# User:John0512

## 2021 AMC 8

2021 AMC 8 problems and solutions. The test has not been held, and will never be held.

## Problems

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## Solutions

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## Results

Highest Score: 0.00

Distinguished Honor Roll: 0.00

Honor Roll: 0.00

Average Score: 0.00

Standard Deviation: 0.00

## Unnamed Theorem

I have something called the Unnamed Theorem (which I did not name as I have not confirmed that this theorem has not existed before).

Claim: Given a set where is a positive integer, the number of ways to choose a subset of then permute said subset is

Proof: The number of ways to choose a subset of size and then permute it is . Therefore, the number of ways to choose any subset of is This is also equal to by symmetry across . This is also Note that is defined as , so our expression becomes We claim that for all positive integers .

Since the reciprocal of a factorial decreases faster than a geometric series, we have that . The right side we can evaluate as , which is always less than or equal to . This means that the terms being subtracted are always strictly less than , so we can simply write it as

Example: How many ways are there 5 distinct clones of mathicorn to each either accept or reject me, then for me to go through the ones that accepted me in some order?

Solution to example: This is equivalent to the Unnamed Theorem for , so our answer is .

Solution 2: Since I am not orz, all 5 clones will reject me, so the answer is . Note that this contradicts with the answer given by the Unnamed Theorem.