For any positive integer
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
, let
![f_k](/media/m/9/8/e/98ee9bcb4ed2badbbbdf0c9e9643f2a2.png)
be the number of elements in the set
![\{ k + 1, k + 2, \ldots, 2k\}](/media/m/2/a/b/2ab6b47f9c6cfdf6187db4e29fc86d20.png)
whose base 2 representation contains exactly three 1s.
(a) Prove that for any positive integer
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
, there exists at least one positive integer
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
such that
![f(k) = m](/media/m/6/c/5/6c586435b7a19c506c5ef5328b0b7af9.png)
.
(b) Determine all positive integers
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
for which there exists exactly one
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
with
![f(k) = m](/media/m/6/c/5/6c586435b7a19c506c5ef5328b0b7af9.png)
.
%V0
For any positive integer $k$, let $f_k$ be the number of elements in the set $\{ k + 1, k + 2, \ldots, 2k\}$ whose base 2 representation contains exactly three 1s.
(a) Prove that for any positive integer $m$, there exists at least one positive integer $k$ such that $f(k) = m$.
(b) Determine all positive integers $m$ for which there exists exactly one $k$ with $f(k) = m$.