SCHWEITZER 2020

We say that two sequences $x,y \colon \mathbb{N} \to \mathbb{N}$ are completely different if $x_n \neq y_n$ holds for all $n\in \mathbb{N}$ . Let $F$ be a function assigning a natural number to every sequence of natural numbers such that $F(x)\neq F(y)$ for any pair of completely different sequences $x$ , $y$ , and for constant sequences we have $F \left((k,k,\dots)\right)=k$ . Prove that there exists $n\in \mathbb{N}$ such that $F(x)=x_{n}$ for all sequences $x$ .

Prove that if $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous periodic function and $\alpha \in \mathbb{R}$ is irrational, then the sequence $\{n\alpha+f(n\alpha)\}_{n=1}^{\infty}$ modulo 1 is dense in $[0,1]$ .

An $n\times n$ matrix $A$ with integer entries is called representative if, for any integer vector $\mathbf{v}$ , there is a finite sequence $0=\mathbf{v}_0,\mathbf{v}_1,\dots,\mathbf{v}_{\ell}=\mathbf{v}$ of integer vectors such that for each $0\leq i <\ell$ , either $\mathbf{v}_{i+1}=A\mathbf{v}_{i}$ or $\mathbf{v}_{i+1}-\mathbf{v}_i$ is an element of the standard basis (i.e. one of its entries is $1$ , the rest are all equal to $0$ ). Show that $A$ is not representative if and only if $A^T$ has a real eigenvector with all non-negative entries and non-negative eigenvalue.

Consider horizontal and vertical segments in the plane that may intersect each other. Let $n$ denote their total number. Suppose that we have $m$ curves starting from the origin that are pairwise disjoint except for their endpoints. Assume that each curve intersects exactly two of the segments, a different pair for each curve. Prove that $m=O(n)$ .

Prove that for a nowhere dense, compact set $K\subset \mathbb{R}^2$ the following are equivalent:

(i) $K=\bigcup_{i=1}^{\infty}K_n$ where $K_n$ is a compact set with connected complement for all $n$ .

(ii) $K$ does not have a nonempty closed subset $S\subseteq K$ such that any neighborhood of any point in $S$ contains a connected component of $\mathbb{R}^2 \setminus S$ .

Does there exist an entire function $F \colon \mathbb{C}\to \mathbb{C}$ such that $F$ is not zero everywhere, $|F(z)|\leq e^{|z|}$ for all $z\in \mathbb{C}$ , $|F(iy)|\leq 1$ for all $y\in \mathbb{R}$ , and $F$ has infinitely many real roots.

Let $p(n)\geq 0$ for all positive integers $n$ . Furthermore, $x(0)=0, v(0)=1$ , and $$x(n)=x(n-1)+v(n-1), v(n)=v(n-1)-p(n)x(n) (n=1,2,\dots).$$ Assume that $v(n)\to 0$ in a decreasing manner as $n \to \infty$ . Prove that the sequence $x(n)$ is bounded if and only if $\sum_{n=1}^{\infty}n\cdot p(n)<\infty$ .

Let $\mathbb{F}_{p}$ denote the $p$ -element field for a prime $p>3$ and let $S$ be the set of functions from $\mathbb{F}_{p}$ to $\mathbb{F}_{p}$ . Find all functions $D\colon S\to S$ satisfying $$D(f\circ g)=(D(f)\circ g)\cdot D(g)$$ for all $f,g \in {S}$ . Here, $\circ$ denotes the function composition, so $(f\circ g)(x)$ is the function $f(g(x))$ , and $\cdot$ denotes multiplication, so $(f\cdot g)=f(x)g(x)$ .

Let $f$ be a polynomial of degree $n$ with integer coefficients and $p$ a prime for which $f$ , considered modulo $p$ , is a degree- $k$ irreducible polynomial over $\mathbb{F}_p$ . Show that $k$ divides the degree of the splitting field of $f$ over $\mathbb{Q}$ .

Given a real number $p>1$ , a continuous function $h\colon [0,\infty)\to [0,\infty)$ , and a smooth vector field $Y\colon \mathbb{R}^n \to \mathbb{R}^n$ with $\mathrm{div}~Y=0$ , prove the following inequality $$\int_{\mathbb{R}^n}h(|x|)|x|^{p}\leq \int_{\mathbb{R}^{n}}h(|x|)|x+Y(x)|^{p}.$$