Difference between revisions of "User:John0512"
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+ | ==2021 AMC 8== | ||
+ | 2021 AMC 8 problems and solutions. The test has not been held, and will never be held. | ||
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+ | ==Problems== | ||
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+ | ==Solutions== | ||
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+ | ==Results== | ||
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+ | Highest Score: 0.00 | ||
+ | Distinguished Honor Roll: 0.00 | ||
+ | Honor Roll: 0.00 | ||
+ | Average Score: 0.00 | ||
+ | Standard Deviation: 0.00 | ||
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==Unnamed Theorem== | ==Unnamed Theorem== | ||
Revision as of 11:53, 26 January 2022
Contents
[hide]2021 AMC 8
2021 AMC 8 problems and solutions. The test has not been held, and will never be held.
Problems
ERROR: Content not found
Solutions
ERROR: Content not found
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.