Herzog Competition - 2024

Herzog competition. MSU Department of Mathematics, November 9 2024

Please, write on a separate paper provided. You may take this list of problems with you.

Question 1

Let $A$ and $B$ be two different matrices of size $n \times n$ with real entries. Given that $A^{4} = B^{4}$ and $A^{3} B = B^{3} A$ can $A^{3} + B^{3}$ be invertible?

Question 2

Find all real polynomials of degree $n \geq 2$ for which there exist real numbers $r_{1} < r_{2} < \dots < r_{n}$ such that $p (r_{i}) = 0$ , $i = 1, \dots n$ and

p^{'} (\frac{r_{i} + r_{i + 1}}{2}) = 0, i = 1, \dots n - 1

where $p^{'} (x)$ denotes the derivative of $p (x)$ .

Question 3

Show that the equation $y^{2} = x^{5} - 4$ has no integer solutions.

Question 4

Let $f : [a, b] \to R$ be a continuous function that is differentiable on $(a, b)$ . Assume that $f$ is nowhere zero on $(a, b)$ . Prove that there exists $θ \in (a, b)$ such that

\frac{f^{'} (θ)}{f (θ)} = \frac{1}{a - θ} + \frac{1}{b - θ}

Question 5

We have 2024 distinct points on a unit circle so that the product of distances from any point on the circle to all the given points does not exceed 2. Prove that the points are vertices of a regular 2024-gon.

Question 6

Let us have a board of size $1 \times N$ (that is, the board composed out of $N$ consecutive squares). Domino tiles (of size $1 \times 2$ ) are placed on this board without overlapping such that each tile occupies two neighbor squares. A configuration of tiles is called admissible if no more tiles can be placed on the board without overlapping. Let $T_{N}$ be the sum over all admissible configurations $p$ taken with weights $f (p) := 6^{k (p)}$ , where $k (p)$ is the number of remaining empty squares in $p$ . Find

{lim}_{N \to \infty} 2^{- N} T_{N} .