CAPS 1999 Problem 4

Find all positive integers $k$ for which the following assertion holds: If $F(x)$ is polynomial with integer coefficients ehich satisfies $F(c) \leq k$ for all $c \in \{0,1, \cdots,k+1 \}$ , then $$F(0)= F(1) = \cdots =F(k+1).$$