How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length $2025$?
$$\textbf{(A) } 2025 \qquad \textbf{(B) } 2026 \qquad \textbf{(C) } 3012 \qquad \textbf{(D) } 3037 \qquad \textbf{(E) } 4050$$
Difficulty: 14/100 — Easy to approach, but requires casework.
Core Concepts: casework, triangle inequality
Challenges: Realizing that the longest side being $2025$ results in two scenarios.
Techniques: Solve inequalities for both cases, avoid double-counting equilateral triangle.
Error-prone Steps: Forgetting or double-counting equilateral case.
Ideal Time:
Experienced: ≤ 25 sec
Intermediate: ≤ 45 sec
Beginner: ≤ 1.5 min