2019 AMC 12B Problems/Problem 9
Contents
[hide]Problem
For how many integral values of can a triangle of positive area be formed having side lengths ?
Solution
Note is a lower bound for , corresponding to a triangle with side lengths . If , , violating the triangle inequality.
Note also that is an upper bound for , corresponding to a triangle with side lengths . If , , again violating the triangle inequality.
It is easy to verify all satisfy and (the third inequality is satisfied trivially). The number of integers strictly between and is .
-DrJoyo
Solution 2
Note that , , and . The second one is redundant, as it's less restrictive in all cases than the last.
Let's raise the first to the power of . . Thus, .
Doing the same for the second nets us: .
Thus, x is an integer strictly between and : .
- Robin's solution
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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All AMC 12 Problems and Solutions |
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