MathDB
IMC 2006, problem 6, day 1

Source: hard!!!

July 22, 2006
functionalgebrapolynomialtrigonometryreal analysisreal analysis unsolved

Problem Statement

Find all sequences a0,a1,…,ana_{0}, a_{1},\ldots, a_{n} of real numbers such that an≠0a_{n}\neq 0, for which the following statement is true: If f:R→Rf: \mathbb{R}\to\mathbb{R} is an nn times differentiable function and x0<x1<…<xnx_{0}<x_{1}<\ldots <x_{n} are real numbers such that f(x0)=f(x1)=…=f(xn)=0f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0 then there is h∈(x0,xn)h\in (x_{0}, x_{n}) for which a0f(h)+a1f′(h)+…+anf(n)(h)=0.a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.
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