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Miklós Schweitzer
1977 Miklós Schweitzer
1977 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
10
1
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Miklos Schweitzer 1977_10
Let the sequence of random variables
{
X
m
,
m
≥
0
}
,
X
0
=
0
\{ X_m, \; m \geq 0\ \}, \; X_0=0
{
X
m
,
m
≥
0
}
,
X
0
=
0
, be an infinite random walk on the set of nonnegative integers with transition probabilities
p
i
=
P
(
X
m
+
1
=
i
+
1
∣
X
m
=
i
)
>
0
,
i
≥
0
p_i=P(X_{m+1}=i+1 \mid X_m=i) >0, \; i \geq 0 \,
p
i
=
P
(
X
m
+
1
=
i
+
1
∣
X
m
=
i
)
>
0
,
i
≥
0
q
i
=
P
(
X
m
+
1
=
i
−
1
∣
X
m
=
i
)
>
0
,
i
>
0.
q_i=P(X_{m+1}=i-1 \mid X_m=i ) >0, \; i>0.
q
i
=
P
(
X
m
+
1
=
i
−
1
∣
X
m
=
i
)
>
0
,
i
>
0.
Prove that for arbitrary
k
>
0
k >0
k
>
0
there is an
α
k
>
1
\alpha_k > 1
α
k
>
1
such that
P
n
(
k
)
=
P
(
max
0
≤
j
≤
n
X
j
=
k
)
P_n(k)=P \left ( \max_{0 \leq j \leq n} X_j =k \right)
P
n
(
k
)
=
P
(
0
≤
j
≤
n
max
X
j
=
k
)
satisfies the limit relation
lim
L
→
∞
1
L
∑
n
=
1
L
P
n
(
k
)
α
k
n
<
∞
.
\lim_{L \rightarrow \infty} \frac 1L \sum_{n=1}^L P_n(k) \alpha_k ^n < \infty.
L
→
∞
lim
L
1
n
=
1
∑
L
P
n
(
k
)
α
k
n
<
∞.
J. Tomko
9
1
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Miklos Schweitzer 1977_9
Suppose that the components of he vector
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
u
<
/
s
p
a
n
>
=
(
u
0
,
…
,
u
n
)
<span class='latex-bold'>u</span>=(u_0,\ldots,u_n)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
u
<
/
s
p
an
>=
(
u
0
,
…
,
u
n
)
are real functions defined on the closed interval
[
a
,
b
]
[a,b]
[
a
,
b
]
with the property that every nontrivial linear combination of them has at most
n
n
n
zeros in
[
a
,
b
]
[a,b]
[
a
,
b
]
. Prove that if
σ
\sigma
σ
is an increasing function on
[
a
,
b
]
[a,b]
[
a
,
b
]
and the rank of the operator
A
(
f
)
=
∫
a
b
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
u
<
/
s
p
a
n
>
(
x
)
f
(
x
)
d
σ
(
x
)
,
f
∈
C
[
a
,
b
]
,
A(f)= \int_{a}^b <span class='latex-bold'>u</span>(x)f(x)d\sigma(x), \;f \in C[a,b]\ ,
A
(
f
)
=
∫
a
b
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
u
<
/
s
p
an
>
(
x
)
f
(
x
)
d
σ
(
x
)
,
f
∈
C
[
a
,
b
]
,
is
r
≤
n
r \leq n
r
≤
n
, then
σ
\sigma
σ
has exactly
r
r
r
points of increase. E. Gesztelyi
8
1
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Miklos Schweitzer 1977_8
Let
p
≥
1
p \geq 1
p
≥
1
be a real number and \mathbb{R}_\plus{}\equal{}(0, \infty). For which continuous functions g : \mathbb{R}_\plus{} \rightarrow \mathbb{R}_\plus{} are following functions all convex? M_n(x)\equal{}\left[ \frac{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}}) x_{i\plus{}1}^p}{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}})} \right ]^\frac 1p , x\equal{}(x_1,\ldots, x_{n\plus{}1}) \in \mathbb{R}_\plus{} ^ {n\plus{}1} , \; n\equal{}1,2,\ldots L. Losonczi
7
1
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Miklos Schweitzer 1977_7
Let
G
G
G
be a locally compact solvable group, let
c
1
,
…
,
c
n
c_1,\ldots, c_n
c
1
,
…
,
c
n
be complex numbers, and assume that the complex-valued functions
f
f
f
and
g
g
g
on
G
G
G
satisfy
∑
k
=
1
n
c
k
f
(
x
y
k
)
=
f
(
x
)
g
(
y
)
for all
x
,
y
∈
G
.
\sum_{k=1}^n c_k f(xy^k)=f(x)g(y) \;\textrm{for all} \;x,y \in G \ \ .
k
=
1
∑
n
c
k
f
(
x
y
k
)
=
f
(
x
)
g
(
y
)
for all
x
,
y
∈
G
.
Prove that if
f
f
f
is a bounded function and
inf
x
∈
G
Re
f
(
x
)
χ
(
x
)
>
0
\inf_{x \in G} \textrm{Re} f(x) \chi(x) >0
x
∈
G
in
f
Re
f
(
x
)
χ
(
x
)
>
0
for some continuous (complex) character
χ
\chi
χ
of
G
G
G
, then
g
g
g
is continuous. L. Szekelyhidi
6
1
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Miklos Schweitzer 1977_6
Let
f
f
f
be a real function defined on the positive half-axis for which f(xy)\equal{}xf(y)\plus{}yf(x) and f(x\plus{}1) \leq f(x) hold for every positive
x
x
x
and
y
y
y
. Show that if f(1/2)\equal{}1/2, then f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x) for every
x
∈
(
0
,
1
)
x\in (0,1)
x
∈
(
0
,
1
)
. Z. Daroczy, Gy. Maksa
5
1
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Miklos Schweitzer 1977_5
Suppose that the automorphism group of the finite undirected graph X\equal{}(P, E) is isomorphic to the quaternion group (of order
8
8
8
). Prove that the adjacency matrix of
X
X
X
has an eigenvalue of multiplicity at least
4
4
4
. ( P\equal{} \{ 1,2,\ldots, n \} is the set of vertices of the graph
X
X
X
. The set of edges
E
E
E
is a subset of the set of all unordered pairs of elements of
P
P
P
. The group of automorphisms of
X
X
X
consists of those permutations of
P
P
P
that map edges to edges. The adjacency matrix M\equal{}[m_{ij}] is the
n
×
n
n \times n
n
×
n
matrix defined by m_{ij}\equal{}1 if
{
i
,
j
}
∈
E
\{ i,j \} \in E
{
i
,
j
}
∈
E
and m_{i,j}\equal{}0 otherwise.) L. Babai
4
1
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Miklos Schweitzer 1977_4
Let
p
>
5
p>5
p
>
5
be a prime number. Prove that every algebraic integer of the
p
p
p
th cyclotomic field can be represented as a sum of (finitely many) distinct units of the ring of algebraic integers of the field. K. Gyory
3
1
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Miklos Schweitzer 1977_3
Prove that if
a
,
x
,
y
a,x,y
a
,
x
,
y
are
p
p
p
-adic integers different from
0
0
0
and
p
∣
x
,
p
a
∣
x
y
p | x, pa | xy
p
∣
x
,
p
a
∣
x
y
, then \frac 1y \frac{(1\plus{}x)^y\minus{}1}{x} \equiv \frac{\log (1\plus{}x)}{x} \;\;\;\; ( \textrm{mod} \; a\ ) \\\\ . L. Redei
2
1
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Miklos Schweitzer 1977_2
Construct on the real projective plane a continuous curve, consisting of simple points, which is not a straight line and is intersected in a single point by every tangent and every secant of a given conic. F. Karteszi
1
1
Hide problems
Miklos Schweitzer 1977_1
Consider the intersection of an ellipsoid with a plane
σ
\sigma
σ
passing through its center
O
O
O
. On the line through the point
O
O
O
perpendicular to
σ
\sigma
σ
, mark the two points at a distance from
O
O
O
equal to the area of the intersection. Determine the loci of the marked points as
σ
\sigma
σ
runs through all such planes. L. Tamassy