# Understand the problem

##### Source of the problem

Kürschák Competition 2018

##### Topic

##### Difficulty Level

##### Suggested Book

# Start with hints

Note that . This implies that . Also, from the Cauchy-Schwarz inequality we get hence . Also, $latex | \overline{v_j}|\cdot |\overline{v_k}|=p^2$. As the dot product is also divisible by , it has to be equal to or 0. It cannot be because the vectors are distinct, hence it is either or 0. Thus the two vectors are either perpendicular or colinear (adding to 0).

Let us plot the vectors in and identify them with their tips (as the tails are at the origin). We join two tips by a straight line segment if they correspond to perpendicular vectors. Show that the number of straight lines is at least . Also prove that, there do not exist 4 vectors such that every possible pair of tips is joined by a line segment.

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