SCHWEITZER 1980

In a lattice, connected the elements $a \wedge b$ and $a \vee b$ by an edge whenever $a$ and $b$ are incomparable. Prove that in the obtained graph every connected component is a sublattice.

M. Ajtai

Let us call a continuous function $f : [a,b] \rightarrow \mathbb{R}^2 \;\textit{reducible}$ if it has a double arc (that is, if there are $a \leq \alpha < \beta \leq \gamma < \delta \leq b$ such that there exists a strictly monotone and continuous $h : [\alpha,\beta] \rightarrow [\gamma,\delta]$ for which $f(t)=f(h(t))$ is satisfied for every $\alpha \leq t \leq \beta$ ); otherwise $f$ is irreducible. Construct irreducible $f : [a,b] \rightarrow \mathbb{R}^2$ and $g : [c,d] \rightarrow \mathbb{R}^2$ such that $f([a,b])=g([c,d])$ and

(a) both $f$ and $g$ are rectifiable but their lengths are different;

(b) $f$ is rectifiable but $g$ is not.

A. Csaszar

Let $f(x)$ be a nonnegative, integrable function on $(0,2\pi)$ whose Fourier series is $f(x)=a_0+\sum_{k=1}^{\infty} a_k \cos (n_k x)$ , where none of the positive integers $n_k$ divides another. Prove that $|a_k| \leq a_0$ .

G. Halasz

Let us divide by straight lines a quadrangle of unit area into $n$ subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most $\pi \sqrt{n}$ (the lines are not allowed to cut the interior of a subpolygon).

G. and L. Fejes-Toth

Suppose that the $T_3$ -space $X$ has no isolated points and that in $X$ any family of pairwise disjoint, nonempty, open sets is countable. Prove that $X$ can be covered by at most continuum many nowhere-dense sets.

I. Juhasz