IMO 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 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 $f(k) = m$ .