IMO shortlist 1993/1 solution

34th IMO 1993 shortlist

Problem 1

Show that there is a finite set of points in the plane such that for any point P in the set we can find 1993 points in the set a distance 1 from P.

Solution

The basic idea is that we can find a circle which passes through a large number of lattice points. The point (±(t²-1)/t²+1), ±2t/(t²+1)) lies on the unit circle, so the set of such points for t = 1,2,...,1993 has 1993 points on each quadrant of the unit circle. Put n = ∏₁¹⁹⁹³(1+t²), then all these points can be written as m/n, where m is an integer with |m| ≤ n.

Now consider the array of points (a/n, b/n), where 0 ≤ a,b ≤ 2n. The unit circle centered at any of these points has at least one of its quadrants passing through the array and hence contains at least 1993 of the points.

34th IMO shortlist 1993