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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(a_1) < S(a_2) < \cdots < S(a_n) \text{ and } S(a_i) = P(a_{i+1}) (i=1,2,\ldots,n). (We let

a_{n+1} = a_1

.)
Problem Committee of the Japan Mathematical Olympiad Foundation

Problem Statement