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2006 AMC 10B 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 x+15>3xx+15 \gt 3x gives x<7.5.x \lt 7.5. The largest integer is x=7,x=7, giving sides 7,7, 21,21, 1515 and perimeter 7+21+15=43.7+21+15=43. Thus, the correct answer is A.

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Concepts: triangle inequality · perimeter · optimization

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