2025 AMC 10A Problems/Problem 3
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length 
?
Solution 1: Casework
Split the problem into two cases:
Case 1: The two sides are both smaller than , which means that they range from 
to 
. There are 
such cases.
Case 2: There are two sides of length , so the last side must be in the range 
to . There are such cases. Keep in mind, an equilateral triangle also counts as an isosceles triangle.
Therefore, the total number of cases is 
-cw
