IT-CESENATICO 2000 Problem 6

Let $p(x)$ be a polynomial with integer coefficients such that $p(0) = 0$ and $0 \le p(1) \le 10^7$ . Suppose that there exist positive integers $a,b$ such that $p(a) = 1999$ and $p(b) = 2001$ . Determine all possible values of $p(1)$ .

(Note: $1999$ is a prime number.)