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Miklós Schweitzer
2023 Miklós Schweitzer
2023 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
1
1
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Set theory from Schweitzer 2023
Prove that if
X
X{}
X
is an infinite set of cardinality
κ
\kappa
κ
then there is a collection
F
\mathcal{F}
F
of subsets of
X
X
X
such that [*]For any
A
⊆
X
A\subseteq X
A
⊆
X
with cardinality
κ
\kappa
κ
there exists
F
∈
F
F\in\mathcal{F}
F
∈
F
for which
A
∩
F
A\cap F
A
∩
F
has cardinality
κ
,
\kappa,
κ
,
and [*]
X
X
X
cannot be written as the union of less than
κ
\kappa
κ
sets from
F
\mathcal{F}
F
which all have cardinalities less than
κ
.
\kappa.
κ
.
9
1
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Polynomial in finite-codimensional subspace of C[-1,1]
Let
C
[
−
1
,
1
]
C[-1,1]
C
[
−
1
,
1
]
be the space of continuous real functions on the interval
[
−
1
,
1
]
[-1,1]
[
−
1
,
1
]
with the usual supremum norm, and let
V
V{}
V
be a closed, finite-codimensional subspace of
C
[
−
1
,
1
]
.
C[-1,1].
C
[
−
1
,
1
]
.
Prove that there exists a polynomial
p
∈
V
p\in V
p
∈
V
with norm at most one, which satisfies
p
′
(
0
)
>
2023.
p'(0)>2023.
p
′
(
0
)
>
2023.
2
1
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Sequence of subsets of Hausdorff space
Let
G
0
,
G
1
,
…
G_0, G_1,\ldots
G
0
,
G
1
,
…
be infinite open subsets of a Hausdorff space. Prove that there exist some infinite pairwise disjoint open sets
V
0
,
V
1
,
…
V_0,V_1,\ldots
V
0
,
V
1
,
…
and some indices
n
0
<
n
1
<
⋯
n_0<n_1<\cdots
n
0
<
n
1
<
⋯
such that
V
i
⊆
G
n
i
V_i\subseteq G_{n_i}
V
i
⊆
G
n
i
for every
i
⩾
0.
i\geqslant 0.
i
⩾
0.
3
1
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Inequality with distances in finite metric space
Let
X
=
{
x
0
,
x
1
,
…
,
x
n
}
X =\{x_0, x_1,\ldots , x_n\}
X
=
{
x
0
,
x
1
,
…
,
x
n
}
be the basis set of a finite metric space, where the points are inductively listed such that
x
k
x_k
x
k
maximizes the product of the distances from the points
{
x
0
,
x
1
,
…
,
x
k
−
1
}
\{x_0, x_1,\ldots , x_{k-1}\}
{
x
0
,
x
1
,
…
,
x
k
−
1
}
for each
1
⩽
k
⩽
n
.
1\leqslant k\leqslant n.
1
⩽
k
⩽
n
.
Prove that if for each
x
∈
X
x\in X
x
∈
X
we let
Π
x
\Pi_x
Π
x
be the product of the distances from
x
x{}
x
to every other point, then
Π
x
n
⩽
2
n
−
1
Π
x
\Pi_{x_n}\leqslant 2^{n-1}\Pi_x
Π
x
n
⩽
2
n
−
1
Π
x
for any
x
∈
X
.
x\in X.
x
∈
X
.
4
1
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Function must be a bivariate polynomial
Determine the pairs of sets
X
,
Y
⊂
R
X,Y\subset\mathbb{R}
X
,
Y
⊂
R
for which the following is true: if
f
(
x
,
y
)
f(x, y)
f
(
x
,
y
)
is a function on
X
×
Y
X\times Y{}
X
×
Y
such that for every
x
∈
X
x\in X
x
∈
X
it is equal to a polynomial in
y
y
y
on
Y
Y
Y
and for every
y
∈
Y
y\in Y
y
∈
Y
it is equal to a polynomial in
x
x
x
on
X
X
X
then
f
f
f
is a bivariate polynomial on
X
×
Y
.
X\times Y.
X
×
Y
.
5
1
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Group theory with functions
Let
G
G{}
G
be an arbitrary finite group, and let
t
n
(
G
)
t_n(G)
t
n
(
G
)
be the number of functions of the form f:G^n\to G, f(x_1,x_2,\ldots,x_n)=a_0x_1a_1\cdots x_na_n (a_0,\ldots,a_n\in G).Determine the limit of
t
n
(
G
)
1
/
n
t_n(G)^{1/n}
t
n
(
G
)
1/
n
as
n
n{}
n
tends to infinity.
8
1
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Complex polynomials
Let
q
q{}
q
be an arbitrary polynomial with complex coefficients which is not identically
0
0
0
and
Γ
q
=
{
z
:
∣
q
(
z
)
∣
=
1
}
\Gamma_q =\{z : |q(z)| = 1\}
Γ
q
=
{
z
:
∣
q
(
z
)
∣
=
1
}
be its contour line. Prove that for every point
z
0
∈
Γ
q
z_0\in\Gamma_q
z
0
∈
Γ
q
there is a polynomial
p
p{}
p
for which
∣
p
(
z
0
)
∣
=
1
|p(z_0)| = 1
∣
p
(
z
0
)
∣
=
1
and
∣
p
(
z
)
∣
<
1
|p(z)|<1
∣
p
(
z
)
∣
<
1
for any
z
∈
Γ
q
∖
{
z
0
}
.
z\in\Gamma_q\setminus\{z_0\}.
z
∈
Γ
q
∖
{
z
0
}
.
11
1
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Expected value of area of figure
Let
K
K{}
K
be an equilateral triangle of unit area, and choose
n
n{}
n
independent random points uniformly from
K
K{}
K
. Let
K
n
K_n
K
n
be the intersection of all translations of
K
K{}
K
that contain all the selected points. Determine the expected value of the area of
K
n
.
K_n.
K
n
.
10
1
Hide problems
Positive semi-definite matrices as exponents
Let
n
⩾
2
n\geqslant2
n
⩾
2
be a natural number. Show that there is no real number
c
c{}
c
for which
exp
(
T
+
S
2
)
⩽
c
⋅
exp
(
T
)
+
exp
(
S
)
2
\exp\left(\frac{T+S}{2}\right)\leqslant c\cdot \frac{\exp(T)+\exp(S)}{2}
exp
(
2
T
+
S
)
⩽
c
⋅
2
exp
(
T
)
+
exp
(
S
)
is satisfied for any self-adjoint
n
×
n
n\times n
n
×
n
complex matrices
T
T{}
T
and
S
S{}
S
. (If
A
A{}
A
and
B
B{}
B
are self-adjoint
n
×
n
n\times n
n
×
n
matrices,
A
⩽
B
A\leqslant B
A
⩽
B
means that
B
−
A
B-A
B
−
A
is positive semi-definite.)
7
1
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Sequence of sets of size 4
Prove that there exist two subsets
D
,
K
D, K
D
,
K
of
N
2
\mathbb{N}^2
N
2
such that for any
4
4
4
-element sets
A
1
,
A
2
A_1, A_2
A
1
,
A
2
we have
∣
A
1
∩
A
2
∣
=
1
|A_1 \cap A_2|=1
∣
A
1
∩
A
2
∣
=
1
if and only if there exist
4
4
4
-element sets
A
3
,
A
4
,
…
A_3, A_4, \ldots
A
3
,
A
4
,
…
, such that
A
i
∩
A
j
=
∅
A_i \cap A_j=\emptyset
A
i
∩
A
j
=
∅
for all
(
i
,
j
)
∈
D
(i, j) \in D
(
i
,
j
)
∈
D
and
∣
A
i
∩
A
j
∣
=
2
|A_i \cap A_j|=2
∣
A
i
∩
A
j
∣
=
2
for all
(
i
,
j
)
∈
K
(i, j) \in K
(
i
,
j
)
∈
K
.
6
1
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Existence of a number with a fixed number of divisors among 1,2,...,n
Prove that for all sufficiently large positive integers
n
n
n
and a positive integer
k
≤
n
k \leq n
k
≤
n
, there exists a positive integer
m
m
m
having exactly
k
k
k
divisors in the set
{
1
,
2
,
…
,
n
}
\{1,2, \ldots, n\}
{
1
,
2
,
…
,
n
}
.