Find all functions f:R->R satisfying the equality f(f(x)+f(y))=(x+y)f(x+y) for all real x and y.
At first glance, it seems to me that the function must be a polynomial.
If the polynomial is of order n, then the LHS is of order n^2 and the RHS is of order n + 1. This rules out most functions, in my mind.
Only f(x) = 0 seems to work.
I will be very interested to see if anybody comes up with anything else.