\(n\) lattice points are drawn on the coordinate plane. These points' pairwise midpoints are then drawn. What is the smallest possible value of \(n\) such that one is guaranteed to have at least \(2014\) of the midpoints also be lattice points?
Details and Assumptions
This problem is inspired by a classic proof problem.
If two pairs of points happen to have a common midpoint, both midpoints should be counted.