2002 AMC 12A Problem 15

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

The eight integers sum to

8\cdot 8 = 64,

and the mode

8

must repeat

Test the largest value

15;

the smallest would then be

7,

and check the remaining sum

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.

Suppose the largest were

15.

The range

8

forces the smallest to be

7.

Because

8

is the mode, it occurs in the sorted list; together with median

8,

this forces the two middle values to be

8, 8.

Then

7 + 8 + 8 + 15 = 38,

so the remaining four values sum to

64 - 38 = 26,

averaging

6.5.

At least one would be below

7,

contradicting the minimum. So

15

is impossible.

Thus, the correct answer is D.

Problem 15 in other years