1994 Miklós Schweitzer

Part of Miklós Schweitzer

Subcontests

(11)

1

analysis

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

k_i

is a natural number,

0 < k_i < p

\sum_{i = 1}^s k_i^{p-2}\equiv0\pmod{p}

1

direct power of group

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.

1

difference between 2 random var

\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}

1

analysis

For a given irrational number

\alpha

y_{1,\alpha} = \alpha

y_{n-1, \alpha}

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.

1

analysis

Prove that a Hausdorff space X is countably compact iff for every open cover

\cal {U}

there is a finite set

A \subset X

\bigcup \{U \in {\cal U} : U \cap A \neq \emptyset \} = X

1

equidistant points in R^3

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

n^\alpha

1

analysis

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.

1

analysis

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.

1

analysis

G_{\delta}

\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.

1

analysis

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}

x,x+h,x-h\in A

|A| \leq \frac{p + 1}{2}

1

mean-free order

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