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Miklós Schweitzer
1978 Miklós Schweitzer
1978 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklos Schweitzer 1978_10
Let
Y
n
Y_n
Y
n
be a binomial random variable with parameters
n
n
n
and
p
p
p
. Assume that a certain set
H
H
H
of positive integers has a density and that this density is equal to
d
d
d
. Prove the following statements: (a) \lim _{n \rightarrow \infty}P(Y_n\in H)\equal{}d if
H
H
H
is an arithmetic progression. (b) The previous limit relation is not valid for arbitrary
H
H
H
. (c) If
H
H
H
is such that
P
(
Y
n
∈
H
)
P(Y_n \in H)
P
(
Y
n
∈
H
)
is convergent, then the limit must be equal to
d
d
d
. L. Posa
9
1
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Miklos Schweitzer 1978_9
Suppose that all subspaces of cardinality at most
ℵ
1
\aleph_1
ℵ
1
of a topological space are second-countable. Prove that the whole space is second-countable. A. Hajnal, I. Juhasz
8
1
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Miklos Schweitzer 1978_8
Let
X
1
,
…
,
X
n
X_1, \ldots ,X_n
X
1
,
…
,
X
n
be
n
n
n
points in the unit square (
n
>
1
n>1
n
>
1
). Let
r
i
r_i
r
i
be the distance of
X
i
X_i
X
i
from the nearest point (other than
X
i
X_i
X
i
). Prove that the inequality r_1^2\plus{} \ldots \plus{}r_n^2 \leq 4. L. Fejes-Toth, E. Szemeredi
7
1
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Miklos Schweitzer 1978_7
Let
T
T
T
be a surjective mapping of the hyperbolic plane onto itself which maps collinear points into collinear points. Prove that
T
T
T
must be an isometry. M. Bognar
6
1
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Miklos Schweitzer 1978_6
Suppose that the function
g
:
(
0
,
1
)
→
R
g : (0,1) \rightarrow \mathbb{R}
g
:
(
0
,
1
)
→
R
can be uniformly approximated by polynomials with nonnegative coefficients. Prove that
g
g
g
must be analytic. Is the statement also true for the interval (\minus{}1,0) instead of
(
0
,
1
)
(0,1)
(
0
,
1
)
? J. Kalina, L. Lempert
5
1
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Miklos Schweitzer 1978_5
Suppose that
R
(
z
)
=
∑
n
=
−
∞
∞
a
n
z
n
R(z)= \sum_{n=-\infty}^{\infty} a_nz^n
R
(
z
)
=
∑
n
=
−
∞
∞
a
n
z
n
converges in a neighborhood of the unit circle
{
z
:
∣
z
∣
=
1
}
\{ z : \;|z|=1\ \}
{
z
:
∣
z
∣
=
1
}
in the complex plane, and
R
(
z
)
=
P
(
z
)
/
Q
(
z
)
R(z)=P(z) / Q(z)
R
(
z
)
=
P
(
z
)
/
Q
(
z
)
is a rational function in this neighborhood, where
P
P
P
and
Q
Q
Q
are polynomials of degree at most
k
k
k
. Prove that there is a constant
c
c
c
independent of
k
k
k
such that
∑
n
=
−
∞
∞
∣
a
n
∣
≤
c
k
2
max
∣
z
∣
=
1
∣
R
(
z
)
∣
.
\sum_{n=-\infty} ^{\infty} |a_n| \leq ck^2 \max_{|z|=1} |R(z)|.
n
=
−
∞
∑
∞
∣
a
n
∣
≤
c
k
2
∣
z
∣
=
1
max
∣
R
(
z
)
∣.
H. S. Shapiro, G. Somorjai
4
1
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Miklos Schweitzer 1978_4
Let
Q
\mathbb{Q}
Q
and
R
\mathbb{R}
R
be the set of rational numbers and the set of real numbers, respectively, and let
f
:
Q
→
R
f : \mathbb{Q} \rightarrow \mathbb{R}
f
:
Q
→
R
be a function with the following property. For every
h
∈
Q
,
x
0
∈
R
h \in \mathbb{Q} , \;x_0 \in \mathbb{R}
h
∈
Q
,
x
0
∈
R
, f(x\plus{}h)\minus{}f(x) \rightarrow 0 as
x
∈
Q
x \in \mathbb{Q}
x
∈
Q
tends to
x
0
x_0
x
0
. Does it follow that
f
f
f
is bounded on some interval? M. Laczkovich
3
1
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Miklos Schweitzer 1978_3
Let
1
<
a
1
<
a
2
<
…
<
a
n
<
x
1<a_1<a_2< \ldots <a_n<x
1
<
a
1
<
a
2
<
…
<
a
n
<
x
be positive integers such that \sum_{i\equal{}1}^n 1/a_i \leq 1. Let
y
y
y
denote the number of positive integers smaller that
x
x
x
not divisible by any of the
a
i
a_i
a
i
. Prove that
y
>
c
x
log
x
y > \frac{cx}{\log x}
y
>
lo
g
x
c
x
with a suitable positive constant
c
c
c
(independent of
x
x
x
and the numbers
a
i
a_i
a
i
). I. Z. Ruzsa
2
1
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Miklos Schweitzer 1978_2
For a distributive lattice
L
L
L
, consider the following two statements: (A) Every ideal of
L
L
L
is the kernel of at least two different homomorphisms. (B)
L
L
L
contains no maximal ideal. Which one of these statements implies the other? (Every homomorphism
φ
\varphi
φ
of
L
L
L
induces an equivalence relation on
L
L
L
:
a
∼
b
a \sim b
a
∼
b
if and only if a \varphi\equal{}b \varphi. We do not consider two homomorphisms different if they imply the same equivalence relation.) J. Varlet, E. Fried
1
1
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Miklos Schweitzer 1978_1
Let
H
\mathcal{H}
H
be a family of finite subsets of an infinite set
X
X
X
such that every finite subset of
X
X
X
can be represented as the union of two disjoint sets from
H
\mathcal{H}
H
. Prove that for every positive integer
k
k
k
there is a subset of
X
X
X
that can be represented in at least
k
k
k
different ways as the union of two disjoint sets from
H
\mathcal{H}
H
. P. Erdos