Problem 1

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

S(a_1)<S(a_2)<\cdots<S(a_n)

and

S(a_i)=P(a_{i+1})

for

i=1,2,\ldots,n

, where

a_{n+1}=a_1

Step 1 of 3: Construct the middle numbers

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a_i:\;2\text{ occurs }k+i-2\text{ times},\quad 1\text{ occurs }2^{k+i-1}-2(k+i-2)\text{ times}\quad(2\le i\le n)

Detailed analysis

Choose a sufficiently large positive integer k. For each index from 2 through n, let the digits of a_i consist only of twos and ones, with the displayed multiplicities. The required number of ones is nonnegative when k is large enough.