MathDB

1

Miklós Schweitzer 1959- Problem 10

2n+1

vertices has at least

3n+1

edges, then the graph contains a circuit having an even number of edges. Prove further that this statemente does not hold for

3n

edges. (By a circuit, we mean a closed line which does not intersect itself.) (C. 5)

9

1

Miklós Schweitzer 1959- Problem 9

9. Let

f(z)= z^n +a_1 z^{n-1}+\dots + a_n

be a polynomial over the field of the complex numbers and let

E_f

denote the closed (not necessarily connected) domain of complex numbers

z

for which

\mid f(z) \mid \leq 1

. Show that there exists a point

z_0 \in E_f

such that

\mid f'(z_0) \mid \geq n

. (F. 5)

8

1

Miklós Schweitzer 1959- Problem 8

8. An Oblique lattice-cubs is a lattice-cube of the three-dimensional fundamental lattice no edge of which is perpendicular to any coordinate axis. Prove that for any integer

h= 8n-1

(

n= 1, 2, \dots

) there existis an oblique lattice-cube with edges of length

h

. Propose a method for finding such a cube. (N. 20)

7

1

Miklós Schweitzer 1959- Problem 7

7. Let

(z_n)_{n=1}^{\infty}

be a sequence of complex numbers tending to zero. Prove that there exists a sequence

(\epsilon_n)_{n=1}^{\infty}

(where

\epsilon_n = +1

or

-1

) such that the series

\sum_{n=1}^{\infty} \epsilon_n z_n

is convergente. (F. 9)

6

1

Miklós Schweitzer 1959- Problem 6

6. Let

T

be a one-to-one mapping of the unit square

E

of the plane into itself. Suppose that

T

and

T^{-1}

are measure-preserving (i.e. if

M \subseteq E

is a measurable set, then

TM

and

T^{-1}M

are also measurable and

\mu (M)= \mu (TM)= \mu (T^{-1}M)

, where

\mu

denotes the Lebesgue measure) and, furthermore, that if

Tx \in N

for almost all points

x

of a measurable set

N \subseteq E

, then either

n

or

E \setminus N

is of measure 0. Prove that, for any measurable set

A \subseteq E

, with

\mu (A)>0

, the function

n(x)

defined by
n(x)=\begin{cases} 0, \mbox{if} T^k x \notin A (k=1, 2, \dots),\\ \min (k: T^k x \in A; k=1,2, \dots ) &\mbox{otherwise} \end{cases}
is measurable and

\int_{A}n(x) d\mu(x) =1

(R. 18)

5

1

Miklós Schweitzer 1959- Problem 5

5. Denote by

c_n

the

n

th positive integer which can be represented in the form

c_n = k^{l} (k,l = 2,3, \dots )

. Prove that

\sum_{n=1}^{\infty}\frac{1}{c_n-1}=1

(N. 18)

4

1

Miklós Schweitzer 1959- Problem 4

4. Consider

n

circles of radius

1

in the planea. Prove that at least one of the circles contains an are of length greater than

\frac{2\pi}{n}

not intersected by any other of these circles. (G. 4)

3

1

Miklós Schweitzer 1959- Problem 3

3.Let

G

be an arbitrary group,

H_1,\dots ,H_n

some (not necessarily distinet) subgroup of

G

and

g_1, \dots , g_n

elements of

G

such that each element of

G

belongs at least to one of the right cosets

H_1 g_1, \dots , H_n g_n

. Show that if, for any

k

, the set-union of the cosets

H_i g_i (i=1, \dots , k-1, k+1, \dots , n)

differs from

G

, then every

H_k (k=1, \dots , n)

is of finite index in

G

. (A. 15)

2

1

Miklós Schweitzer 1959- Problem 2

2. Omit the vertices of a closed rectangle; the configuration obtained in such a way will be called a reduced rectangle. Prove tha the set-union of any system of reduced rectangles with parallel sides is equal to the union of countably many elements of the system. (St. 3)

1

Miklós Schweitzer 1959- Problem 1

1. Let

p_n

be the

n

th prime number. Prove that

\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty

(N.17)

1959 Miklós Schweitzer

Subcontests

Miklós Schweitzer 1959- Problem 10

Miklós Schweitzer 1959- Problem 9

Miklós Schweitzer 1959- Problem 8

Miklós Schweitzer 1959- Problem 7

Miklós Schweitzer 1959- Problem 6

Miklós Schweitzer 1959- Problem 5

Miklós Schweitzer 1959- Problem 4

Miklós Schweitzer 1959- Problem 3

Miklós Schweitzer 1959- Problem 2

Miklós Schweitzer 1959- Problem 1

1959 Mikl&oacute;s Schweitzer

Subcontests

Mikl&oacute;s Schweitzer 1959- Problem 10

Mikl&oacute;s Schweitzer 1959- Problem 9

Mikl&oacute;s Schweitzer 1959- Problem 8

Mikl&oacute;s Schweitzer 1959- Problem 7

Mikl&oacute;s Schweitzer 1959- Problem 6

Mikl&oacute;s Schweitzer 1959- Problem 5

Mikl&oacute;s Schweitzer 1959- Problem 4

Mikl&oacute;s Schweitzer 1959- Problem 3

Mikl&oacute;s Schweitzer 1959- Problem 2

Mikl&oacute;s Schweitzer 1959- Problem 1

1959 Miklós Schweitzer

Miklós Schweitzer 1959- Problem 10

Miklós Schweitzer 1959- Problem 9

Miklós Schweitzer 1959- Problem 8

Miklós Schweitzer 1959- Problem 7

Miklós Schweitzer 1959- Problem 6

Miklós Schweitzer 1959- Problem 5

Miklós Schweitzer 1959- Problem 4

Miklós Schweitzer 1959- Problem 3

Miklós Schweitzer 1959- Problem 2

Miklós Schweitzer 1959- Problem 1