1990 OIM Problems/Problem 1

Problem

Let $f$ be a function defined in the set of integers greater or equal to zero such that:

(i) If $n=2^j-1$ , for all $n=0, 1, 2, \cdots,$ then $f(n)=0$

(ii) If $n \ne 2^j-1$ , for all $n=0, 1, 2, \cdots,$ then $f(n+1)=f(n)-1$

a. Prove that for all integer $n$ , greater or equal to zero, there exist an integer $k$ grater than zero such that $f(n)+n=2^k-1$

b. Calculate $f(2^{1990})$ .

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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