IMO Shortlist 1982 problem 16
Dodao/la:
arhiva2. travnja 2012. Prove that if
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
is a positive integer such that the equation
![x^3-3xy^2+y^3=n](/media/m/4/9/a/49aa0a8b47c4372a9908c02f423bd20d.png)
has a solution in integers
![x,y](/media/m/f/b/6/fb60533620f22cd699e5b58ce9a646a4.png)
, then it has at least three such solutions. Show that the equation has no solutions in integers for
![n=2891](/media/m/d/a/6/da658d87f651d310b067dd9a5b4d428f.png)
.
%V0
Prove that if $n$ is a positive integer such that the equation $$x^3-3xy^2+y^3=n$$ has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.
Izvor: Međunarodna matematička olimpijada, shortlist 1982