2019 AMC 12B Problems/Problem 9
Problem
For how many integral values of can a triangle of positive area be formed having side lengths
?
Solution
For these lengths to form a triangle of positive area, the Triangle Inequality tells us that we need ,
, and
. The second inequality is redundant, as it's always less restrictive than the last inequality.
Let's raise the first inequality to the power of . This gives
. Thus,
.
Doing the same for the second inequality gives (where we are allowed to divide both sides by
since
must be positive in order for the logarithms given in the problem statement to even have real values).
Combining our results, is an integer strictly between
and
, so the number of possible values of
is
.
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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