2014 AMC 10B Problem 18

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 total sum of the list is

110

Minimize the first ten entries while keeping

8

the unique mode and

9

the median

The list has total sum

11\cdot10=110

. To maximize the largest entry, minimize the sum of the other ten entries.

In nondecreasing order, the sixth entry is

9

, and

8

must be the unique mode. If

8

appears twice, the least possible first ten entries sum to

1+2+3+8+8+9

+10+11+12+13

=77

, giving largest entry

33

.

If

8

appears three times, the least possible first ten entries are

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

, with sum

75

, giving largest entry

35

.

If

8

appears four or five times, the least possible sum of the first ten entries is at least

80

, so the largest entry is at most

30

.

Therefore the largest possible entry is

35

, and the correct answer is E .

Problem 18 in other years