2014 APMO 第 1 题

For a positive integer $m$ denote by $S(m)$ and $P(m)$ the sum and product, respectively, of the digits of $m$. Show that for each positive integer $n$, there exist positive integers $a_{1}, a_{2}, \ldots, a_{n}$ satisfying the following conditions:

$$S\left(a_{1}\right)<S\left(a_{2}\right)<\cdots<S\left(a_{n}\right) \text { and } S\left(a_{i}\right)=P\left(a_{i+1}\right) \quad(i=1,2, \ldots, n)$$

(We let $\left.a_{n+1}=a_{1}.\right)$ (Problem Committee of the Japan Mathematical Olympiad Foundation)

对于正整数 $m$，分别用 $S(m)$ 和 $P(m)$ 表示 $m$ 的数字的和与积。证明对于每个正整数$n$，都存在满足以下条件的正整数$a_{1}, a_{2}, \ldots, a_{n}$：

$$

S\left(a_{1}\right)<S\left(a_{2}\right)<\cdots<S\left(a_{n}\right) \text { 和 } S\left(a_{i}\right)=P\left(a_{i+1}\right) \quad(i=1,2, \ldots, n)

$$

(设$\left.a_{n+1}=a_{1}.\right)$(日本数学奥林匹克基金会问题委员会)