KMO 2001 Problem 4

For given positive integers $n$ and $N$ , let $P_n$ be the set of all polynomials $f(x)=a_0+a_1x+\cdots+a_nx^n$ with integer coefficients such that:

[list] (a) $|a_j| \le N$ for $j = 0,1, \cdots ,n$ ; (b) The set $\{ j \mid a_j = N\}$ has at most two elements. [/list]

Find the number of elements of the set $\{f(2N) \mid f(x) \in P_n\}$ .