2004 Putnam B1

Let $P(x) = c_n x^n + c_{n-1} x^{n-1} + \cdots + c_0$ be a polynomial with

integer coefficients. Suppose that $r$ is a rational number such that

$P(r) = 0$. Show that the $n$ numbers

\begin{gather*}

c_n r, \, c_n r^2 + c_{n-1} r, \, c_n r^3 + c_{n-1} r^2 + c_{n-2} r, \\

\dots, \, c_n r^n + c_{n-1} r^{n-1} + \cdots + c_1 r

\end{gather*}

are integers.

设 $P(x) = c_n x^n + c_{n-1} x^{n-1} + \cdots + c_0$ 为多项式

整数系数。假设 $r$ 是一个有理数，使得

$P(r) = 0$。显示 $n$ 个数字

\开始{收集*}

c_n r, \, c_n r^2 + c_{n-1} r, \, c_n r^3 + c_{n-1} r^2 + c_{n-2} r, \\

\dots, \, c_n r^n + c_{n-1} r^{n-1} + \cdots + c_1 r

\结束{聚集*}

是整数。