2019 AMC 12B Problems/Problem 9
Contents
Problem
For how many integral values of can a triangle of positive area be formed having side lengths ?
Solution
The lower bound for x would be x=4, where the sides of the triangle would be (2,1,3). The upper bound for x would be x=4, where the sides of the triangle would be (6,3,3). The number of integers strictly between 4 and 64 is 64 - 4 - 1 = 59
-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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