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User:Cxsmi

Revision as of 11:37, 25 January 2024 by Cxsmi (talk | contribs) (→‎Problem 3 Solutions)

About Me

Hi! I'm just another guy who happens to enjoy math. I often pop onto the AOPS wiki and look for problems to solve, and I sometimes even write solutions for them! I've starred ⭐ a few of my favorite solutions below; please feel free to take a look at any of them. Thanks for visiting my user page, and enjoy your stay!

Solutions

AIME

  1. 1987 AIME Problem 11 Solution 3 ⭐
  2. 2012 AIME I Problem 2 Solution 3

AMC 8

  1. 2012 AMC 8 Problem 19 Solution 6 ⭐
  2. 2002 AMC 8 Problem 17 Solution 3
  3. 2007 AMC 8 Problem 20 Solution 8
  4. 2018 AMC 8 Problem 23 Solution 5 ⭐
  5. 2016 AMC 8 Problem 13 Solution 3
  6. 2017 AMC 8 Problem 9 Solution 2
  7. 2012 AMC 8 Problem 20 Solution 7
  8. 2010 AMC 8 Problem 25 Solution 3

AJHSME

  1. 1997 AJHSME Problem 22 Solution 1
  2. 1985 AJHSME Problem 1 Solution 2
  3. 1985 AJHSME Problem 24 Solution 2 ⭐
  4. 1985 AJHSME Problem 2 Solution 5

AHSME

  1. 1950 AHSME Problem 40 Solution 2
  2. 1950 AHSME Problem 41 Solution 2
  3. 1972 AHSME Problem 16 Solution 2 ⭐
  4. 1950 AHSME Problem 45 Solution 3

AMC 12

  1. 2021 AMC 12B Problem 12 Solution 6⭐

Significant Problems

Here are some problems that, to me, have been significant on my math journey. This section is mainly for myself, but please please feel free to look at the problems if you're interested.

  1. 2017 AMC 10A Problem 19 - First AMC 10 Solution of Difficulty 2 or Higher
  2. 2007 AMC 8 Problem 25 - First AMC 8 Final Five Solution
  3. 1984 AIME Problem 1 - First AIME Solution
  4. 2005 AMC 12B Problem 16 - First AMC 12 Solution of Difficulty 2.5 or Higher
  5. 2016 AMC 10A Problem 21 - First AMC 10 Final Five Solution
  6. 1987 AIME Problem 11 - First AIME Solution of Difficulty 4 or Higher
  7. 2017 AMC 12A Problem 23 - First AMC 12 Final Five Solution
  8. 2009 HMMT February Algebra Problem 1 - First February HMMT Solution

Problems

I enjoy writing problems when I see concepts that interest me. I've written a few below; please feel free to solve them! Also, feel free to add solutions or change any mistakes you see. Note: Unless otherwise stated, all questions are in the AIME format -- that is, the answer is an integer between 1 and 999.

Problem 1

Find the least positive integer $n$ that satisfies the following. The notation $\lfloor{x}\rfloor$ represents the greatest integer less than or equal to $x$.

$\frac{20^{23}}{n} + \frac{24^{23}}{n} = \lfloor{\frac{20^{23}}{n}+\frac{24^{23}}{n}}\rfloor \neq \lfloor{\frac{20^{23}}{n}}\rfloor + \lfloor{\frac{24^{23}}{n}}\rfloor$

Problem 2

Bob has received a test. He must match a list of $2024$ words to a list of $2024$ definitions such that each word matches to exactly one definition and vice versa. He does not know any of these words, so he will guess randomly. If he does so, he will expect to get $n$ matches. Find the remainder when $n$ is divided by $1000$.

Problem 3

A sequence $f(n)$ is defined recursively as $f(1) = f(2) = f(3) = ... = f(2024) = 1$ and

$f(n) = \sum_{k=1} ^{2024} f(n - k)$

for $n \geq 2025$. The value of $f(4049)$ can be written in the form $a(2)^b+1$ for positive integers $a$ and $b$ such that $b$ is maximized. What is the remainder when $a + b$ is divided by $1000$?

Solutions

These are solutions for the problems above. $\textbf{Scroll down at your own risk!}$

Problem 1 Solutions

Solution 1

We split the condition into two separate conditions, as listed below.


$\frac{20^{23}}{n} + \frac{24^{23}}{n} = \lfloor{\frac{20^{23}}{n}+\frac{24^{23}}{n}}\rfloor$

$\frac{20^{23}}{n} + \frac{24^{23}}{n} \neq \lfloor{\frac{20^{23}}{n}}\rfloor + \lfloor{\frac{24^{23}}{n}}\rfloor$


Rearranging the conditions, we find that


$\frac{20^{23}+24^{23}}{n} - \lfloor \frac{20^{23}+24^{23}}{n}\rfloor = 0$

$(\frac{20^{23}}{n} - \lfloor \frac{20^{23}}{n} \rfloor) + (\frac{24^{23}}{n} - \lfloor \frac{24^{23}}{n} \rfloor) \neq 0$


Recalling that $x - \lfloor {x} \rfloor = frac(x)$ where $frac(x)$ represents the fractional part of $x$, we rewrite once more.


$frac(\frac{20^{23}+24^{23}}{n}) = 0$ $\textbf{(1)}$

$frac(\frac{20^{23}}{n}) + frac(\frac{24^{23}}{n}) \neq 0$ $\textbf{(2)}$


We now gain some valuable insight. From $\textbf{(1)}$, we find that $n$ must divide $20^{23} + 24^{23}$. From $\textbf{(2)}$, we find that $n$ cannot divide both $20^{23}$ and $24^{23}$. It is impossible for $n$ to divide only $1$ of $20^{23}$ and $24^{23}$, as this would make $\textbf{(1)}$ false. It must be that $n$ divides neither $20^{23}$ nor $24^{23}$. For both this and $\textbf{(1)}$ to be true simultaneously, we must have that if $20^{23} \equiv a \bmod n$, then $24^{23} \equiv -a \bmod n$. By inspection, this occurs when $n = 22$.We now test the factors of $22$ to see if we can find a smaller value. As both $20^{23}$ and $24^{23}$ are congruent to $0$ mod $2$, $n = 2$ is not a valid solution. However, with $n = 11$, $20^{23} \equiv (-2)^{23} \bmod 11$, while $24^{23} \equiv 2^{23} \bmod 11$. Clearly, $-(-2)^{23} = 2^{23}$, so our final answer is $\boxed{011}$.

Problem 2 Solutions

Solution 1 (Meta-Solving)

We can consider smaller cases. If there is $1$ object to match, Bob will always make exactly $1$ correct match, so he will expect to make $1$ match. If there are $2$ objects, Bob can either match both correctly for a total of $2$ matches or match both incorrectly for a total of $0$ matches. Each of these has a $\frac{1}{2}$ chance of happening, so again Bob will expect to make $1$ match. If there are $3$ objects, we can list out each combination. Suppose we are matching ${1, 2, 3}$ to itself, and the "solution" is the set of ordered pairs ${(1,1),(2,2), (3,3)}$. The possibilities are listed below:


${(1,1), (2,2), (3,3)}$ (3 correct matches)

${(1,2), (2,1), (3,3)}$ (1 correct match)

${(1,3), (2,2), (3,1)}$ (1 correct match)

${(1,1), (2,3), (3,2)}$ (1 correct match)

${(1,2), (2,3), (3,1)}$ (0 correct matches)

${(1,3), (2,1), (3,2)}$ (0 correct matches)


The expected value is $\frac{1}{6} \cdot (3+1+1+1) = 1$. The expected value has remained at 1 for the first three cases, so we can safely assume that it will remain at $1$ for any positive integer number of items to match. Then Bob can expect to make $1$ match, and the remainder when this is divided by $1000$ is $\boxed{001}$.


Note: This is indeed the correct answer, and it can be found more rigorously. Maybe that solution will come in the future :)

Problem 3 Solutions

Solution 1

First, we note the statements below.


$f(n) = f(n-1) + f(n-2) + f(n-3) + ... + f(n - 2023) + f(n-2024)$ $\textbf{(1)}$ $f(n-1) = f(n-2) + f(n-3) + f(n-4) + ... + f(n - 2024) + f(n - 2025)$ $\textbf{(2)}$


We notice that most of the terms telescope if we subtract $\textbf{(2)}$ from $\textbf{(1)}$.


$f(n) - f(n-1) = (f(n-1) + f(n-2) + f(n-3) + ... + f(n - 2023) + f(n-2024)) - (f(n-2) + f(n-3) + f(n-4) + ... + f(n - 2024) + f(n - 2025))$

$= f(n-1) + (f(n-2) - f(n-2)) + (f(n-3) - f(n-3)) + ... + (f(n-2024) - f(n-2024)) - f(n - 2025)$

$= f(n-1) - f(n-2025)$


By adding $f(n-1)$ to both sides, we find that $f(n) = 2f(n-1) - f(n-2025)$. We can verify by finding $f(2026)$; from the original definition, we find that $f(2025) = 2024$, and $f(2026) = 4047$. From our definition, we find that $f(2026) = 2 \cdot f(2025) - f(1) = 2 \cdot 2024 - 1 = 4047$. Of course, this doesn't guarantee that our definition is indeed correct, but it gives us additional verification to our algebraic method. From here, we can consider $f(4049)$. We note that for all $2026 \leq n \leq 4049$, the second part of our definition (the $f(n-2025)$ term) is equal to one. From here, we can list out a few definitions for $f(4049)$ using our formula.


$f(4049) = 2f(4048)-1 = 2^1f(4048) - (2^1-1)$

$= 2(2f(4047)-1)-1 = 4f(4047) - 3 = 2^2f(4047) - (2^2 - 1)$

$= 2(2(2f(4046)-1)-1)-1 = 8f(4046) - 7 = 2^3f(4046) - (2^3 - 1)$

$= 2(2(2(2f(4045)-1)-1)-1)-1 = 16f(4045) - 15 = 2^4f(4045) - (2^4-1)$

It appears that on the interval $2025 \leq n \leq 4049$, $f(4049) = 2^{4049-n}f(n) - (2^{4049-n} - 1) = 2^{4049-n}(f(n)-1) + 1$. ($2024$ is the upper bound because if we tried to calculate $f(2025)$ using our alternate definition, we'd get $2f(2024) - f(0)$, and $f(0)$ is undefined.) Clearly, to maximize $4049 - n$ (to maximize $b$ in the problem), we choose the lower bound. Then we get $f(4049) = 2^{2024}(f(2025)-1) + 1 = 2^{2024}(2024-1) + 1 = 2^{2024}(2023) + 1$, so $a + b = 4047$. The remainder when this is divided by $1000$ is $\boxed{047}$.

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