MathDB

1

Miklos Schweitzer 1970_4

c

is a positive integer and

p

is an odd prime, what is the smallest residue (in absolute value) of

\sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?

J. Suranyi

12

1

Miklos Schweitzer 1970_12

Let

\vartheta_1,...,\vartheta_n

be independent, uniformly distributed, random variables in the unit interval

[0,1]

. Define h(x)\equal{} \frac1n \# \{k: \; \vartheta_k

x_0 \in (0,1)

such that h(x_0)\equal{}x_0, is equal to 1\minus{} \frac1n. G. Tusnady

11

1

Miklos Schweitzer 1970_11

Let

\xi_1,\xi_2,...

be independent random variables such that

E\xi_n=m>0

and

\textrm{Var}(\xi_n)=\sigma^2 < \infty \;(n=1,2,...)\ .

Let

\{a_n \}

be a sequence of positive numbers such that

a_n\rightarrow 0

and

\sum_{n=1}^{\infty} a_n= \infty

. Prove that

P \left( \lim_{n\rightarrow \infty} %Error. "diaplaymath" is a bad command. \sum_{k=1}^n a_k \xi_k =\infty \right)=1.

P. Revesz

10

1

Miklos Schweitzer 1970_10

Prove that for every

\vartheta

,

0<\vartheta<1

, there exist a sequence

\lambda_n

of positive integers and a series

\sum_{n=1}^{\infty} a_n

such that (i)

\lambda_{n+1}-\lambda_n > (\lambda_n)^{\vartheta}

, (ii)

\lim_{r\rightarrow 1^-} \sum_{n=1}^{\infty} a_nr^{\lambda_n}

exists, (iii)

\sum _{n=1}^{\infty} a_n

is divergent. P. Turan

9

1

Miklos Schweitzer 1970_9

Construct a continuous function

f(x)

, periodic with period

2 \pi

, such that the Fourier series of

f(x)

is divergent at x\equal{}0, but the Fourier series of

f^2(x)

is uniformly convergent on

[0,2 \pi].

P. Turan

8

1

Miklos Schweitzer 1970_8

Let

\pi_n(x)

be a polynomial of degree not exceeding

n

with real coefficients such that |\pi_n(x)| \leq \sqrt{1\minus{}x^2} \;\textrm{for}\ \;\minus{}1\leq x \leq 1 \ . Then |\pi'_n(x)| \leq 2(n\minus{}1). P. Turan

7

1

Miklos Schweitzer 1970_7

Let us use the word

N

-measure for nonnegative, finitely additive set functions defined on all subsets of the positive integers, equal to

0

on finite sets, and equal to

1

on the whole set. We say that the system

\Upsilon

of sets determines the

N

-measure

\mu

if any

N

-measure coinciding with

\mu

on all elements of

\Upsilon

is necessarily identical with

\mu

. Prove the existence of an

N

-measure

\mu

that cannot be determined by a system of cardinality less than continuum. I. Juhasz

6

1

Miklos Schweitzer 1970_6

Let a neighborhood basis of a point

x

of the real line consist of all Lebesgue-measurable sets containing

x

whose density at

x

equals

1

. Show that this requirement defines a topology that is regular but not normal. A. Csaszar

5

1

Miklos Schweitzer 1970_5

Prove that two points in a compact metric space can be joined with a rectifiable arc if and only if there exists a positive number

K

such that, for any

\varepsilon>0

, these points can be connected with an

\varepsilon

-chain not longer that

K

. M. Bognar

3

1

Miklos Schweitzer 1970_3

The traffic rules in a regular triangle allow one to move only along segments parallel to one of the altitudes of the triangle. We define the distance between two points of the triangle to be the length of the shortest such path between them. Put \binom{n\plus{}1}{2} points into the triangle in such a way that the minimum distance between pairs of points is maximal. L. Fejes-Toth

2

1

Miklos Schweitzer 1970_2

Let

G

and

H

be countable Abelian

p

-groups (

p

an arbitrary prime). Suppose that for every positive integer

n

, p^nG \not\equal{} p^{n\plus{}1}G . Prove that

H

is a homomorphic image of

G

. M. Makkai

1

Miklos Schweitzer 1970_1

We have 2n\plus{}1 elements in the commutative ring

R

:

\alpha,\alpha_1,...,\alpha_n,\varrho_1,...,\varrho_n .

Let us define the elements \sigma_k\equal{}k\alpha \plus{} \sum_{i\equal{}1}^n \alpha_i\varrho_i^k . Prove that the ideal

(\sigma_0,\sigma_1,...,\sigma_k,...)

can be finitely generated. L. Redei

1970 Miklós Schweitzer

Subcontests

Miklos Schweitzer 1970_4

Miklos Schweitzer 1970_12

Miklos Schweitzer 1970_11

Miklos Schweitzer 1970_10

Miklos Schweitzer 1970_9

Miklos Schweitzer 1970_8

Miklos Schweitzer 1970_7

Miklos Schweitzer 1970_6

Miklos Schweitzer 1970_5

Miklos Schweitzer 1970_3

Miklos Schweitzer 1970_2

Miklos Schweitzer 1970_1