SCHWEITZER 1962

Let $f$ and $g$ be polynomials with rational coefficients, and let $F$ and $G$ denote the sets of values of $f$ and $g$ at rational numbers. Prove that $F = G$ holds if and only if $f(x) = g(ax + b)$ for some suitable rational numbers $a\not = 0$ and $b$ .

E. Fried

Determine the roots of unity in the field of $p$ -adic numbers.

L. Fuchs

Let $A$ and $B$ be two Abelian groups, and define the sum of two homomorphisms $\eta$ and $\chi$ from $A$ to $B$ by $$a( \eta+\chi)=a\eta+a\chi \;\textrm{for all}\ a \in A\ .$$ With this addition, the set of homomorphisms from $A$ to $B$ forms an Abelian group $H$ . Suppose now that $A$ is a $p$ -group ( $p$ a prime number). Prove that in this case $H$ becomes a topological group under the topology defined by taking the subgroups $p^kH \;(k=1,2,...)$ as a neighborhood base of $0$ . Prove that $H$ is complete in this topology and that every connected component of $H$ consists of a single element. When is $H$ compact in this topology? [L. Fuchs]

Show that $$\prod_{1\leq x < y \leq \frac{p-1}{2}} (x^2+y^2) \equiv (-1)^{\lfloor\frac{p+1}{8}\rfloor} \;(\textbf{mod} p\ )$$ for every prime $p\equiv 3 \;(\textbf{mod} 4\ )$ . [J. Suranyi]

Let $f$ be a finite real function of one variable. Let $\overline{D}f$ and $\underline{D}f$ be its upper and lower derivatives, respectively, that is, $$\overline{D}f=\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h+k>0}}} \frac{f(x+h)-f(x-k)}{h+k}$$ , $$\underline{D}f=\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h+k>0}}} \frac{f(x+h)-f(x-k)}{h+k}.$$ Show that $\overline{D}f$ and $\underline{D}f$ are Borel-measurable functions. [A. Csaszar]

Let $E$ be a bounded subset of the real line, and let $\Omega$ be a system of (non degenerate) closed intervals such that for each $x \in E$ there exists an $I \in \Omega$ with left endpoint $x$ . Show that for every $\varepsilon > 0$ there exists a finite number of pairwise non overlapping intervals belonging to $\Omega$ that cover $E$ with the exception of a subset of outer measure less than $\varepsilon$ . [J. Czipszer]

Prove that the function $$f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}$$ (where the positive value of the square root is taken) is monotonically decreasing in the interval $0<\nu<1$ . [P. Turan]

Denote by $M(r,f)$ the maximum modulus on the circle $|z|=r$ of the transcendent entire function $f(z)$ , and by $M_n(r,f)$ that of the $nth$ partial sum of the power series of $f(z)$ . Prove that the existence of an entire function $f_0(z)$ and a corresponding sequence of positive numbers $r_1<r_2<...\rightarrow +\infty$ such that $$\limsup_{n\rightarrow\infty} \frac{M_n(r_n,f_0)}{M(r_n,f_0)}=+\infty$$ [P. Turan]

Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].

From a given triangle of unit area, we choose two points independetly with uniform distribution. The straight line connecting these points divides the triangle. with probability one, into a triangle and a quadrilateral. Calculate the expected values of the areas of these two regions. [A. Renyi]