MathDB
Problems
Contests
Undergraduate contests
Miklós Schweitzer
1973 Miklós Schweitzer
1973 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
Hide problems
Miklos Schweitzer 1973_10
Find the limit distribution of the sequence
η
n
\eta_n
η
n
of random variables with distribution P \left( \eta_n\equal{}\arccos (\cos^2 \frac{(2j\minus{}1) \pi}{2n}) \right)\equal{}\frac 1n \;(j\equal{}1,2,...,n)\ . (
arccos
(
.
)
\arccos(.)
arccos
(
.
)
denotes the main value.) B. Gyires
9
1
Hide problems
Miklos Schweitzer 1973_9
Determine the value of \sup_{ 1 \leq \xi \leq 2} [\log E \xi\minus{}E \log \xi], where
ξ
\xi
ξ
is a random variable and
E
E
E
denotes expectation. Z. Daroczy
8
1
Hide problems
Miklos Schweitzer 1973_8
What is the radius of the largest disc that can be covered by a finite number of closed discs of radius
1
1
1
in such a way that each disc intersects at most three others? L. Fejes-Toth
7
1
Hide problems
Miklos Schweitzer 1973_7
Let us connect consecutive vertices of a regular heptagon inscribed in a unit circle by connected subsets (of the plane of the circle) of diameter less than
1
1
1
. Show that every continuum (in the plane of the circle) of diameter greater than
4
4
4
, containing the center of the circle, intersects one of these connected sets. M. Bognar
6
1
Hide problems
Miklos Schweitzer 1973_6
If
f
f
f
is a nonnegative, continuous, concave function on the closed interval
[
0
,
1
]
[0,1]
[
0
,
1
]
such that
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
, then
∫
0
1
x
f
(
x
)
d
x
≤
2
3
[
∫
0
1
f
(
x
)
d
x
]
2
.
\int_0^1 xf(x)dx \leq \frac 23 \left[ %Error. "diaplaymath" is a bad command. \int_0^1 f(x)dx \right]^2.
∫
0
1
x
f
(
x
)
d
x
≤
3
2
[
∫
0
1
f
(
x
)
d
x
]
2
.
Z. Daroczy
5
1
Hide problems
Miklos Schweitzer 1973_5
Verify that for every
x
>
0
x > 0
x
>
0
, \frac{\Gamma'(x\plus{}1)}{\Gamma (x\plus{}1)} > \log x. P. Medgyessy
4
1
Hide problems
Miklos Schweitzer 1973_4
Let
f
(
n
)
f(n)
f
(
n
)
be that largest integer
k
k
k
such that
n
k
n^k
n
k
divides
n
!
n!
n
!
, and let F(n)\equal{} \max_{2 \leq m \leq n} f(m). Show that \lim_{n\rightarrow \infty} \frac{F(n) \log n}{n \log \log n}\equal{}1. P. Erdos
3
1
Hide problems
Miklos Schweitzer 1973_3
Find a constant
c
>
1
c > 1
c
>
1
with the property that, for arbitrary positive integers
n
n
n
and
k
k
k
such that
n
>
c
k
n>c^k
n
>
c
k
, the number of distinct prime factors of
(
n
k
)
\binom{n}{k}
(
k
n
)
is at least
k
k
k
. P. Erdos
2
1
Hide problems
Miklos Schweitzer 1973_2
Let
R
R
R
be an Artinian ring with unity. Suppose that every idempotent element of
R
R
R
commutes with every element of
R
R
R
whose square is
0
0
0
. Suppose
R
R
R
is the sum of the ideals
A
A
A
and
B
B
B
. Prove that AB\equal{}BA. A. Kertesz
1
1
Hide problems
Miklos Schweitzer 1973_1
We say that the rank of a group
G
G
G
is at most
r
r
r
if every subgroup of
G
G
G
can be generated by at most
r
r
r
elements. Prove that here exists an integer
s
s
s
such that for every finite group
G
G
G
of rank
2
2
2
the commutator series of
G
G
G
has length less than
s
s
s
. J. Erdos