2. In a Cartesian coordinate system, a point whose both vertical and horizontal coordinates are integers is called an integer point. Then, the number of integer points $(x, y)$ that satisfy the inequality $(|x|-1)^{2}+(|y|-1)^{2}<2$ is
A. 16
B. 17
C. 18
D. 25
Answer
As shown in the figure, the equation $(|x|-1)^{2}+(|y|-1)^{2}=2$ corresponds to a figure that is symmetrical about the $x$, $y$ axes, and the origin. In the first quadrant, the figure is a semicircle with $(1,1)$ as the center and $\sqrt{2}$ as the radius. Since there are 16 integer points (excluding the origin) inside the closed figure, the answer is $A$.

