2008/09 AHSMC Part 1 Problem 3

Problem 3 of 16

A polygon has area greater than $0$ . All side lengths are integers, the largest of which is $10$ . The smallest possible perimeter of this polygon is

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Solution

The total length of all but one side in a polygon with positive area must be greater than the length of that side. To minimize the perimeter, choose that side to be the longest. Hence the perimeter is greater than $20$ . Since all side lengths are integers, the minimum is at least $21$ . This minimum value may be achieved with a triangle of side lengths $5$ , $6$ and $10$ . The answer is (D).

University of Alberta

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