AMC 10A 2025 (Problem 3)
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length 2025?
There are two cases.
Case 1:
Since is the longest side, .
Since must be a positive integer,
Since is the longest side, .
Since must be a positive integer,
Case 2:
By the Triangle Inequality:
Since must be a positive integer,
By the Triangle Inequality:
Since must be a positive integer,
We have counted the equilateral triangle with side length in both Case 1 and Case 2.
From Hints 2, 3 and 4:
(D)