Problem

Source: China TST 1986, problem 3

Tags: algebra, Number theoretic functions, China, TST, decimal representation, Digits



Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that: I. There exists positive integer $k$ for which $A_{k+1}=A_k$. II. Find such $A_k$ for $19^{86}.$