1. What is the maximum number of lattice points (i.e. points with integer coordinates) in the plane that can be contained strictly inside a circle of radius 1 ?
Solution: 4. The circle centered at $(1 / 2,1 / 2)$ shows that 4 is achievable. On the other hand, no two points within the circle can be at a mutual distance of 2 or greater. If there are more than four lattice points, classify all such points by the parity of their coordinates: (even, even), (even, odd), (odd, even), or (odd, odd). Then some two points lie in the same class. Since they are distinct, this means either their first or second coordinates must differ by at least 2 , so their distance is at least 2 , a contradiction.