IMO 1989 LL CUB12

Let P(x) be a polynomial such that the following inequalities

imolonglistmathematicsolympiad

IMO 1989 LL CUB12

Origin: CUB

Problem

Let P(x) be a polynomial such that the following inequalities are satisfied: P(0) > 0; P(1) > P(0); P(2) > 2P(1) −P(0); P(3) > 3P(2) −3P(1) + P(0); and also for every natural number n, P(n + 4) > 4P(n + 3) −6P(n + 2) + 4P(n + 1) −P(n). Prove that for every positive natural number n, P(n) is positive.