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.
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.
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 $k = 1, \dots, 9$?