SCHWEITZER 1973

We say that the rank of a group $G$ is at most $r$ if every subgroup of $G$ can be generated by at most $r$ elements. Prove that here exists an integer $s$ such that for every finite group $G$ of rank $2$ the commutator series of $G$ has length less than $s$ .

J. Erdos

Let $R$ be an Artinian ring with unity. Suppose that every idempotent element of $R$ commutes with every element of $R$ whose square is $0$ . Suppose $R$ is the sum of the ideals $A$ and $B$ . Prove that $AB=BA$ .

A. Kertesz

Find a constant $c > 1$ with the property that, for arbitrary positive integers $n$ and $k$ such that $n>c^k$ , the number of distinct prime factors of $\binom{n}{k}$ is at least $k$ .

P. Erdos

Let $f(n)$ be that largest integer $k$ such that $n^k$ divides $n!$ , and let $F(n)= \max_{2 \leq m \leq n} f(m)$ . Show that $$\lim_{n\rightarrow \infty} \frac{F(n) \log n}{n \log \log n}=1.$$

P. Erdos

Verify that for every $x > 0$ , $$\frac{\Gamma'(x+1)}{\Gamma (x+1)} > \log x.$$

P. Medgyessy

If $f$ is a nonnegative, continuous, concave function on the closed interval $[0,1]$ such that $f(0)=1$ , then $$\int_0^1 xf(x)dx \leq \frac 23 \left[ \int_0^1 f(x)dx \right]^2.$$

Z. Daroczy

Let us connect consecutive vertices of a regular heptagon inscribed in a unit circle by connected subsets (of the plane of the circle) of diameter less than $1$ . Show that every continuum (in the plane of the circle) of diameter greater than $4$ , containing the center of the circle, intersects one of these connected sets.

M. Bognar

What is the radius of the largest disc that can be covered by a finite number of closed discs of radius $1$ in such a way that each disc intersects at most three others?

L. Fejes-Toth

Determine the value of $$\sup_{ 1 \leq \xi \leq 2} [\log E \xi-E \log \xi],$$ where $\xi$ is a random variable and $E$ denotes expectation.

Z. Daroczy

Find the limit distribution of the sequence $\eta_n$ of random variables with distribution $$P \left( \eta_n=\arccos (\cos^2 \frac{(2j-1) \pi}{2n}) \right)=\frac 1n \;(j=1,2,...,n)\ .$$ ( $\arccos(.)$ denotes the main value.)

B. Gyires