Thursday, November 17th

Problem
Find all the functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x )-y) f(x+f(y ))=x^2-y^2$ for all $x,y \in \mathbb{R}$.

Solution 
Let $c=f(0)$, then $f(c)=0$. Also, $f(c-y)f(f(y))=-y^2$, and with $y=c$   $f(f(c))=-c=f(0)=c$, thus $c=0$. Let $y=f(x)$, then $x^2-f(x)^2=(x-f(x))(x+f(x))=0$ for all $x$. Since $f(x)=-x$ does not satisfy the equation, the only solution is $f(x)=x$.