IMO 1993 SL 1

Origin: BRA

Problem

Show that there exists a finite set A \subsetR2 such that for every X \inA there are points Y1, Y2, . . . , Y1993 in A such that the distance between X and Yi is equal to 1, for every i.

Solution

First we notice that for a rational point O (i.e., with rational coordinates), there exist 1993 rational points in each quadrant of the unit circle centered at O. In fact, it suffices to take X = . O +  \pmt2 −1 t2 + 1, \pm 2t t2 + 1 

t = 1, 2, . . ., 1993 ; . Now consider the set A = {(i/q, j/q) | i, j = 0, 1, . . ., 2q}, where q = $1993 i=1 (t2 + 1). We claim that A gives a solution for the problem. Indeed, for any P \inA there is a quarter of the unit circle centered at P that is contained in the square [0, 2] \times [0, 2]. As explained above, there are 1993 rational points on this quarter circle, and by definition of q they all belong to A. Remark. Substantially the same problem was proposed by Bulgaria for IMO 71: see (SL71-2), where we give another possible construction of a set A.