SCHWEITZER 1959

Let $p_n$ be the $n$ th prime number. Prove that $\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty$ (N.17)

Omit the vertices of a closed rectangle; the configuration obtained in such a way will be called a reduced rectangle. Prove tha the set-union of any system of reduced rectangles with parallel sides is equal to the union of countably many elements of the system. (St. 3)

**3.**Let $G$ be an arbitrary group, $H_1,\dots ,H_n$ some (not necessarily distinet) subgroup of $G$ and $g_1, \dots , g_n$ elements of $G$ such that each element of $G$ belongs at least to one of the right cosets $H_1 g_1, \dots , H_n g_n$ . Show that if, for any $k$ , the set-union of the cosets $H_i g_i (i=1, \dots , k-1, k+1, \dots , n)$ differs from $G$ , then every $H_k (k=1, \dots , n)$ is of finite index in $G$ . (A. 15)

Consider $n$ circles of radius $1$ in the planea. Prove that at least one of the circles contains an are of length greater than $\frac{2\pi}{n}$ not intersected by any other of these circles. (G. 4)

Denote by $c_n$ the $n$ th positive integer which can be represented in the form $c_n = k^{l} (k,l = 2,3, \dots )$ . Prove that $\sum_{n=1}^{\infty}\frac{1}{c_n-1}=1$ (N. 18)

Let $T$ be a one-to-one mapping of the unit square $E$ of the plane into itself. Suppose that $T$ and $T^{-1}$ are measure-preserving (i.e. if $M \subseteq E$ is a measurable set, then $TM$ and $T^{-1}M$ are also measurable and $\mu (M)= \mu (TM)= \mu (T^{-1}M)$ , where $\mu$ denotes the Lebesgue measure) and, furthermore, that if $Tx \in N$ for almost all points $x$ of a measurable set $N \subseteq E$ , then either $n$ or $E \setminus N$ is of measure 0. Prove that, for any measurable set $A \subseteq E$ , with $\mu (A)>0$ , the function $n(x)$ defined by $$n(x)=\begin{cases} 0, \text{if} T^k x \notin A (k=1, 2, \dots), \min (k: T^k x \in A; k=1,2, \dots ) &\text{otherwise} \end{cases}$$ is measurable and $\int_{A}n(x) d\mu(x) =1$ (R. 18)

Let $(z_n)_{n=1}^{\infty}$ be a sequence of complex numbers tending to zero. Prove that there exists a sequence $(\epsilon_n)_{n=1}^{\infty}$ (where $\epsilon_n = +1$ or $-1$ ) such that the series $\sum_{n=1}^{\infty} \epsilon_n z_n$

is convergente. (F. 9)

An Oblique lattice-cubs is a lattice-cube of the three-dimensional fundamental lattice no edge of which is perpendicular to any coordinate axis. Prove that for any integer $h= 8n-1$ ( $n= 1, 2, \dots$ ) there existis an oblique lattice-cube with edges of length $h$ . Propose a method for finding such a cube. (N. 20)

Let $f(z)= z^n +a_1 z^{n-1}+\dots + a_n$ be a polynomial over the field of the complex numbers and let $E_f$ denote the closed (not necessarily connected) domain of complex numbers $z$ for which $\mid f(z) \mid \leq 1$ . Show that there exists a point $z_0 \in E_f$ such that $\mid f'(z_0) \mid \geq n$ . (F. 5)

Prove that if a graph with $2n+1$ vertices has at least $3n+1$ edges, then the graph contains a circuit having an even number of edges. Prove further that this statemente does not hold for $3n$ edges. (By a circuit, we mean a closed line which does not intersect itself.) (C. 5)