SCHWEITZER 1981

We are given an infinite sequence of $1$ 's and $2$ 's with the following properties:

(1) The first element of the sequence is $1$ .

(2) There are no two consecutive $2$ 's or three consecutive $1$ 's.

(3) If we replace consecutive $1$ 's by a single $2$ , leave the single $1$ 's alone, and delete the original $2$ 's, then we recover the original sequence.

How many $2$ 's are there among the first $n$ elements of the sequence?

P. P. Palfy

Consider the lattice $L$ of the contradictions of a simple graph $G$ (as sets of vertex pairs) with respect to inclusion. Let $n \geq 1$ be an arbitrary integer. Show that the identity $$x \bigwedge \left( \bigvee_{i=0}^n y_i \right) = \bigvee_{j=0}^n \left( x \bigwedge \left( \bigvee_{0 \leq i \leq n, i\not=j\ } y_i \right)\right)$$ holds if and only if $G$ has no cycle of size at least $n+2$ .

A. Huhn

Construct an uncountable Hausdorff space in which the complement of the closure of any nonempty, open set is countable.

A. Hajnal, I. Juhasz

Let $G$ be finite group and $\mathcal{K}$ a conjugacy class of $G$ that generates $G$ . Prove that the following two statements are equivalent:

(1) There exists a positive integer $m$ such that every element of $G$ can be written as a product of $m$ (not necessarily distinct) elements of $\mathcal{K}$ .

(2) $G$ is equal to its own commutator subgroup.

J. Denes

Let $K$ be a convex cone in the $n$ -dimensional real vector space $\mathbb{R}^n$ , and consider the sets $A=K \cup (-K)$ and $B=(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ( $0$ is the origin). Show that one can find two subspaces in $\mathbb{R}^n$ such that together they span $\mathbb{R}^n$ , and one of them lies in $A$ and the other lies in $B$ .

J. Szucs

Let $U$ be a real normed space such that, for an finite-dimensional, real normed space $X,U$ contains a subspace isometrically isomorphic to $X$ . Prove that every (not necessarily closed) subspace $V$ of $U$ of finite codimension has the same property. (We call $V$ of finite codimension if there exists a finite-dimensional subspace $N$ of $U$ such that $V+N=U$ .)

A. Bosznay