SCHWEITZER 1970

We have $2n+1$ elements in the commutative ring $R$ : $$\alpha,\alpha_1,...,\alpha_n,\varrho_1,...,\varrho_n .$$ Let us define the elements $$\sigma_k=k\alpha + \sum_{i=1}^n \alpha_i\varrho_i^k .$$ Prove that the ideal $(\sigma_0,\sigma_1,...,\sigma_k,...)$ can be finitely generated.

L. Redei

Let $G$ and $H$ be countable Abelian $p$ -groups ( $p$ an arbitrary prime). Suppose that for every positive integer $n$ , $$p^nG \not= p^{n+1}G .$$ Prove that $H$ is a homomorphic image of $G$ .

M. Makkai

The traffic rules in a regular triangle allow one to move only along segments parallel to one of the altitudes of the triangle. We define the distance between two points of the triangle to be the length of the shortest such path between them. Put $\binom{n+1}{2}$ points into the triangle in such a way that the minimum distance between pairs of points is maximal.

L. Fejes-Toth

If $c$ is a positive integer and $p$ is an odd prime, what is the smallest residue (in absolute value) of $$\sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod} p\ ) \ ?$$

J. Suranyi

Prove that two points in a compact metric space can be joined with a rectifiable arc if and only if there exists a positive number $K$ such that, for any $\varepsilon>0$ , these points can be connected with an $\varepsilon$ -chain not longer that $K$ .

M. Bognar

Let a neighborhood basis of a point $x$ of the real line consist of all Lebesgue-measurable sets containing $x$ whose density at $x$ equals $1$ . Show that this requirement defines a topology that is regular but not normal.

A. Csaszar

Let us use the word $N$ -measure for nonnegative, finitely additive set functions defined on all subsets of the positive integers, equal to $0$ on finite sets, and equal to $1$ on the whole set. We say that the system $\Upsilon$ of sets determines the $N$ -measure $\mu$ if any $N$ -measure coinciding with $\mu$ on all elements of $\Upsilon$ is necessarily identical with $\mu$ . Prove the existence of an $N$ -measure $\mu$ that cannot be determined by a system of cardinality less than continuum.

I. Juhasz

Let $\pi_n(x)$ be a polynomial of degree not exceeding $n$ with real coefficients such that $$|\pi_n(x)| \leq \sqrt{1-x^2} \;\textrm{for}\ \;-1\leq x \leq 1 \ .$$ Then $$|\pi'_n(x)| \leq 2(n-1).$$

P. Turan

Construct a continuous function $f(x)$ , periodic with period $2 \pi$ , such that the Fourier series of $f(x)$ is divergent at $x=0$ , but the Fourier series of $f^2(x)$ is uniformly convergent on $[0,2 \pi].$

P. Turan

Prove that for every $\vartheta$ , $0<\vartheta<1$ , there exist a sequence $\lambda_n$ of positive integers and a series $\sum_{n=1}^{\infty} a_n$ such that

(i) $\lambda_{n+1}-\lambda_n > (\lambda_n)^{\vartheta}$ ,

(ii) $\lim_{r\rightarrow 1^-} \sum_{n=1}^{\infty} a_nr^{\lambda_n}$ exists,

(iii) $\sum _{n=1}^{\infty} a_n$ is divergent.

P. Turan

Let $\xi_1,\xi_2,...$ be independent random variables such that $E\xi_n=m>0$ and $\textrm{Var}(\xi_n)=\sigma^2 < \infty \;(n=1,2,...)\ .$ Let $\{a_n \}$ be a sequence of positive numbers such that $a_n\rightarrow 0$ and $\sum_{n=1}^{\infty} a_n= \infty$ . Prove that $$P \left( \lim_{n\rightarrow \infty} \sum_{k=1}^n a_k \xi_k =\infty \right)=1.$$

P. Revesz

Let $\vartheta_1,...,\vartheta_n$ be independent, uniformly distributed, random variables in the unit interval $[0,1]$ . Define $$h(x)= \frac1n \# \{k: \vartheta_k<x\ \}.$$ Prove that the probability that there is an $x_0 \in (0,1)$ such that $h(x_0)=x_0$ , is equal to $1- \frac1n.$

G. Tusnady