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Miklós Schweitzer
1963 Miklós Schweitzer
1963 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklos Schweitzer 1963_10
Select
n
n
n
points on a circle independently with uniform distribution. Let
P
n
P_n
P
n
be the probability that the center of the circle is in the interior of the convex hull of these
n
n
n
points. Calculate the probabilities
P
3
P_3
P
3
and
P
4
P_4
P
4
. [A. Renyi]
9
1
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Miklos Schweitzer 1963_9
Let
f
(
t
)
f(t)
f
(
t
)
be a continuous function on the interval
0
≤
t
≤
1
0 \leq t \leq 1
0
≤
t
≤
1
, and define the two sets of points A_t\equal{}\{(t,0): t\in[0,1]\} , B_t\equal{}\{(f(t),1): t\in [0,1]\}. Show that the union of all segments
A
t
B
t
‾
\overline{A_tB_t}
A
t
B
t
is Lebesgue-measurable, and find the minimum of its measure with respect to all functions
f
f
f
. [A. Csaszar]
8
1
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Miklos Schweitzer 1963_8
Let the Fourier series
a
0
2
+
∑
k
≥
1
(
a
k
cos
k
x
+
b
k
sin
k
x
)
\frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)
2
a
0
+
k
≥
1
∑
(
a
k
cos
k
x
+
b
k
sin
k
x
)
of a function
f
(
x
)
f(x)
f
(
x
)
be absolutely convergent, and let
a
k
2
+
b
k
2
≥
a
k
+
1
2
+
b
k
+
1
2
(
k
=
1
,
2
,
.
.
.
)
.
a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .
a
k
2
+
b
k
2
≥
a
k
+
1
2
+
b
k
+
1
2
(
k
=
1
,
2
,
...
)
.
Show that
1
h
∫
0
2
π
(
f
(
x
+
h
)
−
f
(
x
−
h
)
)
2
d
x
(
h
>
0
)
\frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)
h
1
∫
0
2
π
(
f
(
x
+
h
)
−
f
(
x
−
h
)
)
2
d
x
(
h
>
0
)
is uniformly bounded in
h
h
h
. [K. Tandori]
7
1
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Miklos Schweitzer 1963_7
Prove that for every convex function
f
(
x
)
f(x)
f
(
x
)
defined on the interval \minus{}1\leq x \leq 1 and having absolute value at most
1
1
1
, there is a linear function
h
(
x
)
h(x)
h
(
x
)
such that \int_{\minus{}1}^1|f(x)\minus{}h(x)|dx\leq 4\minus{}\sqrt{8}. [L. Fejes-Toth]
6
1
Hide problems
Miklos Schweitzer 1963_6
Show that if
f
(
x
)
f(x)
f
(
x
)
is a real-valued, continuous function on the half-line
0
≤
x
<
∞
0\leq x < \infty
0
≤
x
<
∞
, and
∫
0
∞
f
2
(
x
)
d
x
<
∞
\int_0^{\infty} f^2(x)dx <\infty
∫
0
∞
f
2
(
x
)
d
x
<
∞
then the function g(x)\equal{}f(x)\minus{}2e^{\minus{}x}\int_0^x e^tf(t)dt satisfies \int _0^{\infty}g^2(x)dx\equal{}\int_0^{\infty}f^2(x)dx. [B. Szokefalvi-Nagy]
5
1
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Miklos Schweitzer 1963_5
Let
H
H
H
be a set of real numbers that does not consist of
0
0
0
alone and is closed under addition. Further, let
f
(
x
)
f(x)
f
(
x
)
be a real-valued function defined on
H
H
H
and satisfying the following conditions:
f
(
x
)
≤
f
(
y
)
i
f
x
≤
y
\;f(x)\leq f(y)\ \mathrm{if} \;x \leq y
f
(
x
)
≤
f
(
y
)
if
x
≤
y
and f(x\plus{}y)\equal{}f(x)\plus{}f(y) \;(x,y \in H)\ . Prove that f(x)\equal{}cx on
H
H
H
, where
c
c
c
is a nonnegative number. [M. Hosszu, R. Borges]
4
1
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Miklos Schweitzer 1963_4
Call a polynomial positive reducible if it can be written as a product of two nonconstant polynomials with positive real coefficients. Let
f
(
x
)
f(x)
f
(
x
)
be a polynomial with f(0)\not\equal{}0 such that
f
(
x
n
)
f(x^n)
f
(
x
n
)
is positive reducible for some natural number
n
n
n
. Prove that
f
(
x
)
f(x)
f
(
x
)
itself is positive reducible. [L. Redei]
3
1
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Miklos Schweitzer 1963_3
Let R\equal{}R_1\oplus R_2 be the direct sum of the rings
R
1
R_1
R
1
and
R
2
R_2
R
2
, and let
N
2
N_2
N
2
be the annihilator ideal of
R
2
R_2
R
2
(in
R
2
R_2
R
2
). Prove that
R
1
R_1
R
1
will be an ideal in every ring
R
~
\widetilde{R}
R
containing
R
R
R
as an ideal if and only if the only homomorphism from
R
1
R_1
R
1
to
N
2
N_2
N
2
is the zero homomorphism. [Gy. Hajos]
2
1
Hide problems
Miklos Schweitzer 1963_2
Show that the center of gravity of a convex region in the plane halves at least three chords of the region. [Gy. Hajos]
1
1
Hide problems
Miklos Schweitzer 1963_1
Show that the perimeter of an arbitrary planar section of a tetrahedron is less than the perimeter of one of the faces of the tetrahedron. [Gy. Hajos]