f(f(x)+y)=f(x^2-y)+4f(x)y (Nordic MO 2011)
Posted: Tue Apr 30, 2013 11:03 pm
Find all functions $f:\mathbb R\to \mathbb R$ such that \[f(f(x)+y)=f(x^2-y)+4f(x)y\] for all real $x,y$.
Use $P(x, -f(x)) , P( x, x^2)$ to get $f(x)(f(x)-x^2) = 0$, or $f(x)= x^2 \text{ or } 0 \forall x \in \mathbb R$.
Then use the equation to show $f(y)= 0$ for $y \neq 0$ implies $f(x)=0 \forall x \in \mathbb R$
Then use the equation to show $f(y)= 0$ for $y \neq 0$ implies $f(x)=0 \forall x \in \mathbb R$