SCHWEITZER 1967

Let $$f(x)=a_0+a_1x+a_2x^2+a_{10}x^{10}+a_{11}x^{11}+a_{12}x^{12}+a_{13}x^{13} (a_{13} \not=0)$$ and $$g(x)=b_0+b_1x+b_2x^2+b_{3}x^{3}+b_{11}x^{11}+b_{12}x^{12}+b_{13}x^{13} (b_{3} \not=0)$$ be polynomials over the same field. Prove that the degree of their greatest common divisor is at least $6$ .

L. Redei

Let $K$ be a subset of a group $G$ that is not a union of lift cosets of a proper subgroup. Prove that if $G$ is a torsion group or if $K$ is a finite set, then the subset $$\bigcap _{k \in K} k^{-1}K$$ consists of the identity alone.

L. Redei

Prove that if an infinite, noncommutative group $G$ contains a proper normal subgroup with a commutative factor group, then $G$ also contains an infinite proper normal subgroup.

B. Csakany

Let $a_1,a_2,...,a_N$ be positive real numbers whose sum equals $1$ . For a natural number $i$ , let $n_i$ denote the number of $a_k$ for which $2^{1-i} \geq a_k \geq 2^{-i}$ holds. Prove that $$\sum_{i=1}^{\infty} \sqrt{n_i2^{-i}} \leq 4+\sqrt{\log_2 N}.$$

L. Leinder

Let $f$ be a continuous function on the unit interval $[0,1]$ . Show that $$\lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)$$ and $$\lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).$$

Let $A$ be a family of proper closed subspaces of the Hilbert space $H=l^2$ totally ordered with respect to inclusion (that is , if $L_1,L_2 \in A$ , then either $L_1\subset L_2$ or $L_2\subset L_1$ ). Prove that there exists a vector $x \in H$ not contaied in any of the subspaces $L$ belonging to $A$ .

B. Szokefalvi Nagy

Let $U$ be an $n \times n$ orthogonal matrix. Prove that for any $n \times n$ matrix $A$ , the matrices $$A_m=\frac{1}{m+1} \sum_{j=0}^m U^{-j}AU^j$$ converge entrywise as $m \rightarrow \infty.$

L. Kovacs

Suppose that a bounded subset $S$ of the plane is a union of congruent, homothetic, closed triangles. Show that the boundary of $S$ can be covered by a finite number of rectifiable arcs.

L. Geher

Let $F$ be a surface of nonzero curvature that can be represented around one of its points $P$ by a power series and is symmetric around the normal planes parallel to the principal directions at $P$ . Show that the derivative with respect to the arc length of the curvature of an arbitrary normal section at $P$ vanishes at $P$ . Is it possible to replace the above symmetry condition by a weaker one?

A. Moor

Let $\sigma(S_n,k)$ denote the sum of the $k$ th powers of the lengths of the sides of the convex $n$ -gon $S_n$ inscribed in a unit circle. Show that for any natural number greater than $2$ there exists a real number $k_0$ between $1$ and $2$ such that $\sigma(S_n,k_0)$ attains its maximum for the regular $n$ -gon.

L. Fejes Toth