2000 AMC 10 Problems/Problem 20
Problem
Let ,
, and
be nonnegative integers such that
. What is the maximum value of
?
Solution
The trick is to realize that the sum is similar to the product
.
If we multiply , we get
.
We know that , therefore
.
Therefore the maximum value of is equal to the maximum value of
. Now we will find this maximum.
Suppose that some two of ,
, and
differ by at least
.
Then this triple
is surely not optimal.
Proof: WLOG let . We can then increase the value of
by changing
and
.
Therefore the maximum is achieved in the cases where is a rotation of
. The value of
in this case is
. And thus the maximum of
is
.
See Also
2000 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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