Difference between revisions of "1984 AIME Problems/Problem 4"
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== See also == | == See also == | ||
{{AIME box|year=1984|num-b=3|num-a=5}} | {{AIME box|year=1984|num-b=3|num-a=5}} | ||
+ | * [[AIME Problems and Solutions]] | ||
+ | * [[American Invitational Mathematics Examination]] | ||
+ | * [[Mathematics competition resources]] | ||
[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] |
Revision as of 14:21, 6 May 2007
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 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |