Tuymaada math competition

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.