IMO Shortlist 1994 problem N5

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Dodao/la: arhiva
2. travnja 2012.
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.
Izvor: Međunarodna matematička olimpijada, shortlist 1994