AMC 12B 2024 (Problem 22)
Let be a triangle with integer side lengths such that . What is the least possible perimeter of such a triangle?
By the Law of Sines,
To cancel from both sides, we first need to prove that .Suppose . Since equal sides have equal opposite angles,
By Hints 5 and 6,
Base the casework on .
Case 1: .
This is impossible because every side length must be positive.
Case 2: .
The possible factor pairs are:
This does not satisfy the Triangle Inequality. ✗
Case 3: .
Case 4: .
Case 5: .
Case 6: .
This gives a valid triangle. ✓
Case 7: . By the Triangle Inequality,
Since and are integers,
Therefore,
By Hint 15, a triangle with perimeter exists. Therefore, the least possible perimeter is
(C) 15
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