SCHWEITZER 1977

Consider the intersection of an ellipsoid with a plane $\sigma$ passing through its center $O$ . On the line through the point $O$ perpendicular to $\sigma$ , mark the two points at a distance from $O$ equal to the area of the intersection. Determine the loci of the marked points as $\sigma$ runs through all such planes.

L. Tamassy

Construct on the real projective plane a continuous curve, consisting of simple points, which is not a straight line and is intersected in a single point by every tangent and every secant of a given conic.

F. Karteszi

Prove that if $a,x,y$ are $p$ -adic integers different from $0$ and $p | x, pa | xy$ , then $$\frac 1y \frac{(1+x)^y-1}{x} \equiv \frac{\log (1+x)}{x} \;\;\; ( \textrm{mod} a\ ) .$$

L. Redei

Let $p>5$ be a prime number. Prove that every algebraic integer of the $p$ th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field.

K. Gyory

Suppose that the automorphism group of the finite undirected graph $X=(P, E)$ is isomorphic to the quaternion group (of order $8$ ). Prove that the adjacency matrix of $X$ has an eigenvalue of multiplicity at least $4$ .

( $P= \{ 1,2,\ldots, n \}$ is the set of vertices of the graph $X$ . The set of edges $E$ is a subset of the set of all unordered pairs of elements of $P$ . The group of automorphisms of $X$ consists of those permutations of $P$ that map edges to edges. The adjacency matrix $M=[m_{ij}]$ is the $n \times n$ matrix defined by $m_{ij}=1$ if $\{ i,j \} \in E$ and $m_{i,j}=0$ otherwise.)

L. Babai

Let $G$ be a locally compact solvable group, let $c_1,\ldots, c_n$ be complex numbers, and assume that the complex-valued functions $f$ and $g$ on $G$ satisfy $$\sum_{k=1}^n c_k f(xy^k)=f(x)g(y) \;\textrm{for all} x,y \in G \ \ .$$ Prove that if $f$ is a bounded function and $$\inf_{x \in G} \textrm{Re} f(x) \chi(x) >0$$ for some continuous (complex) character $\chi$ of $G$ , then $g$ is continuous.

L. Szekelyhidi

Let $p \geq 1$ be a real number and $\mathbb{R}_+=(0, \infty)$ . For which continuous functions $g : \mathbb{R}_+ \rightarrow \mathbb{R}_+$ are following functions all convex? $$M_n(x)=\left[ \frac{\sum_{i=1}^n g(\frac{x_i}{x_{i+1}}) x_{i+1}^p}{\sum_{i=1}^n g(\frac{x_i}{x_{i+1}})} \right ]^\frac 1p ,$$ $$x=(x_1,\ldots, x_{n+1}) \in \mathbb{R}_+ ^ {n+1} , n=1,2,\ldots$$

L. Losonczi

Suppose that the components of he vector $\textbf{u}=(u_0,\ldots,u_n)$ are real functions defined on the closed interval $[a,b]$ with the property that every nontrivial linear combination of them has at most $n$ zeros in $[a,b]$ . Prove that if $\sigma$ is an increasing function on $[a,b]$ and the rank of the operator $$A(f)= \int_{a}^b \textbf{u}(x)f(x)d\sigma(x), f \in C[a,b]\ ,$$ is $r \leq n$ , then $\sigma$ has exactly $r$ points of increase.

E. Gesztelyi

Let the sequence of random variables $\{ X_m, m \geq 0\ \}, X_0=0$ , be an infinite random walk on the set of nonnegative integers with transition probabilities $$p_i=P(X_{m+1}=i+1 \mid X_m=i) >0, i \geq 0 \,$$ $$q_i=P(X_{m+1}=i-1 \mid X_m=i ) >0, i>0.$$ Prove that for arbitrary $k >0$ there is an $\alpha_k > 1$ such that $$P_n(k)=P \left ( \max_{0 \leq j \leq n} X_j =k \right)$$ satisfies the limit relation $$\lim_{L \rightarrow \infty} \frac 1L \sum_{n=1}^L P_n(k) \alpha_k ^n < \infty.$$

J. Tomko