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Miklós Schweitzer
1982 Miklós Schweitzer
1982 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklos Schweitzer 1982_10
Let
p
0
,
p
1
,
…
p_0,p_1,\ldots
p
0
,
p
1
,
…
be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let
A
i
A_i
A
i
denote the event that the number
i
i
i
has been selected and that it is in the same place in both lines. Prove that the events A_i \;(i\equal{}1,2,\ldots) are mutually independent, and P(A_i)\equal{}p_i. T. F. Mori
9
1
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Miklos Schweitzer 1982_9
Suppose that
K
K
K
is a compact Hausdorff space and K\equal{} \cup_{n\equal{}0}^{\infty}A_n, where
A
n
A_n
A
n
is metrizable and
A
n
⊂
A
m
A_n \subset A_m
A
n
⊂
A
m
for
n
<
m
n<m
n
<
m
. Prove that
K
K
K
is metrizable. Z. Balogh
8
1
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Miklos Schweitzer 1982_8
Show that for any natural number
n
n
n
and any real number d > 3^n / (3^n\minus{}1), one can find a covering of the unit square with
n
n
n
homothetic triangles with area of the union less than
d
d
d
.
7
1
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Miklos Schweitzer 1982_7
Let
V
V
V
be a bounded, closed, convex set in
R
n
\mathbb{R}^n
R
n
, and denote by
r
r
r
the radius of its circumscribed sphere (that is, the radius of the smallest sphere that contains
V
V
V
). Show that
r
r
r
is the only real number with the following property: for any finite number of points in
V
V
V
, there exists a point in
V
V
V
such that the arithmetic mean of its distances from the other points is equal to
r
r
r
. Gy. Szekeres
6
1
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Miklos Schweitzer 1982_6
For every positive
α
\alpha
α
, natural number
n
n
n
, and at most
α
n
\alpha n
α
n
points
x
i
x_i
x
i
, construct a trigonometric polynomial
P
(
x
)
P(x)
P
(
x
)
of degree at most
n
n
n
for which
P
(
x
i
)
≤
1
,
∫
0
2
π
P
(
x
)
d
x
=
0
,
and
max
P
(
x
)
>
c
n
,
P(x_i) \leq 1, \; \int_0^{2 \pi} P(x)dx=0,\ \; \textrm{and}\ \; \max P(x) > cn\ ,
P
(
x
i
)
≤
1
,
∫
0
2
π
P
(
x
)
d
x
=
0
,
and
max
P
(
x
)
>
c
n
,
where the constant
c
c
c
depends only on
α
\alpha
α
. G. Halasz
5
1
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Miklos Schweitzer 1982_5
Find a perfect set
H
⊂
[
0
,
1
]
H \subset [0,1]
H
⊂
[
0
,
1
]
of positive measure and a continuous function
f
f
f
defined on
[
0
,
1
]
[0,1]
[
0
,
1
]
such that for any twice differentiable function
g
g
g
defined on
[
0
,
1
]
[0,1]
[
0
,
1
]
, the set \{ x \in H : \;f(x)\equal{}g(x)\ \} is finite. M. Laczkovich
4
1
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Miklos Schweitzer 1982_4
Let
f
(
n
)
=
∑
p
∣
n
,
p
α
≤
n
<
p
α
+
1
p
α
.
f(n)= \sum_{p|n , \;p^{\alpha} \leq n < p^{\alpha+1} \ } p^{\alpha} .
f
(
n
)
=
p
∣
n
,
p
α
≤
n
<
p
α
+
1
∑
p
α
.
Prove that
lim sup
n
→
∞
f
(
n
)
log
log
n
n
log
n
=
1.
\limsup_{n \rightarrow \infty}f(n) \frac{ \log \log n}{n \log n}=1 .
n
→
∞
lim
sup
f
(
n
)
n
lo
g
n
lo
g
lo
g
n
=
1.
P. Erdos
3
1
Hide problems
Miklos Schweitzer 1982_3
Let
G
(
V
,
E
)
G(V,E)
G
(
V
,
E
)
be a connected graph, and let
d
G
(
x
,
y
)
d_G(x,y)
d
G
(
x
,
y
)
denote the length of the shortest path joining
x
x
x
and
y
y
y
in
G
G
G
. Let r_G(x)\equal{} \max \{ d_G(x,y) : \; y \in V \ \} for
x
∈
V
x \in V
x
∈
V
, and let r(G)\equal{} \min \{ r_G(x) : \;x \in V\ \}. Show that if
r
(
G
)
≥
2
r(G) \geq 2
r
(
G
)
≥
2
, then
G
G
G
contains a path of length 2r(G)\minus{}2 as an induced subgraph. V. T. Sos
2
1
Hide problems
Miklos Schweitzer 1982_2
Consider the lattice of all algebraically closed subfields of the complex field
C
\mathbb{C}
C
whose transcendency degree (over
Q
\mathbb{Q}
Q
) is finite. Prove that this lattice is not modular. L. Babai
1
1
Hide problems
Miklos Schweitzer 1982_1
A map
F
:
P
(
X
)
→
P
(
X
)
F : P(X) \rightarrow P(X)
F
:
P
(
X
)
→
P
(
X
)
, where
P
(
X
)
P(X)
P
(
X
)
denotes the set of all subsets of
X
X
X
, is called a
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
l
o
s
u
r
e
o
p
e
r
a
t
i
o
n
<
/
s
p
a
n
>
<span class='latex-italic'>closure operation</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
c
l
os
u
reo
p
er
a
t
i
o
n
<
/
s
p
an
>
on
X
X
X
if for arbitrary
A
,
B
⊂
X
A,B \subset X
A
,
B
⊂
X
, the following conditions hold: (i)
A
⊂
F
(
A
)
;
A \subset F(A);
A
⊂
F
(
A
)
;
(ii)
A
⊂
B
⇒
F
(
A
)
⊂
F
(
B
)
;
A \subset B \Rightarrow F(A) \subset F(B);
A
⊂
B
⇒
F
(
A
)
⊂
F
(
B
)
;
(iii) F(F(A))\equal{}F(A). The cardinal number \min \{ |A| : \;A \subset X\ ,\;F(A)\equal{}X\ \} is called the
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
d
e
n
s
i
t
y
<
/
s
p
a
n
>
<span class='latex-italic'>density</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
d
e
n
s
i
t
y
<
/
s
p
an
>
of
F
F
F
and is denoted by
d
(
F
)
d(F)
d
(
F
)
. A set
H
⊂
X
H \subset X
H
⊂
X
is called
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
d
i
s
c
r
e
t
e
<
/
s
p
a
n
>
<span class='latex-italic'>discrete</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
d
i
scre
t
e
<
/
s
p
an
>
with respect to
F
F
F
if u \not \in F(H\minus{}\{ u \}) holds for all
u
∈
H
u \in H
u
∈
H
. Prove that if the density of the closure operation
F
F
F
is a singular cardinal number, then for any nonnegative integer
n
n
n
, there exists a set of size
n
n
n
that is discrete with respect to
F
F
F
. Show that the statement is not true when the existence of an infinite discrete subset is required, even if
F
F
F
is the closure operation of a topological space satisfying the
T
1
T_1
T
1
separation axiom. A. Hajnal