Distinct Circular Sums 1500 pts

Let $n\ge 3$ be an integer. There are $n$ pairwise distinct real numbers.

Prove that they can be arranged around a circle so that, for each $1\le k<n$, the $n$ sums of $k$ consecutive numbers are pairwise distinct.

One arrangement must work for all these lengths. Sums of different lengths may be equal.