Precise Control Over The Number Of Lattice Points

Geometry Level 3

We call two points in the Cartesian plane rivals if the difference between their \(x\) and \(y\) coordinates are both integers. We call two circles in the Cartesian plane enemies if their centers are distinct and rivals of each other.

Let \(S\) be a set of circles in the Cartesian plane such that any two circles in \(S\) are enemies of each other. Also, suppose all circles in \(S\) have exactly \(2014\) lattice points in their interior. What is the maximum number of elements \(S\) can have?

Details and assumptions

  • The circles must have distinct centers.

  • A lattice point is a point whose \(x\) and \(y\) coordinates are both integers.

  • The points lying on the circumference of the circles are counted in the total count of lattice points in the interior. Also, if the center of the circle is a lattice point, it is counted in the total count of lattice points in the interior.

  • As an explicit example, the points \((3.1, 4.3), (5.1, 6.3)\) are rivals of each other, because \[5.1-3.1= 2 \in \mathbb{Z}, 6.3 - 4.3 = 2 \in \mathbb{Z}.\]

  • The circle in the following figure has \(37\) lattice points in its interior (the points marked blue).

Image link: http://s27.postimg.org/an5o0blwj/Untitled.png

Image link: http://s27.postimg.org/an5o0blwj/Untitled.png

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