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 2 of 3: Compute the middle sums and products

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S(a_i)=2^{k+i-1},\qquad P(a_i)=2^{k+i-2}\quad(2\le i\le n)

Detailed analysis

The digit sum is twice the number of twos plus the number of ones, while the digit product is two to the number of twos. The chosen multiplicities therefore give the displayed values.