SCHWEITZER 1972

Let $\mathcal{F}$ be a nonempty family of sets with the following properties:

(a) If $X \in \mathcal{F}$ , then there are some $Y \in \mathcal{F}$ and $Z \in \mathcal{F}$ such that $Y \cap Z =\emptyset$ and $Y \cup Z=X$ .

(b) If $X \in \mathcal{F}$ , and $Y \cup Z =X , Y \cap Z=\emptyset$ , then either $Y \in \mathcal{F}$ or $Z \in \mathcal{F}$ .

Show that there is a decreasing sequence $X_0 \supseteq X_1 \supseteq X_2 \supseteq ...$ of sets $X_n \in \mathcal{F}$ such that $$\bigcap_{n=0}^{\infty} X_n= \emptyset.$$

F. Galvin

Let $\leq$ be a reflexive, antisymmetric relation on a finite set $A$ . Show that this relation can be extended to an appropriate finite superset $B$ of $A$ such that $\leq$ on $B$ remains reflexive, antisymmetric, and any two elements of $B$ have a least upper bound as well as a greatest lower bound. (The relation $\leq$ is extended to $B$ if for $x,y \in A , x \leq y$ holds in $A$ if and only if it holds in $B$ .)

E. Freid

Let $\lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form $$\mu= \sum_{i=1}^{\infty}n_i\lambda_i ,$$ where $n_i \geq 0$ are integers and all but finitely many $n_i$ are $0$ . Let $$L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ M(x)= \sum _{\mu \leq x} 1 \ .$$ (In the latter sum, each $\mu$ occurs as many times as its number of representations in the above form.) Prove that if $$\lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,$$ then $$\lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.$$

G. Halasz

Let $G$ be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that $G$ is finite.

J. Pelikan

We say that the real-valued function $f(x)$ defined on the interval $(0,1)$ is approximately continuous on $(0,1)$ if for any $x_0 \in (0,1)$ and $\varepsilon >0$ the point $x_0$ is a point of interior density $1$ of the set $$H= \{x : \;|f(x)-f(x_0)|< \varepsilon \ \}.$$ Let $F \subset (0,1)$ be a countable closed set, and $g(x)$ a real-valued function defined on $F$ . Prove the existence of an approximately continuous function $f(x)$ defined on $(0,1)$ such that $$f(x)=g(x) \;\textrm{for all}\ x \in F\ .$$

M. Laczkovich, Gy. Petruska

Let $P(z)$ be a polynomial of degree $n$ with complex coefficients, $$P(0)=1, \;\textrm{and}\ \;|P(z)|\leq M\ \;\textrm{for}\ \;|z| \leq 1\ .$$ Prove that every root of $P(z)$ in the closed unit disc has multiplicity at most $c\sqrt{n}$ , where $c=c(M) >0$ is a constant depending only on $M$ .

G. Halasz

Let $f(x,y,z)$ be a nonnegative harmonic function in the unit ball of $\mathbb{R}^3$ for which the inequality $f(x_0,0,0) \leq \varepsilon^2$ holds for some $0\leq x_0 \leq 1$ and $0<\varepsilon<(1-x_0)^2$ . Prove that $f(x,y,z) \leq \varepsilon$ in the ball with center at the origin an radius $(1-3\varepsilon^{1/4}).$

P. Turan

Given four points $A_1,A_2,A_3,A_4$ in the plane in such a way that $A_4$ is the centroid of the $\bigtriangleup A_1A_2A_3$ , find a point $A_5$ in the plane that maximizes the ratio $$\frac{\min_{1 \leq i < j < k \leq 5}T(A_iA_jA_k)}{\max_{1 \leq i < j < k \leq 5}T(A_iA_jA_k)}.$$ ( $T(ABC)$ denotes the area of the triangle $\bigtriangleup ABC.$ )

J. Suranyi

Let $K$ be a compact convex body in the $n$ -dimensional Euclidean space. Let $P_1,P_2,...,P_{n+1}$ be the vertices of a simplex having maximal volume among all simplices inscribed in $K$ . Define the points $P_{n+2},P_{n+3},...$ successively so that $P_k \;(k>n+1)$ is a point of $K$ for which the volume of the convex hull of $P_1,...,P_k$ is maximal. Denote this volume by $V_k$ . Decide, for different values of $n$ , about the truth of the statement "the sequence $V_{n+1},V_{n+2},...$ is concave."

L. Fejes- Toth, E. Makai

Let $\mathcal{T}_1$ and $\mathcal{T}_2$ be second-countable topologies on the set $E$ . We would like to find a real function $\sigma$ defined on $E \times E$ such that $$0 \leq \sigma(x,y) <+\infty, \;\sigma(x,x)=0 \ ,$$ $$\sigma(x,z) \leq \sigma(x,y)+\sigma(y,z) \;(x,y,z \in E) \ ,$$ and, for any $p \in E$ , the sets $$V_1(p,\varepsilon)=\{ x : \;\sigma(x,p)< \varepsilon \ \} \;(\varepsilon >0)$$ form a neighborhood base of $p$ with respect to $\mathcal{T}_1$ , and the sets $$V_2(p,\varepsilon)=\{ x : \;\sigma(p,x)< \varepsilon \ \} \;(\varepsilon >0)$$ form a neighborhood base of $p$ with respect to $\mathcal{T}_2$ . Prove that such a function $\sigma$ exists if and only if, for any $p \in E$ and $\mathcal{T}_i$ -open set $G \ni p \;(i=1,2)$ , there exist a $\mathcal{T}_i$ -open set $G'$ and a $\mathcal{T}_{3-i}$ -closed set $F$ with $p \in G' \subset F \subset G.$

A. Csaszar

We throw $N$ balls into $n$ urns, one by one, independently and uniformly. Let $X_i=X_i(N,n)$ be the total number of balls in the $i$ th urn. Consider the random variable $$y(N,n)=\min_{1 \leq i \leq n}|X_i-\frac Nn|.$$ Verify the following three statements:

(a) If $n \rightarrow \infty$ and $N/n^3 \rightarrow \infty$ , then $$P \left(\frac{y(N,n)}{\frac 1n \sqrt{\frac Nn}}<x \right) \rightarrow 1-e^{-x\sqrt{2/ \pi}} \;\textrm{for all}\ x>0 \ .$$

(b) If $n\rightarrow \infty$ and $N/n^3 \leq K$ ( $K$ constant), then for any $\varepsilon > 0$ there is an $A > 0$ such that $$P(y(N,n) < A) > 1-\varepsilon .$$

(c) If $n \rightarrow \infty$ and $N/n^3 \rightarrow 0$ then $$P(y(N,n) < 1) \rightarrow 1.$$

P. Revesz