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Miklós Schweitzer
1952 Miklós Schweitzer
1952 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklos Schweitzer 1952_10
Let
n
n
n
be a positive integer. Prove that, for 0
n
n
n
is odd and negative if
n
n
n
is even.
9
1
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Miklos Schweitzer 1952_9
Let
C
C
C
denote the set of functions
f
(
x
)
f(x)
f
(
x
)
, integrable (according to either Riemann or Lebesgue) on
(
a
,
b
)
(a,b)
(
a
,
b
)
, with
0
≤
f
(
x
)
≤
1
0\le f(x)\le1
0
≤
f
(
x
)
≤
1
. An element
ϕ
(
x
)
∈
C
\phi(x)\in C
ϕ
(
x
)
∈
C
is said to be an "extreme point" of
C
C
C
if it can not be represented as the arithmetical mean of two different elements of
C
C
C
. Find the extreme points of
C
C
C
and the functions
f
(
x
)
∈
C
f(x)\in C
f
(
x
)
∈
C
which can be obtained as "weak limits" of extreme points
ϕ
n
(
x
)
\phi_n(x)
ϕ
n
(
x
)
of
C
C
C
. (The latter means that \lim_{n\to \infty}\int_a^b \phi_n(x)h(x)\,dx\equal{}\int_a^bf(x)h(x)\,dx holds for every integrable function
h
(
x
)
h(x)
h
(
x
)
.)
8
1
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Miklos Schweitzer 1952_8
For which values of
z
z
z
does the series \sum_{n\equal{}1}^{\infty}c_1c_2\cdots c_n z^n converge, provided that
c
k
>
0
c_k>0
c
k
>
0
and \sum_{k\equal{}1}^{\infty} \frac{c_k}{k}<\infty ?
7
1
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Miklos Schweitzer 1952_7
A point
P
P
P
is performing a random walk on the
X
X
X
-axis. At the instant t\equal{}0,
P
P
P
is at a point
x
0
x_0
x
0
(
∣
x
0
∣
≤
N
|x_0|\le N
∣
x
0
∣
≤
N
, where
x
0
x_0
x
0
and
N
N
N
denote integers,
N
>
0
N>0
N
>
0
). If at an instant
t
t
t
(
t
t
t
being a nonnegative integer),
P
P
P
is at a point of
x
x
x
integer abscissa and
∣
x
∣
<
N
|x|<N
∣
x
∣
<
N
, then by the instant t\plus{}1 it reaches either the point x\plus{}1 or the point x\minus{}1, each with probability
1
2
\frac12
2
1
. If at the instant
t
t
t
,
P
P
P
is at the point x\equal{}N [ x\equal{}\minus{}N], then by the instant t\plus{}1 it is certain to reach the point N\minus{}1 [ \minus{}N\plus{}1]. Denote by
P
k
(
t
)
P_k(t)
P
k
(
t
)
the probability of
P
P
P
being at x\equal{}k at instant
t
t
t
(
k
k
k
is an integer). Find
lim
t
→
∞
P
k
(
2
t
)
\lim_{t\to \infty}P_{k}(2t)
lim
t
→
∞
P
k
(
2
t
)
and \lim_{t\to \infty}P_k(2t\plus{}1) for every fixed
k
k
k
.
6
1
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Miklos Schweitzer 1952_6
Let
2
n
2n
2
n
distinct points on a circle be given. Arrange them into disjoint pairs in an arbitrary way and join the couples by chords. Determine the probability that no two of these
n
n
n
chords intersect. (All possible arrangement into pairs are supposed to have the same probability.)
5
1
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Miklos Schweitzer 1952_5
Let
G
G
G
be anon-commutative group. Consider all the one-to-one mappings
a
→
a
′
a\rightarrow a'
a
→
a
′
of
G
G
G
onto itself such that (ab)'\equal{}b'a' (i.e. the anti-automorphisms of
G
G
G
). Prove that this mappings together with the automorphisms of
G
G
G
constitute a group which contains the group of the automorphisms of
G
G
G
as direct factor.
4
1
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Miklos Schweitzer 1952_4
Let
K
K
K
be a finite field of
p
p
p
elements, where
p
p
p
is a prime. For every polynomial f(x)\equal{}\sum_{i\equal{}0}^na_ix^i (
∈
K
[
x
]
\in K[x]
∈
K
[
x
]
) put \overline{f(x)}\equal{}\sum_{i\equal{}0}^n a_ix^{p^i}. Prove that for any pair of polynomials
f
(
x
)
,
g
(
x
)
∈
K
[
x
]
f(x),g(x)\in K[x]
f
(
x
)
,
g
(
x
)
∈
K
[
x
]
,
f
(
x
)
‾
∣
g
(
x
)
‾
\overline{f(x)}|\overline{g(x)}
f
(
x
)
∣
g
(
x
)
if and only if
f
(
x
)
∣
g
(
x
)
f(x)|g(x)
f
(
x
)
∣
g
(
x
)
.
3
1
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Miklos Schweitzer 1952_3
Prove:If a\equal{}p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{n}^{\alpha_n} is a perfect number, then 2<\prod_{i\equal{}1}^n\frac{p_i}{p_i\minus{}1}<4 ; if moreover,
a
a
a
is odd, then the upper bound
4
4
4
may be reduced to
2
2
3
2\sqrt[3]{2}
2
3
2
.
2
1
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Miklos Schweitzer 1952_2
Is it possible to find three conics in the plane such that any straight line in the plane intersects at least two of the conics and through any point of the plane pass tangents to at least two of them?
1
1
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Miklos Schweitzer 1952_1
Find all convex polyhedra which have no diagonals (that is, for which every segment connecting two vertices lies on the boundary of the polyhedron).