2017 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 10 Easy

Joy has thin rods, one each of every integer length from through . She places the rods with lengths , , and 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? (2017 AMC 10A Problem, Question#10)

Answer:B

The triangle inequality generalizes to all polygons, so and to get . Now, we know that there are numbers between and exclusive, but we must subtract to account for the lengths already used that are between those numbers, which gives .