MathNet 2022-part-02 — 197 problems (EN–VI) Question 102

Let $n$ be an integer such that $n \geqslant 3$. Lucie arranges $n$ blank sheets in a circle, then writes a real number on each sheet. Find all integers $k$ with $1 \leqslant k \leqslant n$ for which the following property holds:
If at least one of the numbers Lucie wrote is nonzero, then there exist $k$ consecutive sheets whose numbers have a nonzero sum.