SCHWEITZER 1968

Consider the endomorphism ring of an Abelian torsion-free (resp. torsion) group $G$ . Prove that this ring is Neumann-regular if and only if $G$ is a discrete direct sum of groups isomorphic to the additive group of the rationals (resp. ,a discrete direct sum of cyclic groups of prime order). (A ring $R$ is called Neumann-regular if for every $\alpha \in R$ there exists a $\beta \in R$ such that $\alpha \beta \alpha=\alpha$ .)

E. Freid

Let $a_1,a_2,...,a_n$ be nonnegative real numbers. Prove that $$( \sum_{i=1}^na_i)( \sum_{i=1}^na_i^{n-1}) \leq n \prod_{i=1}^na_i+ (n-1) ( \sum_{i=1}^na_i^n).$$

J. Suranyi

Let $K$ be a compact topological group, and let $F$ be a set of continuous functions defined on $K$ that has cardinality greater that continuum. Prove that there exist $x_0 \in K$ and $f \not=g \in F$ such that $$f(x_0)=g(x_0)=\max_{x\in K}f(x)=\max_{x \in K}g(x).$$

I. Juhasz

Let $f$ be a complex-valued, completely multiplicative,arithmetical function. Assume that there exists an infinite increasing sequence $N_k$ of natural numbers such that $$f(n)=A_k \not= 0 \;\textrm{provided}\ N_k \leq n \leq N_k+4 \sqrt{N_k}\ .$$ Prove that $f$ is identically $1$ .

I. Katai

Let $k$ be a positive integer, $z$ a complex number, and $\varepsilon <\frac12$ a positive number. Prove that the following inequality holds for infinitely many positive integers $n$ : $$\mid \sum_{0\leq l \leq \frac{n}{k+1}} \binom{n-kl}{l}z^l \mid \geq (\frac 12-\varepsilon)^n.$$

P. Turan

Let $\Psi=\langle A;...\rangle$ be an arbitrary, countable algebraic structure (that is, $\Psi$ can have an arbitrary number of finitary operations and relations). Prove that $\Psi$ has as many as continuum automorphisms if and only if for any finite subset $A'$ of $A$ there is an automorphism $\pi_{A'}$ of $\Psi$ different from the identity automorphism and such that $$(x) \pi_{A'}=x$$ for every $x \in A'$ .

M. Makkai

For every natural number $r$ , the set of $r$ -tuples of natural numbers is partitioned into finitely many classes. Show that if $f(r)$ is a function such that $f(r)\geq 1$ and $\lim _{r\rightarrow \infty} f(r)=+\infty$ , then there exists an infinite set of natural numbers that, for all $r$ , contains $r$ -triples from at most $f(r)$ classes. Show that if $f(r) \not \rightarrow +\infty$ , then there is a family of partitions such that no such infinite set exists.

P. Erdos, A. Hajnal

Let $n$ and $k$ be given natural numbers, and let $A$ be a set such that $$|A| \leq \frac{n(n+1)}{k+1}.$$ For $i=1,2,...,n+1$ , let $A_i$ be sets of size $n$ such that $$|A_i \cap A_j| \leq k \;(i \not=j)\ ,$$ $$A= \bigcup_{i=1}^{n+1} A_i.$$ Determine the cardinality of $A$ .

K. Corradi

Let $f(x)$ be a real function such that $$\lim_{x \rightarrow +\infty} \frac{f(x)}{e^x}=1$$ and $|f''(x)|\leq c|f'(x)|$ for all sufficiently large $x$ . Prove that $$\lim_{x \rightarrow +\infty} \frac{f'(x)}{e^x}=1.$$

P. Erdos

Let $h$ be a triangle of perimeter $1$ , and let $H$ be a triangle of perimeter $\lambda$ homothetic to $h$ . Let $h_1,h_2,...$ be translates of $h$ such that , for all $i$ , $h_i$ is different from $h_{i+2}$ and touches $H$ and $h_{i+1}$ (that is, intersects without overlapping). For which values of $\lambda$ can these triangles be chosen so that the sequence $h_1,h_2,...$ is periodic? If $\lambda \geq 1$ is such a value, then determine the number of different triangles in a periodic chain $h_1,h_2,...$ and also the number of times such a chain goes around the triangle $H$ .

L. Fejes-Toth

Let $A_1,...,A_n$ be arbitrary events in a probability field. Denote by $C_k$ the event that at least $k$ of $A_1,...,A_n$ occur. Prove that $$\prod_{k=1}^n P(C_k) \leq \prod_{k=1}^n P(A_k).$$

A. Renyi