SCHWEITZER 1971

Let $G$ be an infinite compact topological group with a Hausdorff topology. Prove that $G$ contains an element $g \not= 1$ such that the set of all powers of $g$ is either everywhere dense in $G$ or nowhere dense in $G$ .

J. Erdos

Prove that there exists an ordered set in which every uncountable subset contains an uncountable, well-ordered subset and that cannot be represented as a union of a countable family of well-ordered subsets.

A. Hajnal

Let $0<a_k<1$ for $k=1,2,... .$ Give a necessary and sufficient condition for the existence, for every $0<x<1$ , of a permutation $\pi_x$ of the positive integers such that $$x= \sum_{k=1}^{\infty} \frac{a_{\pi_x}(k)}{2^k}.$$

P. Erdos

Suppose that $V$ is a locally compact topological space that admits no countable covering with compact sets. Let $\textbf{C}$ denote the set of all compact subsets of the space $V$ and $\textbf{U}$ the set of open subsets that are not contained in any compact set. Let $f$ be a function from $\textbf{U}$ to $\textbf{C}$ such that $f(U)\subseteq U$ for all $U \in \textbf{U}$ . Prove that either

(i) there exists a nonempty compact set $C$ such that $f(U)$ is not a proper subset of $C$ whenever $C \subseteq U \in \textbf{U}$ ,

(ii) or for some compact set $C$ , the set $$f^{-1}(C)= \bigcup \{U \in \textbf{U}\;: \ f(U)\subseteq C\ \}$$ is an element of $\textbf{U}$ , that is, $f^{-1}(C)$ is not contained in any compact set.

A. Mate

Let $\lambda_1 \leq \lambda_2 \leq...$ be a positive sequence and let $K$ be a constant such that $$\sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).$$ Prove that there exists a constant $K'$ such that $$\sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).$$

L. Leindler

Let $a(x)$ and $r(x)$ be positive continuous functions defined on the interval $[0,\infty)$ , and let $$\liminf_{x \rightarrow \infty} (x-r(x)) >0.$$ Assume that $y(x)$ is a continuous function on the whole real line, that it is differentiable on $[0, \infty)$ , and that it satisfies $$y'(x)=a(x)y(x-r(x))$$ on $[0, \infty)$ . Prove that the limit $$\lim_{x \rightarrow \infty}y(x) \exp \left\{ -\int_0^x a(u)du \right \}$$ exists and is finite.

I. Gyori

Let $n \geq 2$ be an integer, let $S$ be a set of $n$ elements, and let $A_i , 1\leq i \leq m$ , be distinct subsets of $S$ of size at least $2$ such that $$A_i \cap A_j \not= \emptyset, A_i \cap A_k \not= \emptyset, A_j \cap A_k \not= \emptyset, \;\textrm{imply}\ A_i \cap A_j \cap A_k \not= \emptyset \ .$$ Show that $m \leq 2^{n-1}-1$ .

P. Erdos

Show that the edges of a strongly connected bipolar graph can be oriented in such a way that for any edge $e$ there is a simple directed path from pole $p$ to pole $q$ containing $e$ . (A strongly connected bipolar graph is a finite connected graph with two special vertices $p$ and $q$ having the property that there are no points $x,y,x \not = y$ , such that all paths from $x$ to $p$ as well as all paths from $x$ to $q$ contain $y$ .)

A. Adam

Given a positive, monotone function $F(x)$ on $(0, \infty)$ such that $F(x)/x$ is monotone nondecreasing and $F(x)/x^{1+d}$ is monotone nonincreasing for some positive $d$ , let $\lambda_n >0$ and $a_n \geq 0 , n \geq 1$ . Prove that if $$\sum_{n=1}^{\infty} \lambda_n F \left( a_n \sum _{k=1}^n \frac{\lambda_k}{\lambda_n} \right) < \infty,$$ or $$\sum_{n=1}^{\infty} \lambda_n F \left( \sum _{k=1}^n a_k \frac{\lambda_k}{\lambda_n} \right) < \infty,$$ then $\sum_{n=1}^ {\infty} a_n$ is convergent.

L. Leindler

Let $\{\phi_n(x) \}$ be a sequence of functions belonging to $L^2(0,1)$ and having norm less that $1$ such that for any subsequence $\{\phi_{n_k}(x) \}$ the measure of the set $$\{x \in (0,1) : \;|\frac{1}{\sqrt{N}} \sum _{k=1}^N \phi_{n_k}(x)| \geq y\ \}$$ tends to $0$ as $y$ and $N$ tend to infinity. Prove that $\phi_n$ tends to $0$ weakly in the function space $L^2(0,1).$

F. Moricz

Let $C$ be a simple arc with monotone curvature such that $C$ is congruent to its evolute. Show that under appropriate differentiability conditions, $C$ is a part of a cycloid or a logarithmic spiral with polar equation $r=ae^{\vartheta}$ .

J. Szenthe