SCHWEITZER 1956

Solve without use of determinants the following system of linear equations: $\sum_{j=0}{k} \binom{k+\alpha}{j} x_{k-j} =b_k$ ( $k= 0,1, \dots , n$ ),

where $\alpha$ is a fixed real number. (A. 7)

Find the minimum of $max ( |1+z|, |1+z^{2}|)$ if $z$ runs over all complex numbers. (F. 2)

A triangulation of a convex closed polygon is the division into triangles of this poilygon by diagonals not intersecting in the interior of the polygon. Find the number of all triangulations fo a conves n-gon and also the number of those triangulations in which every triangle has at least one side in common with the given n-gon. (C. 4)

Denoting by $a(n)$ the greatest prime factor of the positive integer $n$ , show that $S= \sum_{n=1}^{\infty } \frac{1}{na(n)}$ is convergente. (N. 13)

On a circle consider $n$ points among which there acts a repulsive force inversely proportional to the square of their distance. Prove that the point system is in stable equilibrium if and only if the points form a regular $n$ -gon; in other words, considering the sum of the reciprocal distances of the $\binom{n}{2}$ pairs of points which can be chosen from among the $n$ given points, this sum is minimal if and only if the points lie at the vertices of a regular $n$ -gon. (G. 2)

Show that the number of the faces of a convex polyhedron is even if every face is centrally simmetric. (G. 12)

Let $(a_n)_{n=0}^{\infty}$ be a sequence of real numbers such that, with some positive number $C$ , $\sum_{k=1}^{n}k\mid a_k \mid<n C$ ( $n=1,2, \dots$ )

Putting $s_n= a_0 +a_1+\dots+a_n$ , suppose that $\lim_{n \to \infty }(\frac{s_{0}+s_{1}+\dots+s_n}{n+1})= s$ exists. Prove that $\lim_{n \to \infty }(\frac{s_{0}^2+s_{1}^2+\dots+s_n^2}{n+1})= s^2$ (S. 7)

Let $(a_n)_{n=1}^{\infty}$ be a sequence of positive numbers and suppose that $\sum_{n=1}^{\infty} a_n^2$ is divergent. Let further $0<\epsilon<\frac{1}{2}$ . Show that there exists a sequence $(b_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}b_n^2$ is convergent and $\sum_{n=1}^{N}a_n b_n >(\sum_{n=1}^{N}a_n^2)^{\frac{1}{2}-\epsilon}$ for every positive integer $N$ . (S. 8)

Show that if the trigonometric polynomial $f(\theta)= \sum_{v=1}^{n} a_v \cos v\theta$ monotonically decreases over the closed interval $[0,\pi]$ , then the trigonometric polynomial $g(\theta)=\sum_{v=1}^{n}a_v \sin v\theta$ is non negative in the same interval. (S. 26)

In an urn there are balls of $N$ different colours, $n$ balls of each colour. Balls are drawn and not replaced until one of the colours turns up twice; denote by $V_{N,n}$ the number of the balls drawn and by $M_{N,n}$ the expectation of the random variable $v_{N,n}$ . Find the limit distribution of the random variable $\frac{V_{N,n}}{M_{N,n}}$ if $N \to \infty$ and $n$ is a fixed number. (P. 8)