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. | ||
− | + | ==Problems== | |
+ | ERROR: Content not found | ||
+ | ==Solutions== | ||
+ | ERROR: Content not found | ||
+ | ==Results== | ||
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+ | Highest Score: 0.00 | ||
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+ | Distinguished Honor Roll: 0.00 | ||
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+ | Honor Roll: 0.00 | ||
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+ | Average Score: 0.00 | ||
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+ | Standard Deviation: 0.00 | ||
==Unnamed Theorem== | ==Unnamed Theorem== |
Latest revision as of 03:03, 12 March 2023
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.