Question 10
(CZE) Find all positive integers $k$ for which the following statement is true:
If $F(x)$ is a polynomial with integer coefficients satisfying the condition
$$0\le F(c)\le k\quad\text{for each }c\in\{0,1,\ldots,k+1\},$$
then $F(0)=F(1)=\cdots=F(k+1)$.
If $F(x)$ is a polynomial with integer coefficients satisfying the condition
$$0\le F(c)\le k\quad\text{for each }c\in\{0,1,\ldots,k+1\},$$
then $F(0)=F(1)=\cdots=F(k+1)$.