HMMT 二月 2012 · 冲刺赛 · 第 12 题

HMMT February 2012 — Guts Round — Problem 12

[ 5 ] Knot is ready to face Gammadorf in a card game. In this game, there is a deck with twenty cards numbered from 1 to 20. Each player starts with a five card hand drawn from this deck. In each round, Gammadorf plays a card in his hand, then Knot plays a card in his hand. Whoever played a card with greater value gets a point. At the end of five rounds, the player with the most points wins. If Gammadorf starts with a hand of 1 , 5 , 10 , 15 , 20, how many five-card hands of the fifteen remaining cards can Knot draw which always let Knot win (assuming he plays optimally)? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TH 15 ANNUAL HARVARD-MIT MATHEMATICS TOURNAMENT, 11 FEBRUARY 2012 — GUTS ROUND

解析

[ 5 ] Knot is ready to face Gammadorf in a card game. In this game, there is a deck with twenty cards numbered from 1 to 20. Each player starts with a five card hand drawn from this deck. In each round, Gammadorf plays a card in his hand, then Knot plays a card in his hand. Whoever played a card with greater value gets a point. At the end of five rounds, the player with the most points wins. If Gammadorf starts with a hand of 1 , 5 , 10 , 15 , 20, how many five-card hands of the fifteen remaining cards can Knot draw which always let Knot win (assuming he plays optimally)? Answer: 2982 Knot can only lose if all of his cards are lower than 10; if not he can win by playing the lowest card that beats Gammadorf’s card, or if this is not possible, his lowest card, each turn. ( ) ( ) ( ) 7 15 7 There are = 21 losing hands, so he has − possible winning hands. 5 5 5