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Miklós Schweitzer
1968 Miklós Schweitzer
1968 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
11
1
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Miklos Schweitzer 1968_11
Let
A
1
,
.
.
.
,
A
n
A_1,...,A_n
A
1
,
...
,
A
n
be arbitrary events in a probability field. Denote by
C
k
C_k
C
k
the event that at least
k
k
k
of
A
1
,
.
.
.
,
A
n
A_1,...,A_n
A
1
,
...
,
A
n
occur. Prove that
∏
k
=
1
n
P
(
C
k
)
≤
∏
k
=
1
n
P
(
A
k
)
.
\prod_{k=1}^n P(C_k) \leq \prod_{k=1}^n P(A_k).
k
=
1
∏
n
P
(
C
k
)
≤
k
=
1
∏
n
P
(
A
k
)
.
A. Renyi
10
1
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Miklos Schweitzer 1968_10
Let
h
h
h
be a triangle of perimeter
1
1
1
, and let
H
H
H
be a triangle of perimeter
λ
\lambda
λ
homothetic to
h
h
h
. Let
h
1
,
h
2
,
.
.
.
h_1,h_2,...
h
1
,
h
2
,
...
be translates of
h
h
h
such that , for all
i
i
i
,
h
i
h_i
h
i
is different from h_{i\plus{}2} and touches
H
H
H
and h_{i\plus{}1} (that is, intersects without overlapping). For which values of
λ
\lambda
λ
can these triangles be chosen so that the sequence
h
1
,
h
2
,
.
.
.
h_1,h_2,...
h
1
,
h
2
,
...
is periodic? If
λ
≥
1
\lambda \geq 1
λ
≥
1
is such a value, then determine the number of different triangles in a periodic chain
h
1
,
h
2
,
.
.
.
h_1,h_2,...
h
1
,
h
2
,
...
and also the number of times such a chain goes around the triangle
H
H
H
. L. Fejes-Toth
9
1
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Miklos Schweitzer 1968_9
Let
f
(
x
)
f(x)
f
(
x
)
be a real function such that \lim_{x \rightarrow \plus{}\infty} \frac{f(x)}{e^x}\equal{}1 and
∣
f
′
′
(
x
)
∣
≤
c
∣
f
′
(
x
)
∣
|f''(x)|\leq c|f'(x)|
∣
f
′′
(
x
)
∣
≤
c
∣
f
′
(
x
)
∣
for all sufficiently large
x
x
x
. Prove that \lim_{x \rightarrow \plus{}\infty} \frac{f'(x)}{e^x}\equal{}1. P. Erdos
8
1
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Miklos Schweitzer 1968_8
Let
n
n
n
and
k
k
k
be given natural numbers, and let
A
A
A
be a set such that
∣
A
∣
≤
n
(
n
+
1
)
k
+
1
.
|A| \leq \frac{n(n+1)}{k+1}.
∣
A
∣
≤
k
+
1
n
(
n
+
1
)
.
For
i
=
1
,
2
,
.
.
.
,
n
+
1
i=1,2,...,n+1
i
=
1
,
2
,
...
,
n
+
1
, let
A
i
A_i
A
i
be sets of size
n
n
n
such that
∣
A
i
∩
A
j
∣
≤
k
(
i
≠
j
)
,
|A_i \cap A_j| \leq k \;(i \not=j)\ ,
∣
A
i
∩
A
j
∣
≤
k
(
i
=
j
)
,
A
=
⋃
i
=
1
n
+
1
A
i
.
A= \bigcup_{i=1}^{n+1} A_i.
A
=
i
=
1
⋃
n
+
1
A
i
.
Determine the cardinality of
A
A
A
. K. Corradi
7
1
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Miklos Schweitzer 1968_7
For every natural number
r
r
r
, the set of
r
r
r
-tuples of natural numbers is partitioned into finitely many classes. Show that if
f
(
r
)
f(r)
f
(
r
)
is a function such that
f
(
r
)
≥
1
f(r)\geq 1
f
(
r
)
≥
1
and \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty, then there exists an infinite set of natural numbers that, for all
r
r
r
, contains
r
r
r
-triples from at most
f
(
r
)
f(r)
f
(
r
)
classes. Show that if f(r) \not \rightarrow \plus{}\infty, then there is a family of partitions such that no such infinite set exists. P. Erdos, A. Hajnal
6
1
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Miklos Schweitzer 1968_6
Let \Psi\equal{}\langle A;...\rangle be an arbitrary, countable algebraic structure (that is,
Ψ
\Psi
Ψ
can have an arbitrary number of finitary operations and relations). Prove that
Ψ
\Psi
Ψ
has as many as continuum automorphisms if and only if for any finite subset
A
′
A'
A
′
of
A
A
A
there is an automorphism
π
A
′
\pi_{A'}
π
A
′
of
Ψ
\Psi
Ψ
different from the identity automorphism and such that (x) \pi_{A'}\equal{}x for every
x
∈
A
′
x \in A'
x
∈
A
′
. M. Makkai
5
1
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Miklos Schweitzer 1968_5
Let
k
k
k
be a positive integer,
z
z
z
a complex number, and
ε
<
1
2
\varepsilon <\frac12
ε
<
2
1
a positive number. Prove that the following inequality holds for infinitely many positive integers
n
n
n
:
∣
∑
0
≤
l
≤
n
k
+
1
(
n
−
k
l
l
)
z
l
∣
≥
(
1
2
−
ε
)
n
.
\mid \sum_{0\leq l \leq \frac{n}{k+1}} \binom{n-kl}{l}z^l \mid \geq (\frac 12-\varepsilon)^n.
∣
0
≤
l
≤
k
+
1
n
∑
(
l
n
−
k
l
)
z
l
∣≥
(
2
1
−
ε
)
n
.
P. Turan
4
1
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Miklos Schweitzer 1968_4
Let
f
f
f
be a complex-valued, completely multiplicative,arithmetical function. Assume that there exists an infinite increasing sequence
N
k
N_k
N
k
of natural numbers such that f(n)\equal{}A_k \not\equal{} 0 \;\textrm{provided}\ \; N_k \leq n \leq N_k\plus{}4 \sqrt{N_k}\ . Prove that
f
f
f
is identically
1
1
1
. I. Katai
3
1
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Miklos Schweitzer 1968_3
Let
K
K
K
be a compact topological group, and let
F
F
F
be a set of continuous functions defined on
K
K
K
that has cardinality greater that continuum. Prove that there exist
x
0
∈
K
x_0 \in K
x
0
∈
K
and f \not\equal{}g \in F such that f(x_0)\equal{}g(x_0)\equal{}\max_{x\in K}f(x)\equal{}\max_{x \in K}g(x). I. Juhasz
2
1
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Miklos Schweitzer 1968_2
Let
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
be nonnegative real numbers. Prove that
(
∑
i
=
1
n
a
i
)
(
∑
i
=
1
n
a
i
n
−
1
)
≤
n
∏
i
=
1
n
a
i
+
(
n
−
1
)
(
∑
i
=
1
n
a
i
n
)
.
( \sum_{i=1}^na_i)( \sum_{i=1}^na_i^{n-1}) \leq n \prod_{i=1}^na_i+ (n-1) ( \sum_{i=1}^na_i^n).
(
i
=
1
∑
n
a
i
)
(
i
=
1
∑
n
a
i
n
−
1
)
≤
n
i
=
1
∏
n
a
i
+
(
n
−
1
)
(
i
=
1
∑
n
a
i
n
)
.
J. Suranyi
1
1
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Miklos Schweitzer 1968_1
Consider the endomorphism ring of an Abelian torsion-free (resp. torsion) group
G
G
G
. Prove that this ring is Neumann-regular if and only if
G
G
G
is a discrete direct sum of groups isomorphic to the additive group of the rationals (resp. ,a discrete direct sum of cyclic groups of prime order). (A ring
R
R
R
is called Neumann-regular if for every
α
∈
R
\alpha \in R
α
∈
R
there exists a
β
∈
R
\beta \in R
β
∈
R
such that \alpha \beta \alpha\equal{}\alpha.) E. Freid