IMO Shortlist 1977 problem 2
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 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
be a circle with radius
, that does not pass through any lattice point. An interior boundary point is a lattice point lying inside the circle
that has a neighboring point lying outside
. Similarly, an exterior boundary point is a lattice point lying outside the circle
that has a neighboring point lying inside
. Prove that there are four more exterior boundary points than interior boundary points.
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![r \geq 2](/media/m/1/c/a/1ca33aa1729b44ca8c6cd04194fc5e25.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1977