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Miklós Schweitzer
1981 Miklós Schweitzer
1981 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklos Schweitzer 1981_10
Let
P
P
P
be a probability distribution defined on the Borel sets of the real line. Suppose that
P
P
P
is symmetric with respect to the origin, absolutely continuous with respect to the Lebesgue measure, and its density function
p
p
p
is zero outside the interval [\minus{}1,1] and inside this interval it is between the positive numbers
c
c
c
and
d
d
d
(
c
<
d
c < d
c
<
d
). Prove that there is no distribution whose convolution square equals
P
P
P
. T. F. Mori, G. J. Szekely
9
1
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Miklos Schweitzer 1981_9
Let
n
≥
2
n \geq 2
n
≥
2
be an integer, and let
X
X
X
be a connected Hausdorff space such that every point of
X
X
X
has a neighborhood homeomorphic to the Euclidean space
R
n
\mathbb{R}^n
R
n
. Suppose that any discrete (not necessarily closed ) subspace
D
D
D
of
X
X
X
can be covered by a family of pairwise disjoint, open sets of
X
X
X
so that each of these open sets contains precisely one element of
D
D
D
. Prove that
X
X
X
is a union of at most
ℵ
1
\aleph_1
ℵ
1
compact subspaces. Z. Balogh
8
1
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Miklos Schweitzer 1981_8
Let
W
W
W
be a dense, open subset of the real line
R
\mathbb{R}
R
. Show that the following two statements are equivalent: (1) Every function
f
:
R
→
R
f : \mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
continuous at all points of
R
∖
W
\mathbb{R} \setminus W
R
∖
W
and nondecreasing on every open interval contained in
W
W
W
is nondecreasing on the whole
R
\mathbb{R}
R
. (2)
R
∖
W
\mathbb{R} \setminus W
R
∖
W
is countable. E. Gesztelyi
7
1
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Miklos Schweitzer 1981_7
Let
U
U
U
be a real normed space such that, for an finite-dimensional, real normed space
X
,
U
X,U
X
,
U
contains a subspace isometrically isomorphic to
X
X
X
. Prove that every (not necessarily closed) subspace
V
V
V
of
U
U
U
of finite codimension has the same property. (We call
V
V
V
of finite codimension if there exists a finite-dimensional subspace
N
N
N
of
U
U
U
such that V\plus{}N\equal{}U.) A. Bosznay
6
1
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Miklos Schweitzer 1981_6
Let
f
f
f
be a strictly increasing, continuous function mapping
I
=
[
0
,
1
]
I=[0,1]
I
=
[
0
,
1
]
onto itself. Prove that the following inequality holds for all pairs
x
,
y
∈
I
x,y \in I
x
,
y
∈
I
:
1
−
cos
(
x
y
)
≤
∫
0
x
f
(
t
)
sin
(
t
f
(
t
)
)
d
t
+
∫
0
y
f
−
1
(
t
)
sin
(
t
f
−
1
(
t
)
)
d
t
.
1-\cos (xy) \leq \int_0^xf(t) \sin (tf(t))dt + \int_0^y f^{-1}(t) \sin (tf^{-1}(t)) dt .
1
−
cos
(
x
y
)
≤
∫
0
x
f
(
t
)
sin
(
t
f
(
t
))
d
t
+
∫
0
y
f
−
1
(
t
)
sin
(
t
f
−
1
(
t
))
d
t
.
Zs. Pales
5
1
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Miklos Schweitzer 1981_5
Let
K
K
K
be a convex cone in the
n
n
n
-dimensional real vector space
R
n
\mathbb{R}^n
R
n
, and consider the sets A\equal{}K \cup (\minus{}K) and B\equal{}(\mathbb{R}^n \setminus A) \cup \{ 0 \} (
0
0
0
is the origin). Show that one can find two subspaces in
R
n
\mathbb{R}^n
R
n
such that together they span
R
n
\mathbb{R}^n
R
n
, and one of them lies in
A
A
A
and the other lies in
B
B
B
. J. Szucs
4
1
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Miklos Schweitzer 1981_4
Let
G
G
G
be finite group and
K
\mathcal{K}
K
a conjugacy class of
G
G
G
that generates
G
G
G
. Prove that the following two statements are equivalent: (1) There exists a positive integer
m
m
m
such that every element of
G
G
G
can be written as a product of
m
m
m
(not necessarily distinct) elements of
K
\mathcal{K}
K
. (2)
G
G
G
is equal to its own commutator subgroup. J. Denes
3
1
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Miklos Schweitzer 1981_3
Construct an uncountable Hausdorff space in which the complement of the closure of any nonempty, open set is countable. A. Hajnal, I. Juhasz
2
1
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Miklos Schweitzer 1981_2
Consider the lattice
L
L
L
of the contradictions of a simple graph
G
G
G
(as sets of vertex pairs) with respect to inclusion. Let
n
≥
1
n \geq 1
n
≥
1
be an arbitrary integer. Show that the identity x \bigwedge \left( \bigvee_{i\equal{}0}^n y_i \right) \equal{} \bigvee_{j\equal{}0}^n \left( x \bigwedge \left( \bigvee_{0 \leq i \leq n, \;i\not\equal{}j\ } y_i \right)\right) holds if and only if
G
G
G
has no cycle of size at least n\plus{}2. A. Huhn
1
1
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Miklos Schweitzer 1981_1
We are given an infinite sequence of
1
1
1
's and
2
2
2
's with the following properties: (1) The first element of the sequence is
1
1
1
. (2) There are no two consecutive
2
2
2
's or three consecutive
1
1
1
's. (3) If we replace consecutive
1
1
1
's by a single
2
2
2
, leave the single
1
1
1
's alone, and delete the original
2
2
2
's, then we recover the original sequence. How many
2
2
2
's are there among the first
n
n
n
elements of the sequence? P. P. Palfy