2000 AMC 12 Problems/Problem 12
Problem
Let and
be nonnegative integers such that
. What is the maximum value of
?
Solution 1
It is not hard to see that
Since
, we can rewrite this as
So we wish to maximize
Which is largest when all the factors are equal (consequence of AM-GM). Since
, we set
Which gives us
so the answer is
.
Solution 2 (Nonrigorous)
If you know that to maximize your result you \textit{usually} have to make the numbers as close together as possible, (for example to maximize area for a polygon make it a square) then you can try to make and
as close as possible. In this case, they would all be equal to
, so
, giving you the answer of
.
See also
2000 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 11 |
Followed by Problem 13 |
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All AMC 12 Problems and Solutions |
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