HMMT 十一月 2013 · 冲刺赛 · 第 7 题

英文解析

7. [ 7 ] Marty and three other people took a math test. Everyone got a non-negative integer score. Theaverage score was 20. Marty was told the average score and concluded that everyone else scored belowaverage. What was the minimum possible score Marty could have gotten in order to definitively reachthis conclusion?
   Answer: 61 Suppose for the sake of contradiction Marty obtained a score of 60 or lower. Sincethe mean is 20, the total score of the 4 test takers must be 80. Then there exists the possibility of 2
   students getting 0, and the last student getting a score of 20 or higher. If so, Marty could not haveconcluded with certainty that everyone else scored below average.
   With a score of 61, any of the other three students must have scored points lower or equal to 19 points.
   Thus Marty is able to conclude that everyone else scored below average.