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Miklós Schweitzer
1974 Miklós Schweitzer
1974 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklos Schweitzer 1974_10
Let
μ
\mu
μ
and
ν
\nu
ν
be two probability measures on the Borel sets of the plane. Prove that there are random variables
ξ
1
,
ξ
2
,
η
1
,
η
2
\xi_1, \xi_2, \eta_1, \eta_2
ξ
1
,
ξ
2
,
η
1
,
η
2
such that (a) the distribution of
(
ξ
1
,
ξ
2
)
(\xi_1, \xi_2)
(
ξ
1
,
ξ
2
)
is
μ
\mu
μ
and the distribution of
(
η
1
,
η
2
)
(\eta_1, \eta_2)
(
η
1
,
η
2
)
is
ν
\nu
ν
, (b)
ξ
1
≤
η
1
,
ξ
2
≤
η
2
\xi_1 \leq \eta_1, \xi_2 \leq \eta_2
ξ
1
≤
η
1
,
ξ
2
≤
η
2
almost everywhere, if an only if
μ
(
G
)
≥
ν
(
G
)
\mu(G) \geq \nu(G)
μ
(
G
)
≥
ν
(
G
)
for all sets of the form G\equal{}\cup_{i\equal{}1}^k (\minus{}\infty, x_i) \times (\minus{}\infty, y_i). P. Major
9
1
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Miklos Schweitzer 1974_9
Let
A
A
A
be a closed and bounded set in the plane, and let
C
C
C
denote the set of points at a unit distance from
A
A
A
. Let
p
∈
C
p \in C
p
∈
C
, and assume that the intersection of
A
A
A
with the unit circle
K
K
K
centered at
p
p
p
can be covered by an arc shorter that a semicircle of
K
K
K
. Prove that the intersection of
C
C
C
with a suitable neighborhood of
p
p
p
is a simple arc which
p
p
p
is not an endpoint. M. Bognar
8
1
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Miklos Schweitzer 1974_8
Prove that there exists a topological space
T
T
T
containing the real line as a subset, such that the Lebesgue-measurable functions, and only those, extend continuously over
T
T
T
. Show that the real line cannot be an everywhere-dense subset of such a space
T
T
T
. A. Csaszar
7
1
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Miklos Schweitzer 1974_7
Given a positive integer
m
m
m
and
0
<
δ
<
π
0 < \delta <\pi
0
<
δ
<
π
, construct a trigonometric polynomial f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx) of degree
m
m
m
such that f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m, and \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}, for some universal constant
c
c
c
. G. Halasz
6
1
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Miklos Schweitzer 1974_6
Let f(x)\equal{}\sum_{n\equal{}1}^{\infty} a_n/(x\plus{}n^2), \;(x \geq 0)\ , where \sum_{n\equal{}1}^{\infty} |a_n|n^{\minus{} \alpha} < \infty for some
α
>
2
\alpha > 2
α
>
2
. Let us assume that for some
β
>
1
/
α
\beta > 1/{\alpha}
β
>
1/
α
, we have f(x)\equal{}O(e^{\minus{}x^{\beta}}) as
x
→
∞
x \rightarrow \infty
x
→
∞
. Prove that
a
n
a_n
a
n
is identically
0
0
0
. G. Halasz
5
1
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Miklos Schweitzer 1974_5
Let
{
f
n
}
n
=
0
∞
\{f_n \}_{n=0}^{\infty}
{
f
n
}
n
=
0
∞
be a uniformly bounded sequence of real-valued measurable functions defined on
[
0
,
1
]
[0,1]
[
0
,
1
]
satisfying
∫
0
1
f
n
2
=
1.
\int_0^1 f_n^2=1.
∫
0
1
f
n
2
=
1.
Further, let
{
c
n
}
\{ c_n \}
{
c
n
}
be a sequence of real numbers with
∑
n
=
0
∞
c
n
2
=
+
∞
.
\sum_{n=0}^{\infty} c_n^2= +\infty.
n
=
0
∑
∞
c
n
2
=
+
∞.
Prove that some re-arrangement of the series
∑
n
=
0
∞
c
n
f
n
\sum_{n=0}^{\infty} c_nf_n
∑
n
=
0
∞
c
n
f
n
is divergent on a set of positive measure. J. Komlos
4
1
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Miklos Schweitzer 1974_4
Let
R
R
R
be an infinite ring such that every subring of
R
R
R
different from
{
0
}
\{0 \}
{
0
}
has a finite index in
R
R
R
. (By the index of a subring, we mean the index of its additive group in the additive group of
R
R
R
.) Prove that the additive group of
R
R
R
is cyclic. L. Lovasz, J. Pelikan
3
1
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Miklos Schweitzer 1974_3
Prove that a necessary and sufficient for the existence of a set
S
⊂
{
1
,
2
,
.
.
.
,
n
}
S \subset \{1,2,...,n \}
S
⊂
{
1
,
2
,
...
,
n
}
with the property that the integers 0,1,...,n\minus{}1 all have an odd number of representations in the form x\minus{}y, x,y \in S, is that (2n\minus{}1) has a multiple of the form 2.4^k\minus{}1 L. Lovasz, J. Pelikan
2
1
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Miklos Schweitzer 1974_2
Let
G
G
G
be a
2
2
2
-connected nonbipartite graph on
2
n
2n
2
n
vertices. Show that the vertex set of
G
G
G
can be split into two classes of
n
n
n
elements such that the edges joining the two classes form a connected, spanning subgraph. L. Lovasz
1
1
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Miklos Schweitzer 1974_1
Let
F
\mathcal{F}
F
be a family of subsets of a ground set
X
X
X
such that
∪
F
∈
F
F
=
X
\cup_{F \in \mathcal{F}}F=X
∪
F
∈
F
F
=
X
, and (a) if
A
,
B
∈
F
A,B \in \mathcal{F}
A
,
B
∈
F
, then
A
∪
B
⊆
C
A \cup B \subseteq C
A
∪
B
⊆
C
for some
C
∈
F
;
C \in \mathcal{F};
C
∈
F
;
(b) if
A
n
∈
F
(
n
=
0
,
1
,
.
.
.
)
,
B
∈
F
,
A_n \in \mathcal{F} \;(n=0,1,...)\ , B \in \mathcal{F},
A
n
∈
F
(
n
=
0
,
1
,
...
)
,
B
∈
F
,
and
A
0
⊂
A
1
⊂
.
.
.
,
A_0 \subset A_1 \subset...,
A
0
⊂
A
1
⊂
...
,
then, for some
k
≥
0
,
A
n
∩
B
=
A
k
∩
B
k \geq 0, \;A_n \cap B=A_k \cap B
k
≥
0
,
A
n
∩
B
=
A
k
∩
B
for all
n
≥
k
n \geq k
n
≥
k
. Show that there exist pairwise disjoint sets
X
γ
(
γ
∈
Γ
)
{ X_{\gamma} \;( \gamma \in \Gamma}\ )
X
γ
(
γ
∈
Γ
)
, with
X
=
∪
{
X
γ
:
γ
∈
Γ
}
,
X= \cup \{ X_{\gamma} : \;\gamma \in \Gamma \ \},
X
=
∪
{
X
γ
:
γ
∈
Γ
}
,
such that every
X
γ
X_{\gamma}
X
γ
is contained in some member of
F
\mathcal{F}
F
, and every element of
F
\mathcal{F}
F
is contained in the union of finitely many
X
γ
X_{\gamma}
X
γ
's. A. Hajnal