2014 AMC 12B Problem 11

Problem 11 of 25IntermediateAlgebra

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?

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Solution

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.