Prove that there are no such different positive numbers that satisfy both $a+b=c+d$ and $a^3+b^3=c^3+d^3$. $a,b,c$ and $d$ are all different and positive. Prove that it can't be true that equalities $a+b=c+d$ and $a^3+b^3=c^3+d^3$ are both correct at the same time. I sadly don't know how to start trying to solve this and haven't really tried much, so I'd like to get some ideas. Thanks.

Hint: factor the LHS and the RHS and see what happens.
