Difference between revisions of "2013 AIME I Problems/Problem 8"
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<math>m + n = 5371</math>, so the answer is <math>\boxed{371}</math>. | <math>m + n = 5371</math>, so the answer is <math>\boxed{371}</math>. | ||
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+ | == See also == | ||
+ | {{AIME box|year=2013|n=I|num-b=7|num-a=9}} |
Revision as of 20:44, 16 March 2013
Problem 8
The domain of the function is a closed interval of length , where and are positive integers and . Find the remainder when the smallest possible sum is divided by 1000.
Solution
The domain of the arcsin function is , so .
For to be an integer, must divide , and . To minimize , should be as small as possible because increasing will decrease , the amount you are subtracting, and increase , the amount you are adding; this also leads to a small which clearly minimizes .
We let equal 3, the smallest factor of that isn't . Then we have
, so the answer is .
See also
2013 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |