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Miklós Schweitzer
2012 Miklós Schweitzer
2012 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
11
1
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Miklós Schweitzer 2012 P11
Let
X
1
,
X
2
,
.
.
X_1,X_2,..
X
1
,
X
2
,
..
be independent random variables with the same distribution, and let
S
n
=
X
1
+
X
2
+
.
.
.
+
X
n
,
n
=
1
,
2
,
.
.
.
S_n=X_1+X_2+...+X_n, n=1,2,...
S
n
=
X
1
+
X
2
+
...
+
X
n
,
n
=
1
,
2
,
...
. For what real numbers
c
c
c
is the following statement true:
P
(
∣
S
2
n
2
n
−
c
∣
⩽
∣
S
n
n
−
c
∣
)
⩾
1
2
P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2}
P
(
2
n
S
2
n
−
c
⩽
n
S
n
−
c
)
⩾
2
1
10
1
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Miklós Schweitzer 2012 P10
Let
K
K
K
be a knot in the
3
3
3
-dimensional space (that is a differentiable injection of a circle into
R
3
\mathbb{R}^3
R
3
, and
D
D
D
be the diagram of the knot (that is the projection of it to a plane so that in exception of the transversal double points it is also an injection of a circle). Let us color the complement of
D
D
D
in black and color the diagram
D
D
D
in a chessboard pattern, black and white so that zones which share an arc in common are coloured differently. Define
Γ
B
(
D
)
\Gamma_B(D)
Γ
B
(
D
)
the black graph of the diagram, a graph which has vertices in the black areas and every two vertices which correspond to black areas which have a common point have an edge connecting them.[*]Determine all knots having a diagram
D
D
D
such that
Γ
B
(
D
)
\Gamma_B(D)
Γ
B
(
D
)
has at most
3
3
3
spanning trees. (Two knots are not considered distinct if they can be moved into each other with a one parameter set of the injection of the circle)[/*] [*]Prove that for any knot and any diagram
D
D
D
,
Γ
B
(
D
)
\Gamma_B(D)
Γ
B
(
D
)
has an odd number of spanning trees.[/*]
9
1
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Miklós Schweitzer 2012 P9
Let
D
D
D
be the complex unit disk
D
=
{
z
∈
C
:
∣
z
∣
<
1
}
D=\{z \in \mathbb{C}: |z|<1\}
D
=
{
z
∈
C
:
∣
z
∣
<
1
}
, and
0
<
a
<
1
0<a<1
0
<
a
<
1
a real number. Suppose that
f
:
D
→
C
∖
{
0
}
f:D \to \mathbb{C}\setminus \{0\}
f
:
D
→
C
∖
{
0
}
is a holomorphic function such that
f
(
a
)
=
1
f(a)=1
f
(
a
)
=
1
and
f
(
−
a
)
=
−
1
f(-a)=-1
f
(
−
a
)
=
−
1
. Prove that
sup
z
∈
D
∣
f
(
z
)
∣
⩾
exp
(
1
−
a
2
4
a
π
)
.
\sup_{z \in D} |f(z)| \geqslant \exp\left(\frac{1-a^2}{4a}\pi\right) .
z
∈
D
sup
∣
f
(
z
)
∣
⩾
exp
(
4
a
1
−
a
2
π
)
.
8
1
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Miklós Schweitzer 2012 P8
For any function
f
:
R
2
→
R
f: \mathbb{R}^2\to \mathbb{R}
f
:
R
2
→
R
consider the function
Φ
f
:
R
2
→
[
−
∞
,
∞
]
\Phi_f:\mathbb{R}^2\to [-\infty,\infty]
Φ
f
:
R
2
→
[
−
∞
,
∞
]
for which
Φ
f
(
x
,
y
)
=
lim sup
z
→
y
f
(
x
,
z
)
\Phi_f(x,y)=\limsup_{ z \to y} f(x,z)
Φ
f
(
x
,
y
)
=
lim
sup
z
→
y
f
(
x
,
z
)
for any
(
x
,
y
)
∈
R
2
(x,y) \in \mathbb{R}^2
(
x
,
y
)
∈
R
2
.[*]Is it true that if
f
f
f
is Lebesgue measurable then
Φ
f
\Phi_f
Φ
f
is also Lebesgue measurable?[/*] [*]Is it true that if
f
f
f
is Borel measurable then
Φ
f
\Phi_f
Φ
f
is also Borel measurable?[/*]
7
1
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Miklós Schweitzer 2012 P7
Let
Γ
\Gamma
Γ
be a simple curve, lying inside a circle of radius
r
r
r
, rectifiable and of length
ℓ
\ell
ℓ
. Prove that if
ℓ
>
k
r
π
\ell > kr\pi
ℓ
>
k
r
π
, then there exists a circle of radius
r
r
r
which intersects
Γ
\Gamma
Γ
in at least
k
+
1
k+1
k
+
1
distinct points.
6
1
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Miklós Schweitzer 2012 P6
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be matrices with complex elements such that
[
A
,
B
]
=
C
,
[
B
,
C
]
=
A
[A,B]=C, [B,C]=A
[
A
,
B
]
=
C
,
[
B
,
C
]
=
A
and
[
C
,
A
]
=
B
[C,A]=B
[
C
,
A
]
=
B
, where
[
X
,
Y
]
[X,Y]
[
X
,
Y
]
denotes the
X
Y
−
Y
X
XY-YX
X
Y
−
Y
X
commutator of the matrices. Prove that
e
4
π
A
e^{4 \pi A}
e
4
π
A
is the identity matrix.
5
1
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Miklós Schweitzer 2012 P5
Let
V
1
,
V
2
,
V
3
,
V
4
V_1,V_2,V_3,V_4
V
1
,
V
2
,
V
3
,
V
4
be four dimensional linear subspaces in
R
8
\mathbb{R}^8
R
8
such that the intersection of any two contains only the zero vector. Prove that there exists a linear four dimensional subspace
W
W
W
in
R
8
\mathbb{R}^8
R
8
such that all four vector spaces
W
∩
V
i
W\cap V_i
W
∩
V
i
are two dimensional.
4
1
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Miklós Schweitzer 2012 P4
Let
K
K
K
be a convex shape in the
n
n
n
dimensional space, having unit volume. Let
S
⊂
K
S \subset K
S
⊂
K
be a Lebesgue measurable set with measure at least
1
−
ε
1-\varepsilon
1
−
ε
, where
0
<
ε
<
1
/
3
0<\varepsilon<1/3
0
<
ε
<
1/3
. Prove that dilating
K
K
K
from its centroid by the ratio of
2
ε
ln
(
1
/
ε
)
2\varepsilon \ln(1/\varepsilon)
2
ε
ln
(
1/
ε
)
, the shape obtained contains the centroid of
S
S
S
.
2
1
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Miklós Schweitzer 2012 P2
Call a subset
A
A
A
of the cyclic group
(
Z
n
,
+
)
(\mathbb{Z}_n,+)
(
Z
n
,
+
)
rich if for all
x
,
y
∈
Z
n
x,y \in \mathbb{Z}_n
x
,
y
∈
Z
n
there exists
r
∈
Z
n
r \in \mathbb{Z}_n
r
∈
Z
n
such that
x
−
r
,
x
+
r
,
y
−
r
,
y
+
r
x-r,x+r,y-r,y+r
x
−
r
,
x
+
r
,
y
−
r
,
y
+
r
are all in
A
A
A
. For what
α
\alpha
α
is there a constant
C
α
>
0
C_\alpha>0
C
α
>
0
such that for each odd positive integer
n
n
n
, every rich subset
A
⊂
Z
n
A \subset \mathbb{Z}_n
A
⊂
Z
n
has at least
C
α
n
α
C_\alpha n^\alpha
C
α
n
α
elements?
1
1
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Miklós Schweitzer 2012 P1
Is there any real number
α
\alpha
α
for which there exist two functions
f
,
g
:
N
→
N
f,g: \mathbb{N} \to \mathbb{N}
f
,
g
:
N
→
N
such that
α
=
lim
n
→
∞
f
(
n
)
g
(
n
)
,
\alpha=\lim_{n \to \infty} \frac{f(n)}{g(n)},
α
=
n
→
∞
lim
g
(
n
)
f
(
n
)
,
but the function which associates to
n
n
n
the
n
n
n
-th decimal digit of
α
\alpha
α
is not recursive?
3
1
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Tree with k vertices
There is a simple graph which chromatic number is equal to
k
k
k
. We painted all of the edges of graph using two colors. Prove that there exist a monochromatic tree with
k
k
k
vertices