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.

Another way is just to replace $d$ by $a+b-c$ and then factor $a^3+b^3-c^3-d^3=0$. The result is
$$
- 3(a + b)(a - c)(b - c)=0.
$$
The result follows.
