HMMT 二月 2005 · 冲刺赛 · 第 27 题

英文解析

27. In a chess-playing club, some of the players take lessons from other players. It ispossible (but not necessary) for two players both to take lessons from each other. Itso happens that for any three distinct members of the club, A , B , and C , exactly oneof the following three statements is true: A takes lessons from B ; B takes lessons from
    C ; C takes lessons from A . What is the largest number of players there can be?
    Solution: 4
    If P , Q , R , S , and T are any five distinct players, then consider all pairs A, B ∈
    { P, Q, R, S, T } such that A takes lessons from B . Each pair contributes to exactlythree triples ( A, B, C ) (one for each of the choices of C distinct from A and B ); threetriples ( C, A, B ); and three triples ( B, C, A ). On the other hand, there are 5 × 4 × 3 = 60
    ordered triples of distinct players among these five, and each includes exactly one ofour lesson-taking pairs. That means that there are 60 / 9 such pairs. But this numberisn’t an integer, so there cannot be five distinct people in the club.
    On the other hand, there can be four people, P , Q , R , and S : let P and Q both takelessons from each other, and let R and S both take lessons from each other; it is easyto check that this meets the conditions. Thus the maximum number of players is 4.