Bakayev E.B.

a) Natural numbers $x, x^{2}$ and $x^{3}$ start with the same digit. Is this digit necessarily one?

b) The same question for natural numbers $x, x^{2}, x^{3}, \ldots, x^{2015}$.
a) The numbers 99, $99^{2}=9801, 99^{3}=970299$ start with nine.

b) Obviously, there exists a natural number $k$ such that $10^{-k} \leq 1-\sqrt[2015]{0.9}$. Then the inequality $0.9 \leq\left(1-10^{-k}\right)^{2015}$ $\leq\left(1-10^{-k}\right)^{n}<1$ holds for all $n \leq 2015$. Multiplying by $10^{k n}$, we get $0.9 \cdot 10^{k n} \leq\left(10^{k}-1\right)^{n}<10^{k n}$. This
means that for $x=10^{k}-1$, all numbers of the form $x^{n}$, where $n=1,2, \ldots, 2015$, start with nine.

## Answer

Not necessarily.