4

Part of 2004 Tuymaada Olympiad

Problems(2)

The societies going against the government will be punished.

Source: Tuymaada 2002, day 1, problem 4. - Authors : V. Dolnikov, D. Karpov.

There are many opposition societies in the city of N. Each society consists of

10

members. It is known that for every

2004

societies there is a person belonging to at least

11

of them. Prove that the government can arrest

2003

people so that at least one member of each society is arrested.
Proposed by V.Dolnikov, D.Karpov

combinatorics proposedcombinatorics

Source: tuymaada 2004

It is known that

m

n

are positive integers,

m > n^{n-1}

, and all the numbers

m+1

m+2

m+n

are composite. Prove that there exist such different primes

p_1

p_2

p_n

p_k

m+k

k = 1

n

.
Proposed by C. A. Grimm

number theoryleast common multiplenumber theory proposed