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Miklós Schweitzer
1980 Miklós Schweitzer
1980 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklos Schweitzer 1980_10
Suppose that the
T
3
T_3
T
3
-space
X
X
X
has no isolated points and that in
X
X
X
any family of pairwise disjoint, nonempty, open sets is countable. Prove that
X
X
X
can be covered by at most continuum many nowhere-dense sets. I. Juhasz
9
1
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Miklos Schweitzer 1980_9
Let us divide by straight lines a quadrangle of unit area into
n
n
n
subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most
π
n
\pi \sqrt{n}
π
n
(the lines are not allowed to cut the interior of a subpolygon). G. and L. Fejes-Toth
8
1
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Miklos Schweitzer 1980_8
Let
f
(
x
)
f(x)
f
(
x
)
be a nonnegative, integrable function on
(
0
,
2
π
)
(0,2\pi)
(
0
,
2
π
)
whose Fourier series is f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x), where none of the positive integers
n
k
n_k
n
k
divides another. Prove that
∣
a
k
∣
≤
a
0
|a_k| \leq a_0
∣
a
k
∣
≤
a
0
. G. Halasz
7
1
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Miklos Schweitzer 1980_7
Let
n
≥
2
n \geq 2
n
≥
2
be a natural number and
p
(
x
)
p(x)
p
(
x
)
a real polynomial of degree at most
n
n
n
for which \max _{ \minus{}1 \leq x \leq 1} |p(x)| \leq 1, \; p(\minus{}1)\equal{}p(1)\equal{}0 \ . Prove that then |p'(x)| \leq \frac{n \cos \frac{\pi}{2n}}{\sqrt{1\minus{}x^2 \cos^2 \frac{\pi}{2n}}} \;\;\;\;\; \left( \minus{}\frac{1}{\cos \frac{\pi}{2n}} < x < \frac{1}{\cos \frac{\pi}{2n}} \\\\\ \right). J. Szabados
6
1
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Miklos Schweitzer 1980_6
Let us call a continuous function
f
:
[
a
,
b
]
→
R
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
r
e
d
u
c
i
b
l
e
<
/
s
p
a
n
>
f : [a,b] \rightarrow \mathbb{R}^2 \;<span class='latex-italic'>reducible</span>
f
:
[
a
,
b
]
→
R
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
re
d
u
c
ib
l
e
<
/
s
p
an
>
if it has a double arc (that is, if there are
a
≤
α
<
β
≤
γ
<
δ
≤
b
a \leq \alpha < \beta \leq \gamma < \delta \leq b
a
≤
α
<
β
≤
γ
<
δ
≤
b
such that there exists a strictly monotone and continuous
h
:
[
α
,
β
]
→
[
γ
,
δ
]
h : [\alpha,\beta] \rightarrow [\gamma,\delta]
h
:
[
α
,
β
]
→
[
γ
,
δ
]
for which f(t)\equal{}f(h(t)) is satisfied for every
α
≤
t
≤
β
\alpha \leq t \leq \beta
α
≤
t
≤
β
); otherwise
f
f
f
is irreducible. Construct irreducible
f
:
[
a
,
b
]
→
R
2
f : [a,b] \rightarrow \mathbb{R}^2
f
:
[
a
,
b
]
→
R
2
and
g
:
[
c
,
d
]
→
R
2
g : [c,d] \rightarrow \mathbb{R}^2
g
:
[
c
,
d
]
→
R
2
such that f([a,b])\equal{}g([c,d]) and (a) both
f
f
f
and
g
g
g
are rectifiable but their lengths are different; (b)
f
f
f
is rectifiable but
g
g
g
is not. A. Csaszar
5
1
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Miklos Schweitzer 1980_5
Let
G
G
G
be a transitive subgroup of the symmetric group
S
25
S_{25}
S
25
different from
S
25
S_{25}
S
25
and
A
25
A_{25}
A
25
. Prove that the order of
G
G
G
is not divisible by
23
23
23
. J. Pelikan
4
1
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Miklos Schweitzer 1980_4
Let T \in \textsl{SL}(n,\mathbb{Z}), let
G
G
G
be a nonsingular
n
×
n
n \times n
n
×
n
matrix with integer elements, and put S\equal{}G^{\minus{}1}TG. Prove that there is a natural number
k
k
k
such that S^k \in \textsl{SL}(n,\mathbb{Z}). Gy. Szekeres
3
1
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Miklos Schweitzer 1980_3
In a lattice, connected the elements
a
∧
b
a \wedge b
a
∧
b
and
a
∨
b
a \vee b
a
∨
b
by an edge whenever
a
a
a
and
b
b
b
are incomparable. Prove that in the obtained graph every connected component is a sublattice. M. Ajtai
2
1
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Miklos Schweitzer 1980_2
Let
H
\mathcal{H}
H
be the class of all graphs with at most
2
ℵ
0
2^{\aleph_0}
2
ℵ
0
vertices not containing a complete subgraph of size
ℵ
1
\aleph_1
ℵ
1
. Show that there is no graph
H
∈
H
H \in \mathcal{H}
H
∈
H
such that every graph in
H
\mathcal{H}
H
is a subgraph of
H
H
H
. F. Galvin
1
1
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Miklos Schweitzer 1980_1
For a real number
x
x
x
, let
∥
x
∥
\|x \|
∥
x
∥
denote the distance between
x
x
x
and the closest integer. Let 0 \leq x_n <1 \; (n\equal{}1,2,\ldots)\ , and let
ε
>
0
\varepsilon >0
ε
>
0
. Show that there exist infinitely many pairs
(
n
,
m
)
(n,m)
(
n
,
m
)
of indices such that n \not\equal{} m and \|x_n\minus{}x_m \|< \min \left( \varepsilon , \frac{1}{2|n\minus{}m|} \right). V. T. Sos