Question 34
Let $n$ be a positive integer number. If $S$ is a finite set of vectors in the plane, let $N(S)$ denote the number of two-element subsets $\{\mathbf{v}, \mathbf{v}'\}$ of $S$ such that
$$
4 (\mathbf{v} \cdot \mathbf{v}') + (|\mathbf{v}|^2 - 1)(|\mathbf{v}'|^2 - 1) \lt 0.
$$
Determine the maximum of $N(S)$ when $S$ runs through all $n$-element sets of vectors in the plane.
$$
4 (\mathbf{v} \cdot \mathbf{v}') + (|\mathbf{v}|^2 - 1)(|\mathbf{v}'|^2 - 1) \lt 0.
$$
Determine the maximum of $N(S)$ when $S$ runs through all $n$-element sets of vectors in the plane.