[ Classical inequalities ]

Specify any positive integer $n$ for which

a) $1.001^{n}>10$;

b) $0.999^{n}<0.1$.
a) According to Bernoulli's inequality (see problem $\underline{30899})\left(1+\frac{1}{1000}\right)^{10000}>1+10000 \cdot \frac{1}{1000}>10$.
b) $\left(1-\frac{1}{1000}\right)\left(1+\frac{1}{1000}\right)<1$, therefore, $\left(1-\frac{1}{1000}\right)^{10000}<\left(1+\frac{1}{1000}\right)^{-10000}<1 / 10$.

## Answer

For example, $n=10000$.

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