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Miklós Schweitzer
1972 Miklós Schweitzer
1972 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
11
1
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Miklos Schweitzer 1972_11
We throw
N
N
N
balls into
n
n
n
urns, one by one, independently and uniformly. Let X_i\equal{}X_i(N,n) be the total number of balls in the
i
i
i
th urn. Consider the random variable y(N,n)\equal{}\min_{1 \leq i \leq n}|X_i\minus{}\frac Nn|. Verify the following three statements: (a) If
n
→
∞
n \rightarrow \infty
n
→
∞
and
N
/
n
3
→
∞
N/n^3 \rightarrow \infty
N
/
n
3
→
∞
, then P \left(\frac{y(N,n)}{\frac 1n \sqrt{\frac Nn}}
0 \ . (b) If
n
→
∞
n\rightarrow \infty
n
→
∞
and
N
/
n
3
≤
K
N/n^3 \leq K
N
/
n
3
≤
K
(
K
K
K
constant), then for any
ε
>
0
\varepsilon > 0
ε
>
0
there is an
A
>
0
A > 0
A
>
0
such that P(y(N,n) < A) > 1\minus{}\varepsilon . (c) If
n
→
∞
n \rightarrow \infty
n
→
∞
and
N
/
n
3
→
0
N/n^3 \rightarrow 0
N
/
n
3
→
0
then
P
(
y
(
N
,
n
)
<
1
)
→
1.
P(y(N,n) < 1) \rightarrow 1.
P
(
y
(
N
,
n
)
<
1
)
→
1.
P. Revesz
10
1
Hide problems
Miklos Schweitzer 1972_10
Let
T
1
\mathcal{T}_1
T
1
and
T
2
\mathcal{T}_2
T
2
be second-countable topologies on the set
E
E
E
. We would like to find a real function
σ
\sigma
σ
defined on
E
×
E
E \times E
E
×
E
such that 0 \leq \sigma(x,y) <\plus{}\infty, \;\sigma(x,x)\equal{}0 \ , \sigma(x,z) \leq \sigma(x,y)\plus{}\sigma(y,z) \;(x,y,z \in E) \ , and, for any
p
∈
E
p \in E
p
∈
E
, the sets V_1(p,\varepsilon)\equal{}\{ x : \;\sigma(x,p)< \varepsilon \ \} \;(\varepsilon >0) form a neighborhood base of
p
p
p
with respect to
T
1
\mathcal{T}_1
T
1
, and the sets V_2(p,\varepsilon)\equal{}\{ x : \;\sigma(p,x)< \varepsilon \ \} \;(\varepsilon >0) form a neighborhood base of
p
p
p
with respect to
T
2
\mathcal{T}_2
T
2
. Prove that such a function
σ
\sigma
σ
exists if and only if, for any
p
∈
E
p \in E
p
∈
E
and
T
i
\mathcal{T}_i
T
i
-open set G \ni p \;(i\equal{}1,2) , there exist a
T
i
\mathcal{T}_i
T
i
-open set
G
′
G'
G
′
and a \mathcal{T}_{3\minus{}i}-closed set
F
F
F
with
p
∈
G
′
⊂
F
⊂
G
.
p \in G' \subset F \subset G.
p
∈
G
′
⊂
F
⊂
G
.
A. Csaszar
9
1
Hide problems
Miklos Schweitzer 1972_9
Let
K
K
K
be a compact convex body in the
n
n
n
-dimensional Euclidean space. Let P_1,P_2,...,P_{n\plus{}1} be the vertices of a simplex having maximal volume among all simplices inscribed in
K
K
K
. Define the points P_{n\plus{}2},P_{n\plus{}3},... successively so that P_k \;(k>n\plus{}1) is a point of
K
K
K
for which the volume of the convex hull of
P
1
,
.
.
.
,
P
k
P_1,...,P_k
P
1
,
...
,
P
k
is maximal. Denote this volume by
V
k
V_k
V
k
. Decide, for different values of
n
n
n
, about the truth of the statement "the sequence V_{n\plus{}1},V_{n\plus{}2},... is concave." L. Fejes- Toth, E. Makai
8
1
Hide problems
Miklos Schweitzer 1972_8
Given four points
A
1
,
A
2
,
A
3
,
A
4
A_1,A_2,A_3,A_4
A
1
,
A
2
,
A
3
,
A
4
in the plane in such a way that
A
4
A_4
A
4
is the centroid of the
△
A
1
A
2
A
3
\bigtriangleup A_1A_2A_3
△
A
1
A
2
A
3
, find a point
A
5
A_5
A
5
in the plane that maximizes the ratio
min
1
≤
i
<
j
<
k
≤
5
T
(
A
i
A
j
A
k
)
max
1
≤
i
<
j
<
k
≤
5
T
(
A
i
A
j
A
k
)
.
\frac{\min_{1 \leq i < j < k \leq 5}T(A_iA_jA_k)}{\max_{1 \leq i < j < k \leq 5}T(A_iA_jA_k)}.
max
1
≤
i
<
j
<
k
≤
5
T
(
A
i
A
j
A
k
)
min
1
≤
i
<
j
<
k
≤
5
T
(
A
i
A
j
A
k
)
.
(
T
(
A
B
C
)
T(ABC)
T
(
A
BC
)
denotes the area of the triangle
△
A
B
C
.
\bigtriangleup ABC.
△
A
BC
.
) J. Suranyi
7
1
Hide problems
Miklos Schweitzer 1972_7
Let
f
(
x
,
y
,
z
)
f(x,y,z)
f
(
x
,
y
,
z
)
be a nonnegative harmonic function in the unit ball of
R
3
\mathbb{R}^3
R
3
for which the inequality
f
(
x
0
,
0
,
0
)
≤
ε
2
f(x_0,0,0) \leq \varepsilon^2
f
(
x
0
,
0
,
0
)
≤
ε
2
holds for some
0
≤
x
0
≤
1
0\leq x_0 \leq 1
0
≤
x
0
≤
1
and 0<\varepsilon<(1\minus{}x_0)^2. Prove that
f
(
x
,
y
,
z
)
≤
ε
f(x,y,z) \leq \varepsilon
f
(
x
,
y
,
z
)
≤
ε
in the ball with center at the origin an radius (1\minus{}3\varepsilon^{1/4}). P. Turan
6
1
Hide problems
Miklos Schweitzer 1972_6
Let
P
(
z
)
P(z)
P
(
z
)
be a polynomial of degree
n
n
n
with complex coefficients, P(0)\equal{}1, \;\textrm{and}\ \;|P(z)|\leq M\ \;\textrm{for}\ \;|z| \leq 1\ . Prove that every root of
P
(
z
)
P(z)
P
(
z
)
in the closed unit disc has multiplicity at most
c
n
c\sqrt{n}
c
n
, where c\equal{}c(M) >0 is a constant depending only on
M
M
M
. G. Halasz
5
1
Hide problems
Miklos Schweitzer 1972_5
We say that the real-valued function
f
(
x
)
f(x)
f
(
x
)
defined on the interval
(
0
,
1
)
(0,1)
(
0
,
1
)
is approximately continuous on
(
0
,
1
)
(0,1)
(
0
,
1
)
if for any
x
0
∈
(
0
,
1
)
x_0 \in (0,1)
x
0
∈
(
0
,
1
)
and
ε
>
0
\varepsilon >0
ε
>
0
the point
x
0
x_0
x
0
is a point of interior density
1
1
1
of the set H\equal{} \{x : \;|f(x)\minus{}f(x_0)|< \varepsilon \ \}. Let
F
⊂
(
0
,
1
)
F \subset (0,1)
F
⊂
(
0
,
1
)
be a countable closed set, and
g
(
x
)
g(x)
g
(
x
)
a real-valued function defined on
F
F
F
. Prove the existence of an approximately continuous function
f
(
x
)
f(x)
f
(
x
)
defined on
(
0
,
1
)
(0,1)
(
0
,
1
)
such that f(x)\equal{}g(x) \;\textrm{for all}\ \;x \in F\ . M. Laczkovich, Gy. Petruska
4
1
Hide problems
Miklos Schweitzer 1972_4
Let
G
G
G
be a solvable torsion group in which every Abelian subgroup is finitely generated. Prove that
G
G
G
is finite. J. Pelikan
3
1
Hide problems
Miklos Schweitzer 1972_3
Let
λ
i
(
i
=
1
,
2
,
.
.
.
)
\lambda_i \;(i=1,2,...)
λ
i
(
i
=
1
,
2
,
...
)
be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form
μ
=
∑
i
=
1
∞
n
i
λ
i
,
\mu= \sum_{i=1}^{\infty}n_i\lambda_i ,
μ
=
i
=
1
∑
∞
n
i
λ
i
,
where
n
i
≥
0
n_i \geq 0
n
i
≥
0
are integers and all but finitely many
n
i
n_i
n
i
are
0
0
0
. Let
L
(
x
)
=
∑
λ
i
≤
x
1
and
M
(
x
)
=
∑
μ
≤
x
1
.
L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ .
L
(
x
)
=
λ
i
≤
x
∑
1
and
M
(
x
)
=
μ
≤
x
∑
1
.
(In the latter sum, each
μ
\mu
μ
occurs as many times as its number of representations in the above form.) Prove that if
lim
x
→
∞
L
(
x
+
1
)
L
(
x
)
=
1
,
\lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,
x
→
∞
lim
L
(
x
)
L
(
x
+
1
)
=
1
,
then
lim
x
→
∞
M
(
x
+
1
)
M
(
x
)
=
1.
\lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.
x
→
∞
lim
M
(
x
)
M
(
x
+
1
)
=
1.
G. Halasz
2
1
Hide problems
Miklos Schweitzer 1972_2
Let
≤
\leq
≤
be a reflexive, antisymmetric relation on a finite set
A
A
A
. Show that this relation can be extended to an appropriate finite superset
B
B
B
of
A
A
A
such that
≤
\leq
≤
on
B
B
B
remains reflexive, antisymmetric, and any two elements of
B
B
B
have a least upper bound as well as a greatest lower bound. (The relation
≤
\leq
≤
is extended to
B
B
B
if for
x
,
y
∈
A
,
x
≤
y
x,y \in A , x \leq y
x
,
y
∈
A
,
x
≤
y
holds in
A
A
A
if and only if it holds in
B
B
B
.) E. Freid
1
1
Hide problems
Miklos Schweitzer 1972_1
Let
F
\mathcal{F}
F
be a nonempty family of sets with the following properties: (a) If
X
∈
F
X \in \mathcal{F}
X
∈
F
, then there are some
Y
∈
F
Y \in \mathcal{F}
Y
∈
F
and
Z
∈
F
Z \in \mathcal{F}
Z
∈
F
such that
Y
∩
Z
=
∅
Y \cap Z =\emptyset
Y
∩
Z
=
∅
and
Y
∪
Z
=
X
Y \cup Z=X
Y
∪
Z
=
X
. (b) If
X
∈
F
X \in \mathcal{F}
X
∈
F
, and
Y
∪
Z
=
X
,
Y
∩
Z
=
∅
Y \cup Z =X , Y \cap Z=\emptyset
Y
∪
Z
=
X
,
Y
∩
Z
=
∅
, then either
Y
∈
F
Y \in \mathcal{F}
Y
∈
F
or
Z
∈
F
Z \in \mathcal{F}
Z
∈
F
. Show that there is a decreasing sequence
X
0
⊇
X
1
⊇
X
2
⊇
.
.
.
X_0 \supseteq X_1 \supseteq X_2 \supseteq ...
X
0
⊇
X
1
⊇
X
2
⊇
...
of sets
X
n
∈
F
X_n \in \mathcal{F}
X
n
∈
F
such that
⋂
n
=
0
∞
X
n
=
∅
.
\bigcap_{n=0}^{\infty} X_n= \emptyset.
n
=
0
⋂
∞
X
n
=
∅.
F. Galvin