1994 IMO Problems/Problem 1
Let and be two positive integers. Let , , , be different numbers from the set such that for any two indices and with and , there exists an index such that . Show that .
Let satisfy the given conditions. We will prove that for all
WLOG, let . Assume that for some
This implies, for each because
For each of these values of i, we must have such that is a member of the sequence for each . Because . Combining all of our conditions we have that each of must be distinct integers such that
However, there are distinct , but only integers satisfying the above inequality, so we have a contradiction. Our assumption that was false, so for all such that Summing these inequalities together for gives which rearranges to