Difference between revisions of "2017 AIME I Problems/Problem 12"
(→Solution 2(PIE)) |
|||
Line 13: | Line 13: | ||
==Solution 2(PIE)== | ==Solution 2(PIE)== | ||
− | We cannot have the following pairs or triplets: <math>\{2, 4\}, \{3, 9}, \{2, 3, 6\}, \{2, 5, 10\}</math>. | + | We cannot have the following pairs or triplets: <math>\{2, 4\}, \{3, 9\}, \{2, 3, 6\}, \{2, 5, 10\}</math>. |
Since there are <math>512</math> subsets(<math>1</math> isn't needed) we have the following: | Since there are <math>512</math> subsets(<math>1</math> isn't needed) we have the following: | ||
<math>(512-(384-160+40-4)) \implies \boxed{252}</math>. | <math>(512-(384-160+40-4)) \implies \boxed{252}</math>. | ||
+ | |||
==See Also== | ==See Also== | ||
{{AIME box|year=2017|n=I|num-b=11|num-a=13}} | {{AIME box|year=2017|n=I|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 20:24, 27 June 2017
Problem 12
Call a set product-free if there do not exist (not necessarily distinct) such that . For example, the empty set and the set are product-free, whereas the sets and are not product-free. Find the number of product-free subsets of the set .
Solution 1(Casework)
We shall solve this problem by doing casework on the lowest element of the subset. Note that the number cannot be in the subset because . Let be a product-free set. If the lowest element of is , we consider the set . We see that 5 of these subsets can be a subset of (, , , , and the empty set). Now consider the set . We see that 3 of these subsets can be a subset of (, , and the empty set). Note that cannot be an element of , because is. Now consider the set . All four of these subsets can be a subset of . So if the smallest element of is , there are possible such sets.
If the smallest element of is , the only restriction we have is that is not in . This leaves us such sets.
If the smallest element of is not or , then can be any subset of , including the empty set. This gives us such subsets.
So our answer is .
Solution 2(PIE)
We cannot have the following pairs or triplets: . Since there are subsets( isn't needed) we have the following: .
See Also
2017 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.