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2006 AMC 12B Problem 10

Problem 10 of 25EasierAlgebraGeometry

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15.15. What is the greatest possible perimeter of the triangle?

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Solution

Let the sides be x,x, 3x,3x, and 15.15. The triangle inequality requires x+3x>15,x + 3x \gt 15, so x≥4,x \geq 4, and x+15>3x,x + 15 \gt 3x, so x≤7.x \leq 7. The perimeter 4x+154x + 15 is largest when x=7,x = 7, giving 7+21+15=43.7 + 21 + 15 = 43. Thus, the correct answer is A.

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Concepts: triangle inequality · optimization

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.