HMMT 二月 2000 · 代数 · 第 3 题
HMMT February 2000 — Algebra — Problem 3
题目详情
英文原题
- Fiv e studen ts tak e a test on which an y integer 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
英文解析
- Let the s ores be 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, we w an t to maximize ( a + b + + d + e )
5 5
. T o do this, we m ust maximize d and e and minimize or maximize . One w a y todo 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 we maximize , then
5 5 = d = e = 100, and then the mean is (0 + 0 + 100 + 100 + 100 ) = 60, and the 1
median is 60, with a di eren e of 40 as w ell.5