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A positive integer $n$ is known as an interesting number if $n$ satisfies $$ \left\{\frac{n}{10^k}\right\} > \frac{n}{10^{10}} $$ for all $k=1, 2, \ldots, 9$, where $\{x\}=x - \lfloor x \rfloor$.

Find the number of interesting numbers.

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    What does the Curly brackets stands for? Something is missing perhaps?2011-07-07
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    @Amir Hossein What have you tried so far?2011-07-07
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    @Weaam, the curly brackets are for the *fractional part*, e.g., $\lbrace\pi\rbrace=.14159265\dots$.2011-07-07
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    Amir, can you do a simpler problem, say, $\lbrace n/10^k\rbrace\gt n/10^2$ for $k=1$?2011-07-07

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I'll show how to reduce the problem to a more explicit counting problem.

By definition, $0 \leq \{x\} < 1$. Then $\{x\} > \frac{n}{10^{10}} \implies n <10^{10}$. We may then assume that $n = a_9 a_8 \dots a_0$ so that $$\frac{n}{10^k} = a_9\dots a_k.a_{k-1}\dots a_0$$ In particular, $\{\frac{n}{10^k}\} = 0.a_{k-1}\dots a_0$ and $\frac{n}{10^{10}} = 0.a_9a_8\dots a_0$.

Count the choices of $a_9, a_8, \dots, a_0$ satisfying $0.a_{k-1}...a_0 > 0.a_9a_8\dots a_0$ for all $k = 1, \dots, 9$?