Prove that $\displaystyle a^3+b^3+c^3+d^3+e^3= n$ has an integer solution for any intger n.
this is a very well-known problem! first note that $\displaystyle 6 \mid n^3 - n, \ \forall n \in \mathbb{Z}.$ now we have: $\displaystyle n=n^3 + \left(-1-\frac{n^3-n}{6} \right)^3 + \left(1 - \frac{n^3 - n}{6} \right)^3 + \left(\frac{n^3 -n }{6} \right)^3 + \left(\frac{n^3 -n }{6} \right)^3. \ \ \ \Box$