MathDB

1

Miklos Schweitzer 1975_12

Assume that a face of a convex polyhedron

P

has a common edge with every other face. Show that there exists a simple closed polygon that consists of edges of

P

and passes through all vertices. L .Lovasz

11

1

Miklos Schweitzer 1975_11

Let

X_1,X_2,...,X_n

be (not necessary independent) discrete random variables. Prove that there exist at least

n^2/2

pairs

(i,j)

such that H(X_i\plus{}X_j) \geq \frac 13 \min_{1 \leq k \leq n} \{ H(X_k) \}, where

H(X)

denotes the Shannon entropy of

X

. GY. Katona

10

1

Miklos Schweitzer 1975_10

Prove that an idempotent linear operator of a Hilbert space is self-adjoint if and only if it has norm

0

or

1

. J. Szucs

9

1

Miklos Schweitzer 1975_9

Let

l_0,c,\alpha,g

be positive constants, and let

x(t)

be the solution of the differential equation ([l_0\plus{}ct^{\alpha}] ^2x')'\plus{}g[l_0\plus{}ct^{\alpha}] \sin x\equal{}0, \;t \geq 0,\ \;\minus{}\frac{\pi}{2}

x(t)

is defined on the interval

[t_0,\infty)

; furthermore, if

\alpha >2

then for every x_0 \not\equal{} 0 there exists a

t_0

such that

\liminf_{t \rightarrow \infty} |x(t)| >0.

L. Hatvani

8

1

Miklos Schweitzer 1975_8

Prove that if

\sum_{n=1}^m a_n \leq Na_m \;(m=1,2,...)

holds for a sequence

\{a_n \}

of nonnegative real numbers with some positive integer

N

, then

\alpha_{i+p} \geq p \alpha_i

for

i,p=1,2,...,

where

\alpha_i= \sum_{n=(i-1)N+1}^{iN} a_n \;(i=1,2,...)\ .

L. Leindler

7

1

Miklos Schweitzer 1975_7

Let

a<a'<b<b'

be real numbers and let the real function

f

be continuous on the interval

[a,b']

and differentiable in its interior. Prove that there exist

c \in (a,b), c'\in (a',b')

such that f(b)\minus{}f(a)\equal{}f'(c)(b\minus{}a), f(b')\minus{}f(a')\equal{}f'(c')(b'\minus{}a'), and

c<c'

. B. Szokefalvi Nagy

6

1

Miklos Schweitzer 1975_6

Let

f

be a differentiable real function and let

M

be a positive real number. Prove that if |f(x\plus{}t)\minus{}2f(x)\plus{}f(x\minus{}t)| \leq Mt^2 \; \textrm{for all}\ \;x\ \; \textrm{and}\ \;t\ , then |f'(x\plus{}t)\minus{}f'(x)| \leq M|t|. J. Szabados

5

1

Miklos Schweitzer 1975_5

Let

\{ f_n \}

be a sequence of Lebesgue-integrable functions on

[0,1]

such that for any Lebesgue-measurable subset

E

of

[0,1]

the sequence

\int_E f_n

is convergent. Assume also that \lim_n f_n\equal{}f exists almost everywhere. Prove that

f

is integrable and \int_E f\equal{}\lim_n \int_E f_n. Is the assertion also true if

E

runs only over intervals but we also assume

f_n \geq 0 ?

What happens if

[0,1]

is replaced by [0,\plus{}\infty) ? J. Szucs

4

1

Miklos Schweitzer 1975_4

Prove that the set of rational-valued, multiplicative arithmetical functions and the set of complex rational-valued, multiplicative arithmetical functions form isomorphic groups with the convolution operation

f \circ g

defined by

{ (f \circ g)(n)= %Error. "displatmath" is a bad command. \sum_{d|n} f(d)g(\frac nd}).

(We call a complex number

<span class='latex-italic'>complex rational</span>

, if its real and imaginary parts are both rational.) B. Csakany

3

1

Miklos Schweitzer 1975_3

Let

S

be a semigroup without proper two-sided ideals and suppose that for every

a,b \in S

at least one of the products

ab

and

ba

is equal to one of the elements

a,b

. Prove that either ab\equal{}a for all

a,b \in S

or ab\equal{}b for all

a,b \in S

. L. Megyesi

2

1

Miklos Schweitzer 1975_2

Let

\mathcal{A}_n

denote the set of all mappings

f: \{1,2,\ldots ,n \} \rightarrow \{1,2,\ldots, n \}

such that

f^{-1}(i) :=\{ k \colon f(k)=i\ \} \neq \varnothing

implies

f^{-1}(j) \neq \varnothing, j \in \{1,2,\ldots, i \} .

Prove

|\mathcal{A}_n| = \sum_{k=0}^{\infty} \frac{k^n}{2^{k+1}}.

L. Lovasz

1

Miklos Schweitzer 1975_1

Show that there exists a tournament

(T,\rightarrow)

of cardinality

\aleph_1

containing no transitive subtournament of size

\aleph_1

. ( A structure

(T,\rightarrow)

is a

<span class='latex-italic'>tournament</span>

if

\rightarrow

is a binary, irreflexive, asymmetric and trichotomic relation. The tournament

(T,\rightarrow)

is transitive if

\rightarrow

is transitive, that is, if it orders

T

.) A. Hajnal

1975 Miklós Schweitzer

Subcontests

Miklos Schweitzer 1975_12

Miklos Schweitzer 1975_11

Miklos Schweitzer 1975_10

Miklos Schweitzer 1975_9

Miklos Schweitzer 1975_8

Miklos Schweitzer 1975_7

Miklos Schweitzer 1975_6

Miklos Schweitzer 1975_5

Miklos Schweitzer 1975_4

Miklos Schweitzer 1975_3

Miklos Schweitzer 1975_2

Miklos Schweitzer 1975_1