Difference between revisions of "2021 AIME I Problems/Problem 7"
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Revision as of 19:57, 11 March 2021
Problem
Find the number of pairs 
of positive integers with 
such that there exists a real number 
satisfying![\[\sin(mx)+\sin(nx)=2.\]](http://latex.artofproblemsolving.com/3/d/c/3dc7189bf3226f56ef411df7b580c6778ace2c72.png)
Solution
Since 
, 
means that each of 
and 
must be exactly 
. Then 
and 
must be cycles away, or the difference between them must be multiple of 
. If is , then can be 
. Like this, the table below can be listed:
| Range of
|
Number of Possibilities | |
|---|---|---|
| Case 1 | 
|
|
| Case 2 | 
|
|
| Case 3 | 
|
|
| Case 4 | 
|
|
| Case 5 | 
|
|
| Case 6 | 
|
|
| Case 7 | 
|
|
| Case 8 | 
|
|
In total, there are 
possible solutions.
However the answer is 
, where is the last possible solution?
~Interstigation
See also
| 2021 AIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
