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 .
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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