US-TST 2005 Problem 3

We choose random a unitary polynomial of degree $n$ and coefficients in the set $1,2,...,n!$ . Prove that the probability for this polynomial to be special is between $0.71$ and $0.75$ , where a polynomial $g$ is called special if for every $k>1$ in the sequence $f(1), f(2), f(3),...$ there are infinitely many numbers relatively prime with $k$ .