IMO Shortlist 2011 problem A3
Dodao/la:
arhiva23. lipnja 2013. Determine all pairs
![(f,g)](/media/m/c/b/3/cb3b5845f5cc531664d43d11a74b02e3.png)
of functions from the set of real numbers to itsels that satisfy
![g(f(x+y)) = f(x) + (2x + y)g(y)](/media/m/5/0/c/50c34eb525e6fea97f9fb67c5660b27b.png)
for all real numbers
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
and
![y.](/media/m/9/2/3/923fc0c0f0d7a54598bbd1f90c33b74f.png)
Proposed by Japan
%V0
Determine all pairs $(f,g)$ of functions from the set of real numbers to itsels that satisfy $$g(f(x+y)) = f(x) + (2x + y)g(y)$$ for all real numbers $x$ and $y.$
Proposed by Japan
Izvor: Međunarodna matematička olimpijada, shortlist 2011