<Demonstrate that exist 1000 integers, all different from one another, satisifying the following three properties:their sum is a 21-power, the sum of their squares is a 22-power ad their product is a 23-power.>

I am trying to demonstrate something more general , but I am stuck with a system of congruences...
the generalization could be this one:

<Demonstrate that exist n integers, all different from one another, satisifying the following three properties:their sum is a a-power, the sum of their squares is a b-power ad their product is a c-power, with a, b and c mutually coprime>