International Mathematics Competition
for University Students
2018

IMC2018: Day 2, Problem 6

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.

(Proposed by Alexey Balitskiy, Moscow Institute of Physics and Technology and M.I.T.)

Hint: Consider the vectors \(\displaystyle v_1,v_3,v_5,\ldots\).

  Solution