International Mathematical Olympiad 1994 Problem 3

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 has precisely three 1s.

• (a) Prove that, for each 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 $f(k) = m$.