AMC12每日一题(2000年真题#14)

2000 AMC 12竞赛试题/第14题

Problem

When the mean, median, and mode of the list

are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ?

Solution

• The mean is .

• Arranged in increasing order, the list is , so the median is either  or  depending upon the value of .

• The mode is , since it appears three times.

We apply casework upon the median:

• If the median is  (), then the arithmetic progression must be constant.

• If the median is  (), then the mean can either be  to form an arithmetic progression. Solving for  yields respectively, of which only  works.

• If the median is  (), then the mean can either be  to form an arithmetic progression. Solving for  yields  respectively, of which only  works.

The answer is .