IMC 2013 Problem 8

Suppose that ${{v_1},{v_2},...,{v_d}}$ are unit vectors in ${{{\Bbb R}^d}}$ . Prove that there exists a unitary vector ${u}$ such that ${\left| {u \cdot {v_i}} \right| \leq \frac{1}{{\sqrt d }}}$ for ${i = 1,2,...,d}$ .Note. Here ${ \cdot }$ denotes the usual scalar product on ${{{\Bbb R}^d}}$ .

Proposed by Tomasz Tkocz, University of Warwick.