Question

The mean, median, unique mode, and range of a collection of eight integers are all equal to $8.$ The largest integer that can be an element of this collection is

$11$

$12$

$13$

$14$

$15$

Answer: D

Concepts:mean median (data)mode extremal argument

Difficulty rating: 1660

Solution:

The sum is $8\cdot 8=64.$ The collection $6,6,6,8,8,8,8,14$ has mean, median, unique mode, and range all equal to $8,$ so $14$ is attainable.

If the largest were at least $16,$ the range condition would make the smallest at least $8.$ A mean of $8$ would then force all eight integers to equal $8,$ contradicting the range.

If the largest were $15,$ the range $8$ forces the smallest to be $7,$ so all eight integers are at least $7.$ The other seven then sum to $64-15=49=7\cdot 7,$ forcing every one of them to equal $7.$ But then the median and mode would be $7,$ not $8,$ a contradiction.

Thus, the correct answer is D.

2002 AMC 10A Problem 21

Answer: D

Difficulty rating: 1660

Solution:

Problem 21 in Other Years