Suppose we are given n points in the 2D Euclidean space, described by coordinates (xi, yi), where xi, yi are integers between 0 and n5 for i = 1, 2, . . . n. The goal of this problem is to find a pair (zi, yi) and (Dz, with it j, that are at the smallest distance from each other (i.e., llx-y 2 ((zi-292 (Vi-Uj)2) 1/2 is minimized). Below assume that r > 0 is a positive real a) Call a set of points P to be r-spread, for some r 0, if any pair of points from P are at distance at least r. Fix integers 4 and ly. Consider a set of r-spread points P that lies inside the box [4,4 + 10r] ey, Ly +10r] (i.e., such that for any (x,y) E P, we have that xE [Lx,lx +10r] and y E [ly, Ly +10r]) Prove an upper bound on the number of points in P, i.e., an upper bound on |Pl (note: it is enough to get an asymptotic bound only). b) Fix an integer 1 0 is a positive real a) Call a set of points P to be r-spread, for some r 0, if any pair of points from P are at distance at least r. Fix integers 4 and ly. Consider a set of r-spread points P that lies inside the box [4,4 + 10r] ey, Ly +10r] (i.e., such that for any (x,y) E P, we have that xE [Lx,lx +10r] and y E [ly, Ly +10r]) Prove an upper bound on the number of points in P, i.e., an upper bound on |Pl (note: it is enough to get an asymptotic bound only). b) Fix an integer 1