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Miklós Schweitzer
2022 Miklós Schweitzer
2022 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
8
1
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Miklós Schweitzer 2022 P8
Original in Hungarian; translated with Google translate; polished by myself.Prove that, the signs
ε
n
=
±
1
\varepsilon_n = \pm 1
ε
n
=
±
1
can be chosen such that the function
f
(
s
)
=
∑
n
=
1
∞
ε
n
n
s
:
{
s
∈
C
:
Re
s
>
1
}
→
C
f(s) = \sum_{n = 1}^\infty\frac{\varepsilon_n}{n^s}\colon \{s\in\Bbb C:\operatorname{Re}s > 1\}\to \Bbb C
f
(
s
)
=
∑
n
=
1
∞
n
s
ε
n
:
{
s
∈
C
:
Re
s
>
1
}
→
C
converges to every complex value at every point
ξ
∈
{
s
∈
C
:
Re
s
=
1
}
\xi \in \{s\in\Bbb C:\operatorname{Re}s = 1\}
ξ
∈
{
s
∈
C
:
Re
s
=
1
}
(i.e. for every
ξ
∈
{
s
∈
C
:
Re
s
=
1
}
\xi \in \{s\in\Bbb C:\operatorname{Re}s = 1\}
ξ
∈
{
s
∈
C
:
Re
s
=
1
}
and every
z
∈
C
z \in \Bbb C
z
∈
C
, there exists a sequence
s
n
→
ξ
s_n \to \xi
s
n
→
ξ
,
Re
s
n
>
1
\operatorname{Re}s_n > 1
Re
s
n
>
1
, for which
f
(
s
n
)
→
z
f(s_n) \to z
f
(
s
n
)
→
z
).
3
1
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Miklós Schweitzer 2022 P3
Original in Hungarian; translated with Google translate; polished by myself.Let
f
:
[
0
,
∞
)
→
[
0
,
∞
)
f: [0, \infty) \to [0, \infty)
f
:
[
0
,
∞
)
→
[
0
,
∞
)
be a function that is linear between adjacent integers, and for
n
=
0
,
1
,
…
n = 0, 1, \dots
n
=
0
,
1
,
…
satisfies
f
(
n
)
=
{
0
,
if
2
∣
n
,
4
l
+
1
,
if
2
∤
n
,
4
l
−
1
≤
n
<
4
l
(
l
=
1
,
2
,
…
)
.
f(n) = \begin{cases} 0, & \textrm{if }2\mid n,\\4^l + 1, & \textrm{if }2 \nmid n, 4^{l - 1} \leq n < 4^l(l = 1, 2, \dots).\end{cases}
f
(
n
)
=
{
0
,
4
l
+
1
,
if
2
∣
n
,
if
2
∤
n
,
4
l
−
1
≤
n
<
4
l
(
l
=
1
,
2
,
…
)
.
Let
f
1
(
x
)
=
f
(
x
)
f^1(x) = f(x)
f
1
(
x
)
=
f
(
x
)
, and
f
k
(
x
)
=
f
(
f
k
−
1
(
x
)
)
f^k(x) = f(f^{k - 1}(x))
f
k
(
x
)
=
f
(
f
k
−
1
(
x
))
for all integers
k
≥
2
k \geq 2
k
≥
2
. Determine the values of
lim inf
k
→
∞
f
k
(
x
)
\liminf\nolimits_{k\to\infty}f^k(x)
lim
inf
k
→
∞
f
k
(
x
)
and
lim sup
k
→
∞
f
k
(
x
)
\limsup\nolimits_{k\to\infty}f^k(x)
lim
sup
k
→
∞
f
k
(
x
)
for almost all
x
∈
[
0
,
∞
)
x \in [0, \infty)
x
∈
[
0
,
∞
)
under Lebesgue measure.(Not sure whether the last sentence translates correctly; the original: Határozzuk meg Lebesgue majdnem minden
x
∈
[
0
,
∞
)
x\in [0, \infty)
x
∈
[
0
,
∞
)
-re a
lim inf
k
→
∞
f
k
(
x
)
\liminf\nolimits_{k\to\infty}f^k(x)
lim
inf
k
→
∞
f
k
(
x
)
és
lim sup
k
→
∞
f
k
(
x
)
\limsup\nolimits_{k\to\infty}f^k(x)
lim
sup
k
→
∞
f
k
(
x
)
értékét.)
2
1
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Miklós Schweitzer 2022 P2
Original in Hungarian; translated with Google translate; polished by myself.Let
n
n
n
be a positive integer. Suppose that the sum of the matrices
A
1
,
…
,
A
n
∈
R
n
×
n
A_1, \dots, A_n\in \Bbb R^{n\times n}
A
1
,
…
,
A
n
∈
R
n
×
n
is the identity matrix, but
∑
i
=
1
n
α
i
A
i
\sum\nolimits_{i = 1}^n\alpha_i A_i
∑
i
=
1
n
α
i
A
i
is singular whenever at least one of the coefficients
α
i
∈
R
\alpha_i \in \Bbb R
α
i
∈
R
is zero. a) Show that
∑
i
=
1
n
α
i
A
i
\sum\nolimits_{i = 1}^n\alpha_i A_i
∑
i
=
1
n
α
i
A
i
is nonsingular if
α
i
≠
0
\alpha_i\neq 0
α
i
=
0
for all
i
i
i
. b) Show that if the matrices
A
i
A_i
A
i
are symmetric, then all of them have rank
1
1
1
.
7
1
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2022 MS P7
Point-like figures are placed in the vertices of a regular
k
k
k
-angle, and then we walk with them. In one step, a piece jumps over another piece, i.e. its new location will be a mirror image of its current location to the current location of another piece. In the case of
k
≥
3
k \geq 3
k
≥
3
integers, it is possible to achieve with a series of such steps that the puppets form the vertices of a regular
k
k
k
-angle, different in size from the original?
6
1
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Seventh circular field Q
Let
ϵ
\epsilon
ϵ
be a primitive seventh unit root. Which integers occur in
∣
α
∣
2
|\alpha|^2
∣
α
∣
2
in form, where
α
\alpha
α
is an element of the seventh circular field
Q
(
ϵ
)
\mathbb Q(\epsilon)
Q
(
ϵ
)
?
5
1
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Disjoint segments
Is it possible to select a non-degenerate segment from each line of the plane such that any two selected segments are disjoint?
9
1
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Problem 9 MS
Plane vectors form a group for addition. Show that this group has a generator system of every set
S
S
S
that contains a Borel subset of positive linear measure of a circular arc.
10
1
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Existence of a continuous function
Is there a continuous function
f
:
R
\
Q
→
R
\
Q
f : \mathbb R \backslash \mathbb Q \to \mathbb R \backslash \mathbb Q
f
:
R
\
Q
→
R
\
Q
for which the archetype of every irrational number has a positive Hausdorff dimension?
4
1
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Limit computation
Consider the integral
∫
−
1
1
x
n
f
(
x
)
d
x
\int_{-1}^1 x^nf(x) \; dx
∫
−
1
1
x
n
f
(
x
)
d
x
for every
n
n
n
-th degree polynomial
f
f
f
with integer coefficients. Let
α
n
\alpha_n
α
n
denote the smallest positive real number that such an integral can give. Determine the limit value
lim
n
→
∞
log
α
n
n
.
\lim_{n\to \infty} \frac{\log \alpha_n}n.
n
→
∞
lim
n
lo
g
α
n
.
1
1
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Existence of some defined infinite irregular set
We say that a set
A
⊂
Z
A \subset \mathbb Z
A
⊂
Z
is irregular if, for any different elements
x
,
y
∈
A
x, y \in A
x
,
y
∈
A
, there is no element of the form
x
+
k
(
y
−
x
)
x + k(y -x)
x
+
k
(
y
−
x
)
different from
x
x
x
and
y
y
y
(where
k
k
k
is an integer). Is there an infinite irregular set?