SCHWEITZER 1982

Consider the lattice of all algebraically closed subfields of the complex field $\mathbb{C}$ whose transcendency degree (over $\mathbb{Q}$ ) is finite. Prove that this lattice is not modular.

L. Babai

Find a perfect set $H \subset [0,1]$ of positive measure and a continuous function $f$ defined on $[0,1]$ such that for any twice differentiable function $g$ defined on $[0,1]$ , the set $\{ x \in H : f(x)=g(x)\ \}$ is finite.

M. Laczkovich

For every positive $\alpha$ , natural number $n$ , and at most $\alpha n$ points $x_i$ , construct a trigonometric polynomial $P(x)$ of degree at most $n$ for which $$P(x_i) \leq 1, \int_0^{2 \pi} P(x)dx=0,\ \textrm{and}\ \max P(x) > cn\ ,$$ where the constant $c$ depends only on $\alpha$ .

G. Halasz

Let $V$ be a bounded, closed, convex set in $\mathbb{R}^n$ , and denote by $r$ the radius of its circumscribed sphere (that is, the radius of the smallest sphere that contains $V$ ). Show that $r$ is the only real number with the following property: for any finite number of points in $V$ , there exists a point in $V$ such that the arithmetic mean of its distances from the other points is equal to $r$ .

Gy. Szekeres

Suppose that $K$ is a compact Hausdorff space and $K= \cup_{n=0}^{\infty}A_n$ , where $A_n$ is metrizable and $A_n \subset A_m$ for $n<m$ . Prove that $K$ is metrizable.

Z. Balogh