SEEMOUS 2019

A sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is called "Devin" if for any $f\in C[0,1]$ $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n f(x_i)=\int_0^1 f(x) dx$$ Prove that a sequence $\{x_n\}_{n=1}^{\infty}, 0\leq x_n\leq 1$ is "Devin" if and only if for any non-negative integer $k$ it holds $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n x_i^k=\frac{1}{k+1}.$$ Remark. I left intact the text as it was proposed. Devin is a Bulgarian city and SPA resort, where this competition took place.

Let $A_1, A_2,\dots,A_m\in \mathcal{M}_n(\mathbb{R})$ . Prove that there exist $\varepsilon_1,\varepsilon_2,\dots,\varepsilon_m\in \{-1,1\}$ such that: $$\rm{tr}\left( (\varepsilon_1 A_1+\varepsilon_2A_2+\dots+\varepsilon_m A_m)^2\right)\geq \rm{tr}(A_1^2)+\rm{tr}(A_2^2)+\dots+\rm{tr}(A_m^2)$$

Let $A,B$ be $n\times n$ matrices, $n\geq 2$ , and $B^2=B$ . Prove that: $$\text{rank}\,(AB-BA)\leq \text{rank}\,(AB+BA)$$