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 
.
