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Miklós Schweitzer
1986 Miklós Schweitzer
1986 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklós Schweitzer 1986, Problem 10
Let
X
1
,
X
2
X_1, X_2
X
1
,
X
2
be independent, identically distributed random variables such that
X
i
≥
0
X_i\geq 0
X
i
≥
0
for all
i
i
i
. Let
E
X
i
=
m
\mathrm EX_i=m
E
X
i
=
m
,
V
a
r
(
X
i
)
=
σ
2
<
∞
\mathrm{Var} (X_i)=\sigma ^2<\infty
Var
(
X
i
)
=
σ
2
<
∞
. Show that, for all
0
<
α
≤
1
0<\alpha\leq 1
0
<
α
≤
1
lim
n
→
∞
n
V
a
r
(
[
X
1
+
…
+
X
n
n
]
α
)
=
α
2
σ
2
m
2
(
1
−
α
)
\lim_{n\to\infty} n\,\mathrm{Var} \left( \left[ \frac{X_1+\ldots +X_n}{n}\right] ^\alpha\right)=\frac{\alpha ^ 2 \sigma ^ 2}{m^{2(1-\alpha)}}
n
→
∞
lim
n
Var
(
[
n
X
1
+
…
+
X
n
]
α
)
=
m
2
(
1
−
α
)
α
2
σ
2
[Gy. Michaletzki]
9
1
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Miklós Schweitzer 1986, Problem 9
Consider a latticelike packing of translates of a convex region
K
K
K
. Let
t
t
t
be the area of the fundamental parallelogram of the lattice defining the packing, and let
t
min
(
K
)
t_{\min} (K)
t
m
i
n
(
K
)
denote the minimal value of
t
t
t
taken for all latticelike packings. Is there a natural number
N
N
N
such that for any
n
>
N
n>N
n
>
N
and for any
K
K
K
different from a parallelogram,
n
t
min
(
K
)
nt_{\min} (K)
n
t
m
i
n
(
K
)
is smaller that the area of any convex domain in which
n
n
n
translates to
K
K
K
can be placed without overlapping? (By a latticelike packing of
K
K
K
we mean a set of nonoverlapping translates of
K
K
K
obtained from
K
K
K
by translations with all vectors of a lattice.) [G. and L. Fejes-Toth]
8
1
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Miklós Schweitzer 1986, Problem 8
Let
a
0
=
0
a_0=0
a
0
=
0
,
a
1
,
…
,
a
k
a_1, \ldots, a_k
a
1
,
…
,
a
k
and
b
1
,
…
,
b
k
b_1, \ldots, b_k
b
1
,
…
,
b
k
be arbitrary real numbers. (i) Show that for all sufficiently large
n
n
n
there exist polynomials
p
n
p_n
p
n
of degree at most
n
n
n
for which
p
n
(
i
)
(
−
1
)
=
a
i
,
p
n
(
i
)
(
1
)
=
b
i
,
i
=
0
,
1
,
…
,
k
p_n^{(i)} (-1)=a_i,\,\,\,\,\, p_n^{(i)} (1)=b_i,\,\,\,\,\, i=0, 1, \ldots, k
p
n
(
i
)
(
−
1
)
=
a
i
,
p
n
(
i
)
(
1
)
=
b
i
,
i
=
0
,
1
,
…
,
k
and
max
∣
x
∣
≤
1
∣
p
n
(
x
)
∣
≤
c
n
2
(
∗
)
\max_{|x|\leq 1} |p_n (x)|\leq \frac{c}{n^2}\,\,\,\,\,\,\,\,\,\, (*)
∣
x
∣
≤
1
max
∣
p
n
(
x
)
∣
≤
n
2
c
(
∗
)
where the constant
c
c
c
depends only on the numbers
a
i
,
b
i
a_i, b_i
a
i
,
b
i
.(ii) Prove that, in general, (*) cannot be replaced by the relation
lim
n
→
∞
n
2
⋅
max
∣
x
∣
≤
1
∣
p
n
(
x
)
∣
=
0
\lim_{n\to\infty} n^2\cdot \max_{|x|\leq 1} |p_n (x)| = 0
n
→
∞
lim
n
2
⋅
∣
x
∣
≤
1
max
∣
p
n
(
x
)
∣
=
0
[J. Szabados]
7
1
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Miklós Schweitzer 1986, Problem 7
Prove that the series
∑
p
c
p
f
(
p
x
)
\sum_p c_p f(px)
∑
p
c
p
f
(
p
x
)
, where the summation is over all primes, unconditionally converges in
L
2
[
0
,
1
]
L^2[0,1]
L
2
[
0
,
1
]
for every
1
1
1
-periodic function
f
f
f
whose restriction to
[
0
,
1
]
[0,1]
[
0
,
1
]
is in
L
2
[
0
,
1
]
L^2[0,1]
L
2
[
0
,
1
]
if and only if
∑
p
∣
c
p
∣
<
∞
\sum_p |c_p|<\infty
∑
p
∣
c
p
∣
<
∞
. (Unconditional convergence means convergence for all rearrangements.) [G. Halasz]
6
1
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Miklós Schweitzer 1986, Problem 6
Let
U
U
U
denote the set
{
f
∈
C
[
0
,
1
]
:
∣
f
(
x
)
∣
≤
1
f
o
r
a
l
l
x
∈
[
0
,
1
]
}
\{ f\in C[0, 1] \colon |f(x)|\leq 1\, \mathrm{for}\,\mathrm{all}\, x\in [0, 1]\}
{
f
∈
C
[
0
,
1
]
:
∣
f
(
x
)
∣
≤
1
for
all
x
∈
[
0
,
1
]}
. Prove that there is no topology on
C
[
0
,
1
]
C[0, 1]
C
[
0
,
1
]
that, together with the linear structure of
C
[
0
,
1
]
C[0,1]
C
[
0
,
1
]
, makes
C
[
0
,
1
]
C[0,1]
C
[
0
,
1
]
into a topological vector space in which the set
U
U
U
is compact. (Assume that topological vector spaces are Hausdorff) [V. Totik]
5
1
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Miklós Schweitzer 1986, Problem 5
Prove that existence of a constant
c
c
c
with the following property: for every composite integer
n
n
n
, there exists a group whose order is divisible by
n
n
n
and is less than
n
c
n^c
n
c
, and that contains no element of order
n
n
n
. [P. P. Palfy]
4
1
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Miklós Schweitzer 1986, Problem 4
Determine all real numbers
x
x
x
for which the following statement is true: the field
C
\mathbb C
C
of complex numbers contains a proper subfield
F
F
F
such that adjoining
x
x
x
to
F
F
F
we get
C
\mathbb C
C
. [M. Laczkovich]
3
1
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Miklós Schweitzer 1986, Problem 3
(a) Prove that for every natural number
k
k
k
, there are positive integers
a
1
<
a
2
<
…
<
a
k
a_1<a_2<\ldots <a_k
a
1
<
a
2
<
…
<
a
k
such that
a
i
−
a
j
a_i-a_j
a
i
−
a
j
divides
a
i
a_i
a
i
for all
1
≤
i
,
j
≤
k
,
i
≠
j
1\leq i, j\leq k, i\neq j
1
≤
i
,
j
≤
k
,
i
=
j
.(b) Show that there is an absolute constant
C
>
0
C>0
C
>
0
such that
a
1
>
k
C
k
a_1>k^{Ck}
a
1
>
k
C
k
for every sequence
a
1
,
…
,
a
k
a_1,\ldots, a_k
a
1
,
…
,
a
k
of numbers that satisfy the above divisibility condition. [A. Balogh, I. Z. Ruzsa]
2
1
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Miklós Schweitzer 1986, Problem 2
Show that if
k
≤
n
2
k\leq \frac n2
k
≤
2
n
and
F
\mathcal F
F
is a family
k
×
k
k\times k
k
×
k
submatrices of an
n
×
n
n\times n
n
×
n
matrix such that any two intersect then
∣
F
∣
≤
(
n
−
1
k
−
1
)
2
|\mathcal F|\leq \binom{n-1}{k-1}^2
∣
F
∣
≤
(
k
−
1
n
−
1
)
2
[Gy. Katona]
1
1
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Miklós Schweitzer 1986, Problem 1
If
(
A
,
<
)
(A, <)
(
A
,
<
)
is a partially ordered set, its dimension,
dim
(
A
,
<
)
\dim (A, <)
dim
(
A
,
<
)
, is the least cardinal
κ
\kappa
κ
such that there exist
κ
\kappa
κ
total orderings
{
<
α
:
α
<
κ
}
\{ <_{\alpha} \colon \alpha < \kappa \}
{
<
α
:
α
<
κ
}
on
A
A
A
with
<
=
∩
α
<
κ
<
α
<=\cap_{\alpha < \kappa} <_\alpha
<=
∩
α
<
κ
<
α
. Show that if
dim
(
A
,
<
)
>
ℵ
0
\dim (A, <)>\aleph_0
dim
(
A
,
<
)
>
ℵ
0
, then there exist disjoint
A
0
,
A
1
⊆
A
A_0, A_1\subseteq A
A
0
,
A
1
⊆
A
with
dim
(
A
0
,
<
)
\dim (A_0, <)
dim
(
A
0
,
<
)
,
dim
(
A
1
,
<
)
>
ℵ
0
\dim (A_1, <)>\aleph_0
dim
(
A
1
,
<
)
>
ℵ
0
. [D. Kelly, A. Hajnal, B. Weiss]