Find all positive integers for which the following statement is true: If is a polynomial with integer coefficients satisfying the condition for each , then .
%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 $F(0) = F(1) = \ldots = F(k + 1)$.
Izvor: Međunarodna matematička olimpijada, shortlist 1997
Školjka 
for which the following statement is true: If 
is a polynomial with integer coefficients satisfying the condition 
for each 
, then 
.