1984 USAMO 第 5 题

$P(x)$ is a polynomial of degree $3n$ such that

$$$$
\begin{aligned} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\ && P(3n+1) = 730.\end{aligned}
$$$$

Determine $n$.

$P(x)$ 是 $3n$ 次多项式，使得

$$

$$

\begin{对齐} P(0) = P(3) = \cdots &=& P(3n) = 2, \\ P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\ P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ 和 }\\ && P(3n+1) = 730.\结束{对齐}

$$

$$

确定$n$。