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Miklós Schweitzer
1967 Miklós Schweitzer
1967 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklos Schweitzer 1967_10
Let
σ
(
S
n
,
k
)
\sigma(S_n,k)
σ
(
S
n
,
k
)
denote the sum of the
k
k
k
th powers of the lengths of the sides of the convex
n
n
n
-gon
S
n
S_n
S
n
inscribed in a unit circle. Show that for any natural number greater than
2
2
2
there exists a real number
k
0
k_0
k
0
between
1
1
1
and
2
2
2
such that
σ
(
S
n
,
k
0
)
\sigma(S_n,k_0)
σ
(
S
n
,
k
0
)
attains its maximum for the regular
n
n
n
-gon. L. Fejes Toth
9
1
Hide problems
Miklos Schweitzer 1967_9
Let
F
F
F
be a surface of nonzero curvature that can be represented around one of its points
P
P
P
by a power series and is symmetric around the normal planes parallel to the principal directions at
P
P
P
. Show that the derivative with respect to the arc length of the curvature of an arbitrary normal section at
P
P
P
vanishes at
P
P
P
. Is it possible to replace the above symmetry condition by a weaker one? A. Moor
8
1
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Miklos Schweitzer 1967_8
Suppose that a bounded subset
S
S
S
of the plane is a union of congruent, homothetic, closed triangles. Show that the boundary of
S
S
S
can be covered by a finite number of rectifiable arcs. L. Geher
7
1
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Miklos Schweitzer 1967_7
Let
U
U
U
be an
n
×
n
n \times n
n
×
n
orthogonal matrix. Prove that for any
n
×
n
n \times n
n
×
n
matrix
A
A
A
, the matrices
A
m
=
1
m
+
1
∑
j
=
0
m
U
−
j
A
U
j
A_m=\frac{1}{m+1} \sum_{j=0}^m U^{-j}AU^j
A
m
=
m
+
1
1
j
=
0
∑
m
U
−
j
A
U
j
converge entrywise as
m
→
∞
.
m \rightarrow \infty.
m
→
∞.
L. Kovacs
6
1
Hide problems
Miklos Schweitzer 1967_6
Let
A
A
A
be a family of proper closed subspaces of the Hilbert space H\equal{}l^2 totally ordered with respect to inclusion (that is , if
L
1
,
L
2
∈
A
L_1,L_2 \in A
L
1
,
L
2
∈
A
, then either
L
1
⊂
L
2
L_1\subset L_2
L
1
⊂
L
2
or
L
2
⊂
L
1
L_2\subset L_1
L
2
⊂
L
1
). Prove that there exists a vector
x
∈
H
x \in H
x
∈
H
not contaied in any of the subspaces
L
L
L
belonging to
A
A
A
. B. Szokefalvi Nagy
5
1
Hide problems
Miklos Schweitzer 1967_5
Let
f
f
f
be a continuous function on the unit interval
[
0
,
1
]
[0,1]
[
0
,
1
]
. Show that
lim
n
→
∞
∫
0
1
.
.
.
∫
0
1
f
(
x
1
+
.
.
.
+
x
n
n
)
d
x
1
.
.
.
d
x
n
=
f
(
1
2
)
\lim_{n \rightarrow \infty} \int_0^1... \int_0^1f(\frac{x_1+...+x_n}{n})dx_1...dx_n=f(\frac12)
n
→
∞
lim
∫
0
1
...
∫
0
1
f
(
n
x
1
+
...
+
x
n
)
d
x
1
...
d
x
n
=
f
(
2
1
)
and
lim
n
→
∞
∫
0
1
.
.
.
∫
0
1
f
(
x
1
.
.
.
x
n
n
)
d
x
1
.
.
.
d
x
n
=
f
(
1
e
)
.
\lim_{n \rightarrow \infty} \int_0^1... \int_0^1f (\sqrt[n]{x_1...x_n})dx_1...dx_n=f(\frac1e).
n
→
∞
lim
∫
0
1
...
∫
0
1
f
(
n
x
1
...
x
n
)
d
x
1
...
d
x
n
=
f
(
e
1
)
.
4
1
Hide problems
Miklos Schweitzer 1967_4
Let
a
1
,
a
2
,
.
.
.
,
a
N
a_1,a_2,...,a_N
a
1
,
a
2
,
...
,
a
N
be positive real numbers whose sum equals
1
1
1
. For a natural number
i
i
i
, let
n
i
n_i
n
i
denote the number of
a
k
a_k
a
k
for which
2
1
−
i
≥
a
k
≥
2
−
i
2^{1-i} \geq a_k \geq 2^{-i}
2
1
−
i
≥
a
k
≥
2
−
i
holds. Prove that
∑
i
=
1
∞
n
i
2
−
i
≤
4
+
log
2
N
.
\sum_{i=1}^{\infty} \sqrt{n_i2^{-i}} \leq 4+\sqrt{\log_2 N}.
i
=
1
∑
∞
n
i
2
−
i
≤
4
+
lo
g
2
N
.
L. Leinder
3
1
Hide problems
Miklos Schweitzer 1967_3
Prove that if an infinite, noncommutative group
G
G
G
contains a proper normal subgroup with a commutative factor group, then
G
G
G
also contains an infinite proper normal subgroup. B. Csakany
2
1
Hide problems
Miklos Schweitzer 1967_2
Let
K
K
K
be a subset of a group
G
G
G
that is not a union of lift cosets of a proper subgroup. Prove that if
G
G
G
is a torsion group or if
K
K
K
is a finite set, then the subset
⋂
k
∈
K
k
−
1
K
\bigcap _{k \in K} k^{-1}K
k
∈
K
⋂
k
−
1
K
consists of the identity alone. L. Redei
1
1
Hide problems
Miklos Schweitzer 1967_1
Let f(x)\equal{}a_0\plus{}a_1x\plus{}a_2x^2\plus{}a_{10}x^{10}\plus{}a_{11}x^{11}\plus{}a_{12}x^{12}\plus{}a_{13}x^{13} \; (a_{13} \not\equal{}0) and g(x)\equal{}b_0\plus{}b_1x\plus{}b_2x^2\plus{}b_{3}x^{3}\plus{}b_{11}x^{11}\plus{}b_{12}x^{12}\plus{}b_{13}x^{13} \; (b_{3} \not\equal{}0) be polynomials over the same field. Prove that the degree of their greatest common divisor is at least
6
6
6
. L. Redei