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Miklós Schweitzer
2017 Miklós Schweitzer
2017 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
9
1
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Functional value at x is taken at points converging to x
Let
N
N
N
be a normed linear space with a dense linear subspace
M
M
M
. Prove that if
L
1
,
…
,
L
m
L_1,\ldots,L_m
L
1
,
…
,
L
m
are continuous linear functionals on
N
N
N
, then for all
x
∈
N
x\in N
x
∈
N
there exists a sequence
(
y
n
)
(y_n)
(
y
n
)
in
M
M
M
converging to
x
x
x
satisfying
L
j
(
y
n
)
=
L
j
(
x
)
L_j(y_n)=L_j(x)
L
j
(
y
n
)
=
L
j
(
x
)
for all
j
=
1
,
…
,
m
j=1,\ldots,m
j
=
1
,
…
,
m
and
n
∈
N
n\in \mathbb{N}
n
∈
N
.
7
1
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Set covers <s_n fraction of interval
Characterize all increasing sequences
(
s
n
)
(s_n)
(
s
n
)
of positive reals for which there exists a set
A
⊂
R
A\subset \mathbb{R}
A
⊂
R
with positive measure such that
λ
(
A
∩
I
)
<
s
n
n
\lambda(A\cap I)<\frac{s_n}{n}
λ
(
A
∩
I
)
<
n
s
n
holds for every interval
I
I
I
with length
1
/
n
1/n
1/
n
, where
λ
\lambda
λ
denotes the Lebesgue measure.
4
1
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Sum of n units =0 in number field
Let
K
K
K
be a number field which is neither
Q
\mathbb{Q}
Q
nor a quadratic imaginary extension of
Q
\mathbb{Q}
Q
. Denote by
L
(
K
)
\mathcal{L}(K)
L
(
K
)
the set of integers
n
≥
3
n\ge 3
n
≥
3
for which we can find units
ε
1
,
…
,
ε
n
∈
K
\varepsilon_1,\ldots,\varepsilon_n\in K
ε
1
,
…
,
ε
n
∈
K
for which
ε
1
+
⋯
+
ε
n
=
0
,
\varepsilon_1+\dots+\varepsilon_n=0,
ε
1
+
⋯
+
ε
n
=
0
,
but
∑
i
∈
I
ε
i
≠
0
\displaystyle\sum_{i\in I}\varepsilon_i\neq 0
i
∈
I
∑
ε
i
=
0
for any nonempty proper subset
I
I
I
of
{
1
,
2
,
…
,
n
}
\{1,2,\dots,n\}
{
1
,
2
,
…
,
n
}
. Prove that
L
(
K
)
\mathcal{L}(K)
L
(
K
)
is infinite, and that its smallest element can be bounded from above by a function of the degree and discriminant of
K
K
K
. Further, show that for infinitely many
K
K
K
,
L
(
K
)
\mathcal{L}(K)
L
(
K
)
contains infinitely many even and infinitely many odd elements.
3
1
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Positive degree of algebraic integer at most 2n
For every algebraic integer
α
\alpha
α
define its positive degree
deg
+
(
α
)
\text{deg}^+(\alpha)
deg
+
(
α
)
to be the minimal
k
∈
N
k\in\mathbb{N}
k
∈
N
for which there exists a
k
×
k
k\times k
k
×
k
matrix with non-negative integer entries with eigenvalue
α
\alpha
α
. Prove that for any
n
∈
N
n\in\mathbb{N}
n
∈
N
, every algebraic integer
α
\alpha
α
with degree
n
n
n
satisfies
deg
+
(
α
)
≤
2
n
\text{deg}^+(\alpha)\le 2n
deg
+
(
α
)
≤
2
n
.
2
1
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Field orderable iff sym matrixes diagonalizable over closure
Prove that a field
K
K
K
can be ordered if and only if every
A
∈
M
n
(
K
)
A\in M_n(K)
A
∈
M
n
(
K
)
symmetric matrix can be diagonalized over the algebraic closure of
K
K
K
. (In other words, for all
n
∈
N
n\in\mathbb{N}
n
∈
N
and all
A
∈
M
n
(
K
)
A\in M_n(K)
A
∈
M
n
(
K
)
, there exists an
S
∈
G
L
n
(
K
‾
)
S\in GL_n(\overline{K})
S
∈
G
L
n
(
K
)
for which
S
−
1
A
S
S^{-1}AS
S
−
1
A
S
is diagonal.)
10
1
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Absolute continuity of binary geometric sum random variables
Let
X
1
,
X
2
,
…
X_1,X_2,\ldots
X
1
,
X
2
,
…
be independent and identically distributed random variables with distribution
P
(
X
1
=
0
)
=
P
(
X
1
=
1
)
=
1
2
\mathbb{P}(X_1=0)=\mathbb{P}(X_1=1)=\frac12
P
(
X
1
=
0
)
=
P
(
X
1
=
1
)
=
2
1
. Let
Y
1
Y_1
Y
1
,
Y
2
Y_2
Y
2
,
Y
3
Y_3
Y
3
, and
Y
4
Y_4
Y
4
be independent, identically distributed random variables, where
Y
1
:
=
∑
k
=
1
∞
X
k
1
6
k
Y_1:=\sum_{k=1}^\infty \frac{X_k}{16^k}
Y
1
:=
∑
k
=
1
∞
1
6
k
X
k
. Decide whether the random variables
Y
1
+
2
Y
2
+
4
Y
3
+
8
Y
4
Y_1+2Y_2+4Y_3+8Y_4
Y
1
+
2
Y
2
+
4
Y
3
+
8
Y
4
and
Y
1
+
4
Y
3
Y_1+4Y_3
Y
1
+
4
Y
3
are absolutely continuous.
8
1
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Analysis of dyadic (-1)^{partial sum} sum function
Let the base
2
2
2
representation of
x
∈
[
0
;
1
)
x\in[0;1)
x
∈
[
0
;
1
)
be
x
=
∑
i
=
0
∞
x
i
2
i
+
1
x=\sum_{i=0}^\infty \frac{x_i}{2^{i+1}}
x
=
∑
i
=
0
∞
2
i
+
1
x
i
. (If
x
x
x
is dyadically rational, i.e.
x
∈
{
k
2
n
:
k
,
n
∈
Z
}
x\in\left\{\frac{k}{2^n}\,:\, k,n\in\mathbb{Z}\right\}
x
∈
{
2
n
k
:
k
,
n
∈
Z
}
, then we choose the finite representation.) Define function
f
n
:
[
0
;
1
)
→
Z
f_n:[0;1)\to\mathbb{Z}
f
n
:
[
0
;
1
)
→
Z
by
f
n
(
x
)
=
∑
j
=
0
n
−
1
(
−
1
)
∑
i
=
0
j
x
i
.
f_n(x)=\sum_{j=0}^{n-1}(-1)^{\sum_{i=0}^j x_i}.
f
n
(
x
)
=
j
=
0
∑
n
−
1
(
−
1
)
∑
i
=
0
j
x
i
.
Does there exist a function
φ
:
[
0
;
∞
)
→
[
0
;
∞
)
\varphi:[0;\infty)\to[0;\infty)
φ
:
[
0
;
∞
)
→
[
0
;
∞
)
such that
lim
x
→
∞
φ
(
x
)
=
∞
\lim_{x\to\infty} \varphi(x)=\infty
lim
x
→
∞
φ
(
x
)
=
∞
and
sup
n
∈
N
∫
0
1
φ
(
∣
f
n
(
x
)
∣
)
d
x
<
∞
?
\sup_{n\in\mathbb{N}}\int_0^1 \varphi(|f_n(x)|)\mathrm{d}x<\infty\, ?
n
∈
N
sup
∫
0
1
φ
(
∣
f
n
(
x
)
∣
)
d
x
<
∞
?
6
1
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Technical problem phi(x)+psi(x)=x functions
Let
I
I
I
and
J
J
J
be intervals. Let
φ
,
ψ
:
I
→
R
\varphi,\psi:I\to\mathbb{R}
φ
,
ψ
:
I
→
R
be strictly increasing continuous functions and let
Φ
,
Ψ
:
J
→
R
\Phi,\Psi:J\to\mathbb{R}
Φ
,
Ψ
:
J
→
R
be continuous functions. Suppose that
φ
(
x
)
+
ψ
(
x
)
=
x
\varphi(x)+\psi(x)=x
φ
(
x
)
+
ψ
(
x
)
=
x
and
Φ
(
u
)
+
Ψ
(
u
)
=
u
\Phi(u)+\Psi(u)=u
Φ
(
u
)
+
Ψ
(
u
)
=
u
holds for all
x
∈
I
x\in I
x
∈
I
and
u
∈
J
u\in J
u
∈
J
. Show that if
f
:
I
→
J
f:I\to J
f
:
I
→
J
is a continuous solution of the functional inequality
f
(
φ
(
x
)
+
ψ
(
y
)
)
≤
Φ
(
f
(
x
)
)
+
Ψ
(
f
(
y
)
)
(
x
,
y
∈
I
)
,
f\big(\varphi(x)+\psi(y)\big)\le \Phi\big(f(x)\big)+\Psi\big(f(y)\big)\qquad (x,y\in I),
f
(
φ
(
x
)
+
ψ
(
y
)
)
≤
Φ
(
f
(
x
)
)
+
Ψ
(
f
(
y
)
)
(
x
,
y
∈
I
)
,
then
Φ
∘
f
∘
φ
−
1
\Phi\circ f\circ \varphi^{-1}
Φ
∘
f
∘
φ
−
1
and
Ψ
∘
f
∘
ψ
−
1
\Psi\circ f\circ \psi^{-1}
Ψ
∘
f
∘
ψ
−
1
are convex functions.
5
1
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Places where |p(z)|=1
For every non-constant polynomial
p
p
p
, let
H
p
=
{
z
∈
C
∣
∣
p
(
z
)
∣
=
1
}
H_p=\big\{z\in \mathbb{C} \, \big| \, |p(z)|=1\big\}
H
p
=
{
z
∈
C
∣
p
(
z
)
∣
=
1
}
. Prove that if
H
p
=
H
q
H_p=H_q
H
p
=
H
q
for some polynomials
p
,
q
p,q
p
,
q
, then there exists a polynomial
r
r
r
such that
p
=
r
m
p=r^m
p
=
r
m
and
q
=
ξ
⋅
r
n
q=\xi\cdot r^n
q
=
ξ
⋅
r
n
for some positive integers
m
,
n
m,n
m
,
n
and constant
∣
ξ
∣
=
1
|\xi|=1
∣
ξ
∣
=
1
.
1
1
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Triangulation with no shared sides
Can one divide a square into finitely many triangles such that no two triangles share a side? (The triangles have pairwise disjoint interiors and their union is the square.)