Question 57
For each integer $k \ge 4$, prove that if $F(x)$ is a polynomial with integer coefficients satisfying the condition $0 \le F(c) \le k$ for every $c = 0, 1, \dots, k+1$, then
$$
F(0) = F(1) = \dots = F(k+1).
$$
$$
F(0) = F(1) = \dots = F(k+1).
$$