MathDB
Polynomial inequality

Source: China TST 1991, problem 1

June 27, 2005
algebrapolynomialinequalitiesfunctionalgebra unsolved

Problem Statement

Let real coefficient polynomial f(x)=xn+a1⋅xn−1+…+anf(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n has real roots b1,b2,…,bnb_1, b_2, \ldots, b_n, n≥2,n \geq 2, prove that ∀x≥max{b1,b2,…,bn}\forall x \geq max\{b_1, b_2, \ldots, b_n\}, we have f(x+1)≥2⋅n21x−b1+1x−b2+…+1x−bn.f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.
Back to ProblemsView on AoPS