International Mathematics Competition for University Students

July 22 – 28 2018, Blagoevgrad, Bulgaria


Ivan is Watching You
Ivan's Office

Day 1
    Problem 1
    Problem 2
    Problem 3
    Problem 4
    Problem 5

Day 2
    Problem 6
    Problem 7
    Problem 8
    Problem 9
    Problem 10


    Day 1 questions
    Day 1 solutions
    Day 2 questions
    Day 2 solutions
    Closing Ceremony

Official IMC site

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 \(\displaystyle v_1,v_3,v_5,\ldots\).