AMC10 2006 A

AMC10 2006 A · Q19

AMC10 2006 A · Q19. It mainly tests Casework, Triangles (properties).

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

有多少个不成似的三角形，其角度度数是互不相同的正整数且形成等差数列？

(A) 0 0

(B) 1 1

(D) 89 89

(E) 178 178

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) Let $n-d$, $n$, and $n+d$ be the angles in the triangle. Then $$ 180 = n-d+n+n+d = 3n,\ \text{so}\ n=60. $$ Because the sum of the degree measures of two angles of a triangle is less than 180, we have $$ 180>n+(n+d)=120+d, $$ which implies that $0<d<60$. There are 59 triangles with this property.

（C）设 $n-d$、$n$ 和 $n+d$ 为三角形的三个角，则 $$ 180 = n-d+n+n+d = 3n,\ \text{所以}\ n=60。 $$ 因为三角形任意两个内角的度数和小于 180，我们有 $$ 180>n+(n+d)=120+d， $$ 这推出 $0<d<60$。满足该性质的三角形共有 59 个。

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