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?

$24$

$30$

$31$

$33$

$35$

Solution:

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 .

2014 AMC 10B Problem 18

Solution:

Problem 18 in Other Years