MathDB
(x - a_1)(x - a_2) ... (x - a_n) <= x^n - a^n_1

Source: IMO Shortlist 1996, A2

August 9, 2008
algebraSequenceInequalityIMO Shortlist

Problem Statement

Let a1≥a2≥…≥an a_1 \geq a_2 \geq \ldots \geq a_n be real numbers such that for all integers k>0, k > 0,
a^k_1 \plus{} a^k_2 \plus{} \ldots \plus{} a^k_n \geq 0.
Let p \equal{}\max\{|a_1|, \ldots, |a_n|\}. Prove that p \equal{} a_1 and that
(x \minus{} a_1) \cdot (x \minus{} a_2) \cdots (x \minus{} a_n) \leq x^n \minus{} a^n_1 for all x>a1. x > a_1.
Back to ProblemsView on AoPS