SCHWEITZER 2012

Is there any real number $\alpha$ for which there exist two functions $f,g: \mathbb{N} \to \mathbb{N}$ such that $$\alpha=\lim_{n \to \infty} \frac{f(n)}{g(n)},$$ but the function which associates to $n$ the $n$ -th decimal digit of $\alpha$ is not recursive?

Call a subset $A$ of the cyclic group $(\mathbb{Z}_n,+)$ rich if for all $x,y \in \mathbb{Z}_n$ there exists $r \in \mathbb{Z}_n$ such that $x-r,x+r,y-r,y+r$ are all in $A$ . For what $\alpha$ is there a constant $C_\alpha>0$ such that for each odd positive integer $n$ , every rich subset $A \subset \mathbb{Z}_n$ has at least $C_\alpha n^\alpha$ elements?

Let $K$ be a convex shape in the $n$ dimensional space, having unit volume. Let $S \subset K$ be a Lebesgue measurable set with measure at least $1-\varepsilon$ , where $0<\varepsilon<1/3$ . Prove that dilating $K$ from its centroid by the ratio of $2\varepsilon \ln(1/\varepsilon)$ , the shape obtained contains the centroid of $S$ .

Let $V_1,V_2,V_3,V_4$ be four dimensional linear subspaces in $\mathbb{R}^8$ such that the intersection of any two contains only the zero vector. Prove that there exists a linear four dimensional subspace $W$ in $\mathbb{R}^8$ such that all four vector spaces $W\cap V_i$ are two dimensional.

Let $A,B,C$ be matrices with complex elements such that $[A,B]=C, [B,C]=A$ and $[C,A]=B$ , where $[X,Y]$ denotes the $XY-YX$ commutator of the matrices. Prove that $e^{4 \pi A}$ is the identity matrix.

Let $\Gamma$ be a simple curve, lying inside a circle of radius $r$ , rectifiable and of length $\ell$ . Prove that if $\ell > kr\pi$ , then there exists a circle of radius $r$ which intersects $\Gamma$ in at least $k+1$ distinct points.

For any function $f: \mathbb{R}^2\to \mathbb{R}$ consider the function $\Phi_f:\mathbb{R}^2\to [-\infty,\infty]$ for which $\Phi_f(x,y)=\limsup_{ z \to y} f(x,z)$ for any $(x,y) \in \mathbb{R}^2$ . [list=a] []Is it true that if $f$ is Lebesgue measurable then $\Phi_f$ is also Lebesgue measurable?[/] []Is it true that if $f$ is Borel measurable then $\Phi_f$ is also Borel measurable?[/] [/list]

Let $D$ be the complex unit disk $D=\{z \in \mathbb{C}: |z|<1\}$ , and $0<a<1$ a real number. Suppose that $f:D \to \mathbb{C}\setminus \{0\}$ is a holomorphic function such that $f(a)=1$ and $f(-a)=-1$ . Prove that $$\sup_{z \in D} |f(z)| \geqslant \exp\left(\frac{1-a^2}{4a}\pi\right) .$$

Let $K$ be a knot in the $3$ -dimensional space (that is a differentiable injection of a circle into $\mathbb{R}^3$ , and $D$ be the diagram of the knot (that is the projection of it to a plane so that in exception of the transversal double points it is also an injection of a circle). Let us color the complement of $D$ in black and color the diagram $D$ in a chessboard pattern, black and white so that zones which share an arc in common are coloured differently. Define $\Gamma_B(D)$ the black graph of the diagram, a graph which has vertices in the black areas and every two vertices which correspond to black areas which have a common point have an edge connecting them. [list=a] []Determine all knots having a diagram $D$ such that $\Gamma_B(D)$ has at most $3$ spanning trees. (Two knots are not considered distinct if they can be moved into each other with a one parameter set of the injection of the circle)[/] []Prove that for any knot and any diagram $D$ , $\Gamma_B(D)$ has an odd number of spanning trees.[/] [/list]

Let $X_1,X_2,..$ be independent random variables with the same distribution, and let $S_n=X_1+X_2+...+X_n, n=1,2,...$ . For what real numbers $c$ is the following statement true: $$P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}$$