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Miklós Schweitzer
1984 Miklós Schweitzer
1984 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklós Schweitzer 1984- Problem 10
10. Let
X
1
,
X
2
,
…
X_1, X_2, \dots
X
1
,
X
2
,
…
be independent random variables with the same distribution
P
(
X
i
=
1
)
=
P
(
X
i
=
−
1
)
=
1
2
(
i
=
1
,
2
,
…
)
P(X_i = 1) = P(X_i = -1)=\frac{1}{2}\qquad (i= 1, 2, \dots )
P
(
X
i
=
1
)
=
P
(
X
i
=
−
1
)
=
2
1
(
i
=
1
,
2
,
…
)
Define
S
0
=
0
,
S
n
=
X
1
+
X
2
+
⋯
+
X
n
(
n
=
1
,
2
,
…
S_0=0, Sn=X_1 +X_2+\dots +X_n \qquad (n=1, 2, \dots
S
0
=
0
,
S
n
=
X
1
+
X
2
+
⋯
+
X
n
(
n
=
1
,
2
,
…
),
ξ
(
x
,
n
)
=
∣
{
k
:
0
≤
k
≤
n
,
S
k
=
x
}
∣
(
x
=
0
,
±
1
,
±
2
,
…
\xi (x,n) = \left | \{k : 0 \leq k \leq n, S_k= x \} \right |\qquad (x=0, \pm 1, \pm 2, \dots
ξ
(
x
,
n
)
=
∣
{
k
:
0
≤
k
≤
n
,
S
k
=
x
}
∣
(
x
=
0
,
±
1
,
±
2
,
…
),and
α
(
n
)
=
∣
{
x
:
ξ
(
x
,
n
)
=
a
}
∣
(
n
=
0
,
1
,
…
\alpha(n)= \left | \{ x: \xi(x,n)=a \} \right |\qquad (n=0,1,\dots
α
(
n
)
=
∣
{
x
:
ξ
(
x
,
n
)
=
a
}
∣
(
n
=
0
,
1
,
…
).Prove that
P
(
lim
inf
α
(
n
)
=
0
)
=
1
P(\lim \inf \alpha(n)=0) =1
P
(
lim
in
f
α
(
n
)
=
0
)
=
1
and that there is a number
0
<
c
<
∞
0<c<\infty
0
<
c
<
∞
such that
P
(
lim
inf
α
(
n
)
/
log
n
=
c
)
=
1
P(\lim \inf \alpha(n)/\log n=c) =1
P
(
lim
in
f
α
(
n
)
/
lo
g
n
=
c
)
=
1
(P.24) [P. Révész]
9
1
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Miklós Schweitzer 1984- Problem 9
9. Let
X
0
,
X
1
,
…
X_0, X_1, \dots
X
0
,
X
1
,
…
be independent, indentically distributed, nondegenerate random variables, and let
0
<
α
<
1
0<\alpha <1
0
<
α
<
1
be a real number. Assume that the series
∑
k
=
1
∞
α
k
X
k
\sum_{k=1}^{\infty} \alpha^{k} X_k
∑
k
=
1
∞
α
k
X
k
is convergent with probability one. Prove that the distribution function of the sum is continuous. (P. 23) [T. F. Móri]
8
1
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Miklós Schweitzer 1984- Problem 8
8. Among all point lattices on the plane intersecting every closed convex region of unit width, which on's fundamental parallelogram has the largest area? (G.36) [L. Fejes-Tóth]
7
1
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Miklós Schweitzer 1984- Problem 7
7. Let
V
V
V
be a finite-dimensional subspace of
C
[
0
,
1
]
C[0,1]
C
[
0
,
1
]
such that every nonzero
f
∈
V
f\in V
f
∈
V
attains positive value at some point. Prove that there exists a polynomial
P
P
P
that is strictly positive on
[
0
,
1
]
[0,1]
[
0
,
1
]
and orthogonal to
V
V
V
, that is, for every
f
∈
V
f \in V
f
∈
V
,
∫
0
1
f
(
x
)
P
(
x
)
d
x
=
0
\int_{0}^{1} f(x)P(x)dx =0
∫
0
1
f
(
x
)
P
(
x
)
d
x
=
0
(F.39) [A. Pinkus, V. Totik]
6
1
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Miklós Schweitzer 1984- Problem 6
6. For which Lebesgue-measurable subsets
E
E
E
of the real line does a positive constant
c
c
c
exist for which
sup
−
∞
<
t
<
∞
∣
∫
E
e
i
t
x
f
(
x
)
d
x
∣
≤
c
sup
n
=
0
,
±
1
,
…
∣
∫
E
e
i
n
x
f
(
x
)
d
x
∣
\sup_{-\infty < t<\infty} \left | \int_{E} e^{itx} f(x) dx\right | \leq c \sup_{n=0,\pm 1,\dots } \left | \int_{E} e^{inx} f(x) dx\right |
sup
−
∞
<
t
<
∞
∫
E
e
i
t
x
f
(
x
)
d
x
≤
c
sup
n
=
0
,
±
1
,
…
∫
E
e
in
x
f
(
x
)
d
x
for all integrable functions
f
f
f
on
E
E
E
? (M.17) [G. Halász]
5
1
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Miklós Schweitzer 1984- Problem 5
5. Let
a
0
,
a
1
,
…
a_0 , a_1 , \dots
a
0
,
a
1
,
…
be nonnegative real numbers such that
∑
n
=
0
∞
a
n
=
∞
\sum_{n=0}^{\infty}a_n = \infty
∑
n
=
0
∞
a
n
=
∞
For arbitrary
c
>
0
c>0
c
>
0
, let
n
j
(
c
)
=
min
{
k
:
c
.
j
≤
∑
i
=
0
k
a
i
}
n_{j}(c)= \min \left \{ k : c.j \leq \sum_{i=0}^{k} a_i \right \}
n
j
(
c
)
=
min
{
k
:
c
.
j
≤
∑
i
=
0
k
a
i
}
,
j
=
1
,
2
,
…
j= 1,2, \dots
j
=
1
,
2
,
…
Prove that if
∑
i
=
0
∞
a
i
2
=
∞
\sum_{i=0}^{\infty}a_i^2 = \infty
∑
i
=
0
∞
a
i
2
=
∞
, then there exists a
c
>
0
c>0
c
>
0
for which
∑
j
=
1
∞
a
n
j
(
c
)
=
∞
\sum_{j=1}^{\infty} a_{n_j (c)} = \infty
∑
j
=
1
∞
a
n
j
(
c
)
=
∞
.(S.24) [P. Erdos, I. Joó, L. Székely]
4
1
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Miklós Schweitzer 1984- Problem 4
4. Let
x
1
,
x
2
,
y
1
,
y
2
,
z
1
,
z
2
x_1 , x_2 , y_1 , y_2 , z_1 , z_2
x
1
,
x
2
,
y
1
,
y
2
,
z
1
,
z
2
be transcendental numbers. Suppose that any 3 of them are algebraically independent, and among the 15 four-tuples on
{
x
1
,
x
2
,
y
1
,
y
2
}
\{x_1 , x_2 , y_1, y_2 \}
{
x
1
,
x
2
,
y
1
,
y
2
}
,
{
x
1
,
x
2
,
z
1
,
z
2
}
\{ x_1 , x_2 , z_1 , z_2 \}
{
x
1
,
x
2
,
z
1
,
z
2
}
and
{
y
1
,
y
2
,
z
1
,
z
2
}
\{y_1 , y_2 , z_1 , z_2 \}
{
y
1
,
y
2
,
z
1
,
z
2
}
are algebraically dependent. Prove that there exists a transcendental number
t
t
t
that depends algebraically on each of the pairs
{
x
1
,
x
2
}
\{ x_1 , x_2\}
{
x
1
,
x
2
}
,
{
y
1
,
y
2
}
\{ y_1 , y_2 \}
{
y
1
,
y
2
}
, and
{
z
1
,
z
2
}
\{ z_1 , z_2 \}
{
z
1
,
z
2
}
. (A.37) [L. Lovász]
3
1
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Miklós Schweitzer 1984- Problem 3
3. Let
a
a
a
and
b
b
b
be positive integers such that when dividing them by any prime
p
p
p
, the remainder of
a
a
a
is always less than or equal to the remainder of
b
b
b
. Prove that
a
=
b
a=b
a
=
b
. (N.16) [P. Erdos, P. P. Pálify]
2
1
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Miklós Schweitzer 1984- Problem 2
2. Show that threre exist a compact set
K
⊂
R
K \subset \mathbb{R}
K
⊂
R
and a set
A
⊂
R
A \subset \mathbb{R}
A
⊂
R
of type
F
σ
F_{\sigma}
F
σ
such that the set
{
x
∈
R
:
K
+
x
⊂
A
}
\{ x\in \mathbb{R} : K+x \subset A\}
{
x
∈
R
:
K
+
x
⊂
A
}
is not Borel-measurable (here
K
+
x
=
{
y
+
x
:
y
∈
K
}
K+x = \{y+x : y \in K\}
K
+
x
=
{
y
+
x
:
y
∈
K
}
). (M.16) [M. Laczkovich]
1
1
Hide problems
Miklós Schweitzer 1984- Problem 1
1. Let
k
k
k
be an arbitrary cardinality. Show that there exists a tournament
T
k
=
(
V
n
,
E
n
)
T_k = (V_n , E_n)
T
k
=
(
V
n
,
E
n
)
such that for any coloring
f
:
E
n
→
k
f: E_n \to k
f
:
E
n
→
k
of the edge set
E
n
E_n
E
n
, there are three different vertices
x
0
,
x
1
,
x
2
∈
V
n
x_0 , x_1 , x_2 \in V_n
x
0
,
x
1
,
x
2
∈
V
n
such that
x
0
x
1
,
x
1
x
2
,
x
2
x
0
∈
E
n
x_0 x_1 , x_1 x_2 , x_2 x_0 \in E_n
x
0
x
1
,
x
1
x
2
,
x
2
x
0
∈
E
n
and
∣
{
f
(
x
0
x
1
)
,
f
(
x
1
x
2
)
,
f
(
x
2
x
0
)
}
∣
≤
2
\left | \{ f(x_0 x_1), f(x_1 x_2), f(x_2 x_0)\} \right |\leq 2
∣
{
f
(
x
0
x
1
)
,
f
(
x
1
x
2
)
,
f
(
x
2
x
0
)}
∣
≤
2
(A tounament is a directed graph such that for any vertices
x
,
y
∈
V
n
,
x
≠
y
x, y \in V_n, x \neq y
x
,
y
∈
V
n
,
x
=
y
exactly one of the relations
x
y
∈
E
n
xy \in E_n
x
y
∈
E
n
holds.) (C.19) [A. Hajnal]