This problem appeared in the South-Eastern European Mathematical Olympiad.
Let $n$, $m$ be positive integers. Prove that for any matrices $A_1, \dots, A_m\in M_n(\mathbb{R})$, there exist $\epsilon_1, \dots, \epsilon_m = \pm 1$ such that \begin{align*} \operatorname{tr}\left((\epsilon_1 A_1 + \cdots + \epsilon_m A_m)^2\right) \geq \operatorname{tr}(A_1^2) + \cdots + \operatorname{tr}(A_m^2).\end{align*}
I only managed to prove it for $A_1 = \cdots = A_m$, which is quite trivial.
Since the average of $\operatorname{tr}\left(\left(\sum_i\varepsilon_iA_i\right)^2\right)$ over all $(\varepsilon_1,\ldots,\varepsilon_m)\in\{-1,1\}^m$ is equal to $\operatorname{tr}\left(\sum_iA_i^2\right)$, some $\operatorname{tr}\left(\left(\sum_i\varepsilon_iA_i\right)^2\right)$ must be greater than or equal to $\operatorname{tr}\left(\sum_iA_i^2\right)$.