IMO Shortlist 2004 problem N1
Dodao/la:
arhiva2. travnja 2012. Let
![\tau(n)](/media/m/9/3/5/93573657f80f39012c212de4b60f0b1f.png)
denote the number of positive divisors of the positive integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
. Prove that there exist infinitely many positive integers
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
such that the equation
![\tau(an)=n](/media/m/9/4/c/94cc117b0b33f8f5398fc3ae5c54ba0a.png)
does not have a positive integer solution
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
.
%V0
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $\tau(an)=n$ does not have a positive integer solution $n$.
Izvor: Međunarodna matematička olimpijada, shortlist 2004