Little math problem(s) February 2019

[pier4r]

And another problem. Although I didn't dive deep in it yet. (I may have already exposed it in other threads)

One has a set of positive integers
\(I = \{ i_1,...., i_N \}\)

the question is: can we identify uniquely I if the sum
\(S_1= \sum (i_1, ... , i_N)\)
and
\(S_2 = \sum (i_1^2, ..., i_N^2 )\)
are given?

What if N is fixed (say, 30 elements)? Does it help us to identify I uniquely?
If N is not fixed, does it mean that it is easier to find a I1 and a I2 of different size (and therefore elements) that have the same S1 and S2 ?

The problem here is to find either an argument for uniqueness or a counterexample that uniqueness is not always true.

Inspiration: monthly stats on the bank account.