SEEMOUS 2022

Let $A, B \in \mathcal{M}_n(\mathbb{C})$ be such that $AB^2A = AB$ . Prove that: a) $(AB)^2 = AB.$ b) $(AB - BA)^3 = O_n.$

Let $a, b, c \in \mathbb{R}$ be such that $$a + b + c = a^2 + b^2 + c^2 = 1, \hspace{8px} a^3 + b^3 + c^3 \neq 1.$$ We say that a function $f$ is a Palić function if $f: \mathbb{R} \rightarrow \mathbb{R}$ , $f$ is continuous and satisfies $$f(x) + f(y) + f(z) = f(ax + by + cz) + f(bx + cy + az) + f(cx + ay + bz)$$ for all $x, y, z \in \mathbb{R}.$ Prove that any Palić function is infinitely many times differentiable and find all Palić functions.

Let $\alpha \in \mathbb{C}\setminus \{0\}$ and $A \in \mathcal{M}_n(\mathbb{C})$ , $A \neq O_n$ , be such that $$A^2 + (A^*)^2 = \alpha A\cdot A^*,$$ where $A^* = (\bar A)^T.$ Prove that $\alpha \in \mathbb{R}$ , $|\alpha| \le 2$ . and $A\cdot A^* = A^*\cdot A.$

Let $\mathcal{F}$ be the family of all nonempty finite subsets of $\mathbb{N} \cup \{0\}.$ Find all real numbers $a$ for which the series $$\sum_{A \in \mathcal{F}} \frac{1}{\sum_{k \in A}a^k}$$ is convergent.