2000 AMC 10 Problems/Problem 20
Contents
Problem
Let ,
, and
be nonnegative integers such that
. What is the maximum value of
?
Solution 1
The trick is to realize that the sum is similar to the product
. If we multiply
, we get
We know that
, therefore
and
Now consider the maximal value of this expression. Suppose that some two of
,
, and
differ by at least
. Then this triple
is not optimal. (To see this, WLOG let
We can then increase the value of
by changing
and
.)
Therefore the maximum is achieved when is a rotation of
. The value of
in this case is
and thus the maximum of
is
Solution 2
We could also do it the more rigorous way, and find out that ,
, and
. This makes sense because to get the largest value of
, we need to have
,
, and
be as close together as possible.
. So the answer is
See Also
2000 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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