2022 SEEMOUS

Part of SEEMOUS

Subcontests

(4)

1

SEEMOUS 2022 Problem 4

\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}

1

SEEMOUS 2022 Problem 3

\alpha \in \mathbb{C}\setminus \{0\}

A \in \mathcal{M}_n(\mathbb{C})

A \neq O_n

A^2 + (A^*)^2 = \alpha A\cdot A^*,

A^* = (\bar A)^T.

\alpha \in \mathbb{R}

|\alpha| \le 2

A\cdot A^* = A^*\cdot A.

1

SEEMOUS 2022 Problem 2

a, b, c \in \mathbb{R}

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)

x, y, z \in \mathbb{R}.

Prove that any Palić function is infinitely many times differentiable and find all Palić functions.

1

SEEMOUS 2022 Problem 1

A, B \in \mathcal{M}_n(\mathbb{C})

AB^2A = AB

. Prove that: a)

(AB)^2 = AB.

(AB - BA)^3 = O_n.