Can a cube be divided into three cubes? Given that $a$, $b$, $c$ and $d$ are distinct integers.
Is
$$a^3+b^3+c^3=d^3$$
possible? 

There exist infinitely many solutions $(a,b,c,d), a,b,c,d \in Z$ such that $a^3 + b^3 + c^3 = d^3$. specifically, just consider the 4-tuples of the form $(a,0,0,a)$. Note that the they trivially satisfy the equation. Hence proved.
