SCHWEITZER 2015

Let $K$ be a closed subset of the closed unit ball in $\mathbb{R}^3$ . Suppose there exists a family of chords $\Omega$ of the unit sphere $S^2$ , with the following property: for every $X,Y\in S^2$ , there exist $X',Y'\in S^2$ , as close to $X$ and $Y$ correspondingly, as we want, such that $X'Y'\in \Omega$ and $X'Y'$ is disjoint from $K$ . Verify that there exists a set $H\subset S^2$ , such that $H$ is dense in the unit sphere $S^2$ , and the chords connecting any two points of $H$ are disjoint from $K$ .

EDIT: The statement fixed. See post #4

Let $\{x_n\}$ be a Van Der Corput series,that is,if the binary representation of $n$ is $\sum a_{i}2^{i}$ then $x_n=\sum a_i2^{-i-1}$ .Let $V$ be the set of points on the plane that have the form $(n,x_n)$ .Let $G$ be the graph with vertex set $V$ that is connecting any two points $(p,q)$ if there is a rectangle $R$ which lies in parallel position with the axes and $R\cap V= \{p,q\}$ .Prove that the chromatic number of $G$ is finite.

Let ${A}$ be a finite set and ${\rightarrow}$ be a binary relation on it such that for any ${a,b,c \in A}$ , if ${a\neq b}, {a \rightarrow c}$ and ${b \rightarrow c}$ then either ${a \rightarrow b}$ or ${b \rightarrow a}$ (or possibly both). Let ${B, B \subset A}$ be minimal with the property: for any ${a \in A \setminus B}$ there exists ${b \in B}$ , such that either ${a \rightarrow b}$ or ${b \rightarrow a}$ (or possibly both). Supposing that ${A}$ has at most ${k}$ elements that are pairwise not in relation ${\rightarrow}$ , prove that ${B}$ has at most ${k}$ elements.

Let $a_n$ be a series of positive integers with $a_1=1$ and for any arbitrary prime number $p$ , the set $\{a_1,a_2,\cdots,a_p\}$ is a complete remainder system modulo $p$ . Prove that $\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1$ .

Let $f(x) = x^n+x^{n-1}+\dots+x+1$ for an integer $n\ge 1.$ For which $n$ are there polynomials $g, h$ with real coefficients and degrees smaller than $n$ such that $f(x) = g(h(x)).$

Let $G$ be the permutation group of a finite set $\Omega$ .Consider $S\subset G$ such that $1\in S$ and for any $x,y\in \Omega$ there exists a unique element $\sigma \in S$ such that $\sigma (x)=y$ .Prove that,if the elements of $S \setminus \{1\}$ are conjugate in $G$ ,then $G$ is $2-$ transitive on $\Omega$

We call a bar of width ${w}$ on the surface of the unit sphere ${\Bbb{S}^2}$ , a spherical segment, centered at the origin, which has width ${w}$ and is symmetric with respect to the origin. Prove that there exists a constant ${c>0}$ , such that for any positive integer ${n}$ the surface ${\Bbb{S}^2}$ can be covered with ${n}$ bars of the same width so that any point is contained in no more than ${c\sqrt{n}}$ bars.

Prove that all continuous solutions of the functional equation $\left(f(x)-f(y)\right)\left(f\left(\frac{x+y}{2}\right)-f\left(\sqrt{xy}\right)\right)=0 \ , \ \forall x,y\in (0,+\infty)$ are the constant functions.

For a function ${u}$ defined on ${G \subset \Bbb{C}}$ let us denote by ${Z(u)}$ the neignborhood of unit raduis of the set of roots of ${u}$ . Prove that for any compact set ${K \subset G}$ there exists a constant ${C}$ such that if ${u}$ is an arbitrary real harmonic function on ${G}$ which vanishes in a point of ${K}$ then: $$\sup_{z \in K} |u(z)| \leq C \sup_{Z(u)\cap G}|u(z)|.$$

Let $f:\mathbb{R}\to \mathbb{R}$ be a continuously differentiable,strictly convex function.Let $H$ be a Hilbert space and $A,B$ be bounded,self adjoint linear operators on $H$ .Prove that,if $f(A)-f(B)=f'(B)(A-B)$ then $A=B$ .

For $[0,1]\subset E\subset [0,+\infty)$ where $E$ is composed of a finite number of closed interval,we start a two dimensional Brownian motion from the point $x<0$ terminating when we first hit $E$ .Let $p(x)$ be the probability of the finishing point being in $[0,1]$ .Prove that $p(x)$ is increasing on $[-1,0)$ .