IMO Shortlist 1992 problem 17
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 Let
be the number of digits equal to one in the binary representation of a positive integer
. Prove that:
(a) the inequality
holds;
(b) the above inequality is an equality for infinitely many positive integers, and
(c) there exists a sequence
such that
goes to zero as
goes to
.
Alternative problem: Prove that there exists a sequence a sequence
such that
(d)
;
(e) an arbitrary real number
;
(f) an arbitrary real number
;
as
goes to
.
![\alpha(n)](/media/m/2/b/a/2ba91a633148e5989cf20d1a5d6e6289.png)
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
(a) the inequality
![\alpha(n) \left(n^2 \right) \leq \frac{1}{2} \alpha(n)(\alpha(n) + 1)](/media/m/f/f/5/ff50064eaa3d35fc2e439eeb46823ab8.png)
(b) the above inequality is an equality for infinitely many positive integers, and
(c) there exists a sequence
![\left(n_i \right)^{\infty}_1](/media/m/d/8/0/d80b991d47121e3fd47d5f153ccb2e9b.png)
![\frac{\alpha \left( n^2_i \right)}{\alpha \left(n_i \right)}](/media/m/f/3/2/f32c040d834d495cb656c1c629c99229.png)
goes to zero as
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
![\infty](/media/m/a/2/1/a21bbaf921df8319727cfb5b00c1a35b.png)
Alternative problem: Prove that there exists a sequence a sequence
![\left(n_i \right)^{\infty}_1](/media/m/d/8/0/d80b991d47121e3fd47d5f153ccb2e9b.png)
![\frac{\alpha \left( n^2_i \right)}{\alpha \left(n_i \right)}](/media/m/f/3/2/f32c040d834d495cb656c1c629c99229.png)
(d)
![\infty](/media/m/a/2/1/a21bbaf921df8319727cfb5b00c1a35b.png)
(e) an arbitrary real number
![\gamma \in (0,1)](/media/m/9/3/6/936cd771f07d51aad237a56155bc045f.png)
(f) an arbitrary real number
![\gamma \geq 0](/media/m/c/7/7/c774b426978cbd4bc3ccc62708535719.png)
as
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
![\infty](/media/m/a/2/1/a21bbaf921df8319727cfb5b00c1a35b.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1992