Question

Let $S$ be a list of positive integers—not necessarily distinct—in which the number $68$ appears. The average (arithmetic mean) of the numbers in $S$ is $56.$ However, if $68$ is removed, the average of the remaining numbers drops to $55.$ What is the largest number that can appear in $S?$

Answer

Answer: 649

Concepts:mean optimization

Difficulty rating: 1670

Small Hint:

Let $N$ be the number of entries in the original list

Big Hint:

To maximize one entry, make every unrestricted positive integer as small as possible

Solution:

If the original list has $N$ entries, then $56N-68=55(N-1),$ so $N=13$ and the total of the entries is $56\cdot13=728.$ Besides the entry $68,$ make eleven entries equal to the least positive integer, $1.$ The remaining entry is then $728-68-11=649.$ This construction is valid, and no larger entry is possible.

1984 AIME Problem 4

Answer: 649

Difficulty rating: 1670

Small Hint:

Big Hint:

Solution:

Problem 4 in Other Years