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Miklós Schweitzer
1993 Miklós Schweitzer
1993 Miklós Schweitzer
Part of
Miklós Schweitzer
Subcontests
(9)
8
1
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analysis
Let H be a complex separable Hilbert space and denote
B
(
H
)
B(H)
B
(
H
)
the algebra of bounded linear operators on H. Find all *-subalgebras C of
B
(
H
)
B(H)
B
(
H
)
for which for all
A
∈
B
(
H
)
A \in B(H)
A
∈
B
(
H
)
and
T
∈
C
T \in C
T
∈
C
there exists
S
∈
C
S \in C
S
∈
C
that
T
A
−
A
T
∗
=
T
S
−
S
T
∗
TA-AT^{\ast} = TS-ST^{\ast}
T
A
−
A
T
∗
=
TS
−
S
T
∗
note: *-algebra is also known as involutive algebra.
6
1
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geometry
Let
P
1
,
P
2
,
.
.
.
P_1 , P_2 , ...
P
1
,
P
2
,
...
be arbitrary points and A be a connected compact set in the plane with a diameter greater than 4. Show that for some point P in A ,
P
P
1
‾
⋅
P
P
2
‾
⋯
P
P
n
‾
>
1
\overline {PP_1} \cdot \overline {PP_2} \cdots \overline {PP_n}>1
P
P
1
⋅
P
P
2
⋯
P
P
n
>
1
. Furthermore, prove that this is no longer necessarily true for compact sets of diameter 4.
4
1
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iterated function
Let f be a ternary operation on a set of at least four elements for which (1)
f
(
x
,
x
,
y
)
≡
f
(
x
,
y
,
x
)
≡
f
(
x
,
y
,
y
)
≡
x
f ( x , x , y ) \equiv f ( x , y , x ) \equiv f( x , y , y ) \equiv x
f
(
x
,
x
,
y
)
≡
f
(
x
,
y
,
x
)
≡
f
(
x
,
y
,
y
)
≡
x
(2)
f
(
x
,
y
,
z
)
=
f
(
y
,
z
,
x
)
=
f
(
y
,
x
,
z
)
∈
{
x
,
y
,
z
}
f ( x , y , z ) = f ( y , z , x ) = f ( y , x , z ) \in \{ x , y , z \}
f
(
x
,
y
,
z
)
=
f
(
y
,
z
,
x
)
=
f
(
y
,
x
,
z
)
∈
{
x
,
y
,
z
}
for pairwise distinct x,y,z. Prove that f is a nontrivial composition of g such that g is not a composition of f. (The n-variable operation g is trivial if
g
(
x
1
,
.
.
.
,
x
n
)
≡
x
i
g(x_1, ..., x_n) \equiv x_i
g
(
x
1
,
...
,
x
n
)
≡
x
i
for some i (
1
≤
i
≤
n
1 \leq i \leq n
1
≤
i
≤
n
) )
7
1
Hide problems
analysis
Let H be a Hilbert space over the field of real numbers
R
\Bbb R
R
. Find all
f
:
H
→
R
f: H \to \Bbb R
f
:
H
→
R
continuous functions for which
f
(
x
+
y
+
π
z
)
+
f
(
x
+
2
z
)
+
f
(
y
+
2
z
)
+
f
(
π
z
)
f(x + y + \pi z) + f(x + \sqrt{2} z) + f(y + \sqrt{2} z) + f (\pi z)
f
(
x
+
y
+
π
z
)
+
f
(
x
+
2
z
)
+
f
(
y
+
2
z
)
+
f
(
π
z
)
=
f
(
x
+
y
+
2
z
)
+
f
(
x
+
π
z
)
+
f
(
y
+
π
z
)
+
f
(
2
z
)
= f(x + y + \sqrt{2} z) + f (x + \pi z) + f (y + \pi z) + f(\sqrt{2} z)
=
f
(
x
+
y
+
2
z
)
+
f
(
x
+
π
z
)
+
f
(
y
+
π
z
)
+
f
(
2
z
)
is satisfied for any
x
,
y
,
z
∈
H
x , y , z \in H
x
,
y
,
z
∈
H
.
10
1
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probability inequality
Let
U
1
,
U
2
,
U
3
U_1 , U_2 , U_3
U
1
,
U
2
,
U
3
be iid random variables on [0,1], which in order of magnitude,
U
1
∗
≤
U
2
∗
≤
U
3
∗
U_1^{\ast} \le U_2^{\ast} \leq U_3 ^ {\ast}
U
1
∗
≤
U
2
∗
≤
U
3
∗
. Let
α
,
p
1
,
p
2
,
p
3
∈
[
0
,
1
]
\alpha, p_1 , p_2 , p_3 \in [0,1]
α
,
p
1
,
p
2
,
p
3
∈
[
0
,
1
]
such that
P
(
U
j
∗
≥
p
j
)
=
α
P(U_j ^ {\ast} \ge p_j)= \alpha
P
(
U
j
∗
≥
p
j
)
=
α
( j = 1,2,3). Prove that
P
(
p
1
+
(
p
2
−
p
1
)
U
3
∗
+
(
p
3
−
p
2
)
U
2
∗
+
(
1
−
p
3
)
U
1
∗
≥
1
2
)
≥
1
−
α
P \left( p_1 + (p_2-p_1) U_3^{\ast} + (p_3- p_2) U_2^{\ast} + (1-p_3) U_1^{\ast} \geq \frac{1}{2} \right) \geq 1-\alpha
P
(
p
1
+
(
p
2
−
p
1
)
U
3
∗
+
(
p
3
−
p
2
)
U
2
∗
+
(
1
−
p
3
)
U
1
∗
≥
2
1
)
≥
1
−
α
5
1
Hide problems
analysis
Does the set of real numbers have a well-order
≺
\prec
≺
such that the intersection of the subset
{
(
x
,
y
)
:
x
≺
y
}
\{(x,y) : x\prec y\}
{(
x
,
y
)
:
x
≺
y
}
of the plane with every line is Lebesgue measurable on the line?
1
1
Hide problems
geometry
There are n points in the plane with the property that the distance between any two points is at least 1. Prove that for sufficiently large n , the number of pairs of points whose distance is in
[
t
1
,
t
1
+
1
]
∪
[
t
2
,
t
2
+
1
]
[ t_1 , t_1 + 1] \cup [ t_2 , t_2 + 1]
[
t
1
,
t
1
+
1
]
∪
[
t
2
,
t
2
+
1
]
for some
t
1
,
t
2
t_1, t_2
t
1
,
t
2
, is at most
[
n
2
3
]
[\frac{n^2}{3}]
[
3
n
2
]
, and the bound is sharp.
2
1
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set decomposition
Let A be a subset of natural numbers and let k , r be positive integers. Suppose that for any r different elements selected from A , their greatest common divisor has at most k different prime factors. Prove that A can be partitioned into B and C , where any element of B has at most k + 1 different prime divisors and
∑
n
∈
C
1
n
<
∞
\sum_{n\in C} \frac{1}{n} <\infty
n
∈
C
∑
n
1
<
∞
3
1
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exceptional units
Let K be the field formed by the addition of a root of the polynomial
x
4
−
2
x
2
−
1
x^4 - 2x^2 - 1
x
4
−
2
x
2
−
1
to the rational field. Prove that there are no exceptional units in the ring of integers of K. (A unit
ε
\varepsilon
ε
is called exceptional if
1
−
ε
1-\varepsilon
1
−
ε
is also a unit.)