Problem

Source: IMO Shortlist 1997, Q10

Tags: algebra, polynomial, functional equation, IMO Shortlist



Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k + 1\}$, then $ F(0) = F(1) = \ldots = F(k + 1)$.