AMC10 2025 A
AMC10 2025 A · Q3
AMC10 2025 A · Q3. It mainly tests Triangles (properties), Ratios in geometry.
How many isosceles triangles are there with positive area whose side lengths are all positive integers and whose longest side has length $2025$?
有多少个面积为正的等腰三角形，其边长均为正整数，且最长边长为 2025？
(A)
2025
2025
(B)
2026
2026
(C)
3012
3012
(D)
3037
3037
(E)
4050
4050
Answer
Correct choice: (D)
正确答案：（D）
Solution
You can split the problem into two cases:
Case $1$: The two sides with equal length 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 of equal length.
Therefore, the total number of cases is $1012 + 2025 = \boxed{\text{(D) }3037}$
可以将问题分为两种情况：
情况 1：两条相等边都小于 2025，即从 1013 到 2024，有 1012 种情况。
情况 2：有两条边长为 2025，则第三条边在 1 到 2025 范围内。有 2025 种情况。请注意，等边三角形也算等腰三角形，因为它至少有两条边相等。
因此，总数为 $1012 + 2025 = \boxed{\text{(D) }3037}$
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