2009 IMS

Part of IMS

Subcontests

(7)

1

G is finite

G

be a group such that

G'

is abelian and each normal and abelian subgroup of

G

is finite. Prove that

G

1

Last two students sitting in the right place

Suppose that there are 100 seats in a saloon for 100 students. All students except one know their seat. First student (which is the one who doesn't know his seat) comes to the saloon and sits randomly somewhere. Then others enter the saloon one by one. Every student that enters the saloon and finds his seat vacant, sits there and if he finds his seat occupied he sits somewhere else randomly. Find the probability that last two students sit on their seats.

1

Double Integral on Closed Disc

f: \mathbb R^2\rightarrow \mathbb R

is a non-negative and continuous function that \iint_{\mathbb R^2}f(x,y)dxdy\equal{}1. Prove that there is a closed disc

D

with the least radius possible such that \iint_D f(x,y)dxdy\equal{}\frac12.

1

Infinite graph

In this infinite tree, degree of each vertex is equal to 3. A real number

\lambda

is given. We want to assign a real number to each node in such a way that for each node sum of numbers assigned to its neighbors is equal to

\lambda

times of the number assigned to this node. Find all

\lambda

for which this is possible.

1

Analytic on C excluding countably many points

A\subset \mathbb C

be a closed and countable set. Prove that if the analytic function

f: \mathbb C\backslash A\longrightarrow \mathbb C

is bounded, then

f

1

Idempotent ring

R

be a ring with 1. Every element in

R

can be written as product of idempotent ( u^n\equal{}u for some

n

) elements. Prove that

R

1

All subgroups are normal

G

is a group. Prove that the following are equivalent: 1. All subgroups of

G

are normal. 2. For all

a,b\in G

there is an integer

m

such that (ab)^m\equal{}ba.