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Miklós Schweitzer
1957 Miklós Schweitzer
1957 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(9)
10
1
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Miklós Schweitzer 1957- Problem 10
10. An Abelian group
G
G
G
is said to have the property
(
A
)
(A)
(
A
)
if torsion subgroup of
G
G
G
is a direct summand of
G
G
G
. Show that if
G
G
G
is an Abelian group such that
n
G
nG
n
G
has the property
(
A
)
(A)
(
A
)
for some positive integer
n
n
n
, then
G
G
G
itself has the property
(
A
)
(A)
(
A
)
. (A. 13)
9
1
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Miklós Schweitzer 1957- Problem 9
9. Find all pairs of linear polynomials
f
(
x
)
f(x)
f
(
x
)
,
g
(
x
)
g(x)
g
(
x
)
with integer coefficients for which there exist two polynomials
u
(
x
)
u(x)
u
(
x
)
,
v
(
x
)
v(x)
v
(
x
)
with integer coefficients such that
f
(
x
)
u
(
x
)
+
g
(
x
)
v
(
x
)
=
1
f(x)u(x)+g(x)v(x)=1
f
(
x
)
u
(
x
)
+
g
(
x
)
v
(
x
)
=
1
. (A. 8)
8
1
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Miklós Schweitzer 1957- Problem 8
8. Find all integers
a
>
1
a>1
a
>
1
for which the least (integer) solution
n
n
n
of the congruence
a
n
≡
1
(
m
o
d
p
)
a^{n} \equiv 1 \pmod{p}
a
n
≡
1
(
mod
p
)
differs from 6 (p is any prime number). (N. 9)
7
1
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Miklós Schweitzer 1957- Problem 7
7. Prove that any real number x satysfying the inequalities
0
<
x
≤
1
0<x\leq 1
0
<
x
≤
1
can be represented in the form
x
=
∑
k
=
1
∞
1
n
k
x= \sum_{k=1}^{\infty}\frac{1}{n_k}
x
=
∑
k
=
1
∞
n
k
1
where
(
n
k
)
k
=
1
∞
(n_k)_{k=1}^{\infty}
(
n
k
)
k
=
1
∞
is a sequence of positive integers such that
n
k
+
1
n
k
\frac{n_{k+1}}{n_k}
n
k
n
k
+
1
assumes, for each
k
k
k
, one of the three values
2
,
3
2,3
2
,
3
or
4
4
4
. (N. 14)
6
1
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Miklós Schweitzer 1957- Problem 6
6. Let
f
(
x
)
f(x)
f
(
x
)
be an arbitrary function, differentiable infinitely many times. Then the
n
n
n
th derivative of
f
(
e
x
)
f(e^{x})
f
(
e
x
)
has the form
d
n
d
x
n
f
(
e
x
)
=
∑
k
=
0
n
a
k
n
e
k
x
f
(
k
)
(
e
x
)
\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})
d
x
n
d
n
f
(
e
x
)
=
∑
k
=
0
n
a
kn
e
k
x
f
(
k
)
(
e
x
)
(
n
=
0
,
1
,
2
,
…
n=0,1,2,\dots
n
=
0
,
1
,
2
,
…
).From the coefficients
a
k
n
a_{kn}
a
kn
compose the sequence of polynomials
P
n
(
x
)
=
∑
k
=
0
n
a
k
n
x
k
P_{n}(x)= \sum_{k=0}^{n} a_{kn}x^{k}
P
n
(
x
)
=
∑
k
=
0
n
a
kn
x
k
(
n
=
0
,
1
,
2
,
…
n=0,1,2,\dots
n
=
0
,
1
,
2
,
…
)and find a closed form for the function
F
(
t
,
x
)
=
∑
n
=
0
∞
P
n
(
x
)
n
!
t
n
.
F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.
F
(
t
,
x
)
=
∑
n
=
0
∞
n
!
P
n
(
x
)
t
n
.
(S. 22)
5
1
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Miklós Schweitzer 1957- Problem 5
5. Find the continuous solutions of the functional equation
f
(
x
y
z
)
=
f
(
x
)
+
f
(
y
)
+
f
(
z
)
f(xyz)= f(x)+f(y)+f(z)
f
(
x
yz
)
=
f
(
x
)
+
f
(
y
)
+
f
(
z
)
in the following cases:(a)
x
,
y
,
z
x,y,z
x
,
y
,
z
are arbitrary non-zero real numbers; (b)
a
<
x
,
y
,
z
<
b
(
1
<
a
3
<
b
)
a<x,y,z<b (1<a^{3}<b)
a
<
x
,
y
,
z
<
b
(
1
<
a
3
<
b
)
. (R. 13)
4
1
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Miklós Schweitzer 1957- Problem 4
4. Let
F
ϵ
(
0
<
ϵ
<
1
)
F_{\epsilon} (0<\epsilon<1)
F
ϵ
(
0
<
ϵ
<
1
)
denote the class of non-negative piecewise continuous functions defined on
[
0
,
∞
)
[0,\infty)
[
0
,
∞
)
which satisfy the following condition:
f
(
x
)
f
(
y
)
≤
ϵ
∣
x
−
y
∣
(
x
,
y
≥
0
)
f(x)f(y)\leq \epsilon^{\mid x-y\mid} (x,y \geq 0)
f
(
x
)
f
(
y
)
≤
ϵ
∣
x
−
y
∣
(
x
,
y
≥
0
)
. Find the value of
s
ϵ
=
sup
f
∈
F
ϵ
∫
0
∞
f
(
x
)
d
x
s_{\epsilon}= \sup_{f\in F_{\epsilon}} \int_{0}^{\infty} f(x) dx
s
ϵ
=
sup
f
∈
F
ϵ
∫
0
∞
f
(
x
)
d
x
(R. 5)
3
1
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Miklós Schweitzer 1957- Problem 3
3. Let
A
A
A
be a subset of n-dimensional space containing at least one inner point and suppose that, for every point pair
x
,
y
∈
A
x, y \in A
x
,
y
∈
A
, the subset
A
A
A
contains the mid point of the line segment beteween
x
x
x
and
y
y
y
. Show that
A
A
A
consists of a convex open set and of some of its boundary points. (St. 1)
1
1
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Miklós Schweitzer 1957- Problem 1
1. Let
C
i
j
C_{ij}
C
ij
(
i
,
j
=
1
,
2
,
3
i,j=1,2,3
i
,
j
=
1
,
2
,
3
) be the coefficients of a real non-involutive orthogonal transformation. Prove that the function
w
=
∑
i
,
j
=
1
3
c
i
j
z
i
z
j
ˉ
w= \sum_{i,j=1}^{3} c_{ ij}z_{i}\bar{z_{ j}}
w
=
∑
i
,
j
=
1
3
c
ij
z
i
z
j
ˉ
maps the surface of complex unit sphere
∑
i
=
1
3
z
i
z
i
ˉ
=
1
\sum_{i=1}^{3} z_{i}\bar{z_{i}} = 1
∑
i
=
1
3
z
i
z
i
ˉ
=
1
onto a triangle of the w-plane. (F. 3)