It is well known that for all $n>2$, there is no integer solution for the following:
$$x^n + y^n = z^n $$
I will modify this equation slightly. What if the number of terms on the left-hand side is equal to $n$? For instance $n=5$:
$$a_{1}^{5}+a_{2}^{5}+a_{3}^{5}+a_{4}^{5}+a_{5}^{5}=+a_{6}^{5}$$
Is there solution for such equations, where the $a_k$'s are integer? (I mean non-trivial)
We have the following non-trivial solution $$ 72^5=19^5+43^5+46^5+47^5+67^5. $$ Here $72$ is the smallest positive integer with this property. For more examples see here. In general, Euler's conjecture is about your equation. It was disproved for $n=4,5$ but is still open for $n\ge 6$.