User:John0512
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[hide]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
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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 orz, all 5 clones will reject me, so the answer is . Note that this contradicts with the answer given by the Unnamed Theorem.