1997 IMO 第 2 题

Let $R_{1}, R_{2}, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $R_{1}=(1)$, and if $R_{n-1}=\left(x_{1}, \ldots, x_{s}\right)$, then $$R_{n}=\left(1,2, \ldots, x_{1}, 1,2, \ldots, x_{2}, \ldots, 1,2, \ldots, x_{s}, n\right)$$ For example, $R_{2}=(1,2), R_{3}=(1,1,2,3), R_{4}=(1,1,1,2,1,2,3,4)$. Prove that if $n>1$, then the $k$ th term from the left in $R_{n}$ is equal to 1 if and only if the $k$ th term from the right in $R_{n}$ is different from 1.

令 $R_{1}, R_{2}, \ldots$ 为由以下规则定义的有限正整数序列族：$R_{1}=(1)$，并且如果 $R_{n-1}=\left(x_{1}, \ldots, x_{s}\right)$，则 $$R_{n}=\left(1,2, \ldots, x_{1}, 1,2, \ldots, x_{2}, \ldots, 1,2, \ldots, x_{s}, n\right)$$ 例如，$R_{2}=(1,2)、R_{3}=(1,1,2,3)、R_{4}=(1,1,1,2,1,2,3,4)$。证明如果 $n>1$，则 $R_{n}$ 中从左边算起的第 $k$ 项等于 1 当且仅当 $R_{n}$ 中从右边算起的第 $k$ 项不等于 1。