For any positive integer
, let
be the number of elements in the set
whose base 2 representation contains exactly three 1s.
(a) Prove that for any positive integer
, there exists at least one positive integer
such that
.
(b) Determine all positive integers
for which there exists exactly one
with
.
%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 $f(k) = m$.
Školjka