AMC12 2015 B

AMC12 2015 B · Q10

AMC12 2015 B · Q10. It mainly tests Casework, Triangles (properties).

How many noncongruent integer-sided triangles with positive area and perimeter less than 15 are neither equilateral, isosceles, nor right triangles?

有多少个不全等的、正面积的、周长小于15的整数边三角形既不是等边、等腰，也不是直角三角形？

(A) 3 3

(B) 4 4

(D) 6 6

(E) 7 7

Answer

Correct choice: (C)

正确答案：（C）

Solution

Consider integer-sided triangles with sides $a \leq b \leq c$, $a+b+c < 15$, $a+b > c$, excluding equilateral ($a=b=c$), isosceles ($a=b$ or $b=c$, but since sorted), and right-angled (satisfy $a^2 + b^2 = c^2$ or other permutations, but primarily the latter). Enumerating all possible such triangles and excluding those categories yields 5 triangles.

考虑边长$a \leq b \leq c$的整数边三角形，满足$a+b+c < 15$，$a+b > c$，排除等边（$a=b=c$）、等腰（$a=b$或$b=c$，但已排序）和直角（满足$a^2 + b^2 = c^2$或其他置换，主要后者）。枚举所有可能三角形并排除这些类别，得5个三角形。

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