IMO Shortlist 1999 problem A4
Dodao/la:
arhiva2. travnja 2012. Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers
![x,y](/media/m/f/b/6/fb60533620f22cd699e5b58ce9a646a4.png)
taken from two different subsets, the number
![x^2-xy+y^2](/media/m/c/7/3/c73ed5038a2a34daa0443e34dc7cbeca.png)
belongs to the third subset.
%V0
Prove that the set of positive integers cannot be partitioned into three nonempty subsets such that, for any two integers $x,y$ taken from two different subsets, the number $x^2-xy+y^2$ belongs to the third subset.
Izvor: Međunarodna matematička olimpijada, shortlist 1999