1. How many ordered pairs of integers $(a, b)$ satisfy all of the following inequalities?
$$
\begin{array}{l}
a^{2}+b^{2}<16 \\
a^{2}+b^{2}<8 a \\
a^{2}+b^{2}<8 b
\end{array}
$$
Solution:
6
This is easiest to see by simply graphing the inequalities. They correspond to the (strict) interiors of circles of radius 4 and centers at $(0,0),(4,0),(0,4)$, respectively. So we can see that there are 6 lattice points in their intersection (circled in the figure).