Problem

Source: IMO ShortList 1990, Problem 8 (HUN 1)

Tags: number theory, Digits, sum of digits, Calculate, decimal representation, IMO Shortlist



For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n+1}(k) = f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$