f(f(x)+y)=f(x^2-y)+4f(x)y (Nordic MO 2011)

For discussing Olympiad Level Algebra (and Inequality) problems
Post Reply
User avatar
Phlembac Adib Hasan
Posts:1016
Joined:Tue Nov 22, 2011 7:49 pm
Location:127.0.0.1
Contact:
Contact Phlembac Adib Hasan
f(f(x)+y)=f(x^2-y)+4f(x)y (Nordic MO 2011)

Unread post by Phlembac Adib Hasan » 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 $x,y$.
Top
User avatar
*Mahi*
Posts:1175
Joined:Wed Dec 29, 2010 12:46 pm
Location:23.786228,90.354974
Contact:
Contact *Mahi*

Re: f(f(x)+y)=f(x^2-y)+4f(x)y (Nordic MO 2011)

Unread post by *Mahi* » Wed May 01, 2013 12:41 am

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 $f(x)=0 \forall x \in \mathbb R$
Please read Forum Guide and Rules before you post.

Use $L^AT_EX$, It makes our work a lot easier!

Nur Muhammad Shafiullah | Mahi
Top
Post Reply

Return to “Algebra”