AIME 2006 II · 第 2 题

Solution

By the Triangle Inequality and applying the well-known logarithmic property $\log_{c} a + \log_{c} b = \log_{c} ab$, we have that

$\log_{10} 12 + \log_{10} n > \log_{10} 75$ $\log_{10} 12n > \log_{10} 75$ $12n > 75$ $n > \frac{75}{12} = \frac{25}{4} = 6.25$

Also,

$\log_{10} 12 + \log_{10} 75 > \log_{10} n$ $\log_{10} 12\cdot75 > \log_{10} n$ $n < 900$

Combining these two inequalities:

$$6.25 < n < 900$$ Thus $n$ is in the set $(6.25 , 900)$; the number of positive integer $n$ which satisfies this requirement is $\boxed{893}$.