2017 AMC 10A Problems/Problem 10
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Problem
Joy has thin rods, one each of every integer length from cm through cm. She places the rods with lengths cm, cm, and cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod?
Solution
The triangle inequality generalizes to all polygons, so you can just use and to get , so takes on 19 values. Then, of course, you subtract 2 because 7 and 15 are already used, to get .