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# Understand the problem

Find all functions $f : \mathbb R \rightarrow \mathbb R$ such that $(x + y)(f(x) - f(y)) = (x -y)f(x + y)$ for all $x, y\in \mathbb R$

##### Source of the problem
Singapore Team Selection Test 2008
##### Topic
Functional Equations
Medium
##### Suggested Book
Functional Equations by BJ Venkatachala

Do you really need a hint? Try it first!

Play with choices of $x,y$ belonging to $\{0,1,-1\}$. Show that $f(0)=0$.

Play with choices of $x,y$ from the set $\left\{-\frac{z}{2},\frac{z}{2}+1,\frac{z}{2}-1\right\}$.

Following hint 2, you should be able to get equations that can be added to cancel some terms.

Putting $x=\frac{z}{2}-1,y=\frac{z}{2}+1$
we get $f(\frac{z}{2}-1)-f(\frac{z}{2}+1)=-\frac{2}{z}f(z)$ Putting $z=\frac{z}{2}+1,y=-\frac{z}{2}$ we get $f(\frac{z}{2}+1)-f(-\frac{z}{2})=(z+1)f(1)$ $x=\frac{z}{2},y=\frac{z}{2}-1$ gives $f(-\frac{z}{2})-f(\frac{z}{2}-1)=(z-1)f(-1)$

Adding, we get $f(z)=\frac{f(1)+f(-1)}{2}z^2+\frac{f(1)-f(-1)}{2}z$.