SCHWEITZER 1960

Consider in the plane a set $H$ of pairwise disjoint circles of radius 1 such that, for infinitely many positive integers $n$ , the circle $k_n$ with centre at the origin and of radius $n$ contains at least $cn^2$ elements of the set $H$ . Prove that there exists a straight line which intersects infinitely many of the circles of $H$ . Show further that if we require only that the circles $k_n$ contain o(n²) elements of $H$ , the proposition will be false. (G. 5)

Construct a sequence $(a_n)_{n=1}^{\infty}$ of complex numbers such that, for every $l>0$ , the series $\sum_{n=1}^{\infty} \mid a_n \mid ^{l}$ be divergent, but for almost all $\theta$ in $(0,2\pi)$ , $\prod_{n=1}^{\infty} (1+a_n e^{i\theta})$ be convergent. (S. 11)

Let $f(z)$ with $f(0)=1$ be regular in the unit disk and let $\left [\frac{\partial^2 \mid f(z)\mid}{\partial x\partial y} \right ] _{z=0} =1$ .

Show thatthe area of the image of the unit disk by $w= f(z)$ (taken with multiplicity) is not less than $\frac {1} {2}$ .(f. 6)

Let $\left (H_{\alpha} \right )$ be a system of sets of integers having the property that for any $\alpha _1 \neq \alpha _2 , H_{\alpha _1}\cap H_{\alpha _2}$ is a finite set and $H_{{\alpha} _1} \neq H_{{\alpha} _2}$ . Prove that there exists a system $\left (H_{\alpha} \right )$ of this kind whose cardinality is that of the continuum. Prove further that if none of the intersections of two sets $H_\alpha$ contains more than $K$ elements, then the system $\left (H_{\alpha} \right )$ is countable ( $K$ is an arbitrary fixed integer). (St. 4)

Define the sequence $\{c_n\}_{n=1}^{\infty}$ as follows: $c_1= \frac {1}{2}$ , $c_{n+1}= c_{n}-c_{n}^2$ ( $n\geq 1$ ). Prove that $\lim_{n \to \infty} nc_n= 1$ (S.12)

Let $\{ n_k \}_{k=1}^{\infty}$ be a stricly increasing sequence of positive integers such that $\lim_{k \to \infty} n_k^{\frac {1}{2^k}}= \infty$ Show that the sum of the series $\sum_{k=1}^{\infty} \frac {1}{n_k}$ is an irrational number. (N. 19)

Define the generalized derivative at $x_0$ of the function $f(x)$ by $\lim_{h \to 0} 2 \frac{ \frac{1}{h} \int_{x_0}^{x_0+h} f(t) dt - f(x_0)}{h}$ Show that there exists a function, continuous everywhere, which is nowhere differentiable in this general sense ( R. 8)

Let $f$ be a bounded real function defined on the unit cube $H$ of the $n$ -dimensional space and, for a given $y$ , let $A_y$ and $B_y$ denote the parts of the interior of $H$ on which $f>y$ and $f<y$ , respectively. Show that $f$ is integrable in the Riemannian sense if and only if for every $y$ almost all points of $A_y$ and $B_y$ are inner points. (R. 9)

Let $A_1, \dots , A_n$ and $B$ be ideals of an assoticative ring $R$ such that $B$ is contained in the set-union of the ideals $A_i$ ( $i=1, \dots , n$ ) but not contained in the union of any $n-1$ of the ideals $A_i$ . Show that, for some positive integer $k$ , $B_k$ is contained in the intersection of the ideals $A_i$ . (A. 19)

A car is used by $n$ drivers. Every morning the drivers choose by drawing that one of them who will drive the car that day. Let us define the random variable $\mu (n)$ as the least positive integer such that each driver drives at least one day during the first $\mu (n)$ days. Find the limit distribution of the random variable $\frac {\mu (n) -n \log n}{n}$ as $n \to \infty$ . (P. 9)