1. For which natural number $x$ is the expression $A=\frac{x^{2016}+2 x^{2015}}{x^{2015}-4 x^{2014}}$ an integer?
Solution. We have,

$$
\begin{aligned}
A & =\frac{x^{2016}+2 x^{2015}}{x^{2015}-4 x^{2014}}=\frac{x^{2016}(x+2)}{x^{2015}(x-4)}=\frac{x(x+2)}{x-4} \\
& =\frac{x^{2}+2 x-8 x+8 x+16-16}{x-4}=\frac{(x-4)^{2}+10(x-4)+34}{x-4} \\
& =(x-4)+10+\frac{34}{x-4}
\end{aligned}
$$

Expression $A$ is an integer if and only if $x-4 \in\{ \pm 1, \pm 2, \pm 17, \pm 34\}$, i.e., if and only if $x \in\{-30,-13,2,3,5,6,21,38\}$.