IMC 2013 · Problem 3

Suppose that $v_1, \dots, v_d$ are unit vectors in $\mathbb{R}^d$. Prove that there exists a unit vector $u$ such that

$$|u \cdot v_i| \le 1/\sqrt{d}$$

for $i = 1, 2, \dots, d$.

(Here $\cdot$ denotes the usual scalar product on $\mathbb{R}^d$.)