SCHWEITZER 1994

An ordered set of numbers is mean-free if for all $x < y < z$ , $y \neq \frac{x + z}{2}$ . Is it possible to order the real numbers so it becomes mean-free?

related: https://www.youtube.com/watch?v=ppaXUxsEjMQ

For which finite group G does there exist natural number s with the following property: for any subgroup H of a finite direct power of G, each subgroup of H is produced as an intersection of subgroups of H with index at most s.

not sure of translation.

Let p be an odd prime, A be a non-empty subset of residue classes modulo p, $f:A\to\mathbb R$ . Suppose that f is not constant and satisfies $f(x) \leq \frac{f(x + h) + f(x-h)}{2}$ whenever $x,x+h,x-h\in A$ . Prove that $|A| \leq \frac{p + 1}{2}$ .

For a given irrational number $\alpha$ , $y_{1,\alpha} = \alpha$ . If $y_{n-1, \alpha}$ is given, let $y_{n, \alpha}$ be the first member of the sequence $\big (\{k \alpha \} \big) ^ \infty_{k = 1}$ to fall in the interval $(0, y_{n-1,\alpha})$ ({ x } denotes the fraction of the number x ). Show that there exists an open set $G\subset (0,1)$ , which has a limit point 0 and for all irrational $\alpha$ , infinitely many members of the $(y_{n,\alpha})$ sequence do not belong to G.

Let H be a $G_{\delta}$ subset of $\mathbb R$ whose closure has a positive Lebesgue measure. Prove that the set $H + H + H + H = \{ x + y + z + u : x , y , z , u \in H \}$ contains an interval.

Show that if n is an arbitrary natural number and $\sqrt n \leq K \leq \frac{n}{2}$ , then there exist n distinct integers, $k_j$ ( j = 1, ..., n ) such that $\bigg | \sum_ {j = 1} ^ ne ^ {ik_jt} \bigg | \geq K$ is satisfied on a subset of the interval $(- \pi, \pi)$ with Lebesgue measure at least $\frac{cn}{K^2}$ , where c is a suitable absolute constant.

Prove that there exist $0 < \alpha< \beta<1$ numbers have the following properties. (i) for any sufficiently large n, n points can be specified in $\Bbb R^3$ , so that each point is equidistant from at least $n^\alpha$ other points. (ii) the above statement is no longer true with $n^\beta$ instead of $n^\alpha$

Prove that a Hausdorff space X is countably compact iff for every open cover $\cal {U}$ there is a finite set $A \subset X$ such that $\bigcup \{U \in {\cal U} : U \cap A \neq \emptyset \} = X$ .

Let X be a dense set homeomorphic to $\mathbb R^n$ in the compact Hausdorff space Y. Prove that for $n\geq 2$ , $Y \setminus X$ is connected, and for n=1 it consists of at most two components.

Let $F^2$ be a closed, oriented 2-dimensional smooth surface, $f : F^2 \to F^2$ is a smooth homeomorphism whose order is an odd prime p (i.e., the p-th iterate $f \circ f \circ \cdots \circ f$ is the identity). Then f has a finite number of fixed points: $P_1 , ..., P_s$ . In the tangent plane at the fixed point $P_i$ , a positively directed (i.e., compatible with the direction of the surface) base can be chosen in which f is differentiated by a rotation with positive angle $2\pi k_i/p$ , where $k_i$ is a natural number, $0 < k_i < p$ . Prove that $$\sum_{i = 1}^s k_i^{p-2}\equiv0\pmod{p}$$

$\xi, \xi'$ are iid random variables. let F have the distribution function $\xi+\xi'$ , and G have the uniform distribution over the interval [-1,1]. Prove that $\max | F ( x ) - G ( x ) | \geq 10^{-1994}$ .