# 1994 IMO Problems/Problem 1

Revision as of 23:38, 9 April 2021 by Math31415926535 (talk | contribs) (Created page with "Let <math>a_1, a_2, \dots a_m</math> satisfy the given conditions. We will prove that for all <math>j, 1 \le j \le m,</math> <cmath>a_j+a_{m-j+1} \ge n+1</cmath> WLOG, let <...")

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