The International Olympiad '**"Tuymaada"** is an annual mathematics competition for secondary school students held in Republic of Saka, Russia. The contestants compete individually. The participating teams (national and local teams) can have up to three students. . The contest is held in two days of competitions, in July.

Here are a few interesting math problems from **Tuymaada**:

1. Positive integers a < b are given. Prove that among every b

consecutive positive integers there are two numbers whose product

is divisible by ab.

2. What minimum number of colours is sufficient to colour all positive real

numbers so that every two numbers whose ratio is 4 or 8 have different colours?

3. Several knights are arranged on an infinite chessboard.

No square is attacked by more than one knight (in particular, a square

occupied by a knight can be attacked by one knight but not by two). Sasha

outlined a 14x16 rectangle. What maximum number of knights can

this rectangle contain?

4. Prove that there exists a positive c such that for every positive

integer N among any N positive integers not exceeding 2N there

are two numbers whose greatest common divisor is greater than cN.

5. Seven different odd primes are given. Is it possible that the

difference of 8th powers of every two of them is divisible by each of the

remained numbers?

6. 100 boxers of different strength participate in the Boxing Championship

of Dirtytrickland. Each of them fights each other once. Several boxers

formed a plot: each of them put a leaden horseshoe in his boxing-glove

during one of his fights. When just one of two boxers has a horseshoe,

he wins; otherwise, the stronger boxer wins. It turned out after

the championship that three boxers won more fights than any of the three

strongest participants.

What is the minimum possible number of plotters?

7. Organizers of a mathematical congress found that if they

accomodate any participant in a single room the rest can be accomodated

in double rooms so that two persons living in every double room know each

other.

Prove that every participant can organize a round table on graph theory

for himself and even number of other people so that each participant of

the round table knows both his neighbours.

8.Several rooks stand in the squares of the table shown in the figure.

The rooks beat all the squares (we suppose that a rook beats the square

it stands in). Prove that one can remove several rooks so that not more

than 11 rooks are left and these rooks still beat all the squares.

## Sunday, February 3, 2008

### Tuymaada math competition

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