Do there exist integers $a,b,c$ such that $a^5+b^5+c^5=2016abc$ and $a+b+c=5776$? This question should be solvable without a calculator - I tried playing around with odd/even properties, but didn't get very far.
I also tried looking at the average of $a,b,c$ (about $1900$), but this involved a lot of manual computations and this is supposed to be solvable without a calculator.

No, there does not. In modulo $3$, you have $a+b+c\equiv 1$ and $a^5+b^5+c
^5\equiv 0$. However, $a^5\equiv a$. Therefore, the latter equation reduces to $a+b+c\equiv 0$. This is a contradiction.
