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Miklós Schweitzer
1960 Miklós Schweitzer
1960 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklós Schweitzer 1960- Problem 10
10. A car is used by
n
n
n
drivers. Every morning the drivers choose by drawing that one of them who will drive the car that day. Let us define the random variable
μ
(
n
)
\mu (n)
μ
(
n
)
as the least positive integer such that each driver drives at least one day during the first
μ
(
n
)
\mu (n)
μ
(
n
)
days. Find the limit distribution of the random variable
μ
(
n
)
−
n
log
n
n
\frac {\mu (n) -n \log n}{n}
n
μ
(
n
)
−
n
l
o
g
n
as
n
→
∞
n \to \infty
n
→
∞
. (P. 9)
9
1
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Miklós Schweitzer 1960- Problem 9
9. Let
A
1
,
…
,
A
n
A_1, \dots , A_n
A
1
,
…
,
A
n
and
B
B
B
be ideals of an assoticative ring
R
R
R
such that
B
B
B
is contained in the set-union of the ideals
A
i
A_i
A
i
(
i
=
1
,
…
,
n
i=1, \dots , n
i
=
1
,
…
,
n
) but not contained in the union of any
n
−
1
n-1
n
−
1
of the ideals
A
i
A_i
A
i
. Show that, for some positive integer
k
k
k
,
B
k
B_k
B
k
is contained in the intersection of the ideals
A
i
A_i
A
i
. (A. 19)
8
1
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Miklós Schweitzer 1960- Problem 8
8. Let
f
f
f
be a bounded real function defined on the unit cube
H
H
H
of the
n
n
n
-dimensional space and, for a given
y
y
y
, let
A
y
A_y
A
y
and
B
y
B_y
B
y
denote the parts of the interior of
H
H
H
on which
f
>
y
f>y
f
>
y
and
f
<
y
f<y
f
<
y
, respectively. Show that
f
f
f
is integrable in the Riemannian sense if and only if for every
y
y
y
almost all points of
A
y
A_y
A
y
and
B
y
B_y
B
y
are inner points. (R. 9)
7
1
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Miklós Schweitzer 1960- Problem 7
7. Define the generalized derivative at
x
0
x_0
x
0
of the function
f
(
x
)
f(x)
f
(
x
)
by
lim
h
→
0
2
1
h
∫
x
0
x
0
+
h
f
(
t
)
d
t
−
f
(
x
0
)
h
\lim_{h \to 0} 2 \frac{ \frac{1}{h} \int_{x_0}^{x_0+h} f(t) dt - f(x_0)}{h}
lim
h
→
0
2
h
h
1
∫
x
0
x
0
+
h
f
(
t
)
d
t
−
f
(
x
0
)
Show that there exists a function, continuous everywhere, which is nowhere differentiable in this general sense ( R. 8)
6
1
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Miklós Schweitzer 1960- Problem 6
6. Let
{
n
k
}
k
=
1
∞
\{ n_k \}_{k=1}^{\infty}
{
n
k
}
k
=
1
∞
be a stricly increasing sequence of positive integers such that
lim
k
→
∞
n
k
1
2
k
=
∞
\lim_{k \to \infty} n_k^{\frac {1}{2^k}}= \infty
lim
k
→
∞
n
k
2
k
1
=
∞
Show that the sum of the series
∑
k
=
1
∞
1
n
k
\sum_{k=1}^{\infty} \frac {1}{n_k}
∑
k
=
1
∞
n
k
1
is an irrational number. (N. 19)
5
1
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Miklós Schweitzer 1960- Problem 5
5. Define the sequence
{
c
n
}
n
=
1
∞
\{c_n\}_{n=1}^{\infty}
{
c
n
}
n
=
1
∞
as follows:
c
1
=
1
2
c_1= \frac {1}{2}
c
1
=
2
1
,
c
n
+
1
=
c
n
−
c
n
2
c_{n+1}= c_{n}-c_{n}^2
c
n
+
1
=
c
n
−
c
n
2
(
n
≥
1
n\geq 1
n
≥
1
). Prove that
lim
n
→
∞
n
c
n
=
1
\lim_{n \to \infty} nc_n= 1
lim
n
→
∞
n
c
n
=
1
(S.12)
4
1
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Miklós Schweitzer 1960- Problem 4
4. Let
(
H
α
)
\left (H_{\alpha} \right )
(
H
α
)
be a system of sets of integers having the property that for any
α
1
≠
α
2
,
H
α
1
∩
H
α
2
\alpha _1 \neq \alpha _2 , H_{\alpha _1}\cap H_{\alpha _2}
α
1
=
α
2
,
H
α
1
∩
H
α
2
is a finite set and
H
α
1
≠
H
α
2
H_{{\alpha} _1} \neq H_{{\alpha} _2}
H
α
1
=
H
α
2
. Prove that there exists a system
(
H
α
)
\left (H_{\alpha} \right )
(
H
α
)
of this kind whose cardinality is that of the continuum. Prove further that if none of the intersections of two sets
H
α
H_\alpha
H
α
contains more than
K
K
K
elements, then the system
(
H
α
)
\left (H_{\alpha} \right )
(
H
α
)
is countable (
K
K
K
is an arbitrary fixed integer). (St. 4)
3
1
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Miklós Schweitzer 1960- Problem 3
3. Let
f
(
z
)
f(z)
f
(
z
)
with
f
(
0
)
=
1
f(0)=1
f
(
0
)
=
1
be regular in the unit disk and let
[
∂
2
∣
f
(
z
)
∣
∂
x
∂
y
]
z
=
0
=
1
\left [\frac{\partial^2 \mid f(z)\mid}{\partial x\partial y} \right ] _{z=0} =1
[
∂
x
∂
y
∂
2
∣
f
(
z
)
∣
]
z
=
0
=
1
.Show thatthe area of the image of the unit disk by
w
=
f
(
z
)
w= f(z)
w
=
f
(
z
)
(taken with multiplicity) is not less than
1
2
\frac {1} {2}
2
1
.(f. 6)
2
1
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Miklós Schweitzer 1960- Problem 2
2. Construct a sequence
(
a
n
)
n
=
1
∞
(a_n)_{n=1}^{\infty}
(
a
n
)
n
=
1
∞
of complex numbers such that, for every
l
>
0
l>0
l
>
0
, the series
∑
n
=
1
∞
∣
a
n
∣
l
\sum_{n=1}^{\infty} \mid a_n \mid ^{l}
∑
n
=
1
∞
∣
a
n
∣
l
be divergent, but for almost all
θ
\theta
θ
in
(
0
,
2
π
)
(0,2\pi)
(
0
,
2
π
)
,
∏
n
=
1
∞
(
1
+
a
n
e
i
θ
)
\prod_{n=1}^{\infty} (1+a_n e^{i\theta})
∏
n
=
1
∞
(
1
+
a
n
e
i
θ
)
be convergent. (S. 11)
1
1
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Miklós Schweitzer 1960- Problem 1
1. Consider in the plane a set
H
H
H
of pairwise disjoint circles of radius 1 such that, for infinitely many positive integers
n
n
n
, the circle
k
n
k_n
k
n
with centre at the origin and of radius
n
n
n
contains at least
c
n
2
cn^2
c
n
2
elements of the set
H
H
H
. Prove that there exists a straight line which intersects infinitely many of the circles of
H
H
H
. Show further that if we require only that the circles
k
n
k_n
k
n
contain o(n²) elements of
H
H
H
, the proposition will be false. (G. 5)