1

Part of 2001 Tuymaada Olympiad

Problems(3)

Let us play volleyball, shall we?

Source: Tuymaada 2001, day 1, problem 1.

Ten volleyball teams played a tournament; every two teams met exactly once. The winner of the play gets 1 point, the loser gets 0 (there are no draws in volleyball). If the team that scored

n

x_{n}

n=1, \dots, 10

x_{1}+2x_{2}+\dots+10x_{10}\geq 165

.
Proposed by D. Teryoshin

inequalitiescombinatorics proposedcombinatorics

16 players in a chess tournament, 15 get first place, points of 16th ?

Source: Tuymaada Junior 2001 p1

16

chess players held a tournament among themselves: every two chess players played exactly one game. For victory in the party was given

1

point, for a draw

0.5

points, for defeat

0

points. It turned out that exactly 15 chess players shared the first place. How many points could the sixteenth chess player score?

combinatoricsgamepoints

A special partition of the positive integers.

Source: Tuymaada 2001, day 2, problem 1.

All positive integers are distributed among two disjoint sets

N_{1}

N_{2}

such that no difference of two numbers belonging to the same set is a prime greater than 100.
Find all such distributions.
Proposed by N. Sedrakyan

combinatorics proposedcombinatorics