HMMT 二月 1998 · ADV 赛 · 第 7 题

英文解析

7. The Houson Association of Mathematics Educators decides to hold a grand forum on mathematicseducation and invites a number of politicians from the United States to participate. Around lunchtime the politicians decide to play a game. In this game, players can score 19 points for peggingthe coordinator of the gathering with a spit ball, 9 points for downing an entire cup of the forum’sinterpretation of coffee, or 8 points for quoting more than three consecutive words from the speech
   Senator Bobbo delivered before lunch. What is the product of the two greatest scores that a playercannot score in this game?
   Answer: 1209 . Attainable scores are positive integers that can be written in the form 8 a + 9 b + 19 c ,
   where a , b , and c are nonnegative integers. Consider attainable number of points modulo 8.
   Scores that are 0 (mod 8) can be obtained with 8 a for positive a .
   Scores that are 1 (mod 8) greater than or equal to 9 can be obtained with 9 + 8 a for nonnegative a .
   Scores that are 2 (mod 8) greater than or equal to 18 can be obtained with 9 · 2 + 8 a .
   Scores that are 3 (mod 8) greater than or equal to 19 can be obtained with 19 + 8 a .
   Scores that are 4 (mod 8) greater than or equal to 19 + 9 = 28 can be obtained with 19 + 9 + 8 a .
   Scores that are 5 (mod 8) greater than or equal to 19 + 9 · 2 = 37 can be obtained with 19 + 9 · 2 + 8 a .
   Scores that are 6 (mod 8) greater than or equal to 19 · 2 = 38 can be obtained with 19 · 2 + 8 a .
   Scores that are 7 (mod 8) greater than or equal to 19 · 2 + 9 = 47 can be obtained with 19 · 2 + 9 + 8 a .
   So the largest two unachievable values are 39 and 31. Multiplying them gives 1209. 2