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).

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