14th CanMO 1982 Problem 3 What is the smallest number of points in n-dimensional space R n such that every point of R n is an irrational distance from at least one of the points. Solution
: n = 1 number is 2. For n > 1, number is 3. The result for n = 1 is obvious. The result for n > 1 follows from the fact that we can choose three collinear points A, M, B such that the distance of any point P from one of them is irrational. We can take a plane through P and A, M, B so the number of dimensions is unimportant. Let M be the midpoint of AB. Let PA = a, PB = b, PM = d, and AM = c. Let angle PMA = x. Then the cosine rule gives a 2 = d 2 + c 2 - 2dc cos x, b 2 = d 2 + c 2 + 2dc cos x. So, adding, a 2 + b 2 = 2d 2 + 2c 2 , so d 2 = (a 2 + b 2 )/2 - c 2 . Thus if we take c 2 to be irrational, then if a and b are rational, d 2 , and hence d, is irrational. 14th CanMO 1982 © John Scholes jscholes@kalva.demon.co.uk 21 June 2002