IMO 2007 Shortlist A1
Given a sequence a1,a2,...,an of real numbers. For each i (1 ≤ i ≤ n) define di = max{aj : 1 ≤ j ≤ i} − min{aj : i ≤ j ≤...
Category: Algebra
Problem
Given a sequence a1,a2,...,an of real numbers. For each i (1 ≤ i ≤ n) define di = max{aj : 1 ≤ j ≤ i} − min{aj : i ≤ j ≤ n} and let d = max{di : 1 ≤ i ≤ n}. (a) Prove that for arbitrary real numbers x1 ≤ x2 ≤ ... ≤ xn, max |xi − ai| : 1 ≤ i ≤ n ≥ d . (1) (b) Show that there exists a sequence x1 ≤ x2 ≤ ... ≤ xn of real numbers such that we have equality in (1). (New Zealand)