2009 AIME II Problems/Problem 11

Problem

For certain pairs $(m,n)$ of positive integers with $m\geq n$ there are exactly $50$ distinct positive integers $k$ such that $|\log m - \log k| < \log n$ . Find the sum of all possible values of the product $mn$ .

Solution

We have $\log m - \log k = \log \left( \frac mk \right)$ , hence we can rewrite the inequality as follows: $- \log n < \log \left( \frac mk \right) < \log n$ We can now get rid of the logarithms, obtaining: $\frac 1n < \frac mk < n$ And this can be rewritten in terms of $k$ as $\frac mn < k < mn$

From $k<mn$ it follows that the $50$ solutions for $k$ must be the integers $mn-1, mn-2, \dots, mn-50$ . This will happen if and only if the lower bound on $k$ is in a suitable range -- we must have $mn-51 \leq \frac mn < mn-50$ .

Obviously there is no solution for $n=1$ . For $n>1$ the left inequality can be rewritten as $m\leq\dfrac{51n}{n^2-1}$ , and the right one as $m > \dfrac{50n}{n^2-1}$ .

Remember that we must have $m\geq n$ . However, for $n\geq 8$ we have $\dfrac{51n}{n^2-1} < n$ , and hence $m<n$ , which is a contradiction. This only leaves us with the cases $n\in\{2,3,4,5,6,7\}$ .

For $n=2$ we have $\dfrac{100}3 < m \leq \dfrac{102}3$ with a single integer solution $m=\dfrac{102}3=34$ .
For $n=3$ we have $\dfrac{150}8 < m \leq \dfrac{153}8$ with a single integer solution $m=\dfrac{152}8=19$ .
For $n=4,5,6,7$ our inequality has no integer solutions for $m$ .

Therefore the answer is $34\cdot 2 + 19\cdot 3 = 68 + 57 = \boxed{125}$ .

2009 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 10	Followed by Problem 12
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2009 AIME II Problems/Problem 11

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