Minimize max inner product with n unit vectors in R^n

Question

You are given $n$ unit vectors $u_i \in \mathbb{R}^n$. Is it always true that there is another unit vector $x$ such that $$ \left|\left< u_i, x \right>\right| \leq \frac{1}{\sqrt{n}} \quad \forall i?$$

For $u_i$ the standard basis vectors, then $x = (\frac{1}{\sqrt{n}}, \frac{1}{\sqrt{n}}, \dots, \frac{1}{\sqrt{n}})$ will do. Intuitively as the $u_i$ vectors are less orthogonal, the problem becomes easier. But can anyone provide a proof?

Answer 1

I had to solve this problem for a math competition. You can find it here. It is the $3^{rd}$ problem.

https://www.imc-math.org.uk/imc2013/IMC2013-day2-solutions.pdf

Please let me know if something is not clear.