1995 USAMO 第 1 题

Let $\, p \,$ be an odd prime. The sequence $(a_n)_{n \geq 0}$ is defined as follows: $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ and, for all $\, n \geq p-1, \,$ $\, a_n \,$ is the least positive integer that does not form an arithmetic sequence of length $\, p \,$ with any of the preceding terms. Prove that, for all $\, n, \,$ $\, a_n \,$ is the number obtained by writing $\, n \,$ in base $\, p-1 \,$ and reading the result in base $\, p$.

令 $\, p \,$ 为奇素数。序列 $(a_n)_{n \geq 0}$ 定义如下： $\, a_0 = 0,$ $a_1 = 1, \, \ldots, \, a_{p-2} = p-2 \,$ 并且，对于所有 $\, n \geq p-1, \,$ $\, a_n \,$ 是不与任何以下项形成长度为 $\, p \,$ 的算术序列的最小正整数前面的条款。证明，对于所有$\, n, \,$ $\, a_n \,$ 是将$\, n \,$ 写入基数$\, p-1 \,$ 并读取基数$\, p$ 得到的数。