Find all real functions satisfying $ f \big( y - f ( x ) \big) = f \left( x ^ { 2002 } - y \right) - 2001 y f ( x ) $

Question

The question:

Find all functions $ f $ defined over $ \mathbb R $ satisfying the equality: $ \forall x , y \in \mathbb R $ $$ f \big( y - f ( x ) \big) = f \left( x ^ { 2002 } - y \right) - 2001 y f ( x ) $$

How do I approach (any hints) to solve the problem above?

Answer 1

Set $y=f(x)$ so $f(0)=f(x^{2002}-f(x))-2001f(x)^2$. Set $y=x^{2002}$ to get $f(x^{2002}-f(x))=f(0)-2001x^{2002}f(x)=f(x^{2002}-f(x))-2001f(x)^2-2001x^{2002}f(x)$ and therefore $2001f(x)^2+2001x^{2002}f(x)=0$. Can you continue from here?