SCHWEITZER 1969

Let $G$ be an infinite group generated by nilpotent normal subgroups. Prove that every maximal Abelian normal subgroup of $G$ is infinite. (We call an Abelian normal subgroup maximal if it is not contained in another Abelian normal subgroup.)

P. Erdos

Let $p\geq 7$ be a prime number, $\zeta$ a primitive $p$ th root of unity, $c$ a rational number. Prove that in the additive group generated by the numbers $1,\zeta,\zeta^2,\zeta^3+\zeta^{-3}$ there are only finitely many elements whose norm is equal to $c$ . (The norm is in the $p$ th cyclotomic field.)

K. Gyory

Let $f(x)$ be a nonzero, bounded, real function on an Abelian group $G$ , $g_1,...,g_k$ are given elements of $G$ and $\lambda_1,...,\lambda_k$ are real numbers. Prove that if $$\sum_{i=1}^k \lambda_i f(g_ix) \geq 0$$ holds for all $x \in G$ , then $$\sum_{i=1}^k \lambda_i \geq 0.$$

A. Mate

Show that the following inequality hold for all $k \geq 1$ , real numbers $a_1,a_2,...,a_k$ , and positive numbers $x_1,x_2,...,x_k.$ $$\ln \frac {\sum\limits_{i = 1}^kx_i}{\sum\limits_{i = 1}^kx_i^{1 - a_i}} \leq \frac {\sum\limits_{i = 1}^ka_ix_i \ln x_i}{\sum\limits_{i = 1}^kx_i} .$$ L. Losonczi

Find all continuous real functions $f,g$ and $h$ defined on the set of positive real numbers and satisfying the relation $$f(x+y)+g(xy)=h(x)+h(y)$$ for all $x>0$ and $y>0$ .

Z. Daroczy

Let $x_0$ be a fixed real number, and let $f$ be a regular complex function in the half-plane $\Re z>x_0$ for which there exists a nonnegative function $F \in L_1(- \infty, +\infty)$ satisfying $|f(\alpha+i\beta)| \leq F(\beta)$ whenever $\alpha > x_0$ , $-\infty <\beta < +\infty$ . Prove that $$\int_{\alpha-i \infty} ^{\alpha+i \infty} f(z)dz=0.$$

L. Czach

Prove that if a sequence of Mikusinski operators of the form $\mu e^{-\lambda s}$ ( $\lambda$ and $\mu$ nonnegative real numbers, $s$ the differentiation operator) is convergent in the sense of Mikusinski, then its limit is also of this form.

E. Geaztelyi

Let $f$ and $g$ be continuous positive functions defined on the interval $[0, +\infty)$ , and let $E \subset[0,+\infty)$ be a set of positive measure. Prove that the range of the function defined on $E \times E$ by the relation $$F(x,y)= \int_0^xf(t)dt+ \int_0^y g(t)dt$$ has a nonvoid interior.

L. Losonczi

In $n$ -dimensional Euclidean space, the union of any set of closed balls (of positive radii) is measurable in the sense of Lebesgue.

A. Csaszar

In $n$ -dimensional Euclidean space, the square of the two-dimensional Lebesgue measure of a bounded, closed, (two-dimensional) planar set is equal to the sum of the squares of the measures of the orthogonal projections of the given set on the $n$ -coordinate hyperplanes.

L. Tamassy

Let $A_1,A_2,...$ be a sequence of infinite sets such that $|A_i \cap A_j| \leq 2$ for $i \not=j$ . Show that the sequence of indices can be divided into two disjoint sequences $i_1<i_2<...$ and $j_1<j_2<...$ in such a way that, for some sets $E$ and $F$ , $|A_{i_n} \cap E|=1$ and $|A_{j_n} \cap F|=1$ for $n=1,2,... .$

P. Erdos

Let $A$ and $B$ be nonsingular matrices of order $p$ , and let $\xi$ and $\eta$ be independent random vectors of dimension $p$ . Show that if $\xi,\eta$ and $\xi A+ \eta B$ have the same distribution, if their first and second moments exist, and if their covariance matrix is the identity matrix, then these random vectors are normally distributed.

B. Gyires