Can we find $\sum_i x_i$ given $\{\sum_i x_i^{2n}\}_{n\in \mathbb{N}}$, assuming $\{x_i\}_{i\in\mathbb{N}}$ is a set of positive real numbers?
Perhaps an easier question is, can we find $\sum_i x_i$ given $\{\sum_i x_i^{n}\}_{n\in \mathbb{N}\setminus 1}$? Intuitively it seems impossible without further assumptions on $\{x_i\}_{i\in\mathbb{N}}$. Is that true?