2022 AMC 10A Problems/Problem 14
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Problem
How many ways are there to split the integers through
into
pairs such that in each pair, the greater number is at least
times the lesser number?
Solution
Clearly, the integers from through
must be in different pairs, and
must pair with
Note that can pair with either
or
From here, we consider casework:
- If
pairs with
then
can pair with one of
After that, each of
does not have any restrictions. This case produces
ways.
- If
pairs with
then
can pair with one of
After that, each of
does not have any restrictions. This case produces
ways.
Together, the answer is
~MRENTHUSIASM
Video Solution by OmegaLearn
~ pi_is_3.14
See Also
2022 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2022 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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