IMO Shortlist 2012 problem N4
Dodao/la:
arhiva3. studenoga 2013. An integer
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
is called friendly if the equation
![(m^2+n)(n^2+m)=a(m-n)^3](/media/m/4/4/1/441bbe9e2a61e506c883810950731e08.png)
has a solution over the positive integers.
a) Prove that there are at least
![500](/media/m/1/5/f/15f97bb60a6487c4a235a0af81745b1c.png)
friendly integers in the set
![\{ 1,2,\ldots ,2012\}](/media/m/6/c/e/6ce0ec3650b4797508f86fc87a04d40e.png)
.
b) Decide whether
![a=2](/media/m/d/b/1/db14cf26c022167b9d68c9b827009b66.png)
is friendly.
%V0
An integer $a$ is called friendly if the equation $(m^2+n)(n^2+m)=a(m-n)^3$ has a solution over the positive integers.
a) Prove that there are at least $500$ friendly integers in the set $\{ 1,2,\ldots ,2012\}$.
b) Decide whether $a=2$ is friendly.
Izvor: Međunarodna matematička olimpijada, shortlist 2012