1984 AIME Problems/Problem 4
Problem
Let be a list of positive integers - not necessarily distinct - in which the number
appears. The arithmetic mean of the numbers in
is
. However, if
is removed, the arithmetic mean of the numbers is
. What's the largest number that can appear in
?
Solution
Suppose has
members other than 68, and the sum of these members is
. Then we're given that
and
. Multiplying to clear denominators, we have
and
so
,
and
. Because the sum and number of the elements of
are fixed, if we want to maximize the largest number in
, we should take all but one member of
to be as small as possible. Since all members of
are positive integers, the smallest possible value of a member is 1. Thus the largest possible element is
.