AUSTRIAN-POLISH 2004 Problem 10

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 $|M(P_n)| = m_n.$