MathDB

1

inequality with continuous and smooth vector field

p>1

, a continuous function

h\colon [0,\infty)\to [0,\infty)

, and a smooth vector field

Y\colon \mathbb{R}^n \to \mathbb{R}^n

with

\mathrm{div}~Y=0

, prove the following inequality

\int_{\mathbb{R}^n}h(|x|)|x|^{p}\leq \int_{\mathbb{R}^{n}}h(|x|)|x+Y(x)|^{p}.

10

1

degree of irreducible in F_p divides the degree of the splitting field

Let

f

be a polynomial of degree

n

with integer coefficients and

p

a prime for which

f

, considered modulo

p

, is a degree-

k

irreducible polynomial over

\mathbb{F}_p

. Show that

k

divides the degree of the splitting field of

f

over

\mathbb{Q}

.

9

1

complex subset is not dependent

Let

D\subseteq \mathbb{C}

be a compact set with at least two elements and consider the space \Omega=\bigtimes_{i=1}^{\infty} D with the product topology. For any sequence

(d_n)_{n=0}^{\infty} \in \Omega

let

f_{(d_n)}(z)=\sum_{n=0}^{\infty}d_nz^n

, and for each point

\zeta \in \mathbb{C}

with

|\zeta|=1

we define

S=S(\zeta,(d_n))

to be the set of complex numbers

w

for which there exists a sequence

(z_k)

such that

|z_k|<1

,

z_k \to \zeta

, and

f_{d_n}(z_k) \to w

. Prove that on a residual set of

\Omega

, the set

S

does not depend on the choice of

\zeta

.

8

1

Functional equation on functions on F_p

Let

\mathbb{F}_{p}

denote the

p

-element field for a prime

p>3

and let

S

be the set of functions from

\mathbb{F}_{p}

to

\mathbb{F}_{p}

. Find all functions

D\colon S\to S

satisfying

D(f\circ g)=(D(f)\circ g)\cdot D(g)

for all

f,g \in {S}

. Here,

\circ

denotes the function composition, so

(f\circ g)(x)

is the function

f(g(x))

, and

\cdot

denotes multiplication, so

(f\cdot g)=f(x)g(x)

.

7

1

sequence bounded iff series converges

Let

p(n)\geq 0

for all positive integers

n

. Furthermore,

x(0)=0, v(0)=1

, and

x(n)=x(n-1)+v(n-1), \qquad v(n)=v(n-1)-p(n)x(n) \qquad (n=1,2,\dots).

Assume that

v(n)\to 0

in a decreasing manner as

n \to \infty

. Prove that the sequence

x(n)

is bounded if and only if

\sum_{n=1}^{\infty}n\cdot p(n)<\infty

.

6

1

existence of entire function under certain conditions

Does there exist an entire function

F \colon \mathbb{C}\to \mathbb{C}

such that

F

is not zero everywhere,

|F(z)|\leq e^{|z|}

for all

z\in \mathbb{C}

,

|F(iy)|\leq 1

for all

y\in \mathbb{R}

, and

F

has infinitely many real roots.

5

1

equivalence on nowhere dense compact subset of the real plane

Prove that for a nowhere dense, compact set

K\subset \mathbb{R}^2

the following are equivalent:
(i)

K=\bigcup_{i=1}^{\infty}K_n

where

K_n

is a compact set with connected complement for all

n

.
(ii)

K

does not have a nonempty closed subset

S\subseteq K

such that any neighborhood of any point in

S

contains a connected component of

\mathbb{R}^2 \setminus S

.

4

1

horizontal and vertical segments intersected by curves

Consider horizontal and vertical segments in the plane that may intersect each other. Let

n

denote their total number. Suppose that we have

m

curves starting from the origin that are pairwise disjoint except for their endpoints. Assume that each curve intersects exactly two of the segments, a different pair for each curve. Prove that

m=O(n)

.

3

1

matrix's transpose has has non-negative eigenvector with non-negative value

An

n\times n

matrix

A

with integer entries is called representative if, for any integer vector

\mathbf{v}

, there is a finite sequence

0=\mathbf{v}_0,\mathbf{v}_1,\dots,\mathbf{v}_{\ell}=\mathbf{v}

of integer vectors such that for each

0\leq i <\ell

, either

\mathbf{v}_{i+1}=A\mathbf{v}_{i}

or

\mathbf{v}_{i+1}-\mathbf{v}_i

is an element of the standard basis (i.e. one of its entries is

1

, the rest are all equal to

0

). Show that

A

is not representative if and only if

A^T

has a real eigenvector with all non-negative entries and non-negative eigenvalue.

2

1

periodic continuous implies sequence is dense

Prove that if

f\colon \mathbb{R} \to \mathbb{R}

is a continuous periodic function and

\alpha \in \mathbb{R}

is irrational, then the sequence

\{n\alpha+f(n\alpha)\}_{n=1}^{\infty}

modulo 1 is dense in

[0,1]

.

1

a function on sequences

We say that two sequences

x,y \colon \mathbb{N} \to \mathbb{N}

are completely different if

x_n \neq y_n

holds for all

n\in \mathbb{N}

. Let

F

be a function assigning a natural number to every sequence of natural numbers such that

F(x)\neq F(y)

for any pair of completely different sequences

x

,

y

, and for constant sequences we have

F \left((k,k,\dots)\right)=k

. Prove that there exists

n\in \mathbb{N}

such that

F(x)=x_{n}

for all sequences

x

.

2020 Miklós Schweitzer

Subcontests

inequality with continuous and smooth vector field

degree of irreducible in F_p divides the degree of the splitting field

complex subset is not dependent

Functional equation on functions on F_p

sequence bounded iff series converges

existence of entire function under certain conditions

equivalence on nowhere dense compact subset of the real plane

horizontal and vertical segments intersected by curves

matrix's transpose has has non-negative eigenvector with non-negative value

periodic continuous implies sequence is dense

a function on sequences

2020 Mikl&oacute;s Schweitzer

Subcontests

inequality with continuous and smooth vector field

degree of irreducible in F_p divides the degree of the splitting field

complex subset is not dependent

Functional equation on functions on F_p

sequence bounded iff series converges

existence of entire function under certain conditions

equivalence on nowhere dense compact subset of the real plane

horizontal and vertical segments intersected by curves

matrix's transpose has has non-negative eigenvector with non-negative value

periodic continuous implies sequence is dense

a function on sequences

2020 Miklós Schweitzer