IMC 2018 · Problem 6

Let $k$ be a positive integer. Find the smallest positive integer $n$ for which there exist $k$ nonzero vectors $v_1, \ldots, v_k$ in $\mathbb{R}^n$ such that for every pair $i, j$ of indices with $|i - j| > 1$ the vectors $v_i$ and $v_j$ are orthogonal.