2009 AIME II Problems/Problem 11
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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 • Resources) | ||
| Preceded by Problem 10 | Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
