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Miklós Schweitzer
1985 Miklós Schweitzer
1985 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(12)
12
1
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Miklós Schweitzer 1985, Problem 12
Let
(
Ω
,
A
,
P
)
(\Omega, \mathcal A, P)
(
Ω
,
A
,
P
)
be a probability space, and let
(
X
n
,
F
n
)
(X_n, \mathcal F_n)
(
X
n
,
F
n
)
be an adapted sequence in
(
Ω
,
A
,
P
)
(\Omega, \mathcal A, P)
(
Ω
,
A
,
P
)
(that is, for the
σ
\sigma
σ
-algebras
F
n
\mathcal F_n
F
n
, we have
F
1
⊆
F
2
⊆
⋯
⊆
A
\mathcal F_1\subseteq \mathcal F_2\subseteq \dots \subseteq \mathcal A
F
1
⊆
F
2
⊆
⋯
⊆
A
, and for all
n
n
n
,
X
n
X_n
X
n
is an
F
n
\mathcal F_n
F
n
-measurable and integrable random variable). Assume that
E
(
X
n
+
1
∣
F
n
)
=
1
2
X
n
+
1
2
X
n
−
1
(
n
=
2
,
3
,
…
)
\mathrm E (X_{n+1} \mid \mathcal F_n )=\frac12 X_n+\frac12 X_{n-1}\,\,\,\,\, (n=2, 3, \ldots )
E
(
X
n
+
1
∣
F
n
)
=
2
1
X
n
+
2
1
X
n
−
1
(
n
=
2
,
3
,
…
)
Prove that
s
u
p
n
E
∣
X
n
∣
<
∞
\mathrm{sup}_n \mathrm{E}|X_n|<\infty
sup
n
E
∣
X
n
∣
<
∞
implies that
X
n
X_n
X
n
converges with probability one as
n
→
∞
n\to\infty
n
→
∞
. [I. Fazekas]
11
1
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Miklós Schweitzer 1985, Problem 11
Let
ξ
(
E
,
π
,
B
)
(
π
:
E
→
B
)
\xi (E, \pi, B)\, (\pi\colon E\rightarrow B)
ξ
(
E
,
π
,
B
)
(
π
:
E
→
B
)
be a real vector bundle of finite rank, and let
τ
E
=
V
ξ
⊕
H
ξ
(
∗
)
\tau_E=V\xi \oplus H\xi\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (*)
τ
E
=
V
ξ
⊕
H
ξ
(
∗
)
be the tangent bundle of
E
E
E
, where
V
ξ
=
K
e
r
d
π
V\xi=\mathrm{Ker}\, d\pi
V
ξ
=
Ker
d
π
is the vertical subbundle of
τ
E
\tau_E
τ
E
. Let us denote the projection operators corresponding to the splitting
(
∗
)
(*)
(
∗
)
by
v
v
v
and
h
h
h
. Construct a linear connection
∇
\nabla
∇
on
V
ξ
V\xi
V
ξ
such that
∇
X
∨
Y
−
∇
Y
∨
X
=
v
[
X
,
Y
]
−
v
[
h
X
,
h
Y
]
\nabla_X\lor Y - \nabla_Y \lor X=v[X,Y] - v[hX,hY]
∇
X
∨
Y
−
∇
Y
∨
X
=
v
[
X
,
Y
]
−
v
[
h
X
,
hY
]
(
X
X
X
and
Y
Y
Y
are vector fields on
E
E
E
,
[
.
,
.
]
[.,\, .]
[
.
,
.
]
is the Lie bracket, and all data are of class
C
∞
\mathcal C^\infty
C
∞
. [J. Szilasi]
10
1
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Miklós Schweitzer 1985, Problem 10
Show that any two intervals
A
,
B
⊆
R
A, B\subseteq \mathbb R
A
,
B
⊆
R
of positive lengths can be countably disected into each other, that is, they can be written as countable unions
A
=
A
1
∪
A
2
∪
…
A=A_1\cup A_2\cup\ldots\,
A
=
A
1
∪
A
2
∪
…
and
B
=
B
1
∪
B
2
∪
…
B=B_1\cup B_2\cup\ldots\,
B
=
B
1
∪
B
2
∪
…
of pairwise disjoint sets, where
A
i
A_i
A
i
and
B
i
B_i
B
i
are congruent for every
i
∈
N
i\in \mathbb N
i
∈
N
[Gy. Szabo]
9
1
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Miklós Schweitzer 1985, Problem 9
Let
D
=
{
z
∈
C
:
∣
z
∣
<
1
}
D=\{ z\in \mathbb C\colon |z|<1\}
D
=
{
z
∈
C
:
∣
z
∣
<
1
}
and
D
=
{
w
∈
C
:
∣
w
∣
=
1
}
D=\{ w\in \mathbb C \colon |w|=1\}
D
=
{
w
∈
C
:
∣
w
∣
=
1
}
. Prove that if for a function
f
:
D
×
B
→
C
f\colon D\times B\rightarrow\mathbb C
f
:
D
×
B
→
C
the equality
f
(
a
z
+
b
b
‾
z
+
a
‾
,
a
w
+
b
b
‾
w
+
a
‾
)
=
f
(
z
,
w
)
+
f
(
b
a
‾
,
a
w
+
b
b
‾
w
+
a
‾
)
f\left( \frac{az+b}{\overline{b}z+\overline{a}}, \frac{aw+b}{\overline{b}w+\overline a} \right)=f(z,w)+f\left(\frac{b}{\overline a}, \frac{aw+b}{\overline b w+\overline a} \right)
f
(
b
z
+
a
a
z
+
b
,
b
w
+
a
a
w
+
b
)
=
f
(
z
,
w
)
+
f
(
a
b
,
b
w
+
a
a
w
+
b
)
holds for all
z
∈
D
,
w
∈
B
z\in D, w\in B
z
∈
D
,
w
∈
B
and
a
,
b
∈
C
,
∣
a
∣
2
=
∣
b
∣
2
+
1
a, b\in \mathbb C,|a|^2=|b|^2+1
a
,
b
∈
C
,
∣
a
∣
2
=
∣
b
∣
2
+
1
, then there is a function
L
:
(
0
,
∞
)
→
C
L\colon (0, \infty )\rightarrow \mathbb C
L
:
(
0
,
∞
)
→
C
satisfying
L
(
p
q
)
=
L
(
p
)
+
L
(
q
)
for all
p
,
q
>
0
L(pq)=L(p)+L(q)\,\,\,\text{for all}\,\,\, p,q > 0
L
(
pq
)
=
L
(
p
)
+
L
(
q
)
for all
p
,
q
>
0
such that
f
f
f
can be represented as
f
(
z
,
w
)
=
L
(
1
−
∣
z
∣
2
∣
w
−
z
∣
2
)
for all
z
∈
D
,
w
∈
B
f(z,w)=L\left( \frac{1-|z|^2}{|w-z|^2}\right)\,\,\,\text{for all}\,\,\, z\in D, w\in B
f
(
z
,
w
)
=
L
(
∣
w
−
z
∣
2
1
−
∣
z
∣
2
)
for all
z
∈
D
,
w
∈
B
. [Gy. Maksa]
8
1
Hide problems
Miklós Schweitzer 1985, Problem 8
Let
2
5
+
1
≤
p
<
1
\frac{2}{\sqrt5+1}\leq p < 1
5
+
1
2
≤
p
<
1
, and let the real sequence
{
a
n
}
\{ a_n \}
{
a
n
}
have the following property: for every sequence
{
e
n
}
\{ e_n \}
{
e
n
}
of
0
0
0
's and
±
1
\pm 1
±
1
's for which
∑
n
=
1
∞
e
n
p
n
=
0
\sum_{n=1}^\infty e_np^n=0
∑
n
=
1
∞
e
n
p
n
=
0
, we also have
∑
n
=
1
∞
e
n
a
n
=
0
\sum_{n=1}^\infty e_na_n=0
∑
n
=
1
∞
e
n
a
n
=
0
. Prove that there is a number
c
c
c
such that
a
n
=
c
p
n
a_n=cp^n
a
n
=
c
p
n
for all
n
n
n
. [Z. Daroczy, I. Katai]
7
1
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Miklós Schweitzer 1985, Problem 7
Let
p
1
p_1
p
1
and
p
2
p_2
p
2
be positive real numbers. Prove that there exist functions
f
i
:
R
→
R
f_i\colon \mathbb R \rightarrow \mathbb R
f
i
:
R
→
R
such that the smallest positive period of
f
i
f_i
f
i
is
p
i
(
i
=
1
,
2
)
p_i\, (i=1, 2)
p
i
(
i
=
1
,
2
)
, and
f
1
−
f
2
f_1-f_2
f
1
−
f
2
is also periodic. [J. Riman]
6
1
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Miklós Schweitzer 1985, Problem 6
Determine all finite groups
G
G
G
that have an automorphism
f
f
f
such that
H
⊈
f
(
H
)
H\not\subseteq f(H)
H
⊆
f
(
H
)
for all proper subgroups
H
H
H
of
G
G
G
. [B. Kovacs]
5
1
Hide problems
Miklós Schweitzer 1985, Problem 5
Let
F
(
x
,
y
)
F(x,y)
F
(
x
,
y
)
and
G
(
x
,
y
)
G(x,y)
G
(
x
,
y
)
be relatively prime homogeneous polynomials of degree at least one having integer coefficients. Prove that there exists a number
c
c
c
depending only on the degrees and the maximum of the absolute values of the coefficients of
F
F
F
and
G
G
G
such that
F
(
x
,
y
)
≠
G
(
x
,
y
)
F(x,y)\neq G(x,y)
F
(
x
,
y
)
=
G
(
x
,
y
)
for any integers
x
x
x
and
y
y
y
that are relatively prime and satisfy
max
{
∣
x
∣
,
∣
y
∣
}
>
c
\max \{ |x|,|y|\} > c
max
{
∣
x
∣
,
∣
y
∣
}
>
c
. [K. Gyory]
4
1
Hide problems
Miklós Schweitzer 1985- Problem 4
4. Call a subset
S
S
S
of the set
{
1
,
…
,
n
}
\{1,\dots,n\}
{
1
,
…
,
n
}
exceptional if any pair of distinct elements of
S
S
S
are coprime. Consider an exceptional set with a maximal sum of elements (among all exceptional sets for a fixed
n
n
n
). Prove that if
n
n
n
is sufficiently large, then each element of
S
S
S
has at most two distinct prime divisors. (N.17) [P. Erdos]
3
1
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Miklós Schweitzer 1985- Problem 3
3. Let
k
k
k
and
K
K
K
be concentric circles on the plane, and let
k
k
k
be contained inside
K
K
K
. Assume that
k
k
k
is covered by a finite system of convex angular domains with vertices on
K
K
K
. Prove that the sum of the angles of the domains is not less than the angle under which
k
k
k
can be seen from a point of
K
K
K
. (G.38) [Zs.. Páles]
2
1
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Miklós Schweitzer 1985- Problem 2
2. Let
S
S
S
be a given finite set of hyperplanes in
R
n
\mathbb{R}^n
R
n
, and let
O
O
O
be a point. Show that there exists a compact set
K
⊆
R
n
K \subseteq \mathbb{R}^n
K
⊆
R
n
containing
O
O
O
such that the orthogonal projection of any point of
K
K
K
onto any hyperplane in
S
S
S
is also in
K
K
K
. (G.37) [Gy. Pap]
1
1
Hide problems
Miklós Schweitzer 1985- Problem 1
1. Some proper partitions
P
1
,
…
,
P
n
P_1, \dots , P_n
P
1
,
…
,
P
n
of a finite set
S
S
S
(that is, partitions containing at least two parts) are called independent if no matter how we choose one class from each partition, the intersection of the chosen classes is nonempty. Show that if the inequality\frac{\left | S \right | }{2} < \left |P_1 \right | \dots \left |P_n \right |\qquad (*)holds for some independent partitions, then
P
1
,
…
,
P
n
P_1, \dots , P_n
P
1
,
…
,
P
n
is maximal in the sense that there is no partition
P
P
P
such that
P
,
P
1
,
…
,
P
n
P,P_1, \dots , P_n
P
,
P
1
,
…
,
P
n
are independent. On the other hand, show that inequality
(
∗
)
(*)
(
∗
)
is not necessary for this maximality. (C.20) [E. Gesztelyi]