Problem:
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length 2025?
Answer Choices:
A. 2025
B. 2026
C. 3012
D. 3037
E. 4050
Solution:
We want isosceles triangles with integer sides and longest side 2025. Let the sides be a,a,b.
Case 1: The equal sides are longest.
Then
a=2025,b≤2025.
Since b is a positive integer and b≤2025, we have
b=1,2,…,2025,
giving 2025 triangles.
Case 2: The base is longest.
Now
b=2025,a<2025.
By the triangle inequality
2a>b⇒2a>2025⇒a>22025=1012.5.
So
a=1013,1014,…,2024,
which gives us 2024−1013+1=1012 values of a, and thus 1012 triangles.
Combining our cases, we get a total of
2025+1012=(D) 3037
The problems on this page are the property of the MAA's American Mathematics Competitions