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Value of $f(0)$

Source: Chennai Mathematical Institue Entrance 2018 Problem 3

September 1, 2018
functional equationnumber theorycombinatorics

Problem Statement

Let ff be a function on non-negative integers defined as follows f(2n)=f(f(n))   and   f(2n+1)=f(2n)+1f(2n)=f(f(n))~~~\text{and}~~~f(2n+1)=f(2n)+1 (a) If f(0)=0f(0)=0 , find f(n)f(n) for every nn. (b) Show that f(0)f(0) cannot equal 11. (c) For what non-negative integers kk (if any) can f(0)f(0) equal 2k2^k ?
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