A3 Two circles, each of radius 5 units long, are drawn in the coordinate plane such that their centres $A$ and $C$ have coordinates $(0,0)$ and $(8,0)$ respectively. How many points where both coordinates are integers are within the intersection of these circles (including its boundary)?
Solution: Let $\mathrm{M}$ be the midpoint of $\mathrm{AC}$ and the circumferences of the circles intersect at points $B$ and $D$. Then $\mathrm{AM}=4$ units and $\mathrm{AMB}$ is a right triangle with $\mathrm{AB}=5$.
Nowing the $3-4-5$ Pythagorean triple, $\mathrm{MB}=3$ units. Therefore, there are 7 lattice points on $\mathrm{BD}$ (including the endpoints). There are two more points on the boundary of the intersection: they have coordinates $(3,0)$ and $(5,0)$. The total is 9 lattice points.
Answer: 9 .