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: (an,an1,,a0) and let f(A) denote nk=02nkak. Define A1=f(A),A2=f(A1). Prove that: I. There exists positive integer k for which Ak+1=Ak. II. Find such Ak for 1986.