HMMT 二月 2005 · COMB 赛 · 第 1 题

英文解析

1. A true-false test has ten questions. If you answer five questions “true” and five “false,”
   your score is guaranteed to be at least four. How many answer keys are there for whichthis is true?
   Solution: 22
   Suppose that either nine or ten of the questions have the same answer. Then no matterwhich five questions we pick to have this answer, we will be right at least four times.
   Conversely, suppose that there are at least two questions with each answer; we willshow that we can get a score less than four. By symmetry, assume there are at leastfive questions whose answer is true. Then if we label five of these false, not only willwe get these five wrong, but we will also have answered all the false questions withtrue, for a total of at least seven incorrect. There are 2 ways for all the questions tohave the same answer, and 2 · 10 = 20 ways for one question to have a different answerfrom the others, for a total of 22 ways.