IMO 1989 LL CUB13

Origin: CUB

Problem

Let n be a natural number not greater than 44. Prove that for any function f defined over N2 whose images are in the set {1, 2, . . ., n}, there are four ordered pairs (i, j), (i, k), (l, j), and (l, k) such that f(i, j) = f(i, k) = f(l, j) = f(l, k), where i, j, k, l are chosen in such a way that there are natural numbers n, p that satisfy 1989m \leqi < l < 1989 + 1989m, 1989p \leqj < k < 1989 + 1989p.