics)
4. 2021 Fall AMC 12A Problems/Problem 20 Solution 2 (Number Theory)
5. 2021 Fall AMC 12A Problems/Problem 23 Solution 2.5 (Algebra)

AIME

1. 2011 AIME II Problems/Problem 8 Solution (Complex Numbers)
2. 2017 AIME II Problems/Problem 14 Solution 4 (Combinatorics)
3. 2000 AIME I Problems/Problem 15 Solution 3 (Combinatorics)

Problems that I enjoyed solving

Problem

Let be a positive integer. Prove that there are exactly ordered pairs 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 and be any integer from to . 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 and equaling . 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 .

Then, to find the total number of solutions, we sum each of the generating functions corresponding to each equation and derive the coefficient of .

\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 . 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 , 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 , we can ignore all the powers of in since they are all above . Thus, we must only find the coefficient of in the expression

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

Each (first power of originating from the first series, second power of originating from the infinite geometric series) contribute one factor of . There are of these factors. Therefore, there are exactly ordered pairs of satisfying any of the equations listed in the problem.