JBMO-SL 2021 A2

Let $n > 3$ be a positive integer. Find all integers $k$ such that $1 \le k \le n$ and for which the following property holds: If $x_1, . . . , x_n$ are $n$ real numbers such that $x_i + x_{i + 1} + ... + x_{i + k - 1} = 0$ for all integers $i > 1$ (indexes are taken modulo $n$ ), then $x_1 = . . . = x_n = 0$ .

Proposed by Vincent Jugé and Théo Lenoir, France