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Problems
Contests
Undergraduate contests
Miklós Schweitzer
1995 Miklós Schweitzer
1995 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(10)
12
1
Hide problems
estimation of shift parameter
Let F(x) be a known distribution function, the random variables
η
1
,
η
2
.
.
.
\eta_1 , \eta_2 ...
η
1
,
η
2
...
be independent of the common distribution function
F
(
x
−
ϑ
)
F( x - \vartheta)
F
(
x
−
ϑ
)
, where
ϑ
\vartheta
ϑ
is the shift parameter. Let us call the shift parameter "well estimated" if there exists a positive constant c, so that any of
ε
>
0
\varepsilon> 0
ε
>
0
there exist a Lebesgue measure
ε
\varepsilon
ε
Borel set E ("confidence set") and a Borel-measurable function
t
n
(
x
1
,
.
.
.
,
x
n
)
t_n( x_1 ,. .., x_n )
t
n
(
x
1
,
...
,
x
n
)
( n = 1,2, ...) such that for any
ϑ
\vartheta
ϑ
we have
P
(
ϑ
−
t
n
(
η
1
,
.
.
.
,
η
n
)
∈
E
)
>
1
−
e
−
c
n
(
n
>
n
0
(
ε
,
F
)
)
P ( \vartheta- t_n ( \eta_1 , ..., \eta_n ) \in E )> 1-e^{-cn} \qquad( n > n_0 ( \varepsilon, F ) )
P
(
ϑ
−
t
n
(
η
1
,
...
,
η
n
)
∈
E
)
>
1
−
e
−
c
n
(
n
>
n
0
(
ε
,
F
))
Prove that a) if F is not absolutely continuous, then the shift parameter is "well estimated", b) if F is absolutely continuous and F' is continuous, then it is not "well estimated".
9
1
Hide problems
serpentine in compact set
A serpentine is a sequence of points
P
1
,
.
.
.
,
P
m
P_1 , ..., P_m
P
1
,
...
,
P
m
in a plane, not necessarily all different, such that the distance between
P
i
P_i
P
i
and
P
i
+
1
P_{i+1}
P
i
+
1
is at least 1, and the segments
P
i
P
i
+
1
P_i P_{i +1}
P
i
P
i
+
1
are alternately horizontal and vertical. Construct a compact set in which there is a sequence of serpentines with arbitrary long lengths but there is no closed serpentine (
P
m
=
P
i
P_m = P_i
P
m
=
P
i
for some i < m).
3
1
Hide problems
analysis
Denote
⟨
x
⟩
\langle x\rangle
⟨
x
⟩
the distance of the real number x from the nearest integer. Let f be a linear, 1 periodic, continuous real function. Prove that there exist natural n and real numbers
a
1
,
.
.
.
,
a
n
,
b
1
,
.
.
.
,
b
n
,
c
1
,
.
.
.
,
c
n
a_1 , ..., a_n , b_1 , ..., b_n , c_1 , ..., c_n
a
1
,
...
,
a
n
,
b
1
,
...
,
b
n
,
c
1
,
...
,
c
n
such that
f
(
x
)
=
∑
i
=
1
n
c
i
⟨
a
i
x
+
b
i
⟩
f(x) = \sum_{i = 1}^n c_i \langle a_ix + b_i \rangle
f
(
x
)
=
i
=
1
∑
n
c
i
⟨
a
i
x
+
b
i
⟩
for every x iff there is a k such that
∑
j
=
1
2
k
f
(
x
+
j
2
k
)
\sum_{j = 1}^{2^k} f \left(x+{j\over2^k}\right)
j
=
1
∑
2
k
f
(
x
+
2
k
j
)
is constant.
1
1
Hide problems
identity theorem for harmonic functions
Prove that a harmonic function that is not identically zero in the plane cannot vanish on a two-dimensional positive-measure set.
5
1
Hide problems
combinatorics
Let A be a subset of the set
{
1
,
2
,
.
.
.
,
n
}
\{1,2, ...,n\}
{
1
,
2
,
...
,
n
}
with at least
100
n
100\sqrt n
100
n
elements. Prove that there is a four-element arithmetic sequence in which each element is the sum of two different elements of the set A.
8
1
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analysis
Let P be a finite, partially ordered set with one largest element, which is the only upper bound of the set of minimal elements. Prove that any monotonic function
f
:
P
n
→
P
f : P^n\to P
f
:
P
n
→
P
can be written in the form
g
(
x
1
,
x
2
,
.
.
.
,
x
n
,
c
1
,
.
.
.
,
c
m
)
g( x_1 , x_2 , ..., x_n , c_1 , ..., c_m )
g
(
x
1
,
x
2
,
...
,
x
n
,
c
1
,
...
,
c
m
)
, where
c
i
∈
P
c_i\in P
c
i
∈
P
and g is a monotonic, idempotent function. (g is idempotent iff
g
(
x
,
x
,
.
.
.
,
x
)
=
x
∀
x
∈
P
g(x , x , ..., x) = x\,\forall x\in P
g
(
x
,
x
,
...
,
x
)
=
x
∀
x
∈
P
)
4
1
Hide problems
sum of powers
For odd numbers
a
1
,
.
.
.
,
a
k
a_1 , ..., a_k
a
1
,
...
,
a
k
and even numbers
b
1
,
.
.
.
,
b
k
b_1 , ..., b_k
b
1
,
...
,
b
k
, we know that
∑
j
=
1
k
a
j
n
=
∑
j
=
1
k
b
j
n
\sum_ {j = 1}^k a_j^n = \sum_{j = 1}^k b_j^n
∑
j
=
1
k
a
j
n
=
∑
j
=
1
k
b
j
n
is satisfied for n = 1,2, ..., N. Prove that
k
≥
2
N
k\geq 2^N
k
≥
2
N
and that for
k
=
2
N
k = 2^N
k
=
2
N
there exists a solution
(
a
1
,
.
.
.
,
b
1
,
.
.
.
)
(a_1,...,b_1,...)
(
a
1
,
...
,
b
1
,
...
)
with the above properties.
10
1
Hide problems
analysis
Let
X
=
{
X
1
,
X
2
,
.
.
.
}
X =\{ X_1 , X_2 , ...\}
X
=
{
X
1
,
X
2
,
...
}
be a countable set of points in space. Show that there is a positive sequence
{
a
k
}
\{a_k\}
{
a
k
}
such that for any point
Z
∉
X
Z\not\in X
Z
∈
X
the distance between the point Z and the set
{
X
1
,
X
2
,
.
.
.
,
X
k
}
\{X_1,X_2 , ...,X_k\}
{
X
1
,
X
2
,
...
,
X
k
}
is at least
a
k
a_k
a
k
for infinitely many k.
6
1
Hide problems
embedding in diameter 2 graph
Prove that every finite triangle-free graph can be embedded as an induced subgraph in a finite triangle-free graph of diameter 2.
2
1
Hide problems
find interval where integral is half
Given
f
,
g
∈
L
1
[
0
,
1
]
f,g\in L^1[0,1]
f
,
g
∈
L
1
[
0
,
1
]
and
∫
0
1
f
=
∫
0
1
g
=
1
\int_0^1 f = \int_0^1 g=1
∫
0
1
f
=
∫
0
1
g
=
1
, prove that there exists an interval I st
∫
I
f
=
∫
I
g
=
1
2
\int_I f = \int_I g=\frac12
∫
I
f
=
∫
I
g
=
2
1
.