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Miklós Schweitzer
2008 Miklós Schweitzer
2008 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
11
1
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Miklós Schweitzer 2008, Problem 11
Let
ζ
1
,
…
,
ζ
n
\zeta_1, \ldots, \zeta_n
ζ
1
,
…
,
ζ
n
be (not necessarily independent) random variables with normal distribution for which
E
ζ
j
=
0
E\zeta_j=0
E
ζ
j
=
0
and
E
ζ
j
2
≤
1
E\zeta_j^2\le 1
E
ζ
j
2
≤
1
for all
1
≤
j
≤
n
1\le j\le n
1
≤
j
≤
n
. Prove that
E
(
max
1
≤
j
≤
n
ζ
j
)
≤
2
log
n
E\left( \max_{1\le j\le n} \zeta_j \right)\le\sqrt{2\log n}
E
(
1
≤
j
≤
n
max
ζ
j
)
≤
2
lo
g
n
(translated by Miklós Maróti)
10
1
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Miklós Schweitzer 2008, Problem 10
Let
V
V
V
be the set of non-collinear pairs of vectors in
R
3
\mathbb{R}^3
R
3
, and
H
H
H
be the set of lines passing through the origin. Is is true that for every continuous map
f
:
V
→
H
f\colon V\rightarrow H
f
:
V
→
H
there exists a continuous map
g
:
V
→
R
3
\
{
0
}
g\colon V\rightarrow \mathbb{R}^3\,\backslash\,\{ 0\}
g
:
V
→
R
3
\
{
0
}
such that
g
(
v
)
∈
f
(
v
)
g(v)\in f(v)
g
(
v
)
∈
f
(
v
)
for all
v
∈
V
v\in V
v
∈
V
?(translated by Miklós Maróti)
9
1
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Miklós Schweitzer 2008, Problem 9
For a given
α
>
0
\alpha >0
α
>
0
let us consider the regular, non-vanishing
f
(
z
)
f(z)
f
(
z
)
maps on the unit disc
{
∣
z
∣
<
1
}
\{ |z|<1 \}
{
∣
z
∣
<
1
}
for which
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
and
R
e
z
f
′
(
z
)
f
(
z
)
>
−
α
\mathrm{Re}\, z\frac{f'(z)}{f(z)}>-\alpha
Re
z
f
(
z
)
f
′
(
z
)
>
−
α
(
∣
z
∣
<
1
|z|<1
∣
z
∣
<
1
). Show that the range of
g
(
z
)
=
1
(
1
−
z
)
2
α
g(z)=\frac{1}{(1-z)^{2\alpha}}
g
(
z
)
=
(
1
−
z
)
2
α
1
contains the range of all other such functions. Here we consider that regular branch of
g
(
z
)
g(z)
g
(
z
)
for which
g
(
0
)
=
1
g(0)=1
g
(
0
)
=
1
.(translated by Miklós Maróti)
8
1
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Miklós Schweitzer 2008, Problem 8
Let
S
S
S
be the Sierpiński triangle. What can we say about the Hausdorff dimension of the elevation sets
f
−
1
(
y
)
f^{-1}(y)
f
−
1
(
y
)
for typical continuous real functions defined on
S
S
S
? (A property is satisfied for typical continuous real functions on
S
S
S
if the set of functions not having this property is of the first Baire category in the metric space of continuous
S
→
R
S\rightarrow\mathbb{R}
S
→
R
functions with the supremum norm.)(translated by Miklós Maróti)
7
1
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Miklós Schweitzer 2008, Problem 7
Let
f
:
R
1
→
R
2
f\colon \mathbb{R}^1\rightarrow \mathbb{R}^2
f
:
R
1
→
R
2
be a continuous function such that
f
(
x
)
=
f
(
x
+
1
)
f(x)=f(x+1)
f
(
x
)
=
f
(
x
+
1
)
for all
x
x
x
, and let
t
∈
[
0
,
1
4
]
t\in [0,\frac14]
t
∈
[
0
,
4
1
]
. Prove that there exists
x
∈
R
x\in\mathbb{R}
x
∈
R
such that the vector from
f
(
x
−
t
)
f(x-t)
f
(
x
−
t
)
to
f
(
x
+
t
)
f(x+t)
f
(
x
+
t
)
is perpendicular to the vector from
f
(
x
)
f(x)
f
(
x
)
to
f
(
x
+
1
2
)
f(x+\frac12)
f
(
x
+
2
1
)
.(translated by Miklós Maróti)
6
1
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Miklós Schweitzer 2008, Problem 6
Is it possible to draw circles on the plane so that every line intersects at least one of them but no more than
100
100
100
of them?
5
1
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Miklós Schweitzer 2008, Problem 5
Let
A
A
A
be an infinite subset of the set of natural numbers, and denote by
τ
A
(
n
)
\tau_A(n)
τ
A
(
n
)
the number of divisors of
n
n
n
in
A
A
A
. Construct a set
A
A
A
for which
∑
n
≤
x
τ
A
(
n
)
=
x
+
O
(
log
log
x
)
\sum_{n\le x}\tau_A(n)=x+O(\log\log x)
n
≤
x
∑
τ
A
(
n
)
=
x
+
O
(
lo
g
lo
g
x
)
and show that there is no set for which the error term is
o
(
log
log
x
)
o(\log\log x)
o
(
lo
g
lo
g
x
)
in the above formula.(translated by Miklós Maróti)
4
1
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Miklós Schweitzer 2008, Problem 4
Let
A
A
A
be a subgroup of the symmetric group
S
n
S_n
S
n
, and
G
G
G
be a normal subgroup of
A
A
A
. Show that if
G
G
G
is transitive, then
∣
A
:
G
∣
≤
5
n
−
1
|A\colon G|\le 5^{n-1}
∣
A
:
G
∣
≤
5
n
−
1
(translated by Miklós Maróti)
3
1
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Miklós Schweitzer 2008, Problem 3
A bipartite graph on the sets
{
x
1
,
…
,
x
n
}
\{ x_1,\ldots, x_n \}
{
x
1
,
…
,
x
n
}
and
{
y
1
,
…
,
y
n
}
\{ y_1,\ldots, y_n\}
{
y
1
,
…
,
y
n
}
of vertices (that is the edges are of the form
x
i
y
j
x_iy_j
x
i
y
j
) is called tame if it has no
x
i
y
j
x
k
y
l
x_iy_jx_ky_l
x
i
y
j
x
k
y
l
path (
i
,
j
,
k
,
l
∈
{
1
,
…
,
n
}
i,j,k,l\in\{ 1,\ldots, n\}
i
,
j
,
k
,
l
∈
{
1
,
…
,
n
}
) where
j
<
l
j<l
j
<
l
and
i
+
j
>
k
+
l
i+j>k+l
i
+
j
>
k
+
l
. Calculate the infimum of those real numbers
α
\alpha
α
for which there exists a constant
c
=
c
(
α
)
>
0
c=c(\alpha)>0
c
=
c
(
α
)
>
0
such that for all tame graphs
e
≤
c
n
α
e\le cn^{\alpha}
e
≤
c
n
α
, where
e
e
e
is the number of edges and
n
n
n
is half of the number of vertices.(translated by Miklós Maróti)
2
1
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Miklós Schweitzer 2008, Problem 2
Let
t
≥
3
t\ge 3
t
≥
3
be an integer, and for
1
≤
i
<
j
≤
t
1\le i <j\le t
1
≤
i
<
j
≤
t
let
A
i
j
=
A
j
i
A_{ij}=A_{ji}
A
ij
=
A
ji
be an arbitrary subset of an
n
n
n
-element set
X
X
X
. Prove that there exist
1
≤
i
<
j
≤
t
1\le i < j\le t
1
≤
i
<
j
≤
t
for which
∣
(
X
\
A
i
j
)
∪
⋃
k
≠
i
,
j
(
A
i
k
∩
A
j
k
)
∣
≥
t
−
2
2
t
−
2
n
\left| \left( X\,\backslash\, A_{ij}\right) \cup \bigcup_{k\neq i,j}\left( A_{ik}\cap A_{jk}\right) \right| \ge \frac{t-2}{2t-2}n
(
X
\
A
ij
)
∪
k
=
i
,
j
⋃
(
A
ik
∩
A
jk
)
≥
2
t
−
2
t
−
2
n
(translated by Miklós Maróti)
1
1
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Miklós Schweitzer 2008, Problem 1
Let
H
⊂
P
(
X
)
H \subset P(X)
H
⊂
P
(
X
)
be a system of subsets of
X
X
X
and
κ
>
0
\kappa > 0
κ
>
0
be a cardinal number such that every
x
∈
X
x \in X
x
∈
X
is contained in less than
κ
\kappa
κ
members of
H
H
H
. Prove that there exists an
f
:
X
→
κ
f \colon X \rightarrow \kappa
f
:
X
→
κ
coloring, such that every nonempty
A
∈
H
A \in H
A
∈
H
has a “unique” point, that is, an element
x
∈
A
x \in A
x
∈
A
such that
f
(
x
)
≠
f
(
y
)
f(x) \neq f(y)
f
(
x
)
=
f
(
y
)
for all
x
≠
y
∈
A
x \neq y \in A
x
=
y
∈
A
.(translated by Miklós Maróti)