Find functions $f$ on pairs of positive natural numbers satisfying: 

1. $f(x,x)=x$
2. $f(x,y)=f(y,x)$
3. $(x+y)f(x,y)=yf(x,x+y)$

It is quite easy to find that $f(1,k)=k$ for all $k$ by induction: if $f(k,1)=k$,  

then $(1+k)f(k,1)=(1+k)k=k(1,k+1)$ by condition (3) leading to the result.  

Then also $f(k,k+1)=k(1+k)$.  

After this I'm a bit stuck. $x^2=f(x,x+1)-f(x,x)$ by the above, but I don't know where to go to find a general solution. Hints?