US-TST 2009 Problem 3

For each positive integer $n$ , let $c(n)$ be the largest real number such that $$c(n) \le \left| \frac {f(a) - f(b)}{a - b}\right|$$ for all triples $(f, a, b)$ such that

-- $f$ is a polynomial of degree $n$ taking integers to integers, and -- $a, b$ are integers with $f(a) \neq f(b)$ .

Find $c(n)$ .

Shaunak Kishore.