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](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
be a circle with radius
![r \geq 2](/media/m/1/c/a/1ca33aa1729b44ca8c6cd04194fc5e25.png)
, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
that has a neighboring point lying outside
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
. Similarly, an exterior boundary point is a lattice point lying outside the circle
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
that has a neighboring point lying inside
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
. Prove that there are four more exterior boundary points than interior boundary points.
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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.