From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math,rec.puzzles
Subject: Re: IMO 1
Date: 13 Nov 1998 20:51:16 GMT

In article <yF$cBGA96BT2EwvR@kalva.demon.co.uk>,
John Scholes  <jscholes@kalva.demon.co.uk> wrote:
>Find 1982 points on the circumference of a unit circle such that the
>distance between each pair is rational, or prove it impossible.

SPOILER:

One can find infinitely many such points.

View the points as complex numbers; take any set of them with rational 
real and imaginary parts, e.g. z_n = (n^2-1 + 2*n*I)/(n^2+1); then the
points  p_n=(z_n)^2  are separated by rational distances. (With this sample 
set, dist(p_n, p_m) = 4(nm+1)|n-m|/(n^2+1)/(m^2+1).  )

Known to Euler, I think. Point sets with rational distances _without_ 
collinear or cocyclic subsets are harder to find; none with 7 points is known.

dave