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Miklós Schweitzer
1971 Miklós Schweitzer
1971 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
11
1
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Miklos Schweitzer 1971_11
Let
C
C
C
be a simple arc with monotone curvature such that
C
C
C
is congruent to its evolute. Show that under appropriate differentiability conditions,
C
C
C
is a part of a cycloid or a logarithmic spiral with polar equation r\equal{}ae^{\vartheta}. J. Szenthe
10
1
Hide problems
Miklos Schweitzer 1971_10
Let
{
ϕ
n
(
x
)
}
\{\phi_n(x) \}
{
ϕ
n
(
x
)}
be a sequence of functions belonging to
L
2
(
0
,
1
)
L^2(0,1)
L
2
(
0
,
1
)
and having norm less that
1
1
1
such that for any subsequence
{
ϕ
n
k
(
x
)
}
\{\phi_{n_k}(x) \}
{
ϕ
n
k
(
x
)}
the measure of the set
{
x
∈
(
0
,
1
)
:
∣
1
N
∑
k
=
1
N
ϕ
n
k
(
x
)
∣
≥
y
}
\{x \in (0,1) : \;|\frac{1}{\sqrt{N}} \sum _{k=1}^N \phi_{n_k}(x)| \geq y\ \}
{
x
∈
(
0
,
1
)
:
∣
N
1
k
=
1
∑
N
ϕ
n
k
(
x
)
∣
≥
y
}
tends to
0
0
0
as
y
y
y
and
N
N
N
tend to infinity. Prove that
ϕ
n
\phi_n
ϕ
n
tends to
0
0
0
weakly in the function space
L
2
(
0
,
1
)
.
L^2(0,1).
L
2
(
0
,
1
)
.
F. Moricz
9
1
Hide problems
Miklos Schweitzer 1971_9
Given a positive, monotone function
F
(
x
)
F(x)
F
(
x
)
on
(
0
,
∞
)
(0, \infty)
(
0
,
∞
)
such that
F
(
x
)
/
x
F(x)/x
F
(
x
)
/
x
is monotone nondecreasing and
F
(
x
)
/
x
1
+
d
F(x)/x^{1+d}
F
(
x
)
/
x
1
+
d
is monotone nonincreasing for some positive
d
d
d
, let
λ
n
>
0
\lambda_n >0
λ
n
>
0
and
a
n
≥
0
,
n
≥
1
a_n \geq 0 , \;n \geq 1
a
n
≥
0
,
n
≥
1
. Prove that if
∑
n
=
1
∞
λ
n
F
(
a
n
∑
k
=
1
n
λ
k
λ
n
)
<
∞
,
\sum_{n=1}^{\infty} \lambda_n F \left( a_n \sum _{k=1}^n \frac{\lambda_k}{\lambda_n} \right) < \infty,
n
=
1
∑
∞
λ
n
F
(
a
n
k
=
1
∑
n
λ
n
λ
k
)
<
∞
,
or
∑
n
=
1
∞
λ
n
F
(
∑
k
=
1
n
a
k
λ
k
λ
n
)
<
∞
,
\sum_{n=1}^{\infty} \lambda_n F \left( \sum _{k=1}^n a_k \frac{\lambda_k}{\lambda_n} \right) < \infty,
n
=
1
∑
∞
λ
n
F
(
k
=
1
∑
n
a
k
λ
n
λ
k
)
<
∞
,
then
∑
n
=
1
∞
a
n
\sum_{n=1}^ {\infty} a_n
∑
n
=
1
∞
a
n
is convergent. L. Leindler
8
1
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Miklos Schweitzer 1971_8
Show that the edges of a strongly connected bipolar graph can be oriented in such a way that for any edge
e
e
e
there is a simple directed path from pole
p
p
p
to pole
q
q
q
containing
e
e
e
. (A strongly connected bipolar graph is a finite connected graph with two special vertices
p
p
p
and
q
q
q
having the property that there are no points x,y,x \not \equal{} y, such that all paths from
x
x
x
to
p
p
p
as well as all paths from
x
x
x
to
q
q
q
contain
y
y
y
.) A. Adam
7
1
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Miklos Schweitzer 1971_7
Let
n
≥
2
n \geq 2
n
≥
2
be an integer, let
S
S
S
be a set of
n
n
n
elements, and let
A
i
,
1
≤
i
≤
m
A_i , \; 1\leq i \leq m
A
i
,
1
≤
i
≤
m
, be distinct subsets of
S
S
S
of size at least
2
2
2
such that A_i \cap A_j \not\equal{} \emptyset, A_i \cap A_k \not\equal{} \emptyset, A_j \cap A_k \not\equal{} \emptyset, \;\textrm{imply}\ \;A_i \cap A_j \cap A_k \not\equal{} \emptyset \ . Show that m \leq 2^{n\minus{}1}\minus{}1. P. Erdos
6
1
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Miklos Schweitzer 1971_6
Let
a
(
x
)
a(x)
a
(
x
)
and
r
(
x
)
r(x)
r
(
x
)
be positive continuous functions defined on the interval
[
0
,
∞
)
[0,\infty)
[
0
,
∞
)
, and let
lim inf
x
→
∞
(
x
−
r
(
x
)
)
>
0.
\liminf_{x \rightarrow \infty} (x-r(x)) >0.
x
→
∞
lim
inf
(
x
−
r
(
x
))
>
0.
Assume that
y
(
x
)
y(x)
y
(
x
)
is a continuous function on the whole real line, that it is differentiable on
[
0
,
∞
)
[0, \infty)
[
0
,
∞
)
, and that it satisfies
y
′
(
x
)
=
a
(
x
)
y
(
x
−
r
(
x
)
)
y'(x)=a(x)y(x-r(x))
y
′
(
x
)
=
a
(
x
)
y
(
x
−
r
(
x
))
on
[
0
,
∞
)
[0, \infty)
[
0
,
∞
)
. Prove that the limit
lim
x
→
∞
y
(
x
)
exp
{
−
∫
0
x
a
(
u
)
d
u
}
\lim_{x \rightarrow \infty}y(x) \exp \left\{ -%Error. "diaplaymath" is a bad command. \int_0^x a(u)du \right \}
x
→
∞
lim
y
(
x
)
exp
{
−
∫
0
x
a
(
u
)
d
u
}
exists and is finite. I. Gyori
5
1
Hide problems
Miklos Schweitzer 1971_5
Let
λ
1
≤
λ
2
≤
.
.
.
\lambda_1 \leq \lambda_2 \leq...
λ
1
≤
λ
2
≤
...
be a positive sequence and let
K
K
K
be a constant such that
∑
k
=
1
n
−
1
λ
k
2
<
K
λ
n
2
(
n
=
1
,
2
,
.
.
.
)
.
\sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).
k
=
1
∑
n
−
1
λ
k
2
<
K
λ
n
2
(
n
=
1
,
2
,
...
)
.
Prove that there exists a constant
K
′
K'
K
′
such that
∑
k
=
1
n
−
1
λ
k
<
K
′
λ
n
(
n
=
1
,
2
,
.
.
.
)
.
\sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).
k
=
1
∑
n
−
1
λ
k
<
K
′
λ
n
(
n
=
1
,
2
,
...
)
.
L. Leindler
4
1
Hide problems
Miklos Schweitzer 1971_4
Suppose that
V
V
V
is a locally compact topological space that admits no countable covering with compact sets. Let
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
C
<
/
s
p
a
n
>
<span class='latex-bold'>C</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
C
<
/
s
p
an
>
denote the set of all compact subsets of the space
V
V
V
and
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
a
n
>
<span class='latex-bold'>U</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
an
>
the set of open subsets that are not contained in any compact set. Let
f
f
f
be a function from
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
a
n
>
<span class='latex-bold'>U</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
an
>
to
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
C
<
/
s
p
a
n
>
<span class='latex-bold'>C</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
C
<
/
s
p
an
>
such that
f
(
U
)
⊆
U
f(U)\subseteq U
f
(
U
)
⊆
U
for all
U
∈
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
a
n
>
U \in <span class='latex-bold'>U</span>
U
∈<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
an
>
. Prove that either (i) there exists a nonempty compact set
C
C
C
such that
f
(
U
)
f(U)
f
(
U
)
is not a proper subset of
C
C
C
whenever
C
⊆
U
∈
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
a
n
>
C \subseteq U \in <span class='latex-bold'>U</span>
C
⊆
U
∈<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
an
>
, (ii) or for some compact set
C
C
C
, the set
f
−
1
(
C
)
=
⋃
{
U
∈
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
a
n
>
:
f
(
U
)
⊆
C
}
f^{-1}(C)= \bigcup \{U \in <span class='latex-bold'>U</span>\;: \ \;f(U)\subseteq C\ \}
f
−
1
(
C
)
=
⋃
{
U
∈<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
an
>
:
f
(
U
)
⊆
C
}
is an element of
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
a
n
>
<span class='latex-bold'>U</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
U
<
/
s
p
an
>
, that is,
f
−
1
(
C
)
f^{-1}(C)
f
−
1
(
C
)
is not contained in any compact set. A. Mate
3
1
Hide problems
Miklos Schweitzer 1971_3
Let
0
<
a
k
<
1
0<a_k<1
0
<
a
k
<
1
for
k
=
1
,
2
,
.
.
.
.
k=1,2,... .
k
=
1
,
2
,
....
Give a necessary and sufficient condition for the existence, for every
0
<
x
<
1
0<x<1
0
<
x
<
1
, of a permutation
π
x
\pi_x
π
x
of the positive integers such that
x
=
∑
k
=
1
∞
a
π
x
(
k
)
2
k
.
x= \sum_{k=1}^{\infty} \frac{a_{\pi_x}(k)}{2^k}.
x
=
k
=
1
∑
∞
2
k
a
π
x
(
k
)
.
P. Erdos
2
1
Hide problems
Miklos Schweitzer 1971_2
Prove that there exists an ordered set in which every uncountable subset contains an uncountable, well-ordered subset and that cannot be represented as a union of a countable family of well-ordered subsets. A. Hajnal
1
1
Hide problems
Miklos Schweitzer 1971_1
Let
G
G
G
be an infinite compact topological group with a Hausdorff topology. Prove that
G
G
G
contains an element g \not\equal{} 1 such that the set of all powers of
g
g
g
is either everywhere dense in
G
G
G
or nowhere dense in
G
G
G
. J. Erdos