2

Part of 1997 China Team Selection Test

Problems(2)

China TST 1997 football teams

Source: China TST 1997, problem 2

n

football teams in a round-robin competition where every 2 teams meet once. The winner of each match receives 3 points while the loser receives 0 points. In the case of a draw, both teams receive 1 point each. Let

k

be as follows: 2 \leq k \leq n \minus{} 1. At least how many points must a certain team get in the competition so as to ensure that there are at most k \minus{} 1 teams whose scores are not less than that particular team's score?

combinatorics unsolvedcombinatorics

China TST 1997 5-element subsets

Source: China TST 1997, problem 5

n

be a natural number greater than 6.

X

is a set such that

|X| = n

A_1, A_2, \ldots, A_m

are distinct 5-element subsets of

X

m > \frac{n(n - 1)(n - 2)(n - 3)(4n - 15)}{600}

, prove that there exists

A_{i_1}, A_{i_2}, \ldots, A_{i_6}

(1 \leq i_1 < i_2 < \cdots, i_6 \leq m)

\bigcup_{k = 1}^6 A_{i_k} = 6

combinatorics unsolvedcombinatorics