Školjka

%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 $f(k) = m$.