IMO Shortlist 1992 problem 17


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Let \alpha(n) be the number of digits equal to one in the binary representation of a positive integer n. Prove that:

(a) the inequality \alpha(n) \left(n^2 \right) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1) holds;
(b) the above inequality is an equality for infinitely many positive integers, and
(c) there exists a sequence \left(n_i \right)^{\infty}_1 such that \frac{\alpha \left( n^2_i \right)}{\alpha \left(n_i \right)}
goes to zero as i goes to \infty.


Alternative problem: Prove that there exists a sequence a sequence \left(n_i \right)^{\infty}_1 such that \frac{\alpha \left( n^2_i \right)}{\alpha \left(n_i \right)}

(d) \infty;
(e) an arbitrary real number \gamma \in (0,1);
(f) an arbitrary real number \gamma \geq 0;

as i goes to \infty.
Izvor: Međunarodna matematička olimpijada, shortlist 1992