MathDB

A lattice point in the plane is a point both of whose coordinates are integers. Each lattice point has four neighboring points: upper, lower, left, and right. Let

k

be a circle with radius

r \geq 2

, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle

k

that has a neighboring point lying outside

k

. Similarly, an exterior boundary point is a lattice point lying outside the circle

k

that has a neighboring point lying inside

k

. Prove that there are four more exterior boundary points than interior boundary points.

Problem Statement