Lattice Midpoints

Let $a,b,c,d$ be the number of points that are $(x,y)=(0,0),(0,1),(1,0),(1,1)\mod{2}$ , respectively.

We know that $a+b+c+d=n$ . Also, note that among the points in $a,b,c$ or $d$ , we can pick any two from one group and have the midpoint be a lattice point. If a group has $x$ points, then we can pick $\dfrac{x(x-1)}{2}$ pairs. Each of these pairs will give a midpoint that is a lattice point.

So let $f(x)=\dfrac{x(x-1)}{2}$ . We know that $a+b+c+d=n$ and we want to make sure $f(a)+f(b)+f(c)+f(d)\ge 2014$ . Thus, we want to make sure that $f(a)+f(b)+f(c)+f(d)$ is at its minimum.

By Jensen's Inequality, $\dfrac{f(a)+f(b)+f(c)+f(d)}{4}\ge f\left(\dfrac{a+b+c+d}{4}\right)=f\left(\dfrac{n}{4}\right)$

Thus, $f(a)+f(b)+f(c)+f(d)\ge 4f\left(\dfrac{n}{4}\right)=\dfrac{n(n-4)}{8}$

We want the minimum to exceed $2014$ , so we let $\dfrac{n(n-4)}{8}\ge 2014$ . Solving gives $n\ge 128.949$ so the minimum possible integer value of $n$ is $\boxed{129}$ .