SCHWEITZER 1979

Let $\Gamma$ be a variety of monoids such that not all monoids of $\Gamma$ are groups. Prove that if $A \in \Gamma$ and $B$ is a submonoid of $A$ , there exist monoids $S \in \Gamma$ and $C$ and epimorphisms $\varphi : S \rightarrow A, \;\varphi_1 : S \rightarrow C$ such that $((e)\varphi_1^{-1})\varphi=B$ ( $e$ is the identity element of $C$ ).

L. Marki

Give an example of ten different noncoplanar points $P_1,\ldots ,P_5,Q_1,\ldots ,Q_5$ in $3$ -space such that connecting each $P_i$ to each $Q_j$ by a rigid rod results in a rigid system.

L. Lovasz

Let us defined a pseudo-Riemannian metric on the set of points of the Euclidean space $\mathbb{E}^3$ not lying on the $z$ -axis by the metric tensor $$\left( \begin{array}{ccc} 1 & 0 & 0 0 & 1 & 0 0 & 0 & -\sqrt{x^2+y^2} \end{array} \right),$$ where $(x,y,z)$ is a Cartesian coordinate system $\mathbb{E}^3$ . Show that the orthogonal projections of the geodesic curves of this Riemannian space onto the $(x,y)$ -plane are straight lines or conic sections with focus at the origin

P. Nagy

Let $T$ be a triangulation of an $n$ -dimensional sphere, and to each vertex of $T$ let us assign a nonzero vector of a linear space $V$ . Show that if $T$ has an $n$ -dimensional simplex such that the vectors assigned to the vertices of this simplex are linearly independent, then another such simplex must also exist.

L. Lovasz

Let $K_n(n=1,2,\ldots)$ be periodical continuous functions of period $2 \pi$ , and write $$k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt .$$ Prove that the following statements are equivalent:

(i) $\int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty)$ for all $f \in \mathcal{L}_1[0,2 \pi]$ .

(ii) $k_n(f;0) \rightarrow f(0)$ for all continuous, $2 \pi$ -periodic functions $f$ .

V. Totik

Let us assume that the series of holomorphic functions $\sum_{k=1}^{\infty}f_k(z)$ is absolutely convergent for all $z \in \mathbb{C}$ . Let $H \subseteq \mathbb{C}$ be the set of those points where the above sum funcion is not regular. Prove that $H$ is nowhere dense but not necessarily countable.

L. Kerchy

Prove that if $a_i(i=1,2,3,4)$ are positive constants, $a_2-a_4 > 2$ , and $a_1a_3-a_2 > 2$ , then the solution $(x(t),y(t))$ of the system of differential equations $$\.{x}=a_1-a_2x+a_3xy,$$ $$\.{y}=a_4x-y-a_3xy \;\;\;(x,y \in \mathbb{R})$$ with the initial conditions $x(0)=0, y(0) \geq a_1$ is such that the function $x(t)$ has exactly one strict local maximum on the interval $[0, \infty)$ .

L. Pinter, L. Hatvani

Let $\{\xi_{k \ell} \}_{k,\ell=1}^{\infty}$ be a double sequence of random variables such that

$$\Bbb{E}( \xi_{ij} \xi_{k\ell})= \mathcal{O} \left(\frac{ \log(2|i-k|+2)}{ \log(2|j-\ell|+2)^{2}}\right) \;\;\;(i,j,k,\ell =1,2, \ldots ) \ .$$

Prove that with probability one,

$$\frac{1}{mn} \sum_{k=1}^m \sum_{\ell=1}^n \xi_{k\ell} \rightarrow 0 \;\;\textrm{as} \max (m,n)\rightarrow \infty\ .$$

F. Moricz