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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

Problem Statement