China Girls' Mathematical Olympiad

Question

Let S = (a, b) 1 a, b 5, a, b Z. Let T be the set of integer points in the plane such that for any point P in S, there exists a different point Q in T such that PQ does not contain integer points except P and Q. Find the minimum value of |T|, where |T| denotes the number of elements of the finite set T.

Accepted Answer

We first prove that |T| 1. If |T| = 1, let T = Q(x_0, y_0). We may take point P(x_1, y_1) in S satisfying the conditions: (1) (x_1, y_1) (x_0, y_0), (2) x_1 and x_0 have the same parity, y_1 and y_0 have the same parity. Then, the midpoint of PQ is an integer, which is a contradiction. If |T| = 2, see the following figure satisfying the conditions of the problem: • • • • • • • ◐ • • • • • • • ◐ • • • • • • • • • • • • as desired.