MathDB
Problems
Contests
Undergraduate contests
Miklós Schweitzer
2015 Miklos Schweitzer
2015 Miklos Schweitzer
Part of
Miklós Schweitzer
Subcontests
(11)
2
1
Hide problems
Van Der Corput Series And Graph
Let
{
x
n
}
\{x_n\}
{
x
n
}
be a Van Der Corput series,that is,if the binary representation of
n
n
n
is
∑
a
i
2
i
\sum a_{i}2^{i}
∑
a
i
2
i
then
x
n
=
∑
a
i
2
−
i
−
1
x_n=\sum a_i2^{-i-1}
x
n
=
∑
a
i
2
−
i
−
1
.Let
V
V
V
be the set of points on the plane that have the form
(
n
,
x
n
)
(n,x_n)
(
n
,
x
n
)
.Let
G
G
G
be the graph with vertex set
V
V
V
that is connecting any two points
(
p
,
q
)
(p,q)
(
p
,
q
)
if there is a rectangle
R
R
R
which lies in parallel position with the axes and
R
∩
V
=
{
p
,
q
}
R\cap V= \{p,q\}
R
∩
V
=
{
p
,
q
}
.Prove that the chromatic number of
G
G
G
is finite.
8
1
Hide problems
Functional Equation With Continuity
Prove that all continuous solutions of the functional equation
(
f
(
x
)
−
f
(
y
)
)
(
f
(
x
+
y
2
)
−
f
(
x
y
)
)
=
0
,
∀
x
,
y
∈
(
0
,
+
∞
)
\left(f(x)-f(y)\right)\left(f\left(\frac{x+y}{2}\right)-f\left(\sqrt{xy}\right)\right)=0 \ , \ \forall x,y\in (0,+\infty)
(
f
(
x
)
−
f
(
y
)
)
(
f
(
2
x
+
y
)
−
f
(
x
y
)
)
=
0
,
∀
x
,
y
∈
(
0
,
+
∞
)
are the constant functions.
11
1
Hide problems
Brownian Motion and Increasing Probability
For
[
0
,
1
]
⊂
E
⊂
[
0
,
+
∞
)
[0,1]\subset E\subset [0,+\infty)
[
0
,
1
]
⊂
E
⊂
[
0
,
+
∞
)
where
E
E
E
is composed of a finite number of closed interval,we start a two dimensional Brownian motion from the point
x
<
0
x<0
x
<
0
terminating when we first hit
E
E
E
.Let
p
(
x
)
p(x)
p
(
x
)
be the probability of the finishing point being in
[
0
,
1
]
[0,1]
[
0
,
1
]
.Prove that
p
(
x
)
p(x)
p
(
x
)
is increasing on
[
−
1
,
0
)
[-1,0)
[
−
1
,
0
)
.
10
1
Hide problems
Operators on Hilbert Space
Let
f
:
R
→
R
f:\mathbb{R}\to \mathbb{R}
f
:
R
→
R
be a continuously differentiable,strictly convex function.Let
H
H
H
be a Hilbert space and
A
,
B
A,B
A
,
B
be bounded,self adjoint linear operators on
H
H
H
.Prove that,if
f
(
A
)
−
f
(
B
)
=
f
′
(
B
)
(
A
−
B
)
f(A)-f(B)=f'(B)(A-B)
f
(
A
)
−
f
(
B
)
=
f
′
(
B
)
(
A
−
B
)
then
A
=
B
A=B
A
=
B
.
6
1
Hide problems
Permutation Group Of A Set And Conjucate Elements
Let
G
G
G
be the permutation group of a finite set
Ω
\Omega
Ω
.Consider
S
⊂
G
S\subset G
S
⊂
G
such that
1
∈
S
1\in S
1
∈
S
and for any
x
,
y
∈
Ω
x,y\in \Omega
x
,
y
∈
Ω
there exists a unique element
σ
∈
S
\sigma \in S
σ
∈
S
such that
σ
(
x
)
=
y
\sigma (x)=y
σ
(
x
)
=
y
.Prove that,if the elements of
S
∖
{
1
}
S \setminus \{1\}
S
∖
{
1
}
are conjugate in
G
G
G
,then
G
G
G
is
2
−
2-
2
−
transitive on
Ω
\Omega
Ω
3
1
Hide problems
A directed graph with a property.
Let
A
{A}
A
be a finite set and
→
{\rightarrow}
→
be a binary relation on it such that for any
a
,
b
,
c
∈
A
{a,b,c \in A}
a
,
b
,
c
∈
A
, if
a
≠
b
,
a
→
c
{a\neq b}, {a \rightarrow c}
a
=
b
,
a
→
c
and
b
→
c
{b \rightarrow c}
b
→
c
then either
a
→
b
{a \rightarrow b}
a
→
b
or
b
→
a
{b \rightarrow a}
b
→
a
(or possibly both). Let
B
,
B
⊂
A
{B,\,B \subset A}
B
,
B
⊂
A
be minimal with the property: for any
a
∈
A
∖
B
{a \in A \setminus B}
a
∈
A
∖
B
there exists
b
∈
B
{b \in B}
b
∈
B
, such that either
a
→
b
{a \rightarrow b}
a
→
b
or
b
→
a
{b \rightarrow a}
b
→
a
(or possibly both). Supposing that
A
{A}
A
has at most
k
{k}
k
elements that are pairwise not in relation
→
{\rightarrow}
→
, prove that
B
{B}
B
has at most
k
{k}
k
elements.
7
1
Hide problems
Covering of the unit sphere
We call a bar of width
w
{w}
w
on the surface of the unit sphere
S
2
{\Bbb{S}^2}
S
2
, a spherical segment, centered at the origin, which has width
w
{w}
w
and is symmetric with respect to the origin. Prove that there exists a constant
c
>
0
{c>0}
c
>
0
, such that for any positive integer
n
{n}
n
the surface
S
2
{\Bbb{S}^2}
S
2
can be covered with
n
{n}
n
bars of the same width so that any point is contained in no more than
c
n
{c\sqrt{n}}
c
n
bars.
1
1
Hide problems
Dense system of chords in the unit ball
Let
K
K
K
be a closed subset of the closed unit ball in
R
3
\mathbb{R}^3
R
3
. Suppose there exists a family of chords
Ω
\Omega
Ω
of the unit sphere
S
2
S^2
S
2
, with the following property: for every
X
,
Y
∈
S
2
X,Y\in S^2
X
,
Y
∈
S
2
, there exist
X
′
,
Y
′
∈
S
2
X',Y'\in S^2
X
′
,
Y
′
∈
S
2
, as close to
X
X
X
and
Y
Y
Y
correspondingly, as we want, such that
X
′
Y
′
∈
Ω
X'Y'\in \Omega
X
′
Y
′
∈
Ω
and
X
′
Y
′
X'Y'
X
′
Y
′
is disjoint from
K
K
K
. Verify that there exists a set
H
⊂
S
2
H\subset S^2
H
⊂
S
2
, such that
H
H
H
is dense in the unit sphere
S
2
S^2
S
2
, and the chords connecting any two points of
H
H
H
are disjoint from
K
K
K
.EDIT: The statement fixed. See post #4
9
1
Hide problems
An inequality about harmonic functions.
For a function
u
{u}
u
defined on
G
⊂
C
{G \subset \Bbb{C}}
G
⊂
C
let us denote by
Z
(
u
)
{Z(u)}
Z
(
u
)
the neignborhood of unit raduis of the set of roots of
u
{u}
u
. Prove that for any compact set
K
⊂
G
{K \subset G}
K
⊂
G
there exists a constant
C
{C}
C
such that if
u
{u}
u
is an arbitrary real harmonic function on
G
{G}
G
which vanishes in a point of
K
{K}
K
then:
sup
z
∈
K
∣
u
(
z
)
∣
≤
C
sup
Z
(
u
)
∩
G
∣
u
(
z
)
∣
.
\displaystyle \sup_{z \in K} |u(z)| \leq C \sup_{Z(u)\cap G}|u(z)|.
z
∈
K
sup
∣
u
(
z
)
∣
≤
C
Z
(
u
)
∩
G
sup
∣
u
(
z
)
∣.
5
1
Hide problems
Composition of Polynomials
Let
f
(
x
)
=
x
n
+
x
n
−
1
+
⋯
+
x
+
1
f(x) = x^n+x^{n-1}+\dots+x+1
f
(
x
)
=
x
n
+
x
n
−
1
+
⋯
+
x
+
1
for an integer
n
≥
1.
n\ge 1.
n
≥
1.
For which
n
n
n
are there polynomials
g
,
h
g, h
g
,
h
with real coefficients and degrees smaller than
n
n
n
such that
f
(
x
)
=
g
(
h
(
x
)
)
.
f(x) = g(h(x)).
f
(
x
)
=
g
(
h
(
x
))
.
4
1
Hide problems
A series with special terms and complete residue system
Let
a
n
a_n
a
n
be a series of positive integers with
a
1
=
1
a_1=1
a
1
=
1
and for any arbitrary prime number
p
p
p
, the set
{
a
1
,
a
2
,
⋯
,
a
p
}
\{a_1,a_2,\cdots,a_p\}
{
a
1
,
a
2
,
⋯
,
a
p
}
is a complete remainder system modulo
p
p
p
. Prove that
lim
n
→
∞
a
n
n
=
1
\lim_{n\rightarrow \infty} \cfrac{a_n}{n}=1
lim
n
→
∞
n
a
n
=
1
.