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?
Answer Choices:
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Solution:
Four rods can form a quadrilateral with positive area if and only if the length of the longest rod is less than the sum of the lengths of the other three. Therefore if the fourth rod has length , then must satisfy the inequalities and , that is, . Because is an integer, it must be one of the integers from to , inclusive. However, the rods of lengths and have already been chosen, so the number of rods that Joy can choose is .
The problems on this page are the property of the MAA's American Mathematics Competitions