MathDB
Problems
Contests
Undergraduate contests
Miklós Schweitzer
1954 Miklós Schweitzer
1954 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
Hide problems
Miklós Schweitzer 1954- Problem 10
10. Given a triangle
A
B
C
ABC
A
BC
, construct outwards over the sides
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
similiar isosceles triangles
A
B
C
1
,
B
C
A
1
ABC_{1}, BCA_{1}
A
B
C
1
,
BC
A
1
and
C
A
B
1
CAB_{1}
C
A
B
1
. Prove that the straight lines
A
A
1
.
B
B
1
AA_{1}. BB_{1}
A
A
1
.
B
B
1
and
C
C
1
CC_{1}
C
C
1
are concurrent. Is this statemente true in elliptic and hyperbolic geometry, too? (G. 19)
9
1
Hide problems
Miklós Schweitzer 1954- Problem 9
9. Lep
p
p
p
be a connected non-closed broken line without self-intersection in the plane
φ
\varphi
φ
. Prove that if
v
v
v
is a non-zero vector in
φ
\varphi
φ
and
p
p
p
has a commom point with the broken line
p
+
v
p+v
p
+
v
, then
p
p
p
has a common point with the broken line
p
+
α
v
p+\alpha v
p
+
αv
too, where
α
=
1
n
\alpha =\frac{1}{n}
α
=
n
1
and
n
n
n
is a positive integer. Does a similar statemente hold for other positive values of
α
\alpha
α
? (
p
+
v
p+v
p
+
v
denotes the broken line obtained from
p
p
p
through displacemente by the vector
v
v
v
.) (G. 1)
5
1
Hide problems
Miklós Schweitzer 1954- Problem 5
5. Let
ξ
1
,
ξ
2
,
…
,
ξ
n
,
.
.
.
\xi _{1},\xi _{2},\dots ,\xi _{n},...
ξ
1
,
ξ
2
,
…
,
ξ
n
,
...
be independent random variables of uniform distribution in
(
0
,
1
)
(0,1)
(
0
,
1
)
. Show that the distribution of the random variable
η
n
=
n
∏
k
=
1
n
(
1
−
ξ
k
k
)
(
n
=
1
,
2
,
.
.
.
)
\eta _{n}= \sqrt[]{n}\prod_{k=1}^{n}(1-\frac{\xi _{k}}{k}) (n= 1,2,...)
η
n
=
n
∏
k
=
1
n
(
1
−
k
ξ
k
)
(
n
=
1
,
2
,
...
)
tends to a limit distribution for
n
→
∞
n \to \infty
n
→
∞
. (P. 6)
8
1
Hide problems
Miklós Schweitzer 1954- Problem 8
8. Prove the following generalization of the well-known Chinese remainder theorem: Let
R
R
R
be a ring with unit element and let
A
1
,
A
2
,
…
.
A
n
(
n
⩾
2
)
A_{1},A_{2},\dots . A_{n} (n\geqslant 2)
A
1
,
A
2
,
…
.
A
n
(
n
⩾
2
)
be pairwise relative prime ideals of
R
R
R
. Then, for arbitrary elements
c
1
,
c
2
,
…
,
c
n
c_{1},c_{2}, \dots , c_{n}
c
1
,
c
2
,
…
,
c
n
of
R
R
R
, there exists an element
x
∈
R
x\in R
x
∈
R
such that
x
−
c
k
∈
A
k
(
k
=
1
,
2
,
…
,
n
)
x-c_{k} \in A_{k} (k= 1,2, \dots , n)
x
−
c
k
∈
A
k
(
k
=
1
,
2
,
…
,
n
)
. (A. 17)
7
1
Hide problems
Miklós Schweitzer 1954- Problem 7
7. Find the finite groups having only one proper maximal subgroup. (A.12)
6
1
Hide problems
Miklós Schweitzer 1954- Problem 6
6. Prove or disprove the following two propositions: (i) If
a
a
a
and
b
b
b
are positive integers such that
a
<
b
a<b
a
<
b
, then in any set of
b
b
b
consecutive integers there are two whose product is divisible by
a
b
ab
ab
(ii) If
a
,
b
a,b
a
,
b
and
c
c
c
are positive integers such that
a
<
b
<
c
a<b<c
a
<
b
<
c
, then in any set of
c
c
c
consecutive integers there are three whose product is divisible by
a
b
c
abc
ab
c
. (N.8)
4
1
Hide problems
Miklós Schweitzer 1954- Problem 4
4. Find all functions of two variables defined over the entire plane that satisfy the relations
f
(
x
+
u
,
y
+
u
)
=
f
(
x
,
y
)
+
u
f(x+u,y+u)=f(x,y)+u
f
(
x
+
u
,
y
+
u
)
=
f
(
x
,
y
)
+
u
and
f
(
x
v
,
y
v
)
=
f
(
x
,
y
)
v
f(xv,yv)= f(x,y) v
f
(
xv
,
y
v
)
=
f
(
x
,
y
)
v
for any real numbers
x
,
y
,
u
,
v
x,y,u,v
x
,
y
,
u
,
v
. (R.12)
3
1
Hide problems
Miklós Schweitzer 1954- Problem 3
3. Is there a real-valued function
A
f
Af
A
f
, defined on the space of the functions, continuous on
[
0
,
1
]
[0,1]
[
0
,
1
]
, such that
f
(
x
)
≤
g
(
x
)
f(x)\leq g(x)
f
(
x
)
≤
g
(
x
)
and
f
(
x
)
≢
g
(
x
)
f(x)\not\equiv g(x)
f
(
x
)
≡
g
(
x
)
inply
A
f
<
A
g
Af< Ag
A
f
<
A
g
? Is this also true if the functions
f
(
x
)
f(x)
f
(
x
)
are required to be monotonically increasing (rather than continuous) on
[
0
,
1
]
[0,1]
[
0
,
1
]
? (R.4)
2
1
Hide problems
Miklós Schweitzer 1954- Problem 2
2. Show that the series
∑
n
=
1
∞
1
n
s
i
n
(
a
s
i
n
(
2
n
π
N
)
)
e
b
c
o
s
(
2
n
π
N
)
\sum_{n=1}^{\infty}\frac{1}{n}sin(asin(\frac{2n\pi}{N}))e^{bcos(\frac{2n\pi}{N})}
∑
n
=
1
∞
n
1
s
in
(
a
s
in
(
N
2
nπ
))
e
b
cos
(
N
2
nπ
)
is convergent for every positive integer N and any real numbers a and b. (S. 25)
1
1
Hide problems
Miklós Schweitzer 1954- Problem 1
1. Given a positive integer
r
>
1
r>1
r
>
1
, prove that there exists an infinite number of infinite geometrical series, with positive terms, having the sum 1 and satisfying the following condition: for any positive real numbers
S
1
,
S
2
,
…
,
S
r
S_{1},S_{2},\dots,S_{r}
S
1
,
S
2
,
…
,
S
r
such that
S
1
+
S
2
+
⋯
+
S
r
=
1
S_{1}+S_{2}+\dots+S_{r}=1
S
1
+
S
2
+
⋯
+
S
r
=
1
, any of these infinite geometrical series can be divided into
r
r
r
infinite series(not necessarily geometrical) having the sums
S
1
,
S
2
,
…
,
S
r
S_{1},S_{2},\dots,S_{r}
S
1
,
S
2
,
…
,
S
r
, respectively. (S. 6)