2025 AMC 10 A
Complete problem set with solutions and individual problem pages
Problem 3 Easy
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length ?
- A.
- B.
- C.
- D.
- E.
Answer:D
Consider isosceles triangles with side lengths where .
Scenario 1: .
By triangle inequality: , which gives , so .
Therefore, gives us 1013 valid triangles.
Scenario 2: .
We have , yielding 2025 valid triangles.
Notice that we count the equilateral with side length twice, so the combined total is triangles.
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