Find all positive integers
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
for which the following statement is true: If
![F(x)](/media/m/c/7/6/c76f744fc9b9f5f897e479459d5e1026.png)
is a polynomial with integer coefficients satisfying the condition
![0 \leq F(c) \leq k](/media/m/5/5/0/5508c19a882e519ae232dce50eacbcf0.png)
for each
![c\in \{0,1,\ldots,k + 1\}](/media/m/8/a/2/8a2dc2811d44d3a53e5e2563393152d3.png)
, then
![F(0) = F(1) = \ldots = F(k + 1)](/media/m/a/e/1/ae139813513bd70b34abc86c2156d340.png)
.
%V0
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 $F(0) = F(1) = \ldots = F(k + 1)$.