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 | |
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