MathDB
Problems
Contests
Undergraduate contests
Miklós Schweitzer
1976 Miklós Schweitzer
1976 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
11
1
Hide problems
Miklos Schweitzer 1976_11
Let
ξ
1
,
ξ
2
,
.
.
.
\xi_1,\xi_2,...
ξ
1
,
ξ
2
,
...
be independent, identically distributed random variables with distribution
P
(
ξ
1
=
−
1
)
=
P
(
ξ
1
=
1
)
=
1
2
.
P(\xi_1=-1)=P(\xi_1=1)=\frac 12 .
P
(
ξ
1
=
−
1
)
=
P
(
ξ
1
=
1
)
=
2
1
.
Write
S
n
=
ξ
1
+
ξ
2
+
.
.
.
+
ξ
n
(
n
=
1
,
2
,
.
.
.
)
,
S
0
=
0
,
S_n=\xi_1+\xi_2+...+\xi_n \;(n=1,2,...),\ \;S_0=0\ ,
S
n
=
ξ
1
+
ξ
2
+
...
+
ξ
n
(
n
=
1
,
2
,
...
)
,
S
0
=
0
,
and
T
n
=
1
n
max
0
≤
k
≤
n
S
k
.
T_n= \frac{1}{\sqrt{n}} \max _{ 0 \leq k \leq n}S_k .
T
n
=
n
1
0
≤
k
≤
n
max
S
k
.
Prove that
lim inf
n
→
∞
(
log
n
)
T
n
=
0
\liminf_{n \rightarrow \infty} (\log n)T_n=0
lim
inf
n
→
∞
(
lo
g
n
)
T
n
=
0
with probability one. P. Revesz
10
1
Hide problems
Miklos Schweitzer 1976_10
Suppose that
τ
\tau
τ
is a metrizable topology on a set
X
X
X
of cardinality less than or equal to continuum. Prove that there exists a separable and metrizable topology on
X
X
X
that is coarser that
τ
\tau
τ
. L. Juhasz
9
1
Hide problems
Miklos Schweitzer 1976_9
Let
D
D
D
be a convex subset of the
n
n
n
-dimensional space, and suppose that
D
′
D'
D
′
is obtained from
D
D
D
by applying a positive central dilatation and then a translation. Suppose also that the sum of the volumes of
D
D
D
and
D
′
D'
D
′
is
1
1
1
, and D \cap D'\not\equal{} \emptyset . Determine the supremum of the volume of the convex hull of
D
∪
D
′
D \cup D'
D
∪
D
′
taken for all such pairs of sets
D
,
D
′
D,D'
D
,
D
′
. L. Fejes-Toth, E. Makai
8
1
Hide problems
Miklos Schweitzer 1976_8
Prove that the set of all linearly combinations (with real coefficients) of the system of polynomials \{ x^n\plus{}x^{n^2} \}_{n\equal{}0}^{\infty} is dense in
C
[
0
,
1
]
C[0,1]
C
[
0
,
1
]
. J. Szabados
7
1
Hide problems
Miklos Schweitzer 1976_7
Let
f
1
,
f
2
,
…
,
f
n
f_1,f_2,\dots,f_n
f
1
,
f
2
,
…
,
f
n
be regular functions on a domain of the complex plane, linearly independent over the complex field. Prove that the functions
f
i
f
‾
k
,
1
≤
i
,
k
≤
n
f_i\overline{f}_k, \;1 \leq i,k \leq n
f
i
f
k
,
1
≤
i
,
k
≤
n
, are also linearly independent. L. Lempert
6
1
Hide problems
Miklos Schweitzer 1976_6
Let
0
≤
c
≤
1
0 \leq c \leq 1
0
≤
c
≤
1
, and let
η
\eta
η
denote the order type of the set of rational numbers. Assume that with every rational number
r
r
r
we associate a Lebesgue-measurable subset
H
r
H_r
H
r
of measure
c
c
c
of the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
. Prove the existence of a Lebesgue-measurable set
H
⊂
[
0
,
1
]
H \subset [0,1]
H
⊂
[
0
,
1
]
of measure
c
c
c
such that for every
x
∈
H
x \in H
x
∈
H
the set
{
r
:
x
∈
H
r
}
\{r : \;x \in H_r\ \}
{
r
:
x
∈
H
r
}
contains a subset of type
η
\eta
η
. M. Laczkovich
5
1
Hide problems
Miklos Schweitzer 1976_5
Let S_{\nu}\equal{}\sum_{j\equal{}1}^n b_jz_j^{\nu} \;(\nu\equal{}0,\pm 1, \pm 2 ,...) , where the
b
j
b_j
b
j
are arbitrary and the
z
j
z_j
z
j
are nonzero complex numbers . Prove that
∣
S
0
∣
≤
n
max
0
<
∣
ν
∣
≤
n
∣
S
ν
∣
.
|S_0| \leq n \max_{0<|\nu| \leq n} |S_{\nu}|.
∣
S
0
∣
≤
n
0
<
∣
ν
∣
≤
n
max
∣
S
ν
∣.
G. Halasz
4
1
Hide problems
Miklos Schweitzer 1976_4
Let
Z
\mathbb{Z}
Z
be the ring of rational integers. Construct an integral domain
I
I
I
satisfying the following conditions: a)
Z
⫋
I
\mathbb{Z} \varsubsetneqq I
Z
I
; b) no element of I \minus{} \mathbb{Z} (only in
I
I
I
) is algebraic over
Z
\mathbb{Z}
Z
(that is, not a root of a polynomial with coefficients in
Z
\mathbb{Z}
Z
); c)
I
I
I
only has trivial endomorphisms. E. Fried
3
1
Hide problems
Miklos Schweitzer 1976_3
Let
H
H
H
denote the set of those natural numbers for which
τ
(
n
)
\tau(n)
τ
(
n
)
divides
n
n
n
, where
τ
(
n
)
\tau(n)
τ
(
n
)
is the number of divisors of
n
n
n
. Show that a)
n
!
∈
H
n! \in H
n
!
∈
H
for all sufficiently large
n
n
n
, b)
H
H
H
has density
0
0
0
. P. Erdos
2
1
Hide problems
Miklos Schweitzer 1976_2
Let
G
G
G
be an infinite graph such that for any countably infinite vertex set
A
A
A
there is a vertex
p
p
p
, not in
A
A
A
, joined to infinitely many elements of
A
A
A
. Show that
G
G
G
has a countably infinite vertex set
A
A
A
such that
G
G
G
contains uncountably infinitely many vertices
p
p
p
joined to infinitely many elements of
A
A
A
. P. Erdos, A. Hajnal
1
1
Hide problems
Miklos Schweitzer 1976_1
Assume that
R
R
R
, a recursive, binary relation on
N
\mathbb{N}
N
(the set of natural numbers), orders
N
\mathbb{N}
N
into type
ω
\omega
ω
. Show that if
f
(
n
)
f(n)
f
(
n
)
is the
n
n
n
th element of this order, then
f
f
f
is not necessarily recursive. L. Posa