HMMT 十一月 2009 · GEN2 赛 · 第 10 题

HMMT November 2009 — GEN2 Round — Problem 10

个人笔记

[ 7 ] Five guys join five girls for a night of bridge. Bridge games are always played by a team of twoguys against a team of two girls. The guys and girls want to make sure that every guy and girl playagainst each other an equal number of times. Given that at least one game is played, what is the leastnumber of games necessary to accomplish this?
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解析

[ 7 ] Five guys join five girls for a night of bridge. Bridge games are always played by a team of twoguys against a team of two girls. The guys and girls want to make sure that every guy and girl playagainst each other an equal number of times. Given that at least one game is played, what is the leastnumber of games necessary to accomplish this?
Answer: 25 Suppose that each guy plays each girl t times. Since each guy plays against two girls
5 t 25 tin one game, the total number of games each guy plays is . Then the total number of games is ,
2 4
which is a multiple of 25 and therefore at least 25. To check that 25 games is enough, we arrange theguys and girls in two circles. A good pair of guys is a pair of guys who are adjacent in the circle; agood pair of girls is defined similarly. There are 5 good pairs of guys and girls — making each goodpair of guys play each good pair of girls works.