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Miklós Schweitzer
2007 Miklós Schweitzer
2007 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklós Schweitzer 2007 Problem 10
Let
ζ
1
,
ζ
2
,
…
\zeta_1, \zeta_2,\ldots
ζ
1
,
ζ
2
,
…
be identically distributed, independent real-valued random variables with expected value
0
0
0
. Suppose that the
Λ
(
λ
)
:
=
log
E
exp
(
λ
ζ
i
)
\Lambda (\lambda) := \log \mathbb E \exp (\lambda \zeta_i)
Λ
(
λ
)
:=
lo
g
E
exp
(
λ
ζ
i
)
logarithmic moment-generating function always exists for
λ
∈
R
\lambda\in\mathbb R
λ
∈
R
(
E
\mathbb E
E
is the expected value). Furthermore, let
G
:
R
→
R
G\colon\mathbb R \rightarrow \mathbb R
G
:
R
→
R
be a function such that
G
(
x
)
≤
min
(
∣
x
∣
,
x
2
)
G(x)\leq \min (|x|, x^2)
G
(
x
)
≤
min
(
∣
x
∣
,
x
2
)
. Prove that for small
γ
>
0
\gamma >0
γ
>
0
the following sequence is bounded:
{
E
exp
(
γ
l
G
(
1
l
(
ζ
1
+
…
+
ζ
l
)
)
)
}
l
=
1
∞
\left\{ \mathbb E \exp \left( \gamma l G \left( \frac 1l (\zeta_1+\ldots + \zeta_l)\right)\right)\right\}^{\infty}_{l=1}
{
E
exp
(
γ
lG
(
l
1
(
ζ
1
+
…
+
ζ
l
)
)
)
}
l
=
1
∞
(translated by j___d)
9
1
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Miklós Schweitzer 2007 Problem 9
Let
A
A
A
and
B
B
B
be two triangles on the plane such that the interior of both contains the origin and for each circle
C
r
C_r
C
r
centered at the origin
∣
C
r
∩
A
∣
=
∣
C
r
∩
B
∣
|C_r \cap A|=|C_r\cap B|
∣
C
r
∩
A
∣
=
∣
C
r
∩
B
∣
(where
∣
⋅
∣
|\cdot |
∣
⋅
∣
is the arc-length measure). Prove that
A
A
A
and
B
B
B
are congruent. Does this statement remain true if the origin is on the border of
A
A
A
or
B
B
B
?(translated by Miklós Maróti)
8
1
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Miklós Schweitzer 2007 Problem 8
For an
A
=
{
a
i
}
i
=
0
∞
A=\{ a_i\}^{\infty}_{i=0}
A
=
{
a
i
}
i
=
0
∞
sequence let
S
A
=
{
a
0
,
a
0
+
a
1
,
a
0
+
a
1
+
a
2
,
…
}
SA=\{ a_0, a_0+a_1, a_0+a_1+a_2, \ldots\}
S
A
=
{
a
0
,
a
0
+
a
1
,
a
0
+
a
1
+
a
2
,
…
}
be the sequence of partial sums of the
a
0
+
a
1
+
…
a_0+a_1+\ldots
a
0
+
a
1
+
…
series. Does there exist a non-identically zero sequence
A
A
A
such that all of the sequences
A
,
S
A
,
S
S
A
,
S
S
S
A
,
…
A, SA, SSA, SSSA, \ldots
A
,
S
A
,
SS
A
,
SSS
A
,
…
are convergent?(translated by Miklós Maróti)
7
1
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Miklós Schweitzer 2007 Problem 7
Prove that there exist natural numbers
n
k
,
m
k
,
k
=
0
,
1
,
2
,
…
n_k, m_k, k=0,1,2,\ldots
n
k
,
m
k
,
k
=
0
,
1
,
2
,
…
, such that the numbers
n
k
+
m
k
,
k
=
1
,
2
,
…
n_k+m_k, k=1,2,\ldots
n
k
+
m
k
,
k
=
1
,
2
,
…
are pairwise distinct primes and the set of linear combination of the polynomials
x
n
k
y
m
k
x^{n_k}y^{m_k}
x
n
k
y
m
k
is dense in
C
(
[
0
,
1
]
×
[
0
,
1
]
)
C([0,1] \times [0,1])
C
([
0
,
1
]
×
[
0
,
1
])
under the supremum norm.(translated by Miklós Maróti)
6
1
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Miklós Schweitzer 2007 Problem 6
For which subsets
A
⊂
R
A\subset \mathbb R
A
⊂
R
is it true that whenever
0
≤
x
0
<
x
1
<
⋯
<
x
n
≤
1
0\leq x_0 < x_1 < \cdots < x_n\leq 1
0
≤
x
0
<
x
1
<
⋯
<
x
n
≤
1
,
n
=
1
,
2
,
…
n=1,2, \ldots
n
=
1
,
2
,
…
, then there exist
y
j
∈
A
y_j\in A
y
j
∈
A
numbers, such that
y
j
+
1
−
y
j
>
x
j
+
1
−
x
j
y_{j+1}-y_j>x_{j+1}-x_j
y
j
+
1
−
y
j
>
x
j
+
1
−
x
j
for all
0
≤
j
<
n
0\leq j < n
0
≤
j
<
n
.(translated by Miklós Maróti)
5
1
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Miklós Schweitzer 2007 Problem 5
Let
D
=
{
(
x
,
y
)
∣
x
>
0
,
y
≠
0
}
D=\{ (x,y) \mid x>0, y\neq 0\}
D
=
{(
x
,
y
)
∣
x
>
0
,
y
=
0
}
and let
u
∈
C
1
(
D
‾
)
u\in C^1(\overline {D})
u
∈
C
1
(
D
)
be a bounded function that is harmonic on
D
D
D
and for which
u
=
0
u=0
u
=
0
on the
y
y
y
-axis. Prove that
u
u
u
is identically zero.(translated by Miklós Maróti)
4
1
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Miklós Schweitzer 2007 Problem 4
Let
p
p
p
be a prime number and
a
1
,
…
,
a
p
−
1
a_1, \ldots, a_{p-1}
a
1
,
…
,
a
p
−
1
be not necessarily distinct nonzero elements of the
p
p
p
-element
Z
p
(
m
o
d
p
)
\mathbb Z_p \pmod{p}
Z
p
(
mod
p
)
group. Prove that each element of
Z
p
\mathbb Z_p
Z
p
equals a sum of some of the
a
i
a_i
a
i
's (the empty sum is
0
0
0
).(translated by Miklós Maróti)
3
1
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Miklós Schweitzer 2007 Problem 3
Denote by
ω
(
n
)
\omega (n)
ω
(
n
)
the number of prime divisors of the natural number
n
n
n
(without multiplicities). Let
F
(
x
)
=
max
n
≤
x
ω
(
n
)
G
(
x
)
=
max
n
≤
x
(
ω
(
n
)
+
ω
(
n
2
+
1
)
)
F(x)=\max_{n\leq x} \omega (n) \,\,\,\,\,\,\,\,\,\,\,\,\, G(x)=\max_{n\leq x} \left( \omega (n) + \omega (n^2+1)\right)
F
(
x
)
=
n
≤
x
max
ω
(
n
)
G
(
x
)
=
n
≤
x
max
(
ω
(
n
)
+
ω
(
n
2
+
1
)
)
Prove that
G
(
x
)
−
F
(
x
)
→
∞
G(x)-F(x)\to \infty
G
(
x
)
−
F
(
x
)
→
∞
as
x
→
∞
x\to\infty
x
→
∞
.(translated by Miklós Maróti)
2
1
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Miklós Schweizer 2007 Problem 2
We partition the
n
n
n
-element subsets of an
n
2
+
n
−
1
n^2+n-1
n
2
+
n
−
1
-element set into two classes. Prove that one of the classes contains
n
n
n
-many pairwise disjunct sets.(translated by Miklós Maróti)
1
1
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Miklós Schweitzer 2007 Problem 1
Prove that there exist subfields of
R
\mathbb R
R
that are a) non-measurable and b) of measure zero and continuum cardinality.(translated by Miklós Maróti)