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Austria-Poland 2004 polynomial

Source: Austria-Poland 2004, team competition, problem 10

February 13, 2005

algebrapolynomialalgebra unsolved

Problem Statement

For each polynomial

Q(x)

let

M(Q)

be the set of non-negative integers

x

with

0 < Q(x) < 2004.

We consider polynomials

P_n(x)

of the form

P_n(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_{n-1} \cdot x + 1

with coefficients

a_i \in \{ \pm1\}

for

i = 1, 2, \ldots, n-1.

For each

n = 3^k, k > 0

determine: a.)

m_n

which represents the maximum of elements in

M(P_n)

for all such polynomials

P_n(x)

b.) all polynomials

P_n(x)

for which

|M(P_n)| = m_n.

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