IMO 1987 SL 13

Origin: GDR

Problem

Is it possible to put 1987 points in the Euclidean plane such that the distance between each pair of points is irrational and each three points determine a nondegenerate triangle with rational area?

Solution

We claim that the points Pi(i, i2), i = 1, 2, . . . , 1987, satisfy the conditions. In fact: (i) PiPj =

(i −j)2 + (i2 −j2)2 = |i −j|

1 + (i + j)2. It is known that for each positive integer n, \sqrtn is either an integer or an irrational number. Since i + j <

1 + (i + j)2 < i + j + 1,

1 + (i + j)2 is not an integer, it is irrational, and so is PiPj. (ii) The area A of the triangle PiPjPk, for distinct i, j, k, is given by A =

i2 + j2 (i −j) + j2 + k2 (j −k) + k2 + i2 (k −i)

=

(i −j)(j −k)(k −i) \inQ ∖{0}, also showing that this triangle is nondegenerate.