A list of 5 positive integers has all of the following properties:

- the only integer in the list that occurs more than once is 8 ,
- its median is 9 , and
- its average (mean) is 10 .

What is the largest possible integer that could appear in the list?

(Note: The median of a set of five positive integers is the middle integer when the set is arranged in increasing order.)
(A) 15
(B) 16
(C) 17
(D) 24
(E) 25
We write the list of five numbers in increasing order.

We know that the number 8 occurs at least twice in the list.

Since the median of the list is 9 , then the middle number (that is, the third number) in the list is 9 .

Thus, the list can be written as $a, b, 9, d, e$.

Since 8 occurs more than once and the middle number is 9 , then 8 must occur twice only with $a=b=8$.

Thus, the list can be written as $8,8,9, d, e$.

Since the average is 10 and there are 5 numbers in the list, then the sum of the numbers in the list is $5(10)=50$.

Therefore, $8+8+9+d+e=50$ or $25+d+e=50$ or $d+e=25$.

Since 8 is the only integer that occurs more than once in the list, then $d>9$.

Thus, $10 \leq d<e$ and $d+e=25$.

To make $e$ as large as possible, we make $d$ as small as possible, so we make $d=10$, and so $e=15$.

The list 8, 8, 9, 10, 15 has the desired properties, so the largest possible integer that could appear in the list is 15 .

ANSWER: (A)