AMC 8 2014 Problem 24
AMC8 2014 年第 24 题
统计与数据图表★★★★☆
One day the Beverage Barn sold 252 cans of soda to 100 customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day?
一天 Beverage Barn 向 100 个顾客卖出了 252 罐苏打水，且每个顾客买了至少一罐苏打水。
那么那天每个顾客所买苏打水的罐数的中位数最大可能是多少？
- A.2.5
- B.3.0
- C.3.5
- D.4.0
- E.4.5
正确答案 · Correct Answer
C
解析 · Solution
解法1
常规解法 · 压低前一半
100 人的中位数是排序后第 50、51 名的平均。要中位数最大，就让前 49 人尽量少——各买 1 罐，用掉 49 罐，剩 252-49=203 罐分给后 51 人。因 203=3×51+50，第 50 名最多 3 罐，第 51 名可取 4，中位数 。
巧解 · 用总数反证 4 不可行
若中位数达到 4，则第 50～100 名（共 51 人）至少各 4 罐，加上前 49 人各 1 罐，至少需 49+51×4=253 罐，超过 252。故中位数最多 3.5（且由解法1知可达）。
Solution 1
Elementary · push down the lower half
The median of 100 values is the average of the 50th and 51st smallest. To maximize it, give the smallest 49 customers 1 can each (49 cans), leaving 252-49=203 for the other 51. Since , the 50th customer can have at most 3 and the 51st can be 4, so the median is .
Slick · use the total to rule out 4
A median of 4 would require the 51 customers ranked 50th-100th to have at least 4 cans each, plus 1 each for the first 49: at least cans, more than 252. So the median is at most 3.5, achievable as above.
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