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p(0) = 0, 0 \le p(1) \le 10^7, p(a) = 1999, p(b) = 2001, p(1) ?

Source: 2000 ITAMO p6

January 25, 2020
polynomialInteger Polynomialalgebra

Problem Statement

Let p(x)p(x) be a polynomial with integer coefficients such that p(0)=0p(0) = 0 and 0≤p(1)≤1070 \le p(1) \le 10^7. Suppose that there exist positive integers a,ba,b such that p(a)=1999p(a) = 1999 and p(b)=2001p(b) = 2001. Determine all possible values of p(1)p(1). (Note: 19991999 is a prime number.)
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