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Miklós Schweitzer 1960- Problem 1

Source:

11/18/2015

1. Consider in the plane a set

H

of pairwise disjoint circles of radius 1 such that, for infinitely many positive integers

n

, the circle

k_n

with centre at the origin and of radius

n

contains at least

cn^2

elements of the set

H

. Prove that there exists a straight line which intersects infinitely many of the circles of

H

. Show further that if we require only that the circles

k_n

contain o(n²) elements of

H

, the proposition will be false. (G. 5)

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Miklós Schweitzer 1960- Problem 2

Source:

11/18/2015

2. Construct a sequence

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

of complex numbers such that, for every

l>0

, the series

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

be divergent, but for almost all

\theta

in

(0,2\pi)

,

\prod_{n=1}^{\infty} (1+a_n e^{i\theta})

be convergent. (S. 11)

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Miklós Schweitzer 1960- Problem 7

Source:

11/21/2015

7. Define the generalized derivative at

x_0

of the function

f(x)

by

\lim_{h \to 0} 2 \frac{ \frac{1}{h} \int_{x_0}^{x_0+h} f(t) dt - f(x_0)}{h}

Show that there exists a function, continuous everywhere, which is nowhere differentiable in this general sense ( R. 8)

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Miklós Schweitzer 1960- Problem 4

Source:

11/18/2015

4. Let

\left (H_{\alpha} \right )

be a system of sets of integers having the property that for any

\alpha _1 \neq \alpha _2 , H_{\alpha _1}\cap H_{\alpha _2}

is a finite set and

H_{{\alpha} _1} \neq H_{{\alpha} _2}

. Prove that there exists a system

\left (H_{\alpha} \right )

of this kind whose cardinality is that of the continuum. Prove further that if none of the intersections of two sets

H_\alpha

contains more than

K

elements, then the system

\left (H_{\alpha} \right )

is countable (

K

is an arbitrary fixed integer). (St. 4)

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Miklós Schweitzer 1960- Problem 5

Source:

11/18/2015

5. Define the sequence

\{c_n\}_{n=1}^{\infty}

as follows:

c_1= \frac {1}{2}

,

c_{n+1}= c_{n}-c_{n}^2

(

n\geq 1

). Prove that

\lim_{n \to \infty} nc_n= 1

(S.12)

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Miklós Schweitzer 1960- Problem 9

Source:

11/21/2015

9. Let

A_1, \dots , A_n

and

B

be ideals of an assoticative ring

R

such that

B

is contained in the set-union of the ideals

A_i

(

i=1, \dots , n

) but not contained in the union of any

n-1

of the ideals

A_i

. Show that, for some positive integer

k

,

B_k

is contained in the intersection of the ideals

A_i

. (A. 19)

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Miklós Schweitzer 1960- Problem 8

Source:

11/21/2015

8. Let

f

be a bounded real function defined on the unit cube

H

of the

n

-dimensional space and, for a given

y

, let

A_y

and

B_y

denote the parts of the interior of

H

on which

f>y

and

f<y

, respectively. Show that

f

is integrable in the Riemannian sense if and only if for every

y

almost all points of

A_y

and

B_y

are inner points. (R. 9)

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Miklós Schweitzer 1960- Problem 10

Source:

11/21/2015

10. A car is used by

n

drivers. Every morning the drivers choose by drawing that one of them who will drive the car that day. Let us define the random variable

\mu (n)

as the least positive integer such that each driver drives at least one day during the first

\mu (n)

days. Find the limit distribution of the random variable

\frac {\mu (n) -n \log n}{n}

as

n \to \infty

. (P. 9)

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Miklós Schweitzer 1960- Problem 6

Source:

11/21/2015

6. Let

\{ n_k \}_{k=1}^{\infty}

be a stricly increasing sequence of positive integers such that

\lim_{k \to \infty} n_k^{\frac {1}{2^k}}= \infty

Show that the sum of the series

\sum_{k=1}^{\infty} \frac {1}{n_k}

is an irrational number. (N. 19)

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Miklós Schweitzer 1961- Problem 2

Source:

11/22/2015

2. Show that a ring

R

has a unit element if and only if any

R

-module

G

can be written as a direct sum of

RG

and of the trivial submodule of

G

. (An

R

-module is a linear space with

R

as its scalar domain.

RG

denotes the submodule generated by the elements of the form

rg

(

r \in R, g \in G

). The trivial submodule of

G

consists of the elements

g

of

G

for which

rg=0

holds for every

r \in R

.) (A. 20)

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Miklós Schweitzer 1961- Problem 1

Source:

11/22/2015

1. Let

a

(

\neq e

, the unit element) be an element of finite order of a group

G

and let

t

(

\geq 2

) be a positive integer. Show: if the complex

A= \{ e,a,a^2, \dots , a^{t-1} \}

is not a group, then for every positive integer

k

(

2 \leq k \leq t

) the complex

B= \{ e. a^k, a^{2k}, \dots , a^{(t-1)k} \}

differs from

A

. (A. 16)

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Miklós Schweitzer 1961- Problem 6

Source:

12/1/2015

6. Consider a sequence

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

such that, for any convergent subsequence

\{ a_{n_k} \}

of

\{a_n\}

, the sequence

\{ a_{n_k +1} \}

also is convergent and has the same limit as

\{ a_{n_k}\}

. Prove that the sequence

\{ a_n \}

is either convergent of has infinitely many accumulation points the set of which is dense in itself. Give an example for the second case. (A sequence

x_n \to \infty

or

-\infty

is considered to be convergente, too) (S. 13)

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Miklós Schweitzer 1961- Problem 3

Source:

11/22/2015

3. Let

f(x)= x^n +a_1 x^(n-1)+ \dots + a_n

(

n\geq 1

) be an irreducible polynomial over the field

K

. Show that every non-zero matrix commuting with the matrix

\begin{bmatrix} 0 & 1 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ \dots & \dots & \dots & \dots & \dots & \dots \\ 0 & 0 & 0 & \dots & 0 & 1 \\ -a_n & -a_{n-1} & -a_{n-2} & \dots & -a_2 & -a_1 </br>\end{bmatrix}

is invertible. (A. 4)

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Miklós Schweitzer 1961- Problem 4

Source:

11/22/2015

4. Let

f(x)

be a real- or complex-value integrable function on

(0,1)

with

\mid f(x) \mid \leq 1

. Set

c_k = \int_0^1 f(x) e^{-2 \pi i k x} dx

and construct the following matrices of order

n

:

T= (t_{pq})_{p,q=0}^{n-1}, T^{*}= (t_{pq}^{*})_{p,q =0}^{n-1}

where

t_{pq}= c_{q-p}, t^{*}= \overline {c_{p-q}}

. Further, consider the following hyper-matrix of order

m

:

S= \begin{bmatrix} E & T & T^2 & \dots & T^{m-2} & T^{m-1} \\ T^{*} & E & T & \dots & T^{m-3} & T^{m-2} \\ T^{*2} & T^{*} & E & \dots & T^{m-3} & T^{m-2} \\ \dots & \dots & \dots & \dots & \dots & \dots \\ T^{*m-1} & T^{*m-2} & T^{*m-3} & \dots & T^{*} & E \end{bmatrix}

(

S

is a matrix of order

mn

in the ordinary sense; E denotes the unit matrix of order

n

). Show that for any pair

(m , n)

of positive integers,

S

has only non-negative real eigenvalues. (R. 19)

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Miklós Schweitzer 1961- Problem 5

Source:

11/22/2015

5. Determine the functions

G

defined on the set of all non-zero real numbers the values of which are regular matrices of order

2

, and the functions

f

mapping the two-dimensional real vector space

E_2

into itself, such that for any vector

y \in E_2

and for any regular matrix

X

of order

2

,

f(X_y)= G(det X)Xf(y)

(

det X

denotes the determinant of

X

).(A. 5)

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Miklós Schweitzer 1961- Problem 7

Source:

12/1/2015

7. For the differential equation

\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}= 2\frac{\partial^2 u}{\partial x \partial y}

find all solutions of the form

u(x,y)=f(x)g(y)

. (R. 14)

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Miklós Schweitzer 1961- Problem 9

Source:

12/1/2015

9. Spin a regular coin repeatedly until the heads and tails turned up both reach the number

k

(

k= 1, 2, \dots

); denote by

v_k

the number of the necessary throws. Find the distribution of the random variable

v_k

and the limit-distribution of the random variable

\frac {v_k -2k}{\sqrt {2k}}

as

k \to \infty

. (P. 10)

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Miklós Schweitzer 1961- Problem 10

Source:

12/1/2015

10. Given a straight line

g

in the plane and a point

O

on

g

. Construct, without making use of the Parallel Axiom, the half-line perpendicular to

g

at the point

O

and lying in one of the half-planes defined by

g

, under the following restrictions: The construction must be effected by use of a ruler and of a length standard (i.e. an etalon-segment) only; moreover, all lines and points of the construction must lie in the chosen half-plane. (G. 20)

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Miklós Schweitzer 1961- Problem 8

Source:

12/1/2015

8. Let

f(x)

be a convex function defined on the interval

[0, \frac {1}{2}]

with

f(0)=0

and

f(\frac{1}{2})=1

; Let further

f(x)

be differentiable in

(0, \frac {1}{2})

, and differentiable at

0

and

\frac{1}{2}

from the right and from the left, respectively. Finally, let

f'(0)>1

. Extend

f(x)

to

[0.1]

in the following manner: let

f(x)= f(1-x)

if

x \in (\frac {1} {2}, 1]

. Show that the set of the points

x

for shich the terms of the sequence

x_{n+1}=f(x_n)

(

x_0=x; n = 0, 1, 2, \dots

) are not all different is everywhere dense in

[0,1]

; (R. 10)

college contestsreal analysis

Miklós Schweitzer 1959- Problem 1

Source:

10/30/2015

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)

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Miklós Schweitzer 1959- Problem 2

Source:

10/30/2015

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)

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Miklós Schweitzer 1959- Problem 3

Source:

10/30/2015

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)

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Miklós Schweitzer 1959- Problem 4

Source:

10/30/2015

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)

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Miklós Schweitzer 1959- Problem 5

Source:

10/30/2015

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)

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Miklós Schweitzer 1959- Problem 6

Source:

11/8/2015

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)

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