SCHWEITZER 1957

Let $C_{ij}$ ( $i,j=1,2,3$ ) be the coefficients of a real non-involutive orthogonal transformation. Prove that the function $w= \sum_{i,j=1}^{3} c_{ ij}z_{i}\bar{z_{ j}}$ maps the surface of complex unit sphere $\sum_{i=1}^{3} z_{i}\bar{z_{i}} = 1$ onto a triangle of the w-plane. (F. 3)

Let $A$ be a subset of n-dimensional space containing at least one inner point and suppose that, for every point pair $x, y \in A$ , the subset $A$ contains the mid point of the line segment beteween $x$ and $y$ . Show that $A$ consists of a convex open set and of some of its boundary points. (St. 1)

Let $F_{\epsilon} (0<\epsilon<1)$ denote the class of non-negative piecewise continuous functions defined on $[0,\infty)$ which satisfy the following condition: $f(x)f(y)\leq \epsilon^{\mid x-y\mid} (x,y \geq 0)$ . Find the value of $s_{\epsilon}= \sup_{f\in F_{\epsilon}} \int_{0}^{\infty} f(x) dx$ (R. 5)

Find the continuous solutions of the functional equation $f(xyz)= f(x)+f(y)+f(z)$ in the following cases:

(a) $x,y,z$ are arbitrary non-zero real numbers; (b) $a<x,y,z<b (1<a^{3}<b)$ . (R. 13)

Let $f(x)$ be an arbitrary function, differentiable infinitely many times. Then the $n$ th derivative of $f(e^{x})$ has the form $\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})$ ( $n=0,1,2,\dots$ ).

From the coefficients $a_{kn}$ compose the sequence of polynomials $P_{n}(x)= \sum_{k=0}^{n} a_{kn}x^{k}$ ( $n=0,1,2,\dots$ )

and find a closed form for the function $F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.$ (S. 22)

Prove that any real number x satysfying the inequalities $0<x\leq 1$ can be represented in the form $x= \sum_{k=1}^{\infty}\frac{1}{n_k}$ where $(n_k)_{k=1}^{\infty}$ is a sequence of positive integers such that $\frac{n_{k+1}}{n_k}$ assumes, for each $k$ , one of the three values $2,3$ or $4$ . (N. 14)

Find all integers $a>1$ for which the least (integer) solution $n$ of the congruence $a^{n} \equiv 1 \pmod{p}$ differs from 6 (p is any prime number). (N. 9)

Find all pairs of linear polynomials $f(x)$ , $g(x)$ with integer coefficients for which there exist two polynomials $u(x)$ , $v(x)$ with integer coefficients such that $f(x)u(x)+g(x)v(x)=1$ . (A. 8)

An Abelian group $G$ is said to have the property $(A)$ if torsion subgroup of $G$ is a direct summand of $G$ . Show that if $G$ is an Abelian group such that $nG$ has the property $(A)$ for some positive integer $n$ , then $G$ itself has the property $(A)$ . (A. 13)