1984 AIME Problems/Problem 4
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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 1 (Two Variables)
Suppose that has
numbers other than
and the sum of these numbers is
We are given that
Clearing denominators, we have
Subtracting the equations, we get
from which
and
The sum of the twelve remaining numbers is To maximize the largest number, we minimize the other eleven numbers: We can have eleven
s and one
~JBL (Solution)
~MRENTHUSIASM (Reconstruction)
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 | ||
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