2015 AMC 12A Problem 2

Two of the three sides of a triangle are

20

and

15.

Which of the following numbers is not a possible perimeter of the triangle?

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Solution

By the Triangle Inequality, the third side

s

satisfies

20-15 \lt s \lt 20+15,

that is

5 \lt s \lt 35.

The perimeter is

35 + s,

so it lies strictly between

40

and

70.

Among the choices, only

72

falls outside this range. Thus, the correct answer is E.

Concepts: triangle inequality · bounding to limit cases

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