1. Find the number of integer points $(x, y)$ satisfying the equation $\frac{1}{|x|}+\frac{1}{|y|}=\frac{1}{2017}$.
Solution. Transform the original equation $\frac{1}{|x|}+\frac{1}{|y|}=\frac{1}{2017} \Rightarrow$

$2017(|x|+|y|)-|x||y|=0 \Rightarrow 2017 \cdot|x|+2017 \cdot|y|-|x||y|-2017^{2}=-2017^{2}$, from which it follows that $(|x|-2017)(|y|-2017)=2017^{2}$. Since 2017 is a prime number and $|x|$ and $|y|$ are natural numbers, the last equation is equivalent to the union of systems of equations $\left\{\begin{array}{c}|x|-2017=1 \\ |y|-2017=2017^{2}\end{array},\left\{\begin{array}{c}|x|-2017=2017 \\ |y|-2017=2017\end{array},\left\{\begin{array}{c}|x|-2017=2017^{2} \\ |y|-2017=1\end{array}\right.\right.\right.$. Each system of equations has four distinct solutions that do not coincide with the solutions of the other systems. Therefore, the number of points sought is 12.

Answer. 12.

Recommendations for checking. The factorization of the form $(|x|-2017)(|y|-2017)=2017^{2}$ or an equivalent form within another solution method - 3 points.