$F(x)+G(y)= e^{x+y}?$ Are there functions $F(x)$, $G(y)$, such that
 $F(x)+G(y)=e^{x+y}$ , where $x,y$ are real numbers? I have been trying all elementary functions, and have no clues on what else I could do. 

No such $F$ and $G$ exist.
The fact that $F(0) + G(y) = e^{y}$ for all real $y$ would imply $G(y) = e^{y} - F(0)$.
Similarly, $F(x) = e^{x} - G(0)$ for all real $x$. No matter how $F(0)$ and $G(0)$ are chosen, you're sunk.
