Konigsberg wrote: Let $\mathbb{R}*$ denote the set of nonzero real numbers. Find all functions $f:\mathbb{R}* \rightarrow \mathbb{R}*$ such that $f(x^2+y)=f(f(x))+\frac{f(xy)}{f(x)}$ for every pair of nonzero real numbers $x$ and $y$ with $x^2+y \neq 0$. Let $P(x,y)$ be the assertion $f(x^2+y)=f(f(x))+\frac{f(xy)}{f(x)}$ true whatever are $x,y$ such that $x,y,x^2+y\ne 0$ If $f(x)\ne x^2$ for some $x\ne 0$, then $P(x,f(x)-x^2)$ $\implies$ $f(x(f(x)-x^2))=0$, impossible. So $f(x)=x^2$ $\forall x\ne 0$ which unfortunately is not a solution. So no solution for this functional equation.