cmi-entrance 2018 QB3

cmi-entrance · India · ugmath 15 marks Not Maths

Let $f$ be a function on nonnegative integers defined as follows $$f(2n) = f(f(n)) \quad \text{and} \quad f(2n+1) = f(2n)+1$$
(a) If $f(0) = 0$, find $f(n)$ for every $n$.
(b) Show that $f(0)$ cannot equal 1.
(c) For what nonnegative integers $k$ (if any) can $f(0)$ equal $2^{k}$?

Let $f$ be a function on nonnegative integers defined as follows
$$f(2n) = f(f(n)) \quad \text{and} \quad f(2n+1) = f(2n)+1$$

(a) If $f(0) = 0$, find $f(n)$ for every $n$.

(b) Show that $f(0)$ cannot equal 1.

(c) For what nonnegative integers $k$ (if any) can $f(0)$ equal $2^{k}$?

Paper Questions

QA1 QA10 QA2 QA3 QA4 QA5 QA6 QA7 QA8 QA9 QB1 QB2 QB3 QB4 QB5 QB6