2025 AMC 10A Problem 3

Below is the professionally curated solution for Problem 3 of the 2025 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 10A solutions, or check the answer key.

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Concepts:isosceles triangle triangle inequality casework

Difficulty rating: 1130

How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length $2025?$

$2025$

$2026$

$3012$

$3037$

$4050$

Solution:

Split into two cases. Say two sides both equal $2025.$ Then the third side can be any integer from $1$ to $2025,$ which is $2025$ triangles. Now suppose $2025$ is the unique longest side. The two equal legs $s$ must satisfy $2s \gt 2025$ by the triangle inequality, and $s \le 2024.$ So $s$ runs from $1013$ to $2024,$ giving $1012$ triangles. Adding up, $2025 + 1012 = 3037.$ Thus, D is the correct answer.

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