2014 IMS

Part of IMS

Subcontests

(12)

1

Open subset of complex plane

U

be an open subset of the complex plane

\mathbb{C}

\mathbb{D}=\{z \in \mathbb{C} : |z| \le 1\}

f

be analytic over

U

. Prove that if for every

z

with a complex norm equal to

1

|z|=1

0<Re(\bar{z}f(z))

f

has only one root in

\mathbb{D}

and that's simple.

1

Three variable equation

Let the equation

a^2 + b^2 + 1=abc

\mathbb{N}

c=3

1

n-dimensional vector space

V

n-

dimensional vector space over a field

F

\{e_1,e_2, \cdots ,e_n\}

.Prove that for any

m-

dimensional linear subspace

W

V

, the number of elements of the set

W \cap P

is less than or equal to

2^m

P=\{\lambda_1e_1 + \lambda_2e_2 + \cdots + \lambda_ne_n : \lambda_i=0,1\}

1

Simple graph

G

2n-

vertices simple graph such that in any partition of the set of vertices of

G

n-

V_1

V_2

, the number of edges from a vertex in

V_1

to another vertex in

V_1

is equal to the number of edges from a vertex in

V_2

to another vertex in

V_2

. Prove that all the vertices have equal degrees.

1

Is it convergent?

\sum_{n=1}^{+\infty}\frac{\cos n}{n}(1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}})

convergent? why?

1

Subgroups

G

be a finite group such that for every two subgroups of it like

H

K

H \cong K

H \subseteq K

K \subseteq H

. Prove that we can produce each subgroup of

G

with 2 elements at most.

1

n×n Matrix

A=[a_{ij}]_{n \times n}

n \times n

matrix whose elements are all numbers which belong to set

\{1,2,\cdots ,n\}

. Prove that by swapping the columns of

A

with each other we can produce matrix

B=[b_{ij}]_{n \times n}

K(B) \le n

K(B)

is the number of elements of set

\{(i,j) ; b_{ij} =j\}

1

Group homomorphism

G_1

G_2

be two finite groups such that for any finite group

H

, the number of group homomorphisms from

G_1

H

is equal to the number of group homomorphisms from

G_2

H

G_1

G_2

are Isomorphic.

1

commutative ring

R

be a commutative ring with

1

such that the number of elements of

R

p^3

p

is a prime number. Prove that if the number of elements of

\text{zd}(R)

be in the form of

p^n

n \in \mathbb{N^*}

\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}

R

has exactly one maximal ideal.

1

Convergent sequence

(X,d)

be a nonempty connected metric space such that the limit of every convergent sequence, is a term of that sequence. Prove that

X

has exactly one element.

1

Metric space

(X,d)

be a metric space and

f:X \to X

be a function such that

\forall x,y\in X : d(f(x),f(y))=d(x,y)

\text{a})

Prove that for all

x \in X

\lim_{n \rightarrow +\infty} \frac{d(x,f^n(x))}{n}

f^n(x)

\underbrace{f(f(\cdots f(x)}_{n \text{times}} \cdots ))

\text{b})

Prove that the amount of the limit does not depend on choosing

x

1

Irrational numbers

A

be a subset of the irrational numbers such that the sum of any two distinct elements of it be a rational number. Prove that

A

has two elements at most.