2009 AIME II Problems/Problem 11
Problem
For certain pairs of positive integers with there are exactly distinct positive integers such that . Find the sum of all possible values of the product .
Solution
We have , hence we can rewrite the inequality as follows: We can now get rid of the logarithms, obtaining: And this can be rewritten in terms of as
From it follows that the solutions for must be the integers . This will happen if and only if the lower bound on is in a suitable range -- we must have .
Obviously there is no solution for . For the left inequality can be rewritten as , and the right one as .
Remember that we must have . However, for we have , and hence , which is a contradiction. This only leaves us with the cases .
- For we have with a single integer solution .
- For we have with a single integer solution .
- For our inequality has no integer solutions for .
Therefore the answer is .
See Also
2009 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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