3. Solve the equation $\left[\log _{2015}\left[\log _{2015} x\right]\right]=1$, where $[a]$ represents the integer part of the real number a.
Solution.

The equation is defined for $x \in [2015, \infty)$. (1p)

We have

$$
\begin{array}{r}
\left.\left[\log _{2015}\left[\log _{2015} x\right]\right]=1 \Leftrightarrow 1 \leqslant \log _{2015}\left[\log _{2015} x\right]<2 \quad \mathbf{1} \mathbf{p}\right) \\
\left.\Leftrightarrow 2015 \leqslant\left[\log _{2015} x\right]<2015^{2} . \mathbf{( 1 p}\right) \\
\Leftrightarrow\left[\log _{2015} x\right] \in\left\{2015,2016, \ldots, 2015^{2}-1\right\} \Leftrightarrow \log _{2015} x \in\left[2015,2015^{2}\right)
\end{array}
$$

It follows that the set of solutions to the equation is

$$
S=\left[2015^{2015}, 2015^{2015^{2}}\right)
$$