Since it includes the information that "n= 6", I would interpret this as summing over 6 terms- and assume that the sums run from 1 to 6:
and
So the question is "do there exist 6 numbers, that sum to 25, whose squares sum to 64?"
That is 6 numbers determined by only two equations so it seems to me there ought to be many ways to do that. For example, if we choose to take , , and the equations reduce to or and or . Since so that . .
Was there a requirement that the numbers be integers?