### International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria

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Problem 6. Let $\displaystyle k$ be a positive integer. Find the smallest positive integer $\displaystyle n$ for which there exist $\displaystyle k$ nonzero vectors $\displaystyle v_1,\ldots,v_k$ in $\displaystyle \mathbb R^n$ such that for every pair $\displaystyle i,j$ of indices with $\displaystyle |i-j|> 1$ the vectors $\displaystyle v_i$ and $\displaystyle v_j$ are orthogonal.
Hint: Consider the vectors $\displaystyle v_1,v_3,v_5,\ldots$.