AIME 2009 II · 第 11 题

Solution 1

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 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 $m, which is a contradiction. This only leaves us with the cases$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 $34\cdot 2 + 19\cdot 3 = 68 + 57 = \boxed{125}$.