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Miklós Schweitzer
1979 Miklós Schweitzer
1979 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
11
1
Hide problems
Miklos Schweitzer 1979_11
Let
{
ξ
k
ℓ
}
k
,
ℓ
=
1
∞
\{\xi_{k \ell} \}_{k,\ell=1}^{\infty}
{
ξ
k
ℓ
}
k
,
ℓ
=
1
∞
be a double sequence of random variables such that
E
(
ξ
i
j
ξ
k
ℓ
)
=
O
(
log
(
2
∣
i
−
k
∣
+
2
)
log
(
2
∣
j
−
ℓ
∣
+
2
)
2
)
(
i
,
j
,
k
,
ℓ
=
1
,
2
,
…
)
.
\Bbb{E}( \xi_{ij} \xi_{k\ell})= \mathcal{O} \left(\frac{ \log(2|i-k|+2)}{ \log(2|j-\ell|+2)^{2}}\right) \;\;\;(i,j,k,\ell =1,2, \ldots ) \\\ .
E
(
ξ
ij
ξ
k
ℓ
)
=
O
(
lo
g
(
2∣
j
−
ℓ
∣
+
2
)
2
lo
g
(
2∣
i
−
k
∣
+
2
)
)
(
i
,
j
,
k
,
ℓ
=
1
,
2
,
…
)
.
Prove that with probability one,
1
m
n
∑
k
=
1
m
∑
ℓ
=
1
n
ξ
k
ℓ
→
0
as
max
(
m
,
n
)
→
∞
.
\frac{1}{mn} \sum_{k=1}^m \sum_{\ell=1}^n \xi_{k\ell} \rightarrow 0 \;\;\textrm{as} \; \max (m,n)\rightarrow \infty\ \\ .
mn
1
k
=
1
∑
m
ℓ
=
1
∑
n
ξ
k
ℓ
→
0
as
max
(
m
,
n
)
→
∞
.
F. Moricz
10
1
Hide problems
Miklos Schweitzer 1979_10
Prove that if
a
i
(
i
=
1
,
2
,
3
,
4
)
a_i(i=1,2,3,4)
a
i
(
i
=
1
,
2
,
3
,
4
)
are positive constants,
a
2
−
a
4
>
2
a_2-a_4 > 2
a
2
−
a
4
>
2
, and
a
1
a
3
−
a
2
>
2
a_1a_3-a_2 > 2
a
1
a
3
−
a
2
>
2
, then the solution
(
x
(
t
)
,
y
(
t
)
)
(x(t),y(t))
(
x
(
t
)
,
y
(
t
))
of the system of differential equations
x
˙
=
a
1
−
a
2
x
+
a
3
x
y
,
\.{x}=a_1-a_2x+a_3xy,
x
˙
=
a
1
−
a
2
x
+
a
3
x
y
,
y
˙
=
a
4
x
−
y
−
a
3
x
y
(
x
,
y
∈
R
)
\.{y}=a_4x-y-a_3xy \;\;\;(x,y \in \mathbb{R})
y
˙
=
a
4
x
−
y
−
a
3
x
y
(
x
,
y
∈
R
)
with the initial conditions
x
(
0
)
=
0
,
y
(
0
)
≥
a
1
x(0)=0, y(0) \geq a_1
x
(
0
)
=
0
,
y
(
0
)
≥
a
1
is such that the function
x
(
t
)
x(t)
x
(
t
)
has exactly one strict local maximum on the interval
[
0
,
∞
)
[0, \infty)
[
0
,
∞
)
. L. Pinter, L. Hatvani
9
1
Hide problems
Miklos Schweitzer 1979_9
Let us assume that the series of holomorphic functions
∑
k
=
1
∞
f
k
(
z
)
\sum_{k=1}^{\infty}f_k(z)
∑
k
=
1
∞
f
k
(
z
)
is absolutely convergent for all
z
∈
C
z \in \mathbb{C}
z
∈
C
. Let
H
⊆
C
H \subseteq \mathbb{C}
H
⊆
C
be the set of those points where the above sum funcion is not regular. Prove that
H
H
H
is nowhere dense but not necessarily countable. L. Kerchy
8
1
Hide problems
Miklos Schweitzer 1979_8
Let
K
n
(
n
=
1
,
2
,
…
)
K_n(n=1,2,\ldots)
K
n
(
n
=
1
,
2
,
…
)
be periodical continuous functions of period
2
π
2 \pi
2
π
, and write
k
n
(
f
;
x
)
=
∫
0
2
π
f
(
t
)
K
n
(
x
−
t
)
d
t
.
k_n(f;x)= \int_0^{2\pi}f(t)K_n(x-t)dt .
k
n
(
f
;
x
)
=
∫
0
2
π
f
(
t
)
K
n
(
x
−
t
)
d
t
.
Prove that the following statements are equivalent: (i)
∫
0
2
π
∣
k
n
(
f
;
x
)
−
f
(
x
)
∣
d
x
→
0
(
n
→
∞
)
\int_0^{2\pi}|k_n(f;x)-f(x)|dx \rightarrow 0 \;(n \rightarrow \infty)
∫
0
2
π
∣
k
n
(
f
;
x
)
−
f
(
x
)
∣
d
x
→
0
(
n
→
∞
)
for all
f
∈
L
1
[
0
,
2
π
]
f \in \mathcal{L}_1[0,2 \pi]
f
∈
L
1
[
0
,
2
π
]
. (ii)
k
n
(
f
;
0
)
→
f
(
0
)
k_n(f;0) \rightarrow f(0)
k
n
(
f
;
0
)
→
f
(
0
)
for all continuous,
2
π
2 \pi
2
π
-periodic functions
f
f
f
. V. Totik
7
1
Hide problems
Miklos Schweitzer 1979_7
Let
T
T
T
be a triangulation of an
n
n
n
-dimensional sphere, and to each vertex of
T
T
T
let us assign a nonzero vector of a linear space
V
V
V
. Show that if
T
T
T
has an
n
n
n
-dimensional simplex such that the vectors assigned to the vertices of this simplex are linearly independent, then another such simplex must also exist. L. Lovasz
6
1
Hide problems
Miklos Schweitzer 1979_6
Let us defined a pseudo-Riemannian metric on the set of points of the Euclidean space
E
3
\mathbb{E}^3
E
3
not lying on the
z
z
z
-axis by the metric tensor \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \minus{}\sqrt{x^2\plus{}y^2} \\ \end{array} \right), where
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
is a Cartesian coordinate system
E
3
\mathbb{E}^3
E
3
. Show that the orthogonal projections of the geodesic curves of this Riemannian space onto the
(
x
,
y
)
(x,y)
(
x
,
y
)
-plane are straight lines or conic sections with focus at the origin P. Nagy
5
1
Hide problems
Miklos Schweitzer 1979_5
Give an example of ten different noncoplanar points
P
1
,
…
,
P
5
,
Q
1
,
…
,
Q
5
P_1,\ldots ,P_5,Q_1,\ldots ,Q_5
P
1
,
…
,
P
5
,
Q
1
,
…
,
Q
5
in
3
3
3
-space such that connecting each
P
i
P_i
P
i
to each
Q
j
Q_j
Q
j
by a rigid rod results in a rigid system. L. Lovasz
4
1
Hide problems
Miklos Schweitzer 1979_4
For what values of
n
n
n
does the group \textsl{SO}(n) of all orthogonal transformations of determinant
1
1
1
of the
n
n
n
-dimensional Euclidean space possess a closed regular subgroup?( \textsl{G}<\textsl{SO}(n) is called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
r
e
g
u
l
a
r
<
/
s
p
a
n
>
<span class='latex-italic'>regular</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
re
gu
l
a
r
<
/
s
p
an
>
if for any elements
x
,
y
x,y
x
,
y
of the unit sphere there exists a unique \varphi \in \textsl{G} such that \varphi(x)\equal{}y.) Z. Szabo
3
1
Hide problems
Miklos Schweitzer 1979_3
Let
g
(
n
,
k
)
g(n,k)
g
(
n
,
k
)
denote the number of strongly connected,
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
i
m
p
l
e
<
/
s
p
a
n
>
<span class='latex-italic'>simple</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
s
im
pl
e
<
/
s
p
an
>
directed graphs with
n
n
n
vertices and
k
k
k
edges. (
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
S
i
m
p
l
e
<
/
s
p
a
n
>
<span class='latex-italic'>Simple</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
S
im
pl
e
<
/
s
p
an
>
means no loops or multiple edges.) Show that
∑
k
=
n
n
2
−
n
(
−
1
)
k
g
(
n
,
k
)
=
(
n
−
1
)
!
.
\sum_{k=n}^{n^2-n}(-1)^kg(n,k)=(n-1)!.
k
=
n
∑
n
2
−
n
(
−
1
)
k
g
(
n
,
k
)
=
(
n
−
1
)!
.
A. A. Schrijver
2
1
Hide problems
Miklos Schweitzer 1979_2
Let
Γ
\Gamma
Γ
be a variety of monoids such that not all monoids of
Γ
\Gamma
Γ
are groups. Prove that if
A
∈
Γ
A \in \Gamma
A
∈
Γ
and
B
B
B
is a submonoid of
A
A
A
, there exist monoids
S
∈
Γ
S \in \Gamma
S
∈
Γ
and
C
C
C
and epimorphisms
φ
:
S
→
A
,
φ
1
:
S
→
C
\varphi : S \rightarrow A, \;\varphi_1 : S \rightarrow C
φ
:
S
→
A
,
φ
1
:
S
→
C
such that ((e)\varphi_1^{\minus{}1})\varphi\equal{}B (
e
e
e
is the identity element of
C
C
C
). L. Marki
1
1
Hide problems
Miklos Schweitzer 1979_1
Let the operation
f
f
f
of
k
k
k
variables defined on the set
{
1
,
2
,
…
,
n
}
\{ 1,2,\ldots,n \}
{
1
,
2
,
…
,
n
}
be called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
f
r
i
e
n
d
l
y
<
/
s
p
a
n
>
<span class='latex-italic'>friendly</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
f
r
i
e
n
d
l
y
<
/
s
p
an
>
toward the binary relation
ρ
\rho
ρ
defined on the same set if
f
(
a
1
,
a
2
,
…
,
a
k
)
ρ
f
(
b
1
,
b
2
,
…
,
b
k
)
f(a_1,a_2,\ldots,a_k) \;\rho\ \;f(b_1,b_2,\ldots,b_k)
f
(
a
1
,
a
2
,
…
,
a
k
)
ρ
f
(
b
1
,
b
2
,
…
,
b
k
)
implies
a
i
ρ
b
i
a_i \; \rho \ b_i
a
i
ρ
b
i
for at least one
i
,
1
≤
i
≤
k
i,1\leq i \leq k
i
,
1
≤
i
≤
k
. Show that if the operation
f
f
f
is friendly toward the relations "equal to" and "less than," then it is friendly toward all binary relations. B. Csakany