SCHWEITZER 2010 Problem 5

Given the vectors $v_ {1}, \dots, v_ {n}$ and $w_ {1}, \dots, w_ {n}$ in the plane with the following properties: for every $1 \leq i \leq n$ , $\left | v_{i} -w_{i} \right | \leq 1,$ and for every $1 \leq i <j \leq n$ , $\left | v_{i} -v_{j} \right | \ge 3$ and $v_{i} -w_ {i} \ne v_ {j} -w_ {j}$ . Prove that for sets $V = \left \{v_ {1}, \dots, v_{n } \right \}$ and $W = \left \{w_ {1}, \dots, w_ {n} \right \}$ , the set of $V + (V \cup W)$ must have at least $cn^{3/2}$ elements ,for some universal constant $c>0$ .