SCHWEITZER 1985

Let $F(x,y)$ and $G(x,y)$ be relatively prime homogeneous polynomials of degree at least one having integer coefficients. Prove that there exists a number $c$ depending only on the degrees and the maximum of the absolute values of the coefficients of $F$ and $G$ such that $F(x,y)\neq G(x,y)$ for any integers $x$ and $y$ that are relatively prime and satisfy $\max \{ |x|,|y|\} > c$ . [K. Gyory]

Determine all finite groups $G$ that have an automorphism $f$ such that $H\not\subseteq f(H)$ for all proper subgroups $H$ of $G$ . [B. Kovacs]

Let $p_1$ and $p_2$ be positive real numbers. Prove that there exist functions $f_i\colon \mathbb R \rightarrow \mathbb R$ such that the smallest positive period of $f_i$ is $p_i (i=1, 2)$ , and $f_1-f_2$ is also periodic. [J. Riman]

Let $\frac{2}{\sqrt5+1}\leq p < 1$ , and let the real sequence $\{ a_n \}$ have the following property: for every sequence $\{ e_n \}$ of $0$ 's and $\pm 1$ 's for which $\sum_{n=1}^\infty e_np^n=0$ , we also have $\sum_{n=1}^\infty e_na_n=0$ . Prove that there is a number $c$ such that $a_n=cp^n$ for all $n$ . [Z. Daroczy, I. Katai]

Let $D=\{ z\in \mathbb C\colon |z|<1\}$ and $D=\{ w\in \mathbb C \colon |w|=1\}$ . Prove that if for a function $f\colon D\times B\rightarrow\mathbb C$ the equality $$f\left( \frac{az+b}{\overline{b}z+\overline{a}}, \frac{aw+b}{\overline{b}w+\overline a} \right)=f(z,w)+f\left(\frac{b}{\overline a}, \frac{aw+b}{\overline b w+\overline a} \right)$$ holds for all $z\in D, w\in B$ and $a, b\in \mathbb C,|a|^2=|b|^2+1$ , then there is a function $L\colon (0, \infty )\rightarrow \mathbb C$ satisfying $$L(pq)=L(p)+L(q)\,\,\,\text{for all}\,\, p,q > 0$$ such that $f$ can be represented as $$f(z,w)=L\left( \frac{1-|z|^2}{|w-z|^2}\right)\,\,\,\text{for all}\,\, z\in D, w\in B$$ . [Gy. Maksa]

Show that any two intervals $A, B\subseteq \mathbb R$ of positive lengths can be countably disected into each other, that is, they can be written as countable unions $A=A_1\cup A_2\cup\ldots\,$ and $B=B_1\cup B_2\cup\ldots\,$ of pairwise disjoint sets, where $A_i$ and $B_i$ are congruent for every $i\in \mathbb N$ [Gy. Szabo]

Let $\xi (E, \pi, B) (\pi\colon E\rightarrow B)$ be a real vector bundle of finite rank, and let $$\tau_E=V\xi \oplus H\xi\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (*)$$ be the tangent bundle of $E$ , where $V\xi=\mathrm{Ker} d\pi$ is the vertical subbundle of $\tau_E$ . Let us denote the projection operators corresponding to the splitting $(*)$ by $v$ and $h$ . Construct a linear connection $\nabla$ on $V\xi$ such that $$\nabla_X\lor Y - \nabla_Y \lor X=v[X,Y] - v[hX,hY]$$ ( $X$ and $Y$ are vector fields on $E$ , $[., .]$ is the Lie bracket, and all data are of class $\mathcal C^\infty$ . [J. Szilasi]

Let $(\Omega, \mathcal A, P)$ be a probability space, and let $(X_n, \mathcal F_n)$ be an adapted sequence in $(\Omega, \mathcal A, P)$ (that is, for the $\sigma$ -algebras $\mathcal F_n$ , we have $\mathcal F_1\subseteq \mathcal F_2\subseteq \dots \subseteq \mathcal A$ , and for all $n$ , $X_n$ is an $\mathcal F_n$ -measurable and integrable random variable). Assume that $$\mathrm E (X_{n+1} \mid \mathcal F_n )=\frac12 X_n+\frac12 X_{n-1}\,\,\,\, (n=2, 3, \ldots )$$ Prove that $\mathrm{sup}_n \mathrm{E}|X_n|<\infty$ implies that $X_n$ converges with probability one as $n\to\infty$ . [I. Fazekas]