6

Part of 2006 IMC

Problems(2)

IMC 2006, problem 6, day 1

Source: hard!!!

Find all sequences

a_{0}, a_{1},\ldots, a_{n}

of real numbers such that

a_{n}\neq 0

, for which the following statement is true: If

f: \mathbb{R}\to\mathbb{R}

n

times differentiable function and

x_{0}<x_{1}<\ldots <x_{n}

are real numbers such that

f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0

h\in (x_{0}, x_{n})

a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.

functionalgebrapolynomialtrigonometryreal analysisreal analysis unsolved

IMC 2006 / B6 a.k.a. The Monster

Source: IMC 2006 day 2 problem 6

The scores of this problem were: one time 17/20 (by the runner-up) one time 4/20 (by Andrei Negut) one time 1/20 (by the winner) the rest had zero... just to give an idea of the difficulty. Let

A_{i},B_{i},S_{i}

i=1,2,3

) be invertible real

2\times 2

matrices such that [*]not all

A_{i}

have a common real eigenvector, [*]

A_{i}=S_{i}^{-1}B_{i}S_{i}

i=1,2,3

A_{1}A_{2}A_{3}=B_{1}B_{2}B_{3}=I

. Prove that there is an invertible

2\times 2

S

A_{i}=S^{-1}B_{i}S

i=1,2,3

linear algebramatrixalgebrapolynomialIMCcollege contests