MathDB
Problems
Contests
Undergraduate contests
Miklós Schweitzer
1969 Miklós Schweitzer
1969 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(12)
12
1
Hide problems
Miklos Schweitzer 1969_12
Let
A
A
A
and
B
B
B
be nonsingular matrices of order
p
p
p
, and let
ξ
\xi
ξ
and
η
\eta
η
be independent random vectors of dimension
p
p
p
. Show that if
ξ
,
η
\xi,\eta
ξ
,
η
and \xi A\plus{} \eta B have the same distribution, if their first and second moments exist, and if their covariance matrix is the identity matrix, then these random vectors are normally distributed. B. Gyires
11
1
Hide problems
Miklos Schweitzer 1969_11
Let
A
1
,
A
2
,
.
.
.
A_1,A_2,...
A
1
,
A
2
,
...
be a sequence of infinite sets such that
∣
A
i
∩
A
j
∣
≤
2
|A_i \cap A_j| \leq 2
∣
A
i
∩
A
j
∣
≤
2
for i \not\equal{}j. Show that the sequence of indices can be divided into two disjoint sequences
i
1
<
i
2
<
.
.
.
i_1<i_2<...
i
1
<
i
2
<
...
and
j
1
<
j
2
<
.
.
.
j_1<j_2<...
j
1
<
j
2
<
...
in such a way that, for some sets
E
E
E
and
F
F
F
, |A_{i_n} \cap E|\equal{}1 and |A_{j_n} \cap F|\equal{}1 for n\equal{}1,2,... . P. Erdos
10
1
Hide problems
Miklos Schweitzer 1969_10
In
n
n
n
-dimensional Euclidean space, the square of the two-dimensional Lebesgue measure of a bounded, closed, (two-dimensional) planar set is equal to the sum of the squares of the measures of the orthogonal projections of the given set on the
n
n
n
-coordinate hyperplanes. L. Tamassy
9
1
Hide problems
Miklos Schweitzer 1969_9
In
n
n
n
-dimensional Euclidean space, the union of any set of closed balls (of positive radii) is measurable in the sense of Lebesgue. A. Csaszar
8
1
Hide problems
Miklos Schweitzer 1969_8
Let
f
f
f
and
g
g
g
be continuous positive functions defined on the interval
[
0
,
+
∞
)
[0, +\infty)
[
0
,
+
∞
)
, and let
E
⊂
[
0
,
+
∞
)
E \subset[0,+\infty)
E
⊂
[
0
,
+
∞
)
be a set of positive measure. Prove that the range of the function defined on
E
×
E
E \times E
E
×
E
by the relation
F
(
x
,
y
)
=
∫
0
x
f
(
t
)
d
t
+
∫
0
y
g
(
t
)
d
t
F(x,y)= %Error. "dispalymath" is a bad command. \int_0^xf(t)dt+ %Error. "dispalymath" is a bad command. \int_0^y g(t)dt
F
(
x
,
y
)
=
∫
0
x
f
(
t
)
d
t
+
∫
0
y
g
(
t
)
d
t
has a nonvoid interior. L. Losonczi
7
1
Hide problems
Miklos Schweitzer 1969_7
Prove that if a sequence of Mikusinski operators of the form \mu e^{\minus{}\lambda s} (
λ
\lambda
λ
and
μ
\mu
μ
nonnegative real numbers,
s
s
s
the differentiation operator) is convergent in the sense of Mikusinski, then its limit is also of this form. E. Geaztelyi
6
1
Hide problems
Miklos Schweitzer 1969_6
Let
x
0
x_0
x
0
be a fixed real number, and let
f
f
f
be a regular complex function in the half-plane
ℜ
z
>
x
0
\Re z>x_0
ℜ
z
>
x
0
for which there exists a nonnegative function
F
∈
L
1
(
−
∞
,
+
∞
)
F \in L_1(- \infty, +\infty)
F
∈
L
1
(
−
∞
,
+
∞
)
satisfying
∣
f
(
α
+
i
β
)
∣
≤
F
(
β
)
|f(\alpha+i\beta)| \leq F(\beta)
∣
f
(
α
+
i
β
)
∣
≤
F
(
β
)
whenever
α
>
x
0
\alpha > x_0
α
>
x
0
,
−
∞
<
β
<
+
∞
-\infty <\beta < +\infty
−
∞
<
β
<
+
∞
. Prove that
∫
α
−
i
∞
α
+
i
∞
f
(
z
)
d
z
=
0.
\int_{\alpha-i \infty} ^{\alpha+i \infty} f(z)dz=0.
∫
α
−
i
∞
α
+
i
∞
f
(
z
)
d
z
=
0.
L. Czach
5
1
Hide problems
Miklos Schweitzer 1969_5
Find all continuous real functions
f
,
g
f,g
f
,
g
and
h
h
h
defined on the set of positive real numbers and satisfying the relation f(x\plus{}y)\plus{}g(xy)\equal{}h(x)\plus{}h(y) for all
x
>
0
x>0
x
>
0
and
y
>
0
y>0
y
>
0
. Z. Daroczy
4
1
Hide problems
Miklos Schweitzer 1969_4
Show that the following inequality hold for all
k
≥
1
k \geq 1
k
≥
1
, real numbers
a
1
,
a
2
,
.
.
.
,
a
k
a_1,a_2,...,a_k
a
1
,
a
2
,
...
,
a
k
, and positive numbers
x
1
,
x
2
,
.
.
.
,
x
k
.
x_1,x_2,...,x_k.
x
1
,
x
2
,
...
,
x
k
.
\ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} . L. Losonczi
3
1
Hide problems
Miklos Schweitzer 1969_3
Let
f
(
x
)
f(x)
f
(
x
)
be a nonzero, bounded, real function on an Abelian group
G
G
G
,
g
1
,
.
.
.
,
g
k
g_1,...,g_k
g
1
,
...
,
g
k
are given elements of
G
G
G
and
λ
1
,
.
.
.
,
λ
k
\lambda_1,...,\lambda_k
λ
1
,
...
,
λ
k
are real numbers. Prove that if
∑
i
=
1
k
λ
i
f
(
g
i
x
)
≥
0
\sum_{i=1}^k \lambda_i f(g_ix) \geq 0
i
=
1
∑
k
λ
i
f
(
g
i
x
)
≥
0
holds for all
x
∈
G
x \in G
x
∈
G
, then
∑
i
=
1
k
λ
i
≥
0.
\sum_{i=1}^k \lambda_i \geq 0.
i
=
1
∑
k
λ
i
≥
0.
A. Mate
2
1
Hide problems
Miklos Schweitzer 1969_2
Let
p
≥
7
p\geq 7
p
≥
7
be a prime number,
ζ
\zeta
ζ
a primitive
p
p
p
th root of unity,
c
c
c
a rational number. Prove that in the additive group generated by the numbers 1,\zeta,\zeta^2,\zeta^3\plus{}\zeta^{\minus{}3} there are only finitely many elements whose norm is equal to
c
c
c
. (The norm is in the
p
p
p
th cyclotomic field.) K. Gyory
1
1
Hide problems
Miklos Schweitzer 1969_1
Let
G
G
G
be an infinite group generated by nilpotent normal subgroups. Prove that every maximal Abelian normal subgroup of
G
G
G
is infinite. (We call an Abelian normal subgroup maximal if it is not contained in another Abelian normal subgroup.) P. Erdos