SCHWEITZER 1958

Find the groups every generating system of which contains a basis. (A basis is a set of elements of the group such that the direct product of the cyclic groups generated by them is the group itself.) (A. 14)

Let $A(x)$ denote the number of positive integers $n$ not greater than $x$ and having at least one prime divisor greater than $\sqrt[3]{n}$ . Prove that $\lim_{x\to \infty} \frac {A(x)}{x}$ exists. (N. 15)

Let $n$ be a positive integer having at least one prime factor with expoente $\geq 2$ . Show that $n$ has as many factorizations into an odd number of factors as into an even number of factors. (Factorizations into the same factors arranged in different order are considered different.)(N. 10)

Let $P_1 P_2 P_3 P_4 P_5 P_6$ be a convex hexagon. Denote by $T$ its area and by $t$ the area of the triangle $Q_1 Q_2 Q_3$ , where $Q_1,Q_2$ and $Q_3$ are the midpoints of $P_1P_4,P_2P_5,P_3P_6$ respectively. Prove that $t<\frac{1}{4}T$ . (G. 3)

Prove that neither the closed nor the open interval can be decomposed into finitely many mutually disjoint proper subsets which are all congruent by translation. (St. 2)

Prove that if $a_n \geq 0$ and $\frac{1}{n}\sum_{k=1}^{n} a_k \geq \sum_{k=n+1}^{2n}a_k$ $(n=1, 2, \dots)$ ,

then $\sum_{k=1}^{\infty} a_k$ is convergent and its sum is less than $2ea_1$ . (S. 9)

Let $a_0$ and $a_1$ be arbitrary real numbers, and let $a_{n+1}=a_n + \frac{2}{n+1}a_{n-1}$ $(n= 1, 2, \dots)$ Show that the sequence $\left (\frac{a_n}{n^2} \right )_{n=1}^{\infty}$ is convergent and find its limit. (S. 10)

Let the function $f(x)$ be periodic with the period $1$ , non-negative, concave in the interval $(0,1)$ and continuous at the point $0$ . Prove that $f(nx)\leq nf(x)$ for every real $x$ and positive integer $n$ . (R. 6)

Show that if $f(z) = 1+a_1 z+a_2z^2+\dots$ for $\mid z \mid\leq 1$ and $\frac{1}{2\pi}\int_{0}^{2\pi}\mid f(e^{i\phi}) \mid^{2} d\phi < \left (1+\frac{\mid a_1\mid ^2} {4} \right )^2$ ,

then $f(z)$ has a root in the disc $\mid z \mid \leq 1$ .(F. 4)

Prove that the function $f(x)= \int_{-\infty}^{\infty} \left (\frac{\sin\theta}{\theta} \right )^{2k}\cos (2x\theta) d\theta$ where $k$ is a positive integer, satisfies the following conditions:(i) $f(x)=0$ if $\mid x \mid \geq k$ and $f(x) \geq 0$ elsewhere;(ii) in interval $(l,l+1)$ $(l= -k, -k+1, \dots , k-1)$ the function $f(x)$ is a polynomial of degree $2k-1$ at most. (R. 7)

Let $a_n = (-1)^n (n= 1, 2, \dots , 2N)$ . Denote by $A_{N}(x)$ the number of the sequences $1 \leq i_1 < i_2< \dots <i_N \leq 2N$ such that $a_{i_1}+a_{i_2}+ \dots +a_{i_N}< x \sqrt{\frac{N}{2}} (-\infty < x < \infty)$ . Show that $\lim_{N \to \infty} \frac{A_{N}(x)}{\binom{2N}{N}} = \frac {1}{\sqrt {2\pi}} \int_{-\infty}^{\infty} e^{-\frac{u^2}{2}} du$ .(N. 16)