2006 AMC 10A Problems/Problem 19

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Problem

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

$\mathrm{(A) \ } 0\qquad\mathrm{(B) \ } 1\qquad\mathrm{(C) \ } 59\qquad\mathrm{(D) \ } 89\qquad\mathrm{(E) \ } 178\qquad$

Solution

The sum of the angles of a triangle is $180$ degrees. For an arithmetic progression with an odd number of terms, the middle term is equal to the average of the sum of all of the terms, making it $\frac{180}{2} = 60$ degrees. The minimum possibly value for the smallest angle is $1$ and the highest possible is $59$ (since the numbers are distinct), so there are $59$ possibilities $\Longrightarrow \mathrm{C}$ .

2006 AMC 10A (Problems • Resources)
Preceded by Problem 18	Followed by Problem 20
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2006 AMC 10A Problems/Problem 19

From AoPSWiki

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