1995 AMC 12 Problem 25

Question

A list of five positive integers has mean

12

and range

18.

The mode and median are both

8.

How many different values are possible for the second largest element of the list?

4

6

8

10

12

Sort the five integers; the median and mode force at least two entries to equal

8

Let the smallest entry be

m

and use the range and total sum to express the second largest entry

Write the sorted list as

m,8,8,d,m+18.

(If the fourth entry were

8,

the sum and ordering conditions would be impossible.) Since the total is

5(12)=60,

m+8+8+d+(m+18)=60,

so

d=26-2m.

The conditions

8\le d\le m+18

and

m\le8

give

3\le m\le8.

These six values yield

d=20,

18,

16,

14,

12,

10,

all valid. Thus there are

6

possibilities, and the correct answer is B.

Problem 25 in other years