For what $n$ is the following statement true? If $p(x)$ is an integer-coefficient polynomial of degree $n$ that takes values divisible by 3 at three consecutive integer points, then every coefficient of $p(x)$ is divisible by 3.
Since among three consecutive integers, one is always divisible by 3, if $n \geq 3$, the

$$
p(x)=(x-1) x^{n-2}(x+1)=x^{n}-x^{n-2}
$$

$n$-th degree polynomial takes values divisible by 3 at every integer point, but its coefficients are not all divisible by 3. Therefore, the statement can only hold if $n<3$. In this case,

$$
p(x)=A x^{2}+B x+C
$$

where $A, B, C$ are integers. Let $m$ be the integer for which $p(m-1), p(m), p(m+1)$ are divisible by 3. Then the

$$
\begin{aligned}
& p(m+1)-p(m)=(2 m+1) A+B \\
& p(m)-p(m-1)=(2 m-1) A+B
\end{aligned}
$$

numbers are also divisible by 3, and so is their difference, $2 A$. Therefore, $A$ is divisible by 3, and thus the sum $2(m+1) A+B$ is only divisible by 3 if $B$ is also divisible by 3. However, $p(m)$ is only divisible by 3 if $C$ is divisible by 3. Therefore, the statement is true for $n=0,1,2$.