MathDB

Miklós Schweitzer 1958- Problem 9

Source:

10/23/2015

9. Show that if

f(z) = 1+a_1 z+a_2z^2+\dots

for

\mid z \mid\leq 1

and

\frac{1}{2\pi}\int_{0}^{2\pi}\mid f(e^{i\phi}) \mid^{2} d\phi < \left (1+\frac{\mid a_1\mid ^2} {4} \right )^2

,
then

f(z)

has a root in the disc

\mid z \mid \leq 1

.(F. 4)

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Miklós Schweitzer 1958- Problem 8

Source:

10/23/2015

8. Let the function

f(x)

be periodic with the period

1

, non-negative, concave in the interval

(0,1)

and continuous at the point

0

. Prove that

f(nx)\leq nf(x)

for every real

x

and positive integer

n

. (R. 6)

functioncollege contestsMiklos Schweitzer

Miklós Schweitzer 1958- Problem 10

Source:

10/23/2015

10. Prove that the function

f(x)= \int_{-\infty}^{\infty} \left (\frac{\sin\theta}{\theta} \right )^{2k}\cos (2x\theta) d\theta

where

k

is a positive integer, satisfies the following conditions:
(i)

f(x)=0

if

\mid x \mid \geq k

and

f(x) \geq 0

elsewhere; (ii) in interval

(l,l+1)

(l= -k, -k+1, \dots , k-1)

the function

f(x)

is a polynomial of degree

2k-1

at most. (R. 7)

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Miklós Schweitzer 1958- Problem 7

Source:

10/23/2015

7. Let

a_0

and

a_1

be arbitrary real numbers, and let

a_{n+1}=a_n + \frac{2}{n+1}a_{n-1}

(n= 1, 2, \dots)

Show that the sequence

\left (\frac{a_n}{n^2} \right )_{n=1}^{\infty}

is convergent and find its limit. (S. 10)

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Miklós Schweitzer 1958- Problem 11

Source:

10/23/2015

11. Let

a_n = (-1)^n (n= 1, 2, \dots , 2N)

. Denote by

A_{N}(x)

the number of the sequences

1 \leq i_1 < i_2< \dots <i_N \leq 2N

such that

a_{i_1}+a_{i_2}+ \dots +a_{i_N}< x \sqrt{\frac{N}{2}} (-\infty < x < \infty)

. Show that

\lim_{N \to \infty} \frac{A_{N}(x)}{\binom{2N}{N}} = \frac {1}{\sqrt {2\pi}} \int_{-\infty}^{\infty} e^{-\frac{u^2}{2}} du

.
(N. 16)

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Miklos Schweitzer 1962_5

Source:

9/18/2008

Let

f

be a finite real function of one variable. Let

\overline{D}f

and

\underline{D}f

be its upper and lower derivatives, respectively, that is, \overline{D}f\equal{}\limsup_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k} , \underline{D}f\equal{}\liminf_{{h,k\rightarrow 0}_{{h,k \geq 0}_{h\plus{}k>0}}} \frac{f(x\plus{}h)\minus{}f(x\minus{}k)}{h\plus{}k}. Show that

\overline{D}f

and

\underline{D}f

are Borel-measurable functions. [A. Csaszar]

functioncalculusderivativereal analysisreal analysis unsolved

Miklos Schweitzer 1962_1

Source:

9/18/2008

Let

f

and

g

be polynomials with rational coefficients, and let

F

and

G

denote the sets of values of

f

and

g

at rational numbers. Prove that F \equal{} G holds if and only if f(x) \equal{} g(ax \plus{} b) for some suitable rational numbers a\not \equal{} 0 and

b

. E. Fried

algebrapolynomialsearchadvanced fieldsadvanced fields unsolved

Miklos Schweitzer 1962_2

Source:

9/18/2008

Determine the roots of unity in the field of

p

-adic numbers. L. Fuchs

superior algebrasuperior algebra unsolved

Miklos Schweitzer 1962_3

Source:

9/18/2008

Let

A

and

B

be two Abelian groups, and define the sum of two homomorphisms

\eta

and

\chi

from

A

to

B

by a( \eta\plus{}\chi)\equal{}a\eta\plus{}a\chi \;\textrm{for all}\ \;a \in A\ . With this addition, the set of homomorphisms from

A

to

B

forms an Abelian group

H

. Suppose now that

A

is a

p

-group (

p

a prime number). Prove that in this case

H

becomes a topological group under the topology defined by taking the subgroups p^kH \;(k\equal{}1,2,...) as a neighborhood base of

0

. Prove that

H

is complete in this topology and that every connected component of

H

consists of a single element. When is

H

compact in this topology? [L. Fuchs]

abstract algebratopologygroup theorylimitgeometrygeometric transformationsuperior algebra

Miklos Schweitzer 1962_4

Source:

9/18/2008

Show that \prod_{1\leq x < y \leq \frac{p\minus{}1}{2}} (x^2\plus{}y^2) \equiv (\minus{}1)^{\lfloor\frac{p\plus{}1}{8}\rfloor} \;(mod\;p\ ) for every prime

p\equiv 3 \;(<span class='latex-bold'>mod</span>\;4\ )

. [J. Suranyi]

floor functionmodular arithmeticquadraticssymmetryadvanced fieldsadvanced fields unsolved

Miklos Schweitzer 1962_8

Source:

9/18/2008

Denote by

M(r,f)

the maximum modulus on the circle |z|\equal{}r of the transcendent entire function

f(z)

, and by

M_n(r,f)

that of the

nth

partial sum of the power series of

f(z)

. Prove that the existence of an entire function

f_0(z)

and a corresponding sequence of positive numbers r_1

functioncomplex analysiscomplex analysis unsolved

Miklos Schweitzer 1962_9

Source:

9/18/2008

Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].

geometry3D geometryprismreal analysisreal analysis unsolved

Miklos Schweitzer 1962_6

Source:

9/18/2008

Let

E

be a bounded subset of the real line, and let

\Omega

be a system of (non degenerate) closed intervals such that for each

x \in E

there exists an

I \in \Omega

with left endpoint

x

. Show that for every

\varepsilon > 0

there exists a finite number of pairwise non overlapping intervals belonging to

\Omega

that cover

E

with the exception of a subset of outer measure less than

\varepsilon

. [J. Czipszer]

real analysisreal analysis unsolved

Miklos Schweitzer 1962_7

Source:

9/18/2008

Prove that the function

f(\nu)= \int_1^{\frac{1}{\nu}} \frac{dx}{\sqrt{(x^2-1)(1-\nu^2x^2)}}

(where the positive value of the square root is taken) is monotonically decreasing in the interval

0<\nu<1

. [P. Turan]

functionintegrationtrigonometrycalculusreal analysisreal analysis unsolved

Miklos Schweitzer 1962_10

Source:

9/18/2008

From a given triangle of unit area, we choose two points independetly with uniform distribution. The straight line connecting these points divides the triangle. with probability one, into a triangle and a quadrilateral. Calculate the expected values of the areas of these two regions. [A. Renyi]

geometryprobabilityprobability and stats

Miklós Schweitzer 1957- Problem 6

Source:

10/16/2015

6. Let

f(x)

be an arbitrary function, differentiable infinitely many times. Then the

n

th derivative of

f(e^{x})

has the form

\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})

(

n=0,1,2,\dots

).
From the coefficients

a_{kn}

compose the sequence of polynomials

P_{n}(x)= \sum_{k=0}^{n} a_{kn}x^{k}

(

n=0,1,2,\dots

)
and find a closed form for the function

F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.

(S. 22)

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Miklós Schweitzer 1957- Problem 1

Source:

10/16/2015

1. Let

C_{ij}

(

i,j=1,2,3

) be the coefficients of a real non-involutive orthogonal transformation. Prove that the function

w= \sum_{i,j=1}^{3} c_{ ij}z_{i}\bar{z_{ j}}

maps the surface of complex unit sphere

\sum_{i=1}^{3} z_{i}\bar{z_{i}} = 1

onto a triangle of the w-plane. (F. 3)

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Miklós Schweitzer 1957- Problem 3

Source:

10/16/2015

3. Let

A

be a subset of n-dimensional space containing at least one inner point and suppose that, for every point pair

x, y \in A

, the subset

A

contains the mid point of the line segment beteween

x

and

y

. Show that

A

consists of a convex open set and of some of its boundary points. (St. 1)

college contestsreal analysistopology

Miklós Schweitzer 1957- Problem 4

Source:

10/16/2015

4. Let

F_{\epsilon} (0<\epsilon<1)

denote the class of non-negative piecewise continuous functions defined on

[0,\infty)

which satisfy the following condition:

f(x)f(y)\leq \epsilon^{\mid x-y\mid} (x,y \geq 0)

. Find the value of

s_{\epsilon}= \sup_{f\in F_{\epsilon}} \int_{0}^{\infty} f(x) dx

(R. 5)

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Miklós Schweitzer 1957- Problem 5

Source:

10/16/2015

5. Find the continuous solutions of the functional equation

f(xyz)= f(x)+f(y)+f(z)

in the following cases:
(a)

x,y,z

are arbitrary non-zero real numbers; (b)

a<x,y,z<b (1<a^{3}<b)

.
(R. 13)

functional equationcollege contests

Miklós Schweitzer 1957- Problem 9

Source:

10/18/2015

9. Find all pairs of linear polynomials

f(x)

,

g(x)

with integer coefficients for which there exist two polynomials

u(x)

,

v(x)

with integer coefficients such that

f(x)u(x)+g(x)v(x)=1

. (A. 8)

algebrapolynomialcollege contests

Miklós Schweitzer 1957- Problem 7

Source:

10/18/2015

7. Prove that any real number x satysfying the inequalities

0<x\leq 1

can be represented in the form

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

where

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

is a sequence of positive integers such that

\frac{n_{k+1}}{n_k}

assumes, for each

k

, one of the three values

2,3

or

4

. (N. 14)

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Miklós Schweitzer 1957- Problem 8

Source:

10/18/2015

8. Find all integers

a>1

for which the least (integer) solution

n

of the congruence

a^{n} \equiv 1 \pmod{p}

differs from 6 (p is any prime number). (N. 9)

number theorycollege contests

Miklós Schweitzer 1957- Problem 10

Source:

10/18/2015

10. An Abelian group

G

is said to have the property

(A)

if torsion subgroup of

G

is a direct summand of

G

. Show that if

G

is an Abelian group such that

nG

has the property

(A)

for some positive integer

n

, then

G

itself has the property

(A)

. (A. 13)

abstract algebragroup theorycollege contests

Miklós Schweitzer 1960- Problem 3

Source:

11/18/2015

3. Let

f(z)

with

f(0)=1

be regular in the unit disk and let

\left [\frac{\partial^2 \mid f(z)\mid}{\partial x\partial y} \right ] _{z=0} =1

.
Show thatthe area of the image of the unit disk by

w= f(z)

(taken with multiplicity) is not less than

\frac {1} {2}

.(f. 6)

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