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Four ordered pairs satisfying two "1989"-inequalities

Source: IMO Longlist 1989, Problem 13

September 18, 2008
inequalitiesfunctionalgebra unsolvedalgebra

Problem Statement

Let n≤44,n∈N. n \leq 44, n \in \mathbb{N}. Prove that for any function f f defined over N2 \mathbb{N}^2 whose images are in the set {1,2,…,n}, \{1, 2, \ldots , n\}, there are four ordered pairs (i,j),(i,k),(l,j), (i, j), (i, k), (l, j), and (l,k) (l, k) such that f(i, j) \equal{} f(i, k) \equal{} f(l, j) \equal{} f(l, k), in which i,j,k,l i, j, k, l are chosen in such a way that there are natural numbers m,p m, p that satisfy 1989m \leq i < l < 1989 \plus{} 1989m and 1989p \leq j < k < 1989 \plus{} 1989p.
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