2014 AMC 12B Problem 11

Question

A list of

11

positive integers has a mean of

10,

a median of

9,

and a unique mode of

8.

What is the largest possible value of an integer in the list?

24

30

31

33

35

The

11

numbers sum to

110

Maximize one number by minimizing the other ten, keeping the

6

th value

9

and

8

the unique mode

The list sums to

11 \cdot 10 = 110.

To maximize one entry, minimize the sum of the other ten.

Sorted, the sixth number must be

9

(the median), and

8

must appear more often than any other value. Trying

8

three times, the smallest possible ten numbers are

1,1,8,8,8,9,9,10,10,11,

which sum to

75

and keep

8

the unique mode.

The largest entry is then

110 - 75 = 35.

Thus, the correct answer is E.

Problem 11 in other years