HMMT 二月 2000 · 代数 · 第 3 题

HMMT February 2000 — Algebra — Problem 3

Fiv e studen ts tak e a test on whi h an y in teger s ore from 0 to 100 in lusiv e is p ossible. What is the largest p ossible di eren e b et w een the median and the mean of the s ores? 2000

解析

Let the s ores b e a , b , , d , e , where 0 a b d e 100. So, the mean is 1 1 ( a + b + + d + e ) ; and the median is . So, w e w an t to maximize ( a + b + + d + e ) 5 5 . T o do this, w e m ust maximize d and e and minimize or maximize . One w a y to do this is to let a = b = = 0 and d = e = 100, so the di eren e b et w een the mean 1 200 and the median is (0 + 0 + 0 + 100 + 100 ) 0) = = 40 . If w e maximize , then 5 5 1 = d = e = 100, and then the mean is (0 + 0 + 100 + 100 + 100 ) = 60, and the 5 median is 60, with a di eren e of 40 as w ell.