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Miklós Schweitzer
2004 Miklós Schweitzer
2004 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklós Schweitzer 2004, Problem 10
Let
N
p
\mathcal{N}_p
N
p
stand for a
p
p
p
dimensional random variable of standard normal distribution. For
a
∈
R
p
a\in\mathbb{R}^p
a
∈
R
p
, let
H
p
(
a
)
H_p(a)
H
p
(
a
)
stand for the expectation
E
∣
N
p
+
a
∣
E|\mathcal{N}_p+a|
E
∣
N
p
+
a
∣
. For
p
>
1
p>1
p
>
1
, prove that
H
p
(
a
)
=
(
p
−
1
)
∫
0
∞
H
1
(
∣
a
∣
r
2
+
1
)
r
p
−
2
(
r
2
+
1
)
p
d
r
H_p(a)=(p-1)\int_0^{\infty} H_1\left( \frac{|a|}{\sqrt{r^2+1}}\right) \frac{r^{p-2}}{\sqrt{(r^2+1)^p}} \mathrm{d}r
H
p
(
a
)
=
(
p
−
1
)
∫
0
∞
H
1
(
r
2
+
1
∣
a
∣
)
(
r
2
+
1
)
p
r
p
−
2
d
r
9
1
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Miklós Schweitzer 2004, Problem 9
Let
F
F
F
be a smooth (i.e.
C
∞
C^{\infty}
C
∞
) closed surface. Call a continuous map
f
:
F
→
R
2
f\colon F\rightarrow \mathbb{R}^2
f
:
F
→
R
2
an almost-immersion if there exists a smooth closed embedded curve
γ
\gamma
γ
(possibly disconnected) in
F
F
F
such that
f
f
f
is smooth and of maximal rank (i.e., rank 2) on
F
\
γ
F\backslash \gamma
F
\
γ
and each point
p
∈
γ
p\in\gamma
p
∈
γ
admits local coordinate charts
(
x
,
y
)
(x,y)
(
x
,
y
)
and
(
u
,
v
)
(u,v)
(
u
,
v
)
about
p
p
p
and
f
(
p
)
f(p)
f
(
p
)
, respectively, such taht the coordinates of
p
p
p
and
f
(
p
)
f(p)
f
(
p
)
are zero and the map
f
f
f
is given by
(
x
,
y
)
→
(
u
,
v
)
,
u
=
∣
x
∣
,
v
=
y
(x,y)\rightarrow (u,v), u=|x|, v=y
(
x
,
y
)
→
(
u
,
v
)
,
u
=
∣
x
∣
,
v
=
y
. Determine the genera of those smooth, closed, connected, orientable surfaces
F
F
F
that admit an almost-immersion in the plane with the curve
γ
\gamma
γ
having a given positive number
n
n
n
of connected components.
8
1
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Miklós Schweitzer 2004, Problem 8
Prove that for any
0
<
δ
<
2
π
0<\delta <2\pi
0
<
δ
<
2
π
there exists a number
m
>
1
m>1
m
>
1
such that for any positive integer
n
n
n
and unimodular complex numbers
z
1
,
…
,
z
n
z_1,\ldots, z_n
z
1
,
…
,
z
n
with
z
1
v
+
⋯
+
z
n
v
=
0
z_1^v+\dots+z_n^v=0
z
1
v
+
⋯
+
z
n
v
=
0
for all integer exponents
1
≤
v
≤
m
1\le v\le m
1
≤
v
≤
m
, any arc of length
δ
\delta
δ
of the unit circle contains at least one of the numbers
z
1
,
…
,
z
n
z_1,\ldots, z_n
z
1
,
…
,
z
n
.
7
1
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Miklós Schweitzer 2004, Problem 7
Suppose that the closed subset
K
K
K
of the sphere
S
2
=
{
(
x
,
y
,
z
)
∈
R
3
:
x
2
+
y
2
+
z
2
=
1
}
S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}
S
2
=
{(
x
,
y
,
z
)
∈
R
3
:
x
2
+
y
2
+
z
2
=
1
}
is symmetric with respect to the origin and separates any two antipodal points in
S
2
\
K
S^2 \backslash K
S
2
\
K
. Prove that for any positive
ε
\varepsilon
ε
there exists a homogeneous polynomial
P
P
P
of odd degree such that the Hausdorff distance between
Z
(
P
)
=
{
(
x
,
y
,
z
)
∈
S
2
:
P
(
x
,
y
,
z
)
=
0
}
Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}
Z
(
P
)
=
{(
x
,
y
,
z
)
∈
S
2
:
P
(
x
,
y
,
z
)
=
0
}
and
K
K
K
is less than
ε
\varepsilon
ε
.
6
1
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Miklós Schweitzer 2004, Problem 6
Is is true that if the perfect set
F
⊆
[
0
,
1
]
F\subseteq [0,1]
F
⊆
[
0
,
1
]
is of zero Lebesgue measure then those functions in
C
1
[
0
,
1
]
C^1[0,1]
C
1
[
0
,
1
]
which are one-to-one on
F
F
F
form a dense subset of
C
1
[
0
,
1
]
C^1[0,1]
C
1
[
0
,
1
]
? (We use the metric
d
(
f
,
g
)
=
sup
x
∈
[
0
,
1
]
∣
f
(
x
)
−
g
(
x
)
∣
+
sup
x
∈
[
0
,
1
]
∣
f
′
(
x
)
−
g
′
(
x
)
∣
d(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)| + \sup_{x\in[0,1]} |f'(x)-g'(x)|
d
(
f
,
g
)
=
x
∈
[
0
,
1
]
sup
∣
f
(
x
)
−
g
(
x
)
∣
+
x
∈
[
0
,
1
]
sup
∣
f
′
(
x
)
−
g
′
(
x
)
∣
to define the topology in the space
C
1
[
0
,
1
]
C^1[0,1]
C
1
[
0
,
1
]
of continuously differentiable real functions on
[
0
,
1
]
[0,1]
[
0
,
1
]
.)
5
1
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Miklós Schweitzer 2004, Problem 5
Let
G
G
G
be a non-solvable finite group and let
ε
>
0
\varepsilon > 0
ε
>
0
. Show that there exist a positive integer
k
k
k
and a word
w
∈
F
k
w\in F_k
w
∈
F
k
such that
w
w
w
assumes the value
1
1
1
with probability less than
ε
\varepsilon
ε
when its
k
k
k
arguments are considered to be independent and uniformly distributed random variables with values in
G
G
G
. (We write
F
k
F_k
F
k
for the free group generated by
k
k
k
elements.)
4
1
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Miklós Schweitzer 2004, Problem 4
Determine all totally multiplicative and non-negative functions
f
:
Z
→
Z
f\colon\mathbb{Z}\rightarrow \mathbb{Z}
f
:
Z
→
Z
with the property that if
a
,
b
∈
Z
a, b\in \mathbb{Z}
a
,
b
∈
Z
and
b
≠
0
b\neq 0
b
=
0
, then there exist integers
q
q
q
and
r
r
r
such that
a
−
q
b
+
r
a-qb+r
a
−
q
b
+
r
and
f
(
r
)
<
f
(
b
)
f(r)<f(b)
f
(
r
)
<
f
(
b
)
.
3
1
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Miklós Schweitzer 2004, Problem 3
Prove that there is a constant
c
>
0
c>0
c
>
0
such that for any
n
>
3
n>3
n
>
3
there exists a planar graph
G
G
G
with
n
n
n
vertices such that every straight-edged plane embedding of
G
G
G
has a pair of edges with ratio of lengths at least
c
n
cn
c
n
.
2
1
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Miklós Schweitzer 2004, Problem 2
Write
t
(
G
)
t(G)
t
(
G
)
for the number of complete quadrilaterals in the graph
G
G
G
and
e
G
(
S
)
e_G(S)
e
G
(
S
)
for the number of edges spanned by a subset
S
S
S
of vertices of
G
G
G
. Let
G
1
,
G
2
G_1, G_2
G
1
,
G
2
be two (simple) graphs on a common underlying set
V
V
V
of vertices,
∣
V
∣
−
n
|V|-n
∣
V
∣
−
n
, and assume that
∣
e
G
1
(
S
)
−
e
G
2
(
S
)
∣
<
n
2
1000
|e_{G_1}(S)-e_{G_2}(S)|<\frac{n^2}{1000}
∣
e
G
1
(
S
)
−
e
G
2
(
S
)
∣
<
1000
n
2
holds for any subset
S
⊆
V
S\subseteq V
S
⊆
V
. Prove that
∣
t
(
G
1
)
−
t
(
G
2
)
∣
≤
n
4
1000
|t(G_1)-t(G_2)|\le \frac{n^4}{1000}
∣
t
(
G
1
)
−
t
(
G
2
)
∣
≤
1000
n
4
.
1
1
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Miklós Schweitzer 2004, Problem 1
The Lindelöf number
L
(
X
)
L(X)
L
(
X
)
of a topological space
X
X
X
is the least infinite cardinal
λ
\lambda
λ
with the property that every open covering of
X
X
X
has a subcovering of cardinality at most
λ
\lambda
λ
. Prove that if evert non-countably infinite subset of a first countable space
X
X
X
has a point of condensation, then
L
(
X
)
=
sup
L
(
A
)
L(X)=\sup L(A)
L
(
X
)
=
sup
L
(
A
)
, where
A
A
A
runs over the separable closed subspaces of
X
X
X
. (A point of condensation of a subset
H
⊆
X
H\subseteq X
H
⊆
X
is a point
x
∈
X
x\in X
x
∈
X
such that any neighbourhood of
x
x
x
intersects
H
H
H
in a non-countably infinite set.)