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 $2025$ ?

$\textbf{(A)}~2025\qquad\textbf{(B)}~2026\qquad\textbf{(C)}~3012\qquad\textbf{(D)}~3037\qquad\textbf{(E)}~4050$

Solution 1: Casework

You can split the problem into two cases:
Case $1$ : The two sides are both smaller than $2025$ , which means that they range from $1013$ to $2024$ . There are $1012$ such cases.
Case $2$ : There are two sides of length $2025$ , so the last side must be in the range $1$ to $2025$ . There are $2025$ such cases. Keep in mind, an equilateral triangle also counts as an isosceles triangle, since it has at least 2 sides.
Therefore, the total number of cases is $1012 + 2025 = \boxed{3037}$
~cw, minor edit by sd

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2025 AMC 10A Problems/Problem 3

Solution 1: Casework