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Miklós Schweitzer
2003 Miklós Schweitzer
2003 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklós Schweitzer 2003, Problem 10
Let
X
X
X
and
Y
Y
Y
be independent random variables with "Saint-Petersburg" distribution, i.e. for any
k
=
1
,
2
,
…
k=1,2,\ldots
k
=
1
,
2
,
…
their value is
2
k
2^k
2
k
with probability
1
2
k
\frac{1}{2^k}
2
k
1
. Show that
X
X
X
and
Y
Y
Y
can be realized on a sufficiently big probability space such that there exists another pair of independent "Saint-Petersburg" random variables
(
X
′
,
Y
′
)
(X', Y')
(
X
′
,
Y
′
)
on this space with the property that
X
+
Y
=
2
X
′
+
Y
′
I
(
Y
′
≤
X
′
)
X+Y=2X'+Y'I(Y'\le X')
X
+
Y
=
2
X
′
+
Y
′
I
(
Y
′
≤
X
′
)
almost surely (here
I
(
A
)
I(A)
I
(
A
)
denotes the indicator function of the event
A
A
A
).(translated by L. Erdős)
9
1
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Miklós Schweitzer 2003, Problem 9
Given fi nitely many open half planes on the Euclidean plane. The boundary lines of these half planes divide the plane into convex domains. Find a polynomial
C
(
q
)
C(q)
C
(
q
)
of degree two so that the following holds: for any
q
≥
1
q\ge 1
q
≥
1
integer, if the half planes cover each point of the plane at least
q
q
q
times, then the set of points covered exactly
q
q
q
times is the union of at most
C
(
q
)
C(q)
C
(
q
)
domains.(translated by L. Erdős)
8
1
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Miklós Schweitzer 2003, Problem 8
Let
f
1
,
f
2
,
…
f_1, f_2, \ldots
f
1
,
f
2
,
…
be continuous real functions on the real line. Is it true that if the series
∑
n
=
1
∞
f
n
(
x
)
\sum_{n=1}^{\infty} f_n(x)
∑
n
=
1
∞
f
n
(
x
)
is divergent for every
x
x
x
, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those
{
ϵ
n
}
n
=
1
∞
∈
{
+
1
,
−
1
}
N
\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}
{
ϵ
n
}
n
=
1
∞
∈
{
+
1
,
−
1
}
N
sequences, for which there series
∑
n
=
1
∞
ϵ
n
f
n
(
x
)
\sum_{n=1}^{\infty} \epsilon_nf_n(x)
∑
n
=
1
∞
ϵ
n
f
n
(
x
)
is convergent at least at one point
x
x
x
, forms a subset of first category within the set
{
+
1
,
−
1
}
N
\{+1,-1\}^{\mathbb{N}}
{
+
1
,
−
1
}
N
)?(translated by L. Erdős)
7
1
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Miklós Schweitzer 2003, Problem 7
Let
r
r
r
be a nonnegative continuous function on the real line. Show that there exists a function
f
∈
C
1
(
R
)
f\in C^1(\mathbb{R})
f
∈
C
1
(
R
)
, not identically zero, such that
f
′
(
x
)
=
f
(
x
−
r
(
f
(
x
)
)
)
f'(x)=f(x-r(f(x)))
f
′
(
x
)
=
f
(
x
−
r
(
f
(
x
)))
,
x
∈
R
x\in\mathbb{R}
x
∈
R
.(translated by L. Erdős)
6
1
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Miklós Schweitzer 2003, Problem 6
Show that the recursion
n
=
x
n
(
x
n
−
1
+
x
n
+
x
n
+
1
)
n=x_n(x_{n-1}+x_n+x_{n+1})
n
=
x
n
(
x
n
−
1
+
x
n
+
x
n
+
1
)
,
n
=
1
,
2
,
…
n=1,2,\ldots
n
=
1
,
2
,
…
,
x
0
=
0
x_0=0
x
0
=
0
has exaclty one nonnegative solution.(translated by L. Erdős)
5
1
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Miklós Schweitzer 2003, Problem 5
Let
d
>
1
d>1
d
>
1
be integer and
0
<
r
<
1
2
0<r<\frac12
0
<
r
<
2
1
. Show that there exist finitely many (depending only on
d
,
r
d,r
d
,
r
) nonzero vectors in
R
d
\mathbb{R}^d
R
d
such that if the distance of a straight line in
R
d
\mathbb{R}^d
R
d
from the integer lattice
Z
d
\mathbb{Z}^d
Z
d
is at least
r
r
r
, then this line is orthogonal to one of these finitely many vectors.(translated by L. Erdős)
4
1
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Miklós Schweitzer 2003, Problem 4
Let
{
a
n
,
1
,
…
,
a
n
,
n
}
n
=
1
∞
\{a_{n,1},\ldots, a_{n,n} \}_{n=1}^{\infty}
{
a
n
,
1
,
…
,
a
n
,
n
}
n
=
1
∞
integers such that
a
n
,
i
≠
a
n
,
j
a_{n,i}\neq a_{n,j}
a
n
,
i
=
a
n
,
j
for
1
≤
i
<
j
≤
n
,
n
=
2
,
3
,
…
1\le i<j\le n\, , n=2,3,\ldots
1
≤
i
<
j
≤
n
,
n
=
2
,
3
,
…
and let
⟨
y
⟩
∈
[
0
,
1
)
\left\langle y\right\rangle\in [0,1)
⟨
y
⟩
∈
[
0
,
1
)
denote the fractional part of the real number
y
y
y
. Show that there exists a real sequence
{
x
n
}
n
=
1
∞
\{ x_n\}_{n=1}^{\infty}
{
x
n
}
n
=
1
∞
such that the numbers
⟨
a
n
,
1
x
n
⟩
,
…
,
⟨
a
n
,
n
x
n
⟩
\langle a_{n,1}x_n \rangle, \ldots, \langle a_{n,n}x_n \rangle
⟨
a
n
,
1
x
n
⟩
,
…
,
⟨
a
n
,
n
x
n
⟩
are asymptotically uniformly distributed on the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
.(translated by L. Erdős)
3
1
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Miklós Schweitzer 2003, Problem 3
Let
Z
=
{
z
1
,
…
,
z
n
−
1
}
Z=\{ z_1,\ldots, z_{n-1}\}
Z
=
{
z
1
,
…
,
z
n
−
1
}
,
n
≥
2
n\ge 2
n
≥
2
, be a set of different complex numbers such that
Z
Z
Z
contains the conjugate of any its element. a) Show that there exists a constant
C
C
C
, depending on
Z
Z
Z
, such that for any
ε
∈
(
0
,
1
)
\varepsilon\in (0,1)
ε
∈
(
0
,
1
)
there exists an algebraic integer
x
0
x_0
x
0
of degree
n
n
n
, whose algebraic conjugates
x
1
,
x
2
,
…
,
x
n
−
1
x_1, x_2, \ldots, x_{n-1}
x
1
,
x
2
,
…
,
x
n
−
1
satisfy
∣
x
1
−
z
1
∣
≤
ε
,
…
,
∣
x
n
−
1
−
z
n
−
1
∣
≤
ε
|x_1-z_1|\le \varepsilon, \ldots, |x_{n-1}-z_{n-1}|\le \varepsilon
∣
x
1
−
z
1
∣
≤
ε
,
…
,
∣
x
n
−
1
−
z
n
−
1
∣
≤
ε
and
∣
x
0
∣
≤
C
ε
|x_0|\le \frac{C}{\varepsilon}
∣
x
0
∣
≤
ε
C
. b) Show that there exists a set
Z
=
{
z
1
,
…
,
z
n
−
1
}
Z=\{ z_1, \ldots, z_{n-1}\}
Z
=
{
z
1
,
…
,
z
n
−
1
}
and a positive number
c
n
c_n
c
n
such that for any algebraic integer
x
0
x_0
x
0
of degree
n
n
n
, whose algebraic conjugates satisfy
∣
x
1
−
z
1
∣
≤
ε
,
…
,
∣
x
n
−
1
−
z
n
−
1
∣
≤
ε
|x_1-z_1|\le \varepsilon,\ldots, |x_{n-1}-z_{n-1}|\le \varepsilon
∣
x
1
−
z
1
∣
≤
ε
,
…
,
∣
x
n
−
1
−
z
n
−
1
∣
≤
ε
, it also holds that
∣
x
0
∣
>
c
n
ε
|x_0|>\frac{c_n}{\varepsilon}
∣
x
0
∣
>
ε
c
n
.(translated by L. Erdős)
2
1
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Miklós Schweitzer 2003, Problem 2
Let
p
p
p
be a prime and let
M
M
M
be an
n
×
m
n\times m
n
×
m
matrix with integer entries such that
M
v
≢
0
(
m
o
d
p
)
Mv\not\equiv 0\pmod{p}
M
v
≡
0
(
mod
p
)
for any column vector
v
≠
0
v\neq 0
v
=
0
whose entries are
0
0
0
are
1
1
1
. Show that there exists a row vector
x
x
x
with integer entries such that no entry of
x
M
xM
x
M
is
0
(
m
o
d
p
)
0\pmod{p}
0
(
mod
p
)
.(translated by L. Erdős)
1
1
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Miklós Schweitzer 2003, Problem 1
Let
(
X
,
<
)
(X, <)
(
X
,
<
)
be an arbitrary ordered set. Show that the elements of
X
X
X
can be coloured by two colours in such a way that between any two points of the same colour there is a point of the opposite colour.(translated by L. Erdős)